Embed Size (px)
Citation preview
ASYMPTOTIC STABILITY CRITERION FOR NONLINEAR MONOTONIC SYSTEMS
AND ITS APPLICATIONS (REVIEW)
À.À. Martynyuk
This paper discusses uniform asymptotic stability criteria for nonlinear monotonic systems and their
applications in various problems of nonlinear dynamics and population dynamics
Keywords: nonlinear monotonic systems, asymptotic stability criterion, large-scale systems, cooperative
systems, hybrid systems, population dynamics
Introduction. The comparison method in the qualitative theory of equations is of paramount importance in studies of
large-scale systems. This method is based on differential inequalities such as the Chaplygin–Wazewski inequality and Lyapunov
functions (scalar, vector, or matrix-valued), which play the role of nonlinear transformation of the original system to an equation
(a system or a matrix system) of lower dimension. The comparison method states that if for the system in question, there exists a
Lyapunov function (scalar, vector, or matrix-valued) satisfying appropriate conditions, then various dynamic properties of the
solutions of the original system follow from the respective dynamic properties of the comparison system.
Thus, the stability analysis of a dynamic system involves setting up a Lyapunov function and establishing stability
criteria for the zero solution of the comparison system.
The present review discusses some approaches to solving such problems, establishes a uniform asymptotic stability
criterion for a nonlinear comparison system, and exemplifies its applications in modern nonlinear dynamics.
1. Problem Formulation. Let x t( )be an n-dimensional state vector of some mechanical system with a finite number of
degrees of freedom. Its behavior is described by the perturbed equations of motion
dx
dt
f t x� ( , ), (1)
where x ��, t ��; � is a domain in �n
containing the origin (0� �� �n
); f is vector function such that
A1. f t( , )0 0� for any t ��;
A2. f t x( , ) is uniformly continuous in t for any x ��;
A3. f t x( , ) is locally Lipschitz in x for any t �
�
� .
There are two well-known methods for analyzing the dynamic behavior of the solutions of system (1). One method
involves direct integration of Eqs. (1), which is impossible in the general case. The other method uses an auxiliary (Lyapunov)
function to ascertain the behavior of the solutions of system (1) without direct integration. The latter method was intensively
developed and used to solve various practical problems.
The discovery of the comparison principle in the 20th century was an outstanding achievement in the qualitative theory
of equations. It made it possible to develop new means for the analysis of nonlinear large-scale systems of various designation.
Along with setting up of scalar, vector, or matrix Lyapunov functions, a key element of the comparison method is the
stability criteria for the zero solution of comparison systems. These systems can be either linear or nonlinear and have special
properties among which monotonicity in some cone is the major one.
International Applied Mechanics, Vol. 47, No. 5, November, 2011 (Russian Original Vol. 47, No. 5, September, 2011)
1063-7095/11/4705-0475 ©2011 Springer Science+Business Media, Inc. 475
S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, 3 Nesterov St., Kyiv, Ukraine
03057, e-mail: [email protected]. Translated from Prikladnaya Mekhanika, Vol. 47, No. 5, pp. 3–67, September 2011.
Original article submitted December 16, 2010.
The present review discusses the uniform asymptotic stability criterion for nonlinear monotonic systems and its
development for the last three decades. We will consider the following applications:
(a) nonlinear nonautonomous system;
(b) special automatic control system;
(c) nonlinear switched system;
(d) large-scale (complex) system;
(e) cooperative system;
(f) cooperative system with input;
(g) cooperative hereditary system;
(h) nonlinear systems describing the dynamics of populations.
For these systems of perturbed equations of motion, we will present various sufficient conditions of stability and/or
dissipativity of their solutions.
Some standard notation is given below; the other concepts will be explained later.
Denote ��
� �[ , )0 and � ��
� � �
n nx x{ : }0 . The Euclidean norm of the vector x
n�� is denoted by ||x||. A matrix
An n
�
�
� is nonnegative if aij� 0, 1 i j n, . A vector function g
n n:� � is nonnegative if g x( ) � 0for all x
n�
�
� . The norm
of the matrix An n
�
�
� induced by the Euclidean norm of the vector xn
�� is denoted by ||A||.
2. Comparison Equation with a Scalar Lyapunov Function. In his fundamental work [25], Lyapunov noticed that it
is Poincare’s memoirs [39] that had suggested him the idea of using, for stability analysis, some functions V t x( , ) whose total
derivatives with respect to time along the solutions of Eqs. (1) have certain properties. The description of these functions and
various generalizations can be found in many well-known monographs (see [7, 13, 15, 16, 41] and the references therein);
therefore, these results will be omitted here. Recall that all functionsV t x( , ) that allow ascertaining whether the zero solution of
system (1) is stable or unstable are called Lyapunov functions. These functions are sought in the class of all functions that are
sign-definite and decreasing.
Replacing the sign-definiteness of the total derivative of the Lyapunov function with respect to time along the solutions
of system (1) with some set of conditions guaranteeing that the auxiliary function decreases along the solutions of the system
necessitates considering a scalar equation (comparison equation), which helps, under certain conditions, to ascertain the stability
of the zero solution of system (1).
The upper right Dini derivative of any functionV t x( , ),V Cn
� �
�
( , )� � � , is defined by
D V t x V t h x hf t x V t x h h� � �
� � � � ( , ) limsup{[ ( , ( , )) ( , )] : }1
0
for any ( , )t xn
� �
�
� � .
Let for system (1) there exist a Lyapunov functionV t x( , )and a real continuous function g t u( , )defined for all t t�
0and
| |u � �� such that
D V t x g t V t x�
( , )| ( , ( , ))( )1
, (2)
where D�
is the derivative along the solutions of system (1).
Definition 1. The equation
dy
dt
g t y� ( , ), y t V t x( ) ( , )0 0 0
0� � , (3)
is called a comparison equation for system (1) with a scalar Lyapunov function if the function V t x t( , ( )) and the maximum
solution y t( ) of Eq. (3) are related by
V t x t y t( , ( )) ( ) (4)
for all t t�
0for which the solutions x t x t t x( ) ( , , )�
0 0and y t y t t y( ) ( , , )�
0 0are defined on a common interval.
Thus, in view of (4), the properties of the zero solution of system (1) are related to the properties of the zero solution of
the comparison equation (3).
476
In what follows, we will need the following statement (see [2] and the references therein).
Lemma 1. Let a function g t y( , )with a continuous partial derivative with respect to y be defined in an open domain�1
containing the point y � 0. If the functions v t( ) and y t( ) are continuous and differentiable in [ , ]t t a0 0
� , the function | ( )|v t H�
satisfies the inequality
dv
dt
g t v t� ( , ( )),
and the function y t( ) is a solution of problem (3), then v t y t( ) ( ) for t t t a� �[ , ]0 0
as soon as v t y t( ) ( )0 0
.
This statement is a variant of Chaplygin’s theorem [26] and underlies the method of comparison with a scalar Lyapunov
function.
The following statement is true.
Lemma 2. For system (1) to have a comparison equation
dy
dt
g t y� ( , ),
where y��
� , g:� � �� �
� , g t( , )0 0� , it is sufficient that
(a) there exist a functionV t x( , ),Vn
:� � �� �
� , that is locally Lipschitz in x for any t ��
� ;
(b) for ( , )t x � �
�
� �,
D V t x g t V t x�
( , )| ( , ( , ))( )1
,
where g t y( , )is continuous in the domain ( , )t y � �
�
� �1,�
1�
�
� and has a continuous partial derivative with respect to y for
any t ��
� .
Proof. If conditions (a) and (b) of Lemma 2 are satisfied for system (1), then, in view of Lemma 1, the following
estimate holds:
V t x t y t( , ( )) ( ) for all t t t a� �[ , )0 0
,
where y t( ) is the solution of Eq. (3) such that y t V t x( ) ( , )0 0 0
0� � . From (4) it follows that the dynamic properties of the
solution x � 0of system (1) follow from the respective properties of the solution y � 0of Eq. (3) if the function v t x( , )is subject to
some additional constraints. Therefore, Eq. (3) is the equation of comparison with a scalar Lyapunov function for system (1).
One of the first results in the context with Lemma 2 was obtained in [20].
Theorem 1. Let for the perturbed equations of motion (1) in the domain � ��n
, there exist positive definite and
decreasing (permitting an infinitesimal upper limit) functionV t x( , ),V:� �� �
� � , and a function g t y( , ), g:� � �� �
� ,
g t( , )0 0� , having a continuous partial derivative with respect to y in the domain �1
0� � �
�
{( , ): , }t y t y H� for which
inequality (2) holds in the domain ( , )t x � �
�
� �. Then the stability of the state x � 0of system (1) follows from the stability of
the zero solution of Eq. (3).
Proof. For a positive definite and decreasing function v t x( , ), there exist functions a and b belonging to Hahn’s K-class
such that
a x V t x b x(| | | | ) ( , ) (| | | | ) (5)
for all ( , )t x � �
�
� �. Lemma 2 together with estimate (4) and inequality (5) allows us to establish the correspondence between
the properties of the solutions of Eq. (3) and system (1).
Remark 1. If the function g t y( , ) in (2) is independent of t, i.e., g t y g y( , ) ( )� for all t ��
� , then Theorem 1 remains
valid.
Remark 2. If g t y( , ) � 0 in the conditions of Theorem 1, then we arrive at Lyapunov’s stability theorem [25].
Example 1 [41]. For the system
dx
dt
E t A t x x� �( sin ( , )) , (6)
477
where x��
� , E is a unit matrix and A t x( , ) is antisymmetric matrix continuous in t � 0, | | | |x H� (H > 0 = const), there exists the
comparison equation
dy
dt
y t� 2 sin
with a scalar functionV t x x( , ) | | | |�
2.
Indeed, for the functionV t x x( , ) | | | |�
2, we have DV t x V t x t( , )| ( , )sin
( )62� . Hereafter D
d
dt
� .
Example 2 [2]. Consider the equation
�� ( )� ( ) ( )x a t x b t q x� � � 0, (7)
where a t C( ) ( , )�
� �
� � , b t C( ) ( , )�
� �
1� � , q x( ) is given and continuous for all x� �� �( , ) and � �
�q s ds
x
( )
0
0 for all
x � 0. For Eq. (7), there exists a comparison equation
dy
dt
c t y� ( ) ,
where c t b t b t a t( ) max{�
( ) / ( ), ( )}� �2 .
Indeed, for the function
V t x x b t x x( , , � ) ( ) ( ) � �
1
2
2
we have
DV t x x b t x a t x( , , � )|�
( ) ( ) ( )�( )7
2� �
and for b t b( ) �0
, b0� const > 0, and for any x x, � ( , )� �� � , the following estimate holds:
DV t x x c t V t x x( , , � )| ( ) ( , , � )( )7
.
Example 3. Consider the system of perturbed equations of motion
dx
dt
Px Q x H t x� � �( ) ( , ), (8)
where xn
�� , P is a constant n n� -matrix, Q x( ) are polynomials of at least second order of smallness, and H t x( , ) are bounded
functions on ��
�N , Nn
�� . For Eq. (8), there exists a comparison equation
dy
dt
y y t� � �� � �
2( ),
where �, � are positive constants and �( )t is a bounded nonnegative function.
This equation cannot be integrated analytically, but using it leads to a number of interesting results for both system (8)
and uncertain systems (see [30] and the references therein).
3. System of Comparison with Vector Lyapunov Function. The curse of dimensionality in the sense of Bellman (see
[8]) in the qualitative theory of equations makes it impossible to set up a scalar Lyapunov function for a large-scale system. In
these conditions, Kron developed diakoptics [22] for the analysis of complex systems by parts. This method is based on the
concept of a vector Lyapunov function and corresponding comparison system.
Recall the following concept from the theory of monotonic systems (see [98]).
478
Definition 2. A vector function G t y( , ) ym
�� , Gm m
:� � ��
� , is quasimonotinically increasing if for any pair
( , )t u � �
�
� �1
and ( , )t v � �
�
� �1
and all i = 1, 2, …, m, the estimates G t u G t vi i( , ) ( , )� are true when u v
i i� and u v� .
Together with system (1), we ill consider the vector Lyapunov function
V t x V t x V t xm
( , ) ( ( , ), , ( , ))�
1�
T, V t
i( , )0 0� for all i = 1, 2, …, m, (9)
where “T” denotes transposition. LetVi:� �
� �
� � andV t xi( , ) � 0 for all i = 1, 2, …, m.
For any componentV t xi( , ) of the vector functionV t x( , ), we will define a function D V t x
i
�
( , ) for any ( , ):t xn
� ��
�
and for all i = 1, 2, …, m.
Definition 3. The system of equations
dy
dt
G t y� ( , ), y V t x0 0 0
( , ), (10)
is called a comparison system for system (1) with vector Lyapunov function (9) if the componentsV t xi( , ), i = 1, 2, …, m, of the
functionV t x( , ) and the components of the maximum solution y t t y( , , )0 0
of the system of equations (10) are related by
V t x t t x y t t yi i( , ( , , )) ( , , )
0 0 0 0 , i = 1, 2, …, m,
for all t t�
0for which the solutions x t t x( , , )
0 0and y t t y
i( , , )
0 0exist on a common interval.
The following statement is true.
Lemma 3. For system (1) to have a comparison equation
dy
dt
G t y� ( , ),
where ym
�
�
� , Gm m
:� � �� �
� , G t( , )0 0� , it is sufficient that
(a) there exist a vector function (9) with componentsV t xi( , ) locally Lipschitz in x for all i = 1, 2, …, m;
(b) for ( , )t x � �
�
� �,
D V t x G t V t x�
( , )| ( , ( , ))( )1
, (11)
where G Cm m
� �
� �
( , )� � � , G t( , )0 0� , and G t u( , ) is a function quasimonotonically increasing in u for each t ��
� .
Proof. If conditions (a) and (b) of Lemma 3 are satisfied, the componentsV t xi( , ) of function (9) can be estimated as
V t x t y ti i( , ( )) ( ) (12)
for all t in the common interval of existence of the solutions of the original and comparison systems. Here
y t y t y tm
( ) ( ( ), , ( ))�
1�
Tis the solution of the comparison system (10) satisfying the condition y t V t x
i i( ) ( , )
0 0 00� � for all
i = 1, 2, …, m. Estimates (12) together with additional constraints on the functions
V t x V t x
i mi0
1
( , ) max ( , )�
or
V t x d V t x0
( , ) ( , )�
T, d
i� 0, i = 1, 2, …, m,
allow us to establish the stability conditions for the state x � 0 of system (1) by analyzing the solution y � 0 of the comparison
system (10) for stability.
The following statement is true.
Theorem 2. Let for the system of perturbed equations of motion (1), there exist functionsV t x( , )andG t y( , )specified in
Lemma 3 and being such that
(a) the functionV t x0
( , ) id positive definite and decreasing in ( , )t x � �
�
� �;
479
(b) for ( , )t x � �
�
� �, inequality (11) holds. Then the stability of the state x � 0of system (1) follows from the stability of
the zero solution of the comparison system (10).
The proof of this theorem is well-known (see [23, 33, 41] and the references therein).
Remark 3. If G t y G y( , ) ( )� for all t ��
� in (10), then Theorem 2 remains valid.
Thus, one of the central problems in the comparison method is the stability analysis of the comparison system (10) or
Eq. (3).
Example 4 [33]. Consider the system of equations
dx
dt
e x x t x x x tt1
1 2 1 1
2
2
2 2� � � �
�
sin ( )sin ,
dx
dt
x t e x x x x tt2
1 2 2 1
2
2
2 2� � � �
�
sin ( )sin . (13)
For system (13), there exists a comparison system
dy
dt
P t y� ( ) ,
where P t
e t
e t
t
t
( )
sin
sin
�
�
�
�
�
�
�
�
�
�
�
�
�
2
0
0
. HereV t x x x x x( , ) (( ) , ( ) )� � �
1 2
2
1 2
2 T.
The problem of setting up comparison systems for some classes of perturbed equations of motion is addressed in a great
many publications (see [35] and the references therein).
4. Comparison Principle with a Matrix-Valued Lyapunov Function. A new area of development of Lyapunov’s
direct method in the theory of the stability of motion was defined by introducing auxiliary two-index functions as a suitable
medium for setting up both scalar and vector Lyapunov functions (see [83, 86] and the references therein). This class of auxiliary
functions considerably extends the capabilities of Lyapunov’s direct method by allowing the analysis of the dynamics of
large-scale (complex) systems if the following conditions are satisfied:
B1. Subsystems of a complex system can be not only exponentially stable, but also neutrally stable or unstable.
B2. The functions relating the subsystems can be either stabilizing or destabilizing for the dynamics of an individual
subsystem.
B3. The domain of values of the parameters of the comparison system that guarantees the stability of the zero solution of
the original system is not a “point.”
Some theorems on matrix-valued Lyapunov functions that allow for conditions B1–B3 are formulated below. Using
vector Lyapunov functions in these conditions is known to be difficult (see [99]).
4.1. Uniform Asymptotic Stability. We will use the following two-index system of functions [60, 82] for the dynamic
analysis of the solutions of system (1):
U t x v t xij
( , ) [ ( , )]� (i, j = 1, 2, …, m), (14)
where v t x Cii
n( , ) ( , )� �
� �
� � � , v tii
( , )0 0� , v t x Cij
n( , ) ( , )� �
�
� � � for all i j� , v tij
( , )0 0� . We use the matrix-valued
function (14) to set up the vector function
V t x U t x( , , ) ( , )� �� , ��
�
�m
, (15)
and the scalar function
v t x U t x( , , ) ( , )� � ��
T. (16)
The two-index system of functions (14) is a suitable medium for setting up scalar (16) or vector (15) Lyapunov
functions if at least one of them together with the total derivatives D v t x�
( , , )|( )
�
1and D V t x
�
( , , )|( )
�
1allows ascertaining
whether the zero solution of system (1) is stabile or not. In this case, the following equalities hold:
480
D v t x D U t x� �
�( , , )| ( , )( )
� � �
1
Tand D V t x D U t x
� �
�( , , )| ( , )( )
� �
1,
where D U t x U t x f t x U t x� � �
� � � � ( , ) limsup{[ ( , ( , )) ( , ) : }� � � �
10 is calculated component-wise for all ( , )t x
n� �
�
� � .
Definition 4. The function w is called angular if it is continuous and nondecreasing in its domain of definition
w:� �� �
, w( )0 0� , and w s( ) � 0for s � 0.
All functions with properties specified in Definition 4 will be called functions of W-class.
Let the diagonal (v t xii
( , )) and off-diagonal (v t xij
( , )) elements of the matrix-valued function (14) be estimated as
� � � �
ii i i ii ii i ix v t x x
1
2
2
2(| | | | ) ( , ) (| | | | ) , (17)
� � � � �
ij i i j j ij ij i ix x v t x x
1
2
1 2(| | | | ) (| | | | ) ( , ) (| | | | ) �
2i jx(| | | | ) (18)
for all ( , , )t x xi j
n n
ij
� � �
�
� � � , xn
i�� , x
j
nj
�� , n nii
m
��
�
1, �
ii
, �ii� 0 for all i = 1, 2, …, m; �
ii
, �ii
are arbitrary
constants; ( , )� �
1 2i iW� -class.
Lemma 4. With (17) and (18), function (16) can be estimated as
� � � � �
1 1 2 2
T T(| | | | ) (| | | | ) ( , , ) (| | | | ) (| | | | )x A x v t x x B x (19)
for all ( , )t xn
� �
�
� � , where � � �
1 11 1 1(| | | | ) ( (| | | | ), , (| | | | ))x x x
m m� �
T, � � �
2 21 1 2(| | | | ) ( (| | | | ), , (| | | | ))x x x
m m� �
T,
A Y GY�
T, B Y G Y�
T, Y
m�diag[ , ]� �
1� , G
ij
� [ ]� , Gij
� [ ]� , i, j = 1, 2, …, m.
This statement can be proved through the direct calculation of the upper- and lower-bound estimates for function (16)
taking inequalities (17) and (18) into account.
Following [28], we will formulate the conditions for the uniform asymptotic stability of the zero solution of system (1)
based on the scalar auxiliary function (16).
Theorem 3. Let the vector function f in (1) be continuous in � ��
�
nand such that
(a) there exist a matrix-valued function (14) with elements (17), (18) and a vector ���
�m
such that function (16) is
locally Lipschitz in x for all t ��
� ;
(b) there exist functions �1i
, �2i
, �3i
of W-class and (m m� ) matrices A( )� , B( )� , C( )� such that estimate (19) holds
for all ( , )t x N� �
�
� , Nn
� � and
D v t x x C x m t x�
�( , , )| (| | | | ) ( ) (| | | | ) ( , (| | |( )
� � � � �
1 3 3 3
T| )
for all ( , )t x N� �
�
� , where m t u( , ) satisfies the condition lim
| ( , )|
| | | ||| ||u
m t u
u
�
0
0 uniformly in t ��
� ;
(c) the system of inequalities
A( )� � 0, B( )� � 0, C( )� � 0
has a solution� �
1
* *, ,�
msatisfying the condition 0 ��
i
*for all i = 1, 2, …, m.
Then the solution x � 0of system (1) is uniformly asymptotically stable.
Proof. If conditions (a) and (c) of Theorem 3 are satisfied, function (16) can be estimated as in (19) and is positive
definite and decreasing. Indeed, since m
A( ) � 0and
mB( ) � 0, we have
� � � � �
m MA x x v t x B x( ) (| | | | ) (| | | | ) ( , , ) ( ) (| | | | )
1 1 2
T T
2(| | | | )x
for all ( , )t x N� �
�
� . Since � �
1 2i iW, � -class for i = 1, 2, …, m according to condition (b) of Theorem 3, there exist functions
! !
1 2, �W-class such that ! � �
1 1 1(| | | | ) ( ) (| | | | ) (| | | | )x A x x
m
Tand ! � �
2 2 2(| | | | ) ( ) (| | | | ) (| | | | )x B x x
m
Tfor all
( , )t x N� �
�
� .
481
Then
! � !
1 2(| | | | ) ( , , ) (| | | | )x v t x x (20)
for all ( , )t x N� �
�
� . This condition is known [66] to be necessary and sufficient for function (16) to be positive definite and to
decrease.
Next, if conditions (b) and (c) of Theorem 3 are satisfied, then for any 0 1� �" , we can select #( )$ � 0 such that
| ( , (| | | | )| ( ) (| | | | ) (| | | | )m t x C x xM
� " � �
3 3 3 �
T
for all ( , ) ( )t x S� �
�
� # ; therefore,
D v t x C w xM
�
�( , , )| ( ) ( ) (| | | | )( )
� "
1 31 , (21)
where w x x x3 3 3
(| | | | ) (| | | | ) (| | | | )� � �
T, w W
3� -class. Since
MC( ) � 0, D v t x
�
( , , )|( )
�
1is a negative definite function for all
( , ) ( )t x S� �
�
� # , S N( )# � . Hence, conditions (20) and (21) guarantee the uniform asymptotic stability of the zero solution of
system (1).
Corollary 1. If M
C( ) 0 in the conditions of Theorem 3, then the zero solution x � 0 of systems (1) is uniformly
stable, and if M
C( ) 0and B( )� � 0, then the state x � 0 is stable.
Remark 4. To find the elements v t x v t xii ii i
( , ) ( , )� and v t x v t x xij ij i j
( , ) ( , , )� , i j� of function (14), system (1) is
decomposed into
dx
dt
f t x r t x xi
i i i m� �( , ) ( , , , )
1� , i = 1, 2, …, m,
where xi
ni
�� , fi
n ni i:� � �
�
� , ri
n n nm i:� � � �
�
� � � 1
� . In this case, the functions v t xii i
( , ) are related to the
independent subsystems
dx
dt
f t xi
i i� ( , ), i = 1, 2, …, m,
and the functions v t x xij i j
( , , ), i j� , can be found from a special partial differential equation. This makes it possible to take into
account conditions B1–B3.
4.2. Stability of a Nonlinear Autonomous System. Let the behavior of some large-scale system be described by the
equations
dx
dt
f x� ( ), x t x( )0 0
� , (22)
where xn
�� , fn n
:� � , f ( )0 0� . System (22) is decomposed into
dx
dt
f x r xi
i i i� �( ) ( )
*, i = 1, 2, …, m, (23)
where xi
ni
�� , fi
n ni i:� � , r
i
n ni
*:� � , n n
ii
m
��
�
1, f r
i i( ) ( )
*0 0 0� � .
Let the independent subsystems
dx
dt
f xi
i i� ( ), i = 1, 2, …, m,
be associated with the diagonal elements v xii i
( ), i = 1, 2, …, m. The matrices of the functionU x( )and the elements v x xij i j
( , )for
all i j� , i, j = 1, 2, …, m, describe the action of the ith subsystem on the jth subsystem in (23).
482
Let the elements v xii i
( )and v x xij i j
( , )of the matrix functionU x vij
( ) [ ( )]� % , i, j = 1, 2, …, m, can be estimated as in (17)
and (18) with functions � �
1 2i i i i ix x x(| | | | ) (| | | | ) | | | |� � for all i = 1, 2, …, m. Then estimates (19) for the function
v x U x( , ) ( )� � ��
Tbecome
� � � � �
1 1 2 2
T T(| | | | ) (| | | | ) ( , ) (| | | | ) (| | | | )x A x v x x B x , (24)
where � �
1 1 2 2(| | | | ) (| | | | , | | | | , , | | | | ) (| | | | )x x x x x
m� �� .
By virtue of (23), the function v x( , )� is such that
D v x D v
i
m
i j ij
j
m
�
�
�
�
� %� �
( , )| ( )|( ) ( )
� � �
23
1
23
1
. (25)
Estimating the total derivatives D vij
�
%( )|( )23
, it is easy to derive expressions for the elements cij
( , )� � such that the
following inequalities hold:
� � � �
i j ij i j ijD v x x c�
% ( )| | | | | | | | | ( , )( )23
(26)
for all i, j = 1, 2, …, m, cij
( , )0 0 0� , and c cij ji� for all i, j = 1, 2, …, m. The symbol � indicates the presence of some real
constants in the expressions for cij
( , )� � .
Taking into account estimates (26) for expression (25), we obtain the estimate
D v x x C x�
( , )| (| | | | ) ( , ) (| | | | )( )
� � � � �
23 1 1
T, (27)
where C cij
( , ) [ ( , )]� � � �� is a constant (m m� )-matrix. The statement below follows from (24) and (27).
Theorem 4. Let for system (22) there exist a matrix function U xn m m
( ):� �
�
and a vector ���
�m
such that
estimates (24), (27) hold and the system of inequalities
A( )� � 0, B( )� � 0, C( , )� � � 0
have a solution � �
1
* *, ,�
msuch that 0 � �� &
i
*for any & � 0 for all x
n�� . Then the zero solution of system (22) is
asymptotically stable in the large.
Proof. If the conditions of Theorem 4 are satisfied, the function v x( , )� is positive definite, decreasing, and radially
bounded. The total derivative D v x�
( , )� along the solutions of system (23) is negative definite in the large. These conditions are
sufficient for the state x � 0of system (22) to be asymptotically stable in the large.
Example 5. Following [60, 83], we will consider a mechanical system consisting of three pendulums of equal mass m
and length l. The pendulums are connected with springs of stiffness c attached by their ends to the bars of the pendulums at a
height h from the point of suspension. When at rest, the springs are not deformed. Let the angles of deviation of the pendulums
�
i, i = 1, 2, 3, from the equilibrium position be generalized coordinates. The masses of the bars and springs are neglected.
The motion of this system is described by the equations
ml mgl h c h ci i i j i k
2 2 20�� ( ) ( )� � � � � �� � � � � � , (28)
i, j, k = 1, 2, 3, i j� , k j� , k i� . Let us decompose system (28) into three free subsystems that describe the oscillations of
uncoupled pendulums:
ml mgli i
20��� �� � , i = 1, 2, 3.
When the deviations �i, i = 1, 2, 3, are small and quantities of the third order of smallness are neglected, the total energy of the ith
free subsystem is given by
483
E ml
mgl
i i i i( ) �� � �� �
1
2 2
2 2, i = 1, 2, 3. (29)
The motion of the ith and jth subsystems, independent of the third subsystem, is described by the equations
ml mgl h ci i i j
2 20�� ( )� � � �� � � � ,
ml mgl h cj j j i
2 20�� ( )� � � �� � � � , (30)
i, j = 1, 2, 3, i j� . Calculating the total energy of system (30) and subtracting the energy of the free subsystems i and j (29), we
obtain the energy of coupling between the ith and jth subsystems:
E h cij i j i j
( , ) ( )� � � �� �
1
2
2 2, i, j = 1, 2, 3, i j� .
We choose a matrixU ( )� , � � � �� [ , , ]1 2 3
, with the following elements:
v Eii i i i
( ) ( )� �� , v Eij i j ij i j
( , ) ( , )� � � ��
1
2
, i, j = 1, 2, 3, i j� .
The function v U( , ) ( )� � � � ��
T,��
�
�3
, is the total energy of system (28). Thus, the matrixU ( )� is of energy nature.
Note that D v�
�( , )|( )
� �
280 and the function v( , )� � is positive definite. Hence, the state �
i� 0, i = 1, 2, 3, of system
(28) is stable.
Many papers and monographs (see [82, 83, 86, 87]) report on the results of development of the method of matrix-valued
functions and formulate constructive sufficient stability conditions for the following classes of perturbed equations of motion:
(i) continuous-time (independent and nonautonomous) systems [86];
(ii) discrete-time systems [83];
(iii) systems with Ito and Katz–Krasovsky random parameters [86];
(iv) singularly perturbed systems, including those with structural perturbations [86];
(v) hereditary systems [87];
(vi) systems of the form (1) and impulsive systems with uncertain parameters [30];
(vii) dynamic equations [9] on time scales [30];
(viii) equations defined on metric spaces [85].
There are several ways to set up matrix auxiliary functions for these classes of perturbed equations of motion.
5. Comparison System for Matrix Differential Equations. The study of matrix differential equations is motivated by
their application for modeling some processes in optimal control theory, differential-game theory, theory of the stability of
motion, and other fields of applied mathematics. One of the fruitful methods of the qualitative theory of these equations is the
generalized Lyapunov’s direct method based on matrix-valued functions (see [29, 95] and the references therein).
Theorems 1 and 3 underlie the comparison principle for matrix systems. This principle allows us to analyze the dynamic
properties of the solutions of system (31) based on a matrix-valued function in a way similar to the analysis of systems of
ordinary differential equations based on matrix-valued Lyapunov functions (see [82]).
Consider the matrix system of equations
dX
dt
F t X� ( , ), X t X( )0 0
� , (31)
where Xn n
�
�
� and F is a matrix-valued function that ensures the existence and uniqueness of the solution X t X t t X( ) ( ; , )�
0 0
of system (31) on the interval T t0 0� �[ , ), t
00� , for ( , )t X T N
0 0 0� � (N
n n�
�
� is an open connected domain). Let
F t X( , ) � 0for all t t� �[ , )0
if Xn n
� �
�
0 � .
Assume that a matrix-valued function U Tn n n n
:0�
� �
� � has been constructed in some way for system (31). For
( , )t X T N� �
0, we define matrix-valued functions
484
D U t x U t h X hF t X U t X h h�
� �
� � � � ( , ) lim inf{[ ( , ( , )) ( , )] : }1
0 , (32)
D U t x U t h X hF t X U t X h h� � �
� � � � ( , ) lim inf{[ ( , ( , )) ( , )] : }1
0 , (33)
which are the lower left and upper right Dini derivatives of the matrix-valued functionU t X( , )along the solutions of system (31).
Note that the functionU t X XX( , ) �T
, where Xn n
�
�
� , can be used as an elementary matrix-valued functionU t X( , ).
Then Eqs. (32) and (33) take the form
D U t X D U t X U X t XX XX XF t X F t�
�
� � � � � �( , ) ( , )�
( ( ))� �
( , ) ( ,T T T
X X)T
.
Let there exist a matrix-valued function G t U( , ) such that
D U t X G t U t X�
( , ) ( , ( , )) (34)
for all ( , )t X T N� �
0, where G C T
n n n n� �
� �
( , )0
� � .
The matrix differential inequality (34) is associated with the matrix differential equation
dY
dt
G t Y� ( , ), Y t Yn n
( )0 0
� �
�
� . (35)
Definition 5. The matrix differential equation (35) is a comparison system for the matrix differential equation (31) if the
set of solutions of Eq. (35) includes a solution Y t( ) related to the solutions X t( ) of matrix system (31) asU t X Y( , )0 0 0
and
U t X t Y t( , ( )) ( ) for all t t�
0.
Note that for each X t( ), there is a separate solution Y t( ).
The set of indices ' �{( , ):i j i = 1, 2, …, n; | j = 1, 2, …, n} is divided into subsets P and Q so that P Q( �) and
P Q* �' and the matrices [ ]Ylk
and [ ]Ymr
are square for any ( , )l k P� and ( , )m r Q� .
Definition 6. The matrix-valued function G t Yn n n n
( , ):� � ��
� �
� :
(i) is mixedly quasimonotonic if the set of indices ' �{( , ):i j i = 1, 2, …, n; j = 1, 2, …, n} can be divided into subsets P
and Q such that
(a) for any ( , )l k P� and ( , )m r Q� , the matrix-valued functionG t Y( , )is nondecreasing inYlk
and nonincreasing inYmr
or
(b) for any ( , )l k P� and ( , )m r Q� , the matrix-valued functionG t Y( , )is nonincreasing inYlk
and nondecreasing inYmr
;
(ii) is quasimonotonic nondecreasing in Y if G t Y( , ) is nondecreasing in Yij
for all ( , )i j �', and quasimonotonic
nonincreasing if G t Y( , ) is nonincreasing in Yij
for all ( , )i j �'.
The following notation is used below:
Y YP ij� [ ] is an n n� -matrix with arbitrary elements Y
lk, ( , )l k P� , Y
mr� 0, ( , )m r Q� ;
Y YQ ij� [ ] is an n n� -matrix with arbitrary elements Y
mr, ( , )m r Q� , Y
lk� 0, ( , )l k P� .
Along with the matrix equation (31), we will consider the matrix equations
dY
dt
G t YP
P� ( , ), Y t Y
P P( )
0 0� , (36)
dY
dt
G t YQ
Q� ( , ), Y t Y
Q Q( )
0 0� , (37)
where G t Y G t YP P
( , ) ( , )� and G t Y G t YQ Q
( , ) ( , )� .
Let Y tP
( ), Y tQ
( ) be maximum and Y tP
( ), Y tQ
( ) be minimum solutions of Eqs. (36) and (37), respectively.
Definition 7. The solution Y t( ) of the matrix comparison equation (35) is called:
(à) P-maximum–Q-minimum if
Y t Y tP P
( ) ( ) and Y t Y tQ Q
( ) ( )� (38)
485
for all t t t a� �[ , )0 0
;
(b) P-minimum–Q-maximum if
Y t Y tP P
( ) ( )� and Y t Y tQ Q
( ) ( ) (39)
for all t t t a� �[ , )0 0
.
If inequalities (38) hold, then the solution Y t( ) of system (35) is maximum when P � ' and minimum when P �).
If inequalities (39) hold, then the solution Y t( ) of system (35) is minimum when P � ' and maximum when P �).
Hereafter the inequalities between the matrices are element-wise.
Considering some results of [95], we will show that the following statements hold.
Theorem 5. Let systems (31) and (35) be such that
(i) there exists a matrix-valued functionU t X C Tn n n n
( , ) ( , )� �
� �
0� � locally Lipschitz in X;
(ii) there exists a matrix-valued function G t Y C Tn n n n
( , ) ( , )� �
� �
0� � mixedly quasimonotonic and such that
(a) for t t�
0, the following estimates hold:
U t X t Y tP P
( , ( )) ( )0 0 0
� , U t X t Y tQ Q
( , ( )) ( )0 0 0
� ;
(b) for t t�
0, the following differential inequalities are valid:
D U t X t G t U t XP P�
( , ( )) ( , ( , )), D Y t X t G t Y tP P�
�( , ( )) ( , ( )), (40)
D Y t G t Y tQ Q�
( ) ( , ( )), D U t X t G t U t XQ Q�
�( , ( )) ( , ( , )). (41)
Then the differential inequalities
U t X t Y tP P
( , ( )) ( )� , U t X t Y tQ Q
( , ( )) ( )� (42)
hold for all t t t a� �[ , )0 0
.
Proof. Denote M t Y t U t X tP P P
( ) ( ) ( , ( ))� � , M t U t X t Y tQ Q Q
( ) ( , ( )) ( )� � . According to condition (ii)(a) of
Theorem 5, we have M tij
( )0
0� for any division of the set of indices ( , )i j �'. Let estimates (42) be invalid. Then the set
T t t t a Mi j
n
ij� � �
�
{ [ , ): }, 0 01
0�
is not empty.
Let t T1� inf , t t
1 0� , and T be closed. Assume that for t T
1� , there exists a pair of indices ( , )l m P� such that
M tlm
( )1
0� and M tij
( )1
0� for ( , ) ( , )i j l m� and D M tlm�
( )1
0.
According to (40), we have
D M t D Y t U tlm lm lm� �
� � ( ) ( ( ) ( ))1 0 0
0,
D Y t D U t G t U t X tlm lm lm� �
( ) ( ) ( , ( , ( )))1 1 1 1 1
whence follows the inequality
G t Y t G t U t X tlm lm
( , ( )) ( , ( , ( )))1 1 1
� .
At the same time, M tlm
( )1
0� and, hence, Y t U tlm lm
( ) ( )1 1
� and M tij
( )1
0� for any ( , ) ( , )i j l m� . The mixed
quasimonotonicity of the matrix-valued function G t Y( , ) leads to the inequality
G t U t X t G t Y tlm lm
( , ( , ( ))) ( , ( ))1 1 1 1 1
,
which is contradictory. Similar reasoning for ( , )l m Q� leads to the contradiction
G t Y t G t U t X tlm lm
( , ( )) ( , ( , ( )))1 1 1 1 1
�
because of the mixed quasimonotonicity of the functionG t Y( , ). Thus, we have proved that the set T is empty. This completes the
proof of Theorem 5.
Define the norm of the matrix Y:
486
| | | | | |
,
/
Y Yij
i j
n
�
�
�
�
�
�
�
�
�
�
�
2
1
1 2
.
Theorem 6. Let the matrix-valued function G t y( , ) be defined and bounded in the domain
+ � � � � {( , ): [ , ), | | | | |}t Y t t t a Y Y B0 0 0
, i.e., there exists a constant M � 0such that | | ( , )| |G t Y M for all ( , )t Y �+. If G t Y( , )
is mixedly quasimonotonic, the comparison system (35) has a P-maximum–Q-minimum solution and a P-minimum–
Q-maximum solution for all t t t� �[ , )0 0
� where� � �min( , / ( ))a B M B2 2 .
Proof. For a given value of B, we choose 0 2� �e Bij
/ for all ( , )i j �1, 2, …, n so that | | | | /E B� 2. To decompose the set
' into subsets P and Q, we will consider the original problems
dY
dt
G t Y EP
P P� �( , ) , Y t Y E
P P P( )
0 0� � ,
dY
dt
G t Y EQ
Q Q� �( , ) , Y t Y E
Q Q Q( )
0 0� � .
Denote G t Y G t Y E G t Y eE ij ij
( , ) ( , ) [ ( , ) ]� , - , for all (i, j) = 1, 2, …, n. The matrix-valued function G t YE
( , ) is
defined on the set+E
t Y t t t a Y Y E B� � � � , {( , ): [ , ), | | ( )| | / }0 0 0
2 and the following estimate is valid:
| | ( , )| | | | ( , )| | | | | | | | | |G t Y G t Y E M E M
B M B
E � � � �
�
2
2
2
.
According to Peano’s theorem, the original problem (35) has a solution Y t E( , ) on [ , )t t0 0
�� , where
� � �min( , / ( ))a B M B2 2 , for p P� and q Q� .
Let matrices 02 1
� � E E E be given such that
Y t E Y t E Y t E Y t EP P P P P P
( , ) ( ) ( , ) ( )0 2 0 2 0 1 0 1
� � � � � ,
Y t E Y t EQ Q
( , ) ( , )0 2 0 1
� or Y t E Y t EQ Q Q Q
( ) ( )0 2 0 1
� � � .
Moreover, the following inequalities hold:
dY t E
dt
G t Y t E EP
P P
( , )
( , ( , ))2
2 2 � ,
dY t E
dt
G t Y t E EQ
Q Q
( , )
( , ( , ))2
2 2� � ,
dY t E
dt
G t Y t E E G t Y t E E
dYP
P P P P
( , )
( , ( , )) ( , ( , ))1
1 2 2 2� � � � �
Pt E
dt
( , )2
,
dY t E
dt
G t Y t E E G t Y t E E
dYQ
Q Q Q Q
( , )
( , ( , )) ( , ( , ))1
1 2 2 2� � � �
Qt E
dt
( , )2
,
whence follows that Y t E Y t EP P
( , ) ( , )2 1
� and Y t E Y t EQ Q
( , ) ( , )2 1
� for all t t t� �[ , )0 0
� . The reasoning is similar for any
matrix element of the sequence En
, n = 1, 2, … . Hence, lim ( , ) ( )
En
n
Y t E Y t
�
0
uniformly in t t t� �[ , )0 0
� , where Y t( ) is the
solution of the original problem (35).
To show thatY t( )is a P-maximum–Q-minimum solution of the comparison equation (35) on[ , )t t0 0
�� , it is necessary
to prove that the inequalities Y t Y tP P
( ) ( ) and Y t Y tQ Q
( ) ( )� hold for all t t t� �[ , )0 0
� .
Let Y t( ) be some solution of system (35) on [ , )t t0 0
�� . The sequence of inequalities
Y t Y t EP P
( ) ( , )0 0
� , Y t Y t EQ Q
( ) ( , )0 0
� ,
487
dY t
dt
G t Y t EP
P P
( )
( , ( ))� � ,
dY t
dt
G t Y t EQ
Q Q
( )
( , ( ))� � ,
dY t E
dt
G t Y t E EP
P P
( , )
( , ( , ))� � ,
dY t E
dt
G t Y t E EQ
Q Q
( , )
( , ( , )) � ,
where | | | |EP
, | | | | /E BQ
� 2, yield Y t Y t EP P
( ) ( , )� and Y t Y t EQ Q
( ) ( , )� for t t t� �[ , )0 0
� and
Y t Y t E Y tP
EP P
( ) lim ( , ) ( ) �
0
, Y t Y t E Y tQ
EQ Q
( ) lim ( , ) ( )� �
0
for all t t t� �[ , )0 0
� .
The existence of the P-minimum–Q-maximum solution of system (35) is proved in a similar way.
Theorem 7. Let systems (31) and (35) be such that
(i) there exists a matrix-valued functionU t X C Tn n n n
( , ) ( , )� �
� �
0� � locally Lipschitz in X;
(ii) there exists a matrix-valued functionG t U C Tn n n n
( , ) ( , )� �
� �
0� � quasimonotonically nondecreasing in U for all
t ��
� such that D U t X t G t U t X�
( , ( )) ( , ( , )) for all ( , )t Xn n
� �
�
�
� � ;
(iii) there exist a solution X t( ) of system (31) and the maximum solution Y t( ) of the comparison system (35) for all
t t�
0.
Then if
U t X Y( , )0 0 0
� , (43)
then the estimate
U t X t Y t t Y( , ( )) ( ; , )�
0 0(44)
holds for all t t�
0.
Proof. Let X t X t t X( ) ( ; , )�
0 0be any solution of system (31) for which U t X Y( , )
0 0 0� . Since the matrix-valued
function M t U t X t t X( ) ( , ( ; , ))�
0 0is locally Lipschitz in X, we have
M t M t K t X t F t X( ) ( ) | | ( ) ( ) ( , )| |� � � � �� � �
� � � �U t X t F t X t U t X t( , ( ) ( , ( ))) ( , ( ))� � , (45)
where � � 0 is indefinitely small and K is a constant matrix. From (45) it follows that
D M t D U t X t G t U t X t G t M t� �
�( ) ( , ( )) ( , ( , ( )) ( , ( )).
Let us first show that the matrix inequality (44) holds component-wise on the interval [ , )t t0 0
� . , where . is an
indefinitely small positive number. If inequality (43) is valid, this follows from the continuity of the elements of the matrices
U t X t( , ( )) and Y t( ).
Let there exist a pair of indices ( , )r s �' such thatU t X Y trs rs
( , ) ( )0 0 0
� . Since the matrix-valued function G t U( , ) is
quasimonotonic, we obtain
D U t X G t U t X G t Y t D Y trs rs rs rs
� �
� �( , ) ( , ( , )) ( , ( )) (0 0 0 0 0 0 0
).
Then D U t X D Y trs rs
� �
� �( , ) ( )0 0 0
0and, consequently, there exists . � 0such that the estimateU t X t Y trs rs
( , ( )) ( )0
� is valid
for all t t t� �[ , )0 0
. and for ( , )r s �'.
Let us show that inequality (44) holds on the interval [ , )t t0 0
�� . Let t t t t*
inf{ [ , )}� � �
0 0� for which inequality (44)
becomes an equality for at least one pair of indices ( , )r s �'. Let U t X t Y trs rs
( , ( )) ( )* * *
� and inequality (44) hold for
t t t� ( , )*
0and for all ( , )i j �1, 2, …, n. As above, it can easily be shown that D U t X t D Y t
rs rs
� �
� �( , ( )) ( )* * *
0. In this case,
488
there exists � � 0such thatU t X t Y trs rs
( , ( )) ( )� for t t t� �[ , )* *
� . But this contradicts the choice of t*
. Hence, inequality (44)
is valid for all t t t� �[ , )0 0
� .
The application of Theorems 5–7 actually involves setting up a matrix-valued function for the class of matrix equations
under consideration. We will outline a procedure for setting up such a function for one class of matrix differential equations.
5.1. Lyapunov Function for the Class of Linear Nonstationary Matrix Systems. Consider the matrix system of
equations
dX
dt
A X A t X1
11 1 12 2� � ( ) , X t X
1 0 10( ) � ,
dX
dt
A X A t X2
22 2 21 1� � ( ) , X t X
2 0 20( ) � , (46)
where X Xn n
1 2, �
�
� ; A An n
11 22, �
�
� are constant matrices; and A t12
( ), A t21
( ) are quasiperiodic matrix functions. Let
A t A ek iw t
k N
N
k
12 12( )
( )�
��
�, A t A e
k iw t
k N
N
k
21 21( )
( )�
��
�,
where wk��, w w
k k�
� � , wk� 0, A
k
12
( ), A
k n n
21
( )�
�
� (� is the set of complex numbers), A Ak k
12 12
( ) ( )�
� , A Ak k
21 21
( ) ( )�
� ,
A12
00
( )� , A
21
00
( )� .
The direct sum X X X
X
X
� / �
�
�
�
�
�
�
�
�1 2
1
2
0
0
is called the state vector of system (46).
Denoting by �1
the set of positive semi-definite matrices that form the closed convex cone of the real linear space of
symmetric n n� -matrices, we obtain the set � ��1
2as the cone of symmetric 2 2n n� -matrices of the phase space � of system
(46). Here
(i) if A ��, � 0, then A ��;
(ii) it A ��, B ��, then A B� ��;
(iii) if A ��, � �A �, then A � 0.
Since the phase space of system (46) forms a subset of the set of real 2 2n n� -matrices with semiorder induced by the
cone of positive semi-definite matrices of the corresponding order, it is also semiordered with the same order relation [18]: A B�
�
if A B� ��. Also A B�
�
if the matrix A B� is positive definite, i.e., A B� �
�
0, where 0 is a zero matrix of the same order.
The cone � is bodily (the set �
�0
0� �{ : }X X of its interior points is not empty), generating, and normal.
Let U ��, U � 0. An element X �� is called measurable if the estimate � � �U X U
� �
is valid for some � � 0. The
minimum value of � is called the U-norm [76] of the element X and is denoted by | | | |XU
.
A unit ball in the set�U
of all U-measurable elements of the space�, which are block-diagonal symmetric matrices, is a
will be conical segment 0� 1 � � U U X U X U, { : }
� �
.
Let us set up a matrix-valued function for system (46):
U t X X P t X
X
X
P P t
P t P
( , ) ( )
( )
( )
� �
�
�
�
�
�
�
�
�
T
T
T T
1
2
11 12
12 22
0
0
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
X
X
1
2
0
0
,
where P11
1
0�
�
, P22
1
0�
�
, P t Cn n
12
1( ) ( , )�
�
� � . If
P P t P P t22 12 11
1
12
1
�
�
�
T( ) ( ), t t�
0, (47)
489
then the matrixU t X( , ) is positive semidefinite in view of the symmetry of the matrix P t( ).
Introduce a two-component (cone-valued) Lyapunov function:
V t X H H V t X H V t X H( , , , ) ( , , ) ( , , )1 2 1 1 2 2
� / , (48)
whereV t X H H U t X H1 1 1 1
( , , ) ( , )�
T,V t X H H U t X H
2 2 2 2( , , ) ( , )�
Tand H
1, H
n n
2
2�
�
are some matrix parameters. With (47),
we obtain the following inequality for function (48):
V t X H1 1
1
0( , , ) �
�
, V t X H2 2
1
0( , , ) �
�
, V t X H H( , , , )1 2
0�
�
, t t�
0. (49)
Let us find the total derivative of function (48) with respect to time along the solutions of system (46):
dV
dt
H
d
dt
X P X X P t X X P t Xi
i
( )
( ( ) ( )
24
1 11 1 1 12 2 2 12 1� � �
T T T T T� X P X H
i2 22 2 46
T)|
( )
� � � �H X A P P A P t A t A t P ti
T T T T T
1 11 11 11 11 12 21 21 12( ( ) ( ) ( ) ( ))X
1
�
�
�
�
� � � �X A P P A P t A t A t P t XT
2 22 22 22 22 12 12 12 12 2
T T T( ( ) ( ) ( ) ( ))
� � � � �X
dP
dt
A P t P t A P A t A t P1
12
11 12 12 22 11 12 21
T T T( ) ( ) ( ) ( )
22 2
�
�
�
�
�
�
�
�
X
� � � � �X
dP
dt
A P t P t A P A t A2
12
22 12 12 11 22 21 12
T
T
T T T T( ) ( ) ( ) (t P X H
i)
11 1
�
�
�
�
�
�
�
�
�
�
�
�
.
Assuming that the matrix function P t12
( ) satisfies the linear differential equation
dP
dt
A P t P t A P A t A t P12
11 12 12 22 11 12 21 22� � � � �
T T( ) ( ) ( ) ( ) ,
we seek its solution in the following form, given constraints on the matrices A11
and A22
(see [24]):
P t P ek iw t
k N
N
k
12 12( )
( )�
��
�,
where Pk n n
12
( )�
�
� . It can be verified that
dV
dt
dV
dt
dV
dt( ) ( ) ( )46
1
46
2
46
� / , (50)
where
dV
dt
H X S t XHi
i i
( )
( )
46
�
T T, i = 1, 2, (51)
S t S t S t( ) ( ) ( )� /
1 2,
S t A P P A P t A t A t P t1 11 11 11 11 12 21 21 12
( ) ( ) ( ) ( ) ( )� � � �
T T T,
S t A P P A P t A t A t P t2 22 22 22 22 12 12 12 12
( ) ( ) ( ) ( ) ( )� � � �
T T T.
490
If
S t1
1
0( )
�
, S t2
1
0( )
�
,
the expression
dV
dt( )46
is negative semidefinite with respect to the cone � for all t t�
0.
Along with (46), we will consider a comparison system for (46):
dY
dt
G t Y� ( , ), Y t Y( )0 0
� , (52)
where G T:0� � � is a continuous function guaranteeing the existence of a unique solution such that G t( , )0 0� .
Definition 8. The solution Y - 0 of Eq. (52) is stable in � if for any $ � 0, t0
0� , there exists . � 0 such that
Y t t Y( ; , )0 0
��$
follows from Y0��
.
for all t0
0� , where �$
$ $� 0� 1U U, .
Definition 9. The solution Y - 0 of Eq. (52) is uniformly stable in � if for any $ � 0, there exists . � 0 such that
Y t t Y( ; , )0 0
��$
follows from Y0��
.
for all t t�
0, where t
00� is some initial time.
Definition 10. The solutionY - 0of Eq. (52) is uniformly asymptotically stable in � if it is uniformly stable in � and for
some h � 0, lim | | ( ; , )| |
tU
Y t t X
�
�
0 00 (uniformly in t
0) follows from Y
h0�� .
Theorem 8. Let the system of equations (46) be such that there exist constant symmetric n n� -matrices P11
, P22
and
constant 2n n� -matrices H1, H
2such that
(i) all successive principal minors of the matrices P11
, P22
are positive;
(ii) all successive principal minors of the matrix P P t P P t22 12 11
1
12�
�T( ) ( ) are positive for all t t�
0;
(iii)U t X( , ) � 0 if and only if X � 0;
(iv) rank H H n1 2� �rank ;
(v) all successive principal minors of the matrices �S t1
( ), �S t2
( ) are nonnegative for all t t�
0.
Then the solution X � 0of system (46) is uniformly stable.
Proof. Let the cone-valued functionV t X H H( , , , )1 2
defined by (48) be an auxiliary function. It follows from conditions
(i)–(iv) of Theorem 8 and estimates (47), (49) that this function is positive semidefinite with respect to the cone �. Since the
negative semidefiniteness of the derivative dV dt/ follows from condition (v) of Theorem 8 and expressions (50), (51) for the
total derivative of the functionV t X H H( , , , )1 2
along the solutions of system (46), it is natural to useG t Y( , ) - 0as the right-hand
side of the comparison system (52) follows and to set Y0
0�
�
. According to Wazewski’s lemma (see [98]) and Theorem 7, we
have
01 2 0 0 0 1 2 0
-
� �
V t X t H H Y t t V t X H H Y( , ( ), , ) ( ; , ( , , , )) . (53)
Thus, since the state Y - 0 is uniformly stable, for any $ � 0 there exists . $ $
0( ) � such that that for any
| | | | | | ( ; , ( , , , ))| |Y Y t t V t X H HU U0 0 0 0 0 1 2 0� � . ,
we have | | ( ; , ( , , , ))| |Y t t V t X H HU0 0 0 0 1 2� $. By continuity, for each $ there exists . $( ) � 0 such that the inequality below
follows from the inequality | | | |X0
� .:
| | ( , , , ))| | | | | |V t X H H Y
C
0 0 1 2 0
2 2
� �
$
2
,
where2 and C are some finite positive constants determined below. After inequality (53), considering that the cone � is normal
(| | ( , , , ))| | | | | |V t X H H Y1 2 0
2 , where 2 is the normality constant), we obtain the estimate | | ( , , , ))| |V t X H H C1 2
2 2� $ . Also
| | ( , , , ))| | | | ( , , )| | | | ( )/ /
V t X H H V t X H H X P t Pi i i
T T
1 2
1 2 1� �
2 1 2 2( ) | | | | ( ) | |
/t XH P t XH
i i� , i = 1, 2,
491
where | | | |% is a spectral matrix norm, so that
| | ( ) | |/
P t XH Ci
1 2� $. (54)
It is easy to show that the scalar function
�( ) | | ( ) | |/
X P t XHi
�
1 2, t t�
0, (55)
has the following properties:
(i) �( )X � 0 if and only if X � 0;
(ii) � �( ) | | ( )cX c X� for all complex numbers c and any matrix X ��;
(iii) � � �( ) ( ) ( )X X X X� 3 � 3 for all X X, 3 ��.
This means that the function�( )% defined by (55) specifies, on the phase space of system (46), a matrix norm equivalent
(due to the finite dimension of the space �) to the norm | | | |% . If there exists a positive constant C such that | | | | ( ) /X X C � , then
the estimate | | ( ; , )| |X t t X0 0
� $ is valid in view of (54). Since the number $ is random and the choice of . is independent of time
t0
, deriving this estimate from the condition | | | |X � .completes proving the uniform stability of the state X � 0of system (46).
Theorem 9. Let the system of equations (46) be such that there exist constant symmetric n n� -matrices P11
, P22
and
constant 2n n� -matrices H1, H
2such that
(i) all successive principal minors of the matrices P11
, P22
are positive;
(ii) all successive principal minors of the matrix P P t P P t22 22 11
1
12�
�T( ) ( ) are positive for all t t�
0;
(iii)U t X( , ) � 0 if and only if X � 0;
(iv) rankH1
= rankH2
= n;
(v) all successive principal minors of the matrix �S t1
( ), �S t2
( ) are positive for all t t�
0.
Then the solution X � 0of system (46) is uniformly asympotically stable.
Proof. It follows from conditions (i)–(iv) of Theorem 9 and estimates (47), (49) that the function V t X H H( , , , )1 2
is
positive semidefinite.
Let $ be a positive number. According to Theorem 8, there exists a number . � 4 such that | | ( ; , )| |X t t X0 0
� $ follows
from | | | |X0
� . for all t t�
0. Assume that for t t�
0, the solution X t t X( ; , )
0 0is beyond an open ball J
.
of radius .centered at
the point X Xn n
1 20� � �
�
� . Since the solution X t t X( ; , )0 0
is bounded for all t t�
0, the semitrajectory X t t X( ; , )
0 0for
t t�
0lies within some spherical segment D J J
R� \
.
within which
dV
dt
�
�
0 according to condition (v) of Theorem 9.
Using the concept and determining property of the cone � �*
{ :{ , }� � �5 5 X 0 for all X ��} conjugate to � (see
[18]), we can state that the inequality ( , / )6 dV dt � 0, where6��*
, holds at all points of the compact set D, whence
sup ( , / )
|| ||6
6
�
�
� � �
1
0
X D
dV dt "
and the estimate ( , / )6 dV dt �" is valid for all6��*
, X D� . Therefore
( , / ) ( )6 dV ds ds t t
t
t
0
0� � �" .
Finally, we have ( , ( , , , )) ( , ( , , , )) ( )6 6V t X H H V t X H H t t1 2 0 0 1 2 0
� �" . This conclusion is contradictory because
the function V is negative definite when t increases without limit, though we must have ( , ( , , , ))6 V t X H H1 2
0� . Thus, the
solution X t t X( ; , )0 0
falls into the ball J.
at some time, but the number . is selected so that, after finding itself in J.
, the
solution X t t X( ; , )0 0
can no longer leave J$
. Since $ is an arbitrary number, we have lim lim | | ( ; , )| |
t
X t t X
�
�
0 00. The theorem
is proved.
Example 6 [76]. As an application of this way of constructing a matrix Lyapunov function, we will consider the linear
system of differential equations
492
dX
dt
X e XJt1
1 2� �� �
2
,
dX
dt
X e XJt2
2 1� �
�
� .
2
, (56)
where Xi�
�
�2 2
, i = 1, 2, �, �, �, ., 2��, 2 � 0, J �
�
�
�
�
�
�
�
0 1
1 0
.
Let us first set up a matrix-valued function U t X X P t X( , ) ( )�
Tfor system (56), assuming that
P t
I P t
P t I
( )
( )
( )
�
�
�
�
�
�
�
�
�
12
12
T, where I is a unit matrix of the second order and P t
12( ) satisfies the matrix differential equation
dP
dt
P t eJt12
12� � � � �( ) ( ) ( )� � � .
2
.
This equation has the following bounded solution:
P t I J eJt
122 2
( ) (( ) )� �
�
�
� �
� .
7� � �8 2
� � 2
2
.
Assuming that H H I I1 2� � ( , )
Tand considering (48), we fold the matrix U to obtain the cone-valued function
V t X H H V t X H V t X H( , , , ) ( , , ) ( , , )1 2 1 1 2 2
� / , where
V V X X X X X P t X X P t X1 2 1 1 2 2 1 12 2 2 12 1� � � � �
T T T T T( ) ( ) .
Thus, conditions (i)–(iv) of Theorem 9, which ensure the positive definiteness of the Lyapunov function for system
(56), are reduced to the inequality
( ) ( )� � 2 � .� � � � �
2 2 20.
The negative definiteness of the total derivative (50), (51) of this function along the solutions of system (56) is
equivalent to the negative definiteness of the matrices
S I1
2 2
2� �
� �
� �
�
�
�
�
�
�
�
�
�
. � . � �
� � 2
( )( )
( )
, S I2
2 2
2� �
� �
� �
�
�
�
�
�
�
�
�
�
� � . � �
� � 2
( )( )
( )
.
Therefore, condition (v) of Theorem 9 is reduced to the compatibility condition for the system of inequalities
� � � 2 . � . � �(( ) ) ( )( )� � � � � �
2 20,
� � � 2 � � . � �(( ) ) ( )( )� � � � � �
2 20.
Fixing, for example, values of�, �,2, it is possible to find the domain of uniform asymptotic stability of the system in the
parameter space ( ; )� . .
Remark 5. Example 6 of a linear system of the eighth order with periodic coefficients observed in the phase space
formed by a set of quasidiagonal real matrices of the fourth order with blocks of equal dimension (n n1 2
2� � ) demonstrates the
possibility of the stability analysis of matrix systems of differential equations based on Lyapunov functions without the need to
“vectorize” matrix systems. The procedure of applying matrix-valued functions (see [86, etc.]) permits generalizing matrix
differential equations based on the comparison principle with a matrix-valued Lyapunov function. The constructivity of the
procedure of setting up Lyapunov functions allows us to solve some applied problems modeled by matrix systems of differential
equations with evolving relations among their subsystems.
493
6. Linear Monotonic Systems. Linear monotonic systems play a significant role in the method of vector Lyapunov
functions as comparison systems. The way they are constructed depends on the type of stability, the form of the components of
the vector Lyapunov function, and the procedure of estimating the total time derivative of these components along the solutions
of the subsystems of the complex system. There are satisfactory methods to set up linear comparison systems (see [35] and the
references therein). In what follows, we will discuss the general stability criterion for linear autonomous comparison systems.
6.1. Autonomous Systems. Consider an autonomous linear system
du
dt
Pu� , (57)
where um
�� , P is an (m m� )-matrix, P Pij
� [ ], i j m, [ , ]� 1 .
Definition 11. A real (m m� )-matrix W is called a Metzler matrix (M-matrix) if
w
i j
i j i j mij
� �
� � �
9
:
;
0
0 1 2
, ,
, , , , , , .�
Definition 12. The autonomous linear system (57) is called a comparison system if and only if pij� 0, i j� (i j m, [ , ]� 1 ).
Lemma 5. Let P be an (m m� )-matrix such that pij� 0, i j� (i j m, [ , ]� 1 ). If w t
m:[ , ]
0� � is such that
dw
dt
Pv� ,
and v tm
:[ ,
~
]0� � is the solition of the Cauchy problem
dv
dt
Pv� , v t v( )0 0
� ,
then the estimate w t t w v t t v( ; , ) ( ; , )0 0 0 0
holds for all t t t� ([ , ) [ ,
~
)0 0� � only if w v
0 0 .
The proof of Lemma 5 can be found in [7]. Linear comparison systems have some equivalence properties (see [12, 99]
and the references therein).
Lemma 6. Let P in (57) be an M-matrix. Then the following statements are equivalent:
(a) the matrix P is stable (all Re ( ) .
jP � � � 0, where
jare the roots of the equation det( )P E� � 0);
(b) the Sevastyanov–Kotelyanskii conditions
( )� �1 0
11 12 1
21 22 2
1 2
k
k
k
k k kk
p p p
p p p
p p p
�
�
� � � �
�
, k = 1, 2, …, m
are satisfied;
(c) if the vector b b b bm
� ( , , , )1 2
�
Tsatisfies the equation P b c
T� � , then always b � 0 for any c � 0;
(d) there exists a diagonal matrix B with positive diagonal such that the matrix P B BPT
� is negative definite. In the
specific case where B E� (E is a unit (m m� )-matrix), the matrix P PT� is negative definite.
Theorem 10. Let P in (57) be an M-matrix. The isolated state u � 0 of the comparison system (57) is asymptotically
stable if and only if there exists a positive vector y y y ym0 10 20 0
� ( , , )�
Tsuch that
P yT
00� , y
00� .
Proof. Necessity. Let the state u � 0of system (57) be asymptotically stable. Then the matrix P is stable (i.e., the roots of
the characteristic equation det( )P E� � 0 satisfy the condition Re ( ) .
jP � � � 0, < �j m[ , ]1 ). Since statements (a) and (c) of
Lemma 6 are equivalent, there exists a vector y0
0� such that P yT
00� .
494
Sufficiency. Let there exists a vector y0
0� such that P yT
00� . According to condition (d) of Lemma 6, there exists a
diagonal matrix B with positive diagonal such that the matrix P B BPT
� is negative definite. Let us selectV y By�
T(y > 0) as a
Lyapunov function. After some transformations, we get
D v y P B BP y�
� �
T T( ) , < � �
�
( )ym
0 � .
Since the matrix P B BPT
� is negative definite, we have D v�
� 0if y0
0� . This proves the asymptotic stability of the
state u � 0of system (57).
Lemma 6 and Theorem 10 are effectively tested conditions for the asymptotic stability of the state x � 0 of perturbed
linear autonomous equations of motion based on a Lyapunov function.
Let us discuss one criterion of exponential stability for the state u � 0of system (57). The matrix P of system (57) can be
represented as P Q R� � , whereQ P P� �( ) /T
2, R P P� � �( ) /T
2. We use the matrices Q and R to construct a (m m�
2)-matrix
H R QR Q R Q Rm
�
�
( , , , , )2 1
� whose columns are the matrices R QR Q Rm
, , ,�
�1. Let R � 0if all diagonal minors of the matrix R
are greater than zero:
#
k
k
k kk
p p
p p
� �
11 1
1
0
�
� � �
�
, k = 1, 2, …, m,
and R � 0 if #k� 0, k = 1, 2, …, m.
Theorem 11. The isolated state u � 0of the comparison system (57) is exponentially stable if and only if R � 0or R � 0
and the rank of the matrix H is equal to m.
6.2. Reducible Comparison Systems. The necessary and sufficient conditions for the asymptotic stability of the state
u � 0 of system (57) cannot be directly generalized to nonstationary linear comparison systems. At the same time, some
nonstationary systems, including those reducible in the sense of Lyapunov can be analyzed. Definition 13. The matrix P t( ) is a
nonautonomous M-matrix if and only if
p
t t i j
t t i j i j mij
� < � � �
� < � � � �
9
:
;
0
0 1
0
0
, [ , ), ,
, [ , ), , , [ , ].
Definition 14. The nonautonomous linear system
du
dt
P t u� ( ) (58)
is called a reducible comparison system if P t( ) is a nonautonomous M-matrix and some Lyapunov transform can reduce it to the
system
dy
dt
By� (59)
with a constant M-matrix B.
Recall that the Lyapunov transform is the linear transform u L t v� ( ) , where L t( ) is a Lyapunov (m m� )-matrix.
Definition 15. A matrix L t C t( ) ([ , ))� �
1
0with generally complex elements is a Lyapunov matrix if
(a) L t( ) and dL t dt( ) / are bounded for all t t� �[ , )0
:
sup{| | ( )| | , [ , )}L t t t< � � � ��
0, sup{| | ( ) / | | , [ , )}dL t dt t t< � � � ��
0;
(b) |det ( )|L t a� � 0.
The following Erugin’s statement is true.
Theorem 12. A linear differential system is reducible if and only if its fundamental matrix X t( )can be represented as
495
X t L t Bt( ) ( )exp[ ]� . (60)
Proof. Necessity. Let the Lyapunov transform u L t y� ( ) have reduced system (58) to the form (59), where B is a matrix
constant. The fundamental matrix of system (59) has the form
Y t e CtB
( ) � ,
where C is a constant matrix (det C � 0). Since the fundamental matrix of system (59) is
U t L t Y t L t C Bt( ) ( ) ( ) ( ) exp[ ]� � ,
assuming that C E� , we obtain formula (60).
Sufficiency. Let formula (60) holds. Then L t X t Bt( ) ( )exp[ ]� � . In (58), we will perform the transformation
u X t Bt y� �( )exp[ ] .
Carrying out simple calculations, we find
X t e
dy
dt
X t e ByBt tB
( ) ( )� �
� .
Then
dy
dt
By� .
Hence, the comparison system (58) is reducible.
Since the constant matrix B in (59) is an M-matrix, Theorem 10 can be used to analyze it for stability.
Theorem 13. The state u � 0of the nonautonomous linear system (58) is asymptotically stable if
(a) system (58) is reducible;
(b) there exists a vector y y y ym0 10 20 0
� ( , , , )�
Tsuch that
B yT
00� , y
00� .
Proof. Since system (58) is reducible, there exists a Lyapunov transform that reduces it to the form (59). Theorem 10 is
applicable to system (59).
If the Lyapunov transform u L t y� ( ) reduces the nonautonomous comparison system (58) to the system
dy
dt
� 0
with a zero matrix B, the family of trajectories of system (58) in the space � �� �
�
mpermits biunique mapping onto the family of
parallel straight lines in the space � �� �
�
m.
Theorem 14. Let the comparison system (58) be such that
(i) all solutions u t( ) are bounded on [ , )t0� ;
(ii) for some constant a � ��, the following inequality holds:
SpP s ds a
t
t
( )
0
�� .
Then system (58) can be transformed into system (59) with a zero matrix B.
The proof of this theorem is similar to those in [13, 27]. The following statement is true.
Theorem 15. If the M-matrix P t( ) in (58) is absolutely integrable, i.e.,
496
| | ( )| |P s ds k
t0
�
�� � ��,
then this system is reduced to system (59) with a zero matrix B.
Definition 16. The nonautonomous M-matrix P with continuous bounded elements p tij
( ) has a quasidominant
diagonal if there are constants di
such that
d p t d p tj jj i ij
i
i j
m
| ( )| | ( )|�
�
�
�
1
, < � �t t[ , )0
, < �j m[ , ]1 .
Theorem 16. The state u � 0 of the nonautonomous linear system (58) is exponentially stable if the M-matrix P has
continuous bounded elements and a quasidominant diagonal.
Proof. Let us use the function 2( ) | |u d ui i
i
m
�
�
�
1
for which there exists an estimate d dt u2 =2/ ( ) � for all t t� �[ , )0
.
Here = � 0and can be found from the condition
| ( )| | ( )|p t d d p tjj j i ij
i
i j
m
� �
�
�
�
�
1
1
=,
which is valid because the M-matrix P has a quasidominant diagonal. Since | | | | | |u u and7 8
| | | |
/
u uii
m
�
��
2
1
1 2
, | | | |u uii
m
�
��
1,
we have
| ( )| | |exp[ ( )]u t u t t � �� =
0 0.
Here � �
�
d dM m
1, d d i m
M i� �max{ , [ , ]}1 , d d i m
m i� �min{ , [ , ]}1 . Theorem 16 is proved.
7. Stability of Nonlinear Monotonic Systems. For system (57) to be asymptotically stable, it is necessary and
sufficient that there exist a positive vector � such that P� � 0 (Theorem 10). This criterion is of peculiar interest. Its advantage
over the other criteria is that it can be extended to nonlinear autonomous comparison systems. Such an extension was done in
[31, 32].
Consider a nonlinear system
dy
dt
g y� ( ), (61)
where y y ym
� ( , )1
�
Tand g y( )is a vector function defined and continuous in the domainG y y H
m� � �{ : | | | | }� , 0 � � ��H .
Let system (61) have the following properties:
C1. A unique solution passes through each point of the domain G.
C2. g ( )0 0� and for all y G� such that y � 0, we have g y( ) � 0.
C3. System (61) is a Wazewski system, i.e., the components of the vector function g y( ) are quasimonotonically
increasing functions.
Recall that for the differentiable function g y( ) to quasimonotonically increase, it is necessary and sufficient that
>
>
g
y
j
i
� 0 for i j� , i, j = 1, 2, …, m.
Remark 6. If the condition C2 is satisfied, system (61) has an equilibrium position y � 0 that is isolated (there are no
other equilibrium positions in the domain G).
By virtue of the assumptions, the nonnegative cone ��
mis an invariant set for system (61).
497
Let us establish the asymptotic stability conditions for the zero solution in ��
m.
Let K y y i mm
i
00 1� � � �{ : , , , }� � , i.e., K
0is the interior of the cone K
�
.
Definition 17 [45]. The Martynyuk–Obolenskii (MO) condition is satisfied for system (61) if there exists a positive
vector ��G such that g ( )� � 0.
Theorem 17. For the zero solution of system (61) to be asymptotically stable in a nonnegative cone, it is necessary and
sufficient that this system satisfy the MO-condition.
The proof of this theorem is based on the following auxiliary statements.
Theorem 18 [42]. Let a continuous tangent vector field w y( )be given on a setQ y y r rr
m� � � � �
�
{ : | | | | , }� const 0 and
be such that the inequality z w yT
( ) � 0, where >Q Kr
\0
, holds for any y and z such that y Gr
�> , z Qr
�> , y zT
� 0. Then there
exists a point~y Q
r� such that w y(
~) � 0.
We choose a number r, 0 � �r H, and denote by Ar
a set of points y belonging toQr
and being such that for each y Ar
� ,
there exists a number � satisfying the condition g y y( ) � � . It is obvious that if for y Qr
� , there exists �( )y , then
�( )
( )
| | | |
y
y g y
y
�
T
2
. (62)
Lemma 7. For any r H� ( , )0 , the set Ar
is nonempty and is a compact set, while function (62) is continuous on Ar
and
equal to zero at no point of the set.
Proof. Let us first show that Ar�). Consider the vector field
w y g y
y g y
y
y( ) ( )
( )
| | | |
� �
T
2
. (63)
It is easy to verify that if y Qr
� , then y w yT
( ) � 0, i.e., the vector field (63) is tangent to Qr.
Let y Qr
�> , z Qr
�> , y zT
� 0. Then there exists an ordered set (i ip1
, ,� ), where 11 2
� � � i i i mp
� , 1 p m,
such that y1
0� for i i ip
�{ , , }1
� and yi� 0for i i i
p?{ , , }
1� . Since the right-hand sides of system (61) ae quasimonotonic, we
have g yi( ) � 0for all i i i
p?{ , , }
1� , and since the vectors y and z are orthogonal to each other, z
i� 0 for i i i
p�{ , , }
1� . Then
z w y z g y z g yi i
i i ip
T T( ) ( ) ( )
{ , , }
� � �
?
�
1
0
�
.
According to Theorem 18, there exists a vector~y Q
r� such that w y(
~) � 0. The last equality can be written as
g y y y(~
) (~
)~
� � , where �(~
)~
/| |~
| |y y y�
T 2.
Thus, we have proved that the set Ar
is not empty.
The vector function g y( ) is continuous in the domain G. Therefore, Ar
is a compact set, and the function �( )y is
continuous on Ar. Moreover, since g y y y( ) ( )� � and the equilibrium position y � 0is isolated, we have �( )y � 0on the set A
r.
Lemma 8. Let r H� ( , )0 . If
� max
( )
| | | |
� � �
�y Ar
y g y
y
T
2
0,
then there exists a point ��Qr
such that � � 0, g ( )� � 0.
Proof. If B K Ar
� ( �)
0, then for any point ��B we have � � 0, � �( ) � 0 and g ( ) ( )� � � �� . Hence, �( )q � 0.
Let B �). Then A Qr r� > . For each point y A
r� , we construct a neighborhoodU
ysuch that the following inequalities
hold for all z Uy
� :
| | | |z y
r
� �
2
, z g z
rT
( )
�
�
�
2
2
.
498
The sets Uy
form an open covering of the compact set Ar. According to [27], a finite subcovering U U
l1, ,� can be
isolated in this open covering. Let us set up continuous functions� �
1( ), , ( )y y
l� such that�
sy( ) � 0for y U
s� and�
sy( ) � 0
for y Us
? , s = 1, …, l.
Let em
� �( , , )1 1�
T� . Consider the vector field
w y g y
y g y
y y e y
y e y
s
s
ls
s
( ) ( )
( )
( )
( )� �
�
�
�
�
�
�
�
�
�
�
�
�
�
T
T�
�
1
1
l
�
�
�
�
�
�
�
�
�
. (64)
The vector field w y( ) has the following properties:
(a) w y( ) is tangent to Qr, i.e., y w y
T( ) � 0for all y Q
r� ;
(b) the function w y( ) is continuous on Qr;
(c) if y U Qss
l
r�
�
�
�
�
�
�(
�1
> , z Qr
�> , and z yT
� 0, then
z w y z g y
y g y z e
y y e ys
s
l
T T
T T
T
( ) ( )
( ( ))( )
( )
� �
�
�
�
�
�
�
�
�
�
�
��
1
�
s
s
l
y( )
�
��
1
0.
Since z eT
� 0 for y Uss
l
�
�1�
, we have y g yT
( ) � 0, �
ss
l
y( )�
��
10, y y e y
s
s
l
T�
�
�
�
�
�
�
�
�
�
�
�� ( )
1
0, and, as was shown in
proving Lemma 7, the inequality y g yT
( ) � 0holds for all y z Qr
, �> such that y zT
� 0;
(d) if y Q Ur ss
l
�
�
�
�
�
�
�
�
> \1
, z Qr
�> , and z yT
� 0, then z w y z g yT T
( ) ( )� � 0.
Thus, the conditions of Theorem 18 are satisfied for the vector field (64). Then, there exists a point ��Qr
such that
g
g
e
e
s
s
ls
s
l
( )
( )
( )
( )�
� �
� � � �
� � ��
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
T
T
1
1
�
�
�
�
�
. (65)
The point � does not belong to the set U Qss
l
r�
�
�
�
�
�
�(
1�
> because � �
Tg ( ) � 0 for ��
�
Uss
l
1�
, and since (61) is a
Wazewski system, at least one coordinate of the vector g ( )� must be nonnegative for � >� Qr. However, the point � cannot
belong to the set >Q Ur ss
l
\�
�
�
�
�
�
�
1�
either. Indeed, all the functions�s
y( ), s = 1, …, l, are equal to zero on this set, and relation (65)
becomes
g
g
( )
( )
| | | |
�
� �
�
��
T
2
.
On one hand, �� Ar, and on the other hand, the sets U U
l1,� form a covering A
rand, hence, �? A
r. We arrive at a
contradiction.
Thus, � >? Qr, and � �
Tg ( ) � 0. Hence, � � 0and g ( )� � 0.
Lemma 9. The sets D y G g ym�
�
� � * �{ : ( ) }� 0 , D y G g ym�
�
� � * { : ( ) }� 0 are positively invariant sets for
system (61).
499
Proof. Let~y D�
�
, and y t y( ,~
)be the solution of system (61) satisfying the initial condition y y y( ,~
)~
0 � . Denote by[ , )0 @
the right maximum interval of existence of this solution. Let us first prove that y t y ym
( ,~
) (~
)�
�
� , where
� ��
� � �
m my y y y(~
) { :~}.
Consider the function h y g y g y( ) ( ) (~
)� � . We have h y g y( ) ( )� , h y(~
) � 0. Hence, the system
dv
dt
h v� ( )
has a solution~
( )~
v t y- . Also
dv t
dt
g v t
~( )
(~
( )) ,
dy t y
dt
g y t y
( ,~
)
( ( ,~
))- , y y v y( ,~
)~
( )~
0 0� � .
But then, according to [41], the estimate y t y v t y( ,~
)~
( )~
� � holds for all t �[ , )0 @ , i.e., y t y ym
( ,~
) (~
)�
�
� .
Let us now choose two times t1
and t2
such that 01 2
� �t t @. We get y t t y y( ,~
)~
2 1� � . Then y t y y t y( ,
~) ( ,
~)
2 1� .
Hence, all the components of the vector y t y( ,~
) are nondecreasing functions on the interval [ , )0 @ . Therefore, q y t y( ( ,~
)) � 0 for
t �[ , )0 @ , i.e., D�
is a positively invariant set.
It can similarly be shown that if~y D�
�
, then the solution y t y( ,~
) remains in the set K y y y ym�
� � (~
) { :~}� 0 with
time. Then this solution can be extended to the interval [ , )0 � � and all its components on this interval are nonincreasing
functions.
Lemma 10. Let~y D�
�
, | |~
| | /y H� 2. Then for the solution y t y( ,~
) of system (61) emerging from the point~y at t � 0,
there exists T � 0such that | | ( ,~
)| | /y T y H� 2.
Proof. Suppose that the solution y t y( ,~
) remains in the domain | | | | /y H� 2with time. Then it is defined on the interval
[ , )0 � � .
In proving Lemma 9, we showed that all the components of the vector y t y( ,~
)are nondecreasing functions. Then, there
exists lim ( ,~
)
t
y t y z
��
� , where 0 � | | | | /z H 2. Thus, the set of w-limiting points of the trajectory of the solution consists of one
point z. But then the point z should be an equilibrium position for system (61) (see [27]), which contradicts the property C2 of the
system.
Lemma 11. Let there exists a vector � such that � � 4, | | | |� � H, g (�8 � 4. Then the set K�
(�8 is a positively invariant set
for system (61) and is in the attraction domain of the equilibrium position y � 0.
Proof. Consider the solution y t( , )� emerging from the point � at t � 0. According to Lemma 9, this solution is defined
on the interval [ , )0 � � , satisfies the condition y t K( , ( )�8 ��
�
for t � 0, and all its components on this interval are nonincreasing
functions. Then y t( , )� 0as t ��.
With Lemma 9, if � ( )y K�
�
� , then y t y t t( , � ) ( , ) � � for all t � 0. Then y t y K( , � ) ( )�
�
� for t � ��[ , )0 and
y t y( , � ) 0as t ��.
Proof of Theorem 17. Necessity. Let the zero solution of system (61) be asymptotically stable in a nonnegative cone. If
there exists a number r r H, 0 � � , such that
� max
( )
| | | |
� � �
�y Ar
y g y
y
T
2
0,
then, according to Lemma 8, there exists a vector � such that | | | |� � r, � � 4, and g ( )� � 0.
Assume now that the inequality �� � 0 holds for any r H� ( , )0 . Then there is a point~y � 0 in an indefinitely small
neighborhood of the origin of coordinates such that~y � 0, g y y y(
~) (
~)~
� �� 0. Consider the solution y t y( ,~
) emerging from this
point at t � 0. Without loss of generality, we assume that | |~
| | /y H� 2. According to Lemma 10, the solution falls onto the sphere
| | | | /y H� 2at some time t T� . Then, the equilibrium position y � 0of system (61) is unstable.
Sufficiency. Let system (61) satisfy the MO-condition. We choose a positive vector � such that | | | |� � H, g (�8 � 4, and
consider the solution y t( ,�8 emerging from the point � at t � 0.
500
According to Lemma 11, the set K�
( )� is in the attraction domain of the equilibrium position y � 0.
Specify a positive number $. For the solution y t( ,�8, we find T � 0 such that | | ( ,y t �8A A � $ for t T� . The solutions of
system (61) have the property of integral continuity (see [27]). Therefore, for the numbers $ � 0and T � 0, it is possible to select
. � 0such that for 00
y( )
� and | | | |( )
y0
� . the following inequality holds on the interval [ , ]0 T :
| | ( , )| |( )
y t y0
� $. (66)
But then estimate (66) will also be valid for t T� because | | ( , )| | | | ( , | |( )
y t y y t0
�8 � $ for all t T� . Then the zero solution of
system (61) is asymptotically stable in a nonnegative cone. Theorem 17 is proved.
Corollary 1. If the zero solution of system (61) is stable in the cone �m
�
, then it is asymptotically stable in this cone.
Indeed, in proving the necessity of Theorem 17, we showed that if the zero solution is stable in the cone �m
�
, it satisfies
the MO-condition.
From the proof of the sufficiency of the theorem, it follows that the zero solution is asymptotically stable.
With Lemma 10, it is easy to prove the following theorem.
Theorem 19. Let the following conditions be satisfied:
(i) system (61) has properties C1 and C3;
(ii) there exists a sequence of points ym
m
( )�
�
� such that ym( )
0 for m�, ym( )
� 0, g ym
( )( )
� 0;
(iii) g ( )0 0� , g y( ) � 0for y y Gm
m� (
�
� ( )( )
.
Then the zero solution of system (61) is unstable in �m
�
.
Remark 7. Theorem 17 allows us to reduce the analysis of the zero solution of Wazewski’s nonlinear autonomous
system for stability in �m
�
to testing the solvability of an auxiliary system of inequalities in a positive cone.
Example 7. Consider a proportional-plus-floating control system:
dx
dt
xi
i i� � �B &, i = 1, 2, …, n,
d
dt
a x p fi i
i
n
&
& &� � �
�
�( )
1
, (67)
whereBi� 0, p � 0, & &f ( ) � 0for & � 0, f ( )0 0� . Let us estimate the ranges of parameters in which the state x � 0, & � 0of system
(67) is asymptotically stable. Replacing variable y xi i�
22/ , i n�[ , ]1 , z � &
22/ , we reduce system (67) to the form
dy
dt
x x xi
i i i i i
i
� � � � �B & B
&
B
2 2
21
2
1
2
, i n�[ , ]1 ,
dz
dt
a x p f a
x
p
a
i i
i
n
i i
i
i
n
i� � � � �
� �
� �& & & & B
B
2
1
2
12
2( ) | |
| |
ii
n
f
�
�
�
�
�
�
�
�
�
�
�
1
2&
C
& &( ).
According to Theorem 17, the asymptotic stability of the state x � 0, & � 0 follows from the asymptotic stability of the
state y � 0, z � 0of the system
dy
dt
y zi
i i
i
� � �B
B
1
,
dz
dt
a y p
a
z g zi i i
i
n
i
ii
n
� � �
�
�
�
�
�
�
�
�
�
� �
� �| |
| |
( )B
B
1 1
2 ,
where g z f f( ) min{ ( ), ( )}� � �& & & & .
501
The state y � 0, z � 0 is asymptotically stable if and only if the following inequalities are simultaneous:
y
z
i
i
0
2
�
B
, yi
00� , i = 1, 2, …, n,
2
1
0 0 0
1
p
a
z q z a yi
ii
n
i i i
i
n
�
�
�
�
�
�
�
�
�
� �
� �
� �
| |
( ) | |
B
B , z0
0� .
Performing some transformations of this system, we find the estimate
| |a
pi
ii
n
B
�
�
1
, (68)
which defines the range of parameters in which the state y � 0, z � 0 is asymptotically stable.
Remark 8. In [36], the range of parameters for the system of equations (67) with n � 4 is estimated as follows:
1
42
2
1
4 2
min( )
| |
ii
i
i
a
p
B
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�. (69)
It is obvious that when n � 4 estimate (68) defines a wider range of parameters than estimate (69) does. Example 8.
Consider the system
dy
dt
p yi
ij j
m
j
kl
j
i
�
�
�
�
�
�
�
�
�
�
�
1
, i = 1, 2, …, k, (70)
where mj
and li
are odd natural numbers, pij
are constant coefficients, pij� 0 for i j� , i, j = 1, 2, …, k.
Let P pij i j
k�
�
( ), 1
. It is easy to verify that if det P � 0, then system (70) has properties C1–C3, where H is any positive
number. Then, the zero solution of this system is asymptotically stable in a nonnegative cone if and only if there exist positive
numbers � �
1,�
ksatisfying the inequalities
pij j
m
j
k
j
�
�
��
1
0, i = 1, …, k,
whence it follows that for the zero solution of system (70) to be asymptotically stable in a nonnegative cone, it is necessary and
sufficient that the matrix P satisfy the Sevastyanov–Kotelyanskii conditions.
Example 9. Consider the system
y p y p y
y p y p y
l l
m m
1 11 1 12 2
2 21 1 22 2
� �
� �
9
:
;
,
,
(71)
where l and m are odd natural numbers, pij
are constant coefficients, p12
0� , p21
0� .
With Theorem 17, it is easy to show that p11
0� , p p p pm l m l
11 22 12 21� are the necessary and sufficient conditions for the
asymptotic stability of the zero solution of system (71) in a nonnegative cone.
Remark 9. To apply Theorem 17, it is necessary that no equilibrium positions, except y � 0(property C2) exist in some
neighborhood of the origin of coordinates.
It is clear that the isolation of the equilibrium position is a necessary condition for its asymptotic stability. Note that if
the system has properties C1 and C3 and satisfies the MO-condition, and a positive vector � such that g ( )� � 0 exists in any
indefinitely small neighborhood of the point y � 0, then the equilibrium position may not be isolated. Hence, these conditions are
not sufficient for the zero solution to be asymptotically stable in a nonnegative cone.
502
Let us consider an example proposed by Kosov. Consider the scalar equation
dy
dt
g y� ( ), (72)
where g y
y
y
y
y
( )
sin , ,
, .
�
�
�
9
:
D
;
D
31
0
0 0
This equation has properties C1 and C3. Moreover, for any . � 0, there exists a number �such that 0 � � � .and g ( )� � 0.
However, Eq. (72) has a sequence of equilibrium positions y mm�1/ ( )= , m = 1, 2, …, that converges to zero. Then the zero
solution of system (72) is asymptotically stable in a nonnegative cone.
8. Nonlinear Comparison Systems with a Parameter. Consider the system of equations
dy
dt
y y yi
i m�+ ( , , , , )
1 2� � , i = 1, 2, …, m, (73)
that satisfies the following conditions:
(i) the functions +
i my y y( , , , , )
1 2� � are polynomials, +
i( , , , , )0 0 0� � , ( , , , )y y y
m1 2� are variables, m � 2,
� �7� �
1, , )�
pare variable parameters, p � 2, from the compact domain �;
(2) the polynomials
+
i
j
m
y
y y y
>
�( , , , , )1 2
� , i j� , < �i j m, [ , ]1 , are nonnegative for all ���;
(iii) general asymptotic (in)stability criterion for the comparison system (73) follows from the following finite system
of relations:
f y ym
( , , , )1
0� � � , g y ym
( , , , )1
0� � � , h y ym
( , , , )1
0� � � ,
k y ym
( , , , )1
0� � � , l y ym
( , , , )1
0� � � , m y ym
( , , , )1
0� � . (74)
The functions f, g, k, h, l, m are derived from the right-hand side of system (73).
The analysis of relations (74) is intended to determine the values of � �� at which these inequalities ensure the
asymptotic stability (instability) of the state y � 0of system (73).
Theorem 20. By performing a finite number of algebraic operations on system (74), it is possible to find a finite number
of systems E E
1, ,�
rof equalities and inequalities of the same type containing � and not containing y y
m1,� and having the
following property: for system (74) to have at least one real solution� � � �
1 1� �, ,�
p pfor real values y y
m m1 1� �F F, ,� , it
is necessary and sufficient that the numbers � �
1, ,�
p(substituted for� �
1, ,�
p) satisfy at least one of the systems of equalities
and inequalities E E
1, ,�
r.
Example 10. Consider a linear comparison system
du
dt
Pu� , pij� 0, i j� , < �i j m, [ , ]1 . (75)
The set mentioned in Theorem 20 is well-known. It includes Sevastyanov–Kotelyanskii inequalities
( )� �1 0
11 12 1
1 2
k
k
k k kk
p p p
p p p
�
� � � �
�
, k = 1, 2, …, m. (76)
These inequalities are known to be necessary and sufficient conditions for the asymptotic stability of the state u � 0of
system (75).
Example 11. Inequalities (76) are necessary and sufficient for the asymptotic stability of the state u � 0 of the
comparison system
503
du
dt
p ui
ij j
l
j
mm
j
j
�
�
�
�
�
�
�
�
�
�
�
�
�
2 1
1
2 1
, i, j = 1, 2, …, m,
where pij� 0for i j� ; l
iand m
iare natural numbers.
Asymptotic-stability conditions for invariant sets were established in [32] for system (61) with a nonisolated singular
point. This result is illustrated by the following example.
Example 12. Consider the system
du
dt
u ui
i
l
i
l
i
li i i
� � �
� �
�
�
B
2 1 2 1
1
2 1
, i = 1, 2, …, m – 1,
du
dt
a u um
i i
l
m m
l
i
m
i m� �
��
�
�
�| |
2 1 2 1
1
1
B , (77)
where Bi� 0, l
i, < �i m[ , ]1 , are arbitrary natural numbers; | |a
10� . If
B
B
m i
ss i
ml
i
m
a
m
�
�
�
�
�
�
�
�
�
�
�
�
�
�
G�| |
11
2 1
1
1
,
then the “straight line” described by the equation
u ui
ss i
m
m�
�
�
�
�
�
�
�
�
�
�
G
11
B
, i = 1, 2, …, m – 1,
is asymptotically stable. The state u � 0of system (77) is asymptotically stable if the perturbations of the parametersBk
and | |ai
,
< � �i m[ , ]1 1, are such that
| |ai
i
m
ss i
m
m
�
�
�
�
� G
�
�
�
�
�
�
�
�
�
1
1 1
1
B
B .
Let m = 2 in (77). Then this system becomes:
du
dt
a u a u
l l1
11 1
2 1
12 2
2 11 1
� �
� �
,
du
dt
a u a u
l l2
21 1
2 1
22 2
2 12 2
� �
� �
, (78)
where a12
, a21
0� ; l1
and l2
are natural numbers. For the state u1
0� , u2
0� of system (78) to be asymptotically stable, it is
necessary and sufficient that
a11
0� ,
a a
a a
l l
l l
11
2 1
12
2 1
21
2 1
22
2 1
1 2
1 2
0
� �
� �
� .
9. Some Applications of Nonlinear Comparison Systems. The stability criterion for nonlinear comparison systems
(61) has several developments. We will discuss some of them.
9.1. Conditions for the Uniform Asymptotic Stability of the Zero Solution of System (1). Let us apply the auxiliary
vector function (15) and Theorem 17 to establish the stability conditions for system (1).
Definition 18. The autonomous system
504
dy
dt
g y ym
� ( , , )1
� , y t y( )0 0
0� � (79)
is called a comparison system for system (1) if its maximum ( y t yM
( , )0
) and minimum ( y t ym
( , )0
) solutions are related,
through functions (15) and (16), to the solutions of the original system (1) as
y t y V t x t t x y t ym M
( , ) ( , ( ; , ), ) ( , )0 0 0 0
� , (80)
e y t y v t x t t x e y t ym M
T T( , ) ( , ( ; , ), ) ( , )
0 0 0 0 � , (81)
for all t ��
� , en
� �
�
( , , )1 1�
T� . Inequality (80) hold component-wise.
Theorem 21. Assume that the vector function f in (1) is continuous on � ��
�
n, the vector function g in (79) satisfies
conditions C1–C3 and,
(a) there exists a matrix-valued function (14) with elements (17), (18) and a vector���
�m
such that
D V t x G V t x V t xm
�
( , , )| ( ( , , ), , ( , , ))( )
� � �
1 1�
for all ( , )t xn
� �
�
� � ;
(b) the matrices A and B in (19) are positive definite;
(c) for any . � 0, the system of inequalities
Gj m( , , )� �
10� � , j = 1, 2, …, m,
has a solution � �
1, ,�
msuch that 0 � �� .
jfor all j = 1, 2, …, m.
Then the zero solution of system (1) is uniformly asymptotically stable.
Proof. With condition (a) of Theorem 21, it is easy to obtain the comparison system (79). Condition (c) of Theorem 21 is
necessary and sufficient for the isolated zero solution of system (79) to be uniformly asymptotically stabile (see Theorem 17).
Let us now use the comparison principle and prove Theorem 21. If the isolated zero solution of system (79) is uniformly
asymptotically stable, then for any . .
1 10: � � � ��r and $ � 0there existsT T� �( )$ 0such that for any t t
1 0� and initial values
e u t uT
( , )0 1
. , em
� �
�
( , , )1 1� � the following estimate is valid:
e u t u A am
T( , ) ( ) ( )
0 $ (82)
for all t t T� �
1( )$ , where 0 �
mA( ) is the minimum eigenvalue of the matrix A in (19) and the function
a x x x(| | | | ) (| | | | ) (| | | | ) � �
Tis function of W-class for all x N� .
Note that estimate (19) leads to the sequence of inequalities
v t x x B x B x xM
( , , ) (| | | | ) (| | | | ) ( ) (| | | | ) (| | | | )� � � � �
T T
MB b x( ) (| | | | )
for all ( , )t x N� �
�
� , where b x x x(| | | | ) (| | | | ) (| | | | )� � �
Tfor x N� , b x W(| | | | )� -class. Let us choose . from the condition
. .
MB b( ) ( )
1. If t t
1 0� and | | ( )| |x t
1 ., then u t v t x B b x t
M( ) ( , , ) ( ) (| | ( )| | )
1 1 1� � , where
MB( ) � 0due to condition (b)
of Theorem 21.
If inequality (82) holds for all t t T� �
1( )$ , then, according to the comparison principle, we have
v t x V t x e u t u( , , ) ( , , ) ( , )� � ��
T T.
Therefore, $
m mA a x t A a( ) (| | ( )| | ) ( ) ( )� for all t t T� �
1( )$ and, hence, for all t t T� �
1( )$ if t t
1 0� and | | ( )| |x t
1 ..
Theorem 21 is proved.
Corollary 1. Let f Cn n
� �
�
( , )� � � in (1) and
505
(a) there exists a matrix-valued function (14) with elements (17), (18), a vector���
�m
, and a constant (m m� )-matrix P
with nonnegative off-diagonal elements such that
D V t x PV t x�
( , , )| ( , , )( )
� �
1
for all ( , )t xn
� �
�
� � ;
(b) the (m m� )-matrices A and B in (19) are positive definite;
(c) the system of inequalities
pij j
j
m
�
�
��
1
0, i = 1, 2, …, m,
has a solution � �
1, ,�
msuch that 0 � �
jfor all j = 1, 2, …, m.
Then the zero solution of system (1) is uniformly asymptotically stable.
Corollary 2. If system (79) describes the disturbed motion of some real system with a finite number of degrees of
freedom, the vector function g satisfies conditions C1–C3, and condition (iii) of Theorem 21 is satisfied, then the isolated zero
solution of system (79) is uniformly asymptotically stable.
Despite the fact that condition (iii) of Theorem 21 is necessary and sufficient for the uniform asymptotic stability of the
isolated zero solution of the comparison system (79), the accuracy of estimating the domain of parameters of system (1) that
ensures the uniform asymptotic stability of its zero solution depends on the “finenenss” of the majorants of the components of the
total derivative of the vector function along the solutions of system (1).
9.2. Absolute Stability Criterion. Consider the system
dx
dt
a f x b f x f xs
s s s sj n n
j
k
s
j
sn
js
� �
�
�( ) ( ( )
( )( )
1 1
1
1
��
� , s = 1, …, n, (83)
where as
and bsj
are constant coefficients, f xs s( ) are functions defined and continuous for x
s� �� ��( , ) and satisfy the
conditions x f xs s s
( ) � 0for xs� 0, �
si
j( )are nonnegative rational numbers with odd denominators. Let �
si
j
i
n ( )
��
�
10, j = 1, …,
ks, s = 1, …, n. Then system (83) has a zero solution.
System (83) is a generalization of the system
� ( )x f x� P , (84)
where x � ( , , )x xn1
�
T; f x( ) ( ( ), , ( ))� f x f x
n n1 1�
T; P is a constant matrix. Such systems are used in studying automatic
control systems and modeling neural networks [14, 49].
Let the coefficients as
and bsj
satisfy the inequalities
as� 0, b
sj� 0, j = 1, …, k
s, s = 1, …, n. (85)
Definition 19. System (83) is absolutely stable if the zero solution of this system is asymptotically stable for any
admissible functions f x f xn n1 1
( ), , ( )� .
Let us analyze system (83) for absolute stability.
Along with Eqs. (83), we will consider the system of inequalities
a b qs s sj n
j
k
s
j
sn
js
� �
��
� �
�
�1
1
10
( )( )
� , s = 1, …, n.
We choose the following Lyapunov function for system (83):
506
V f ds
s
n
s
x
s
s
�
�
� � @ @
�
1 0
( ) , (86)
where s
are positive constants, and �
sare positive rational numbers with odd numerators and denominators. Function (86) is
positive definite.
Theorem 22. System (83) is absolutely stable if and only if there exists a Lyapunov function in the form (86) satisfying
the conditions of Lyapunov’s theorem of asymptotic stability.
Proof. Sufficiency is obvious. Necessity was proved by Aleksandrov and Platonov [45] using the MO-condition.
Corollary 1. For system (83) to be absolutely stable, it is necessary and sufficient that it satisfy the MO-condition in an
indefinitely small neighborhood of the origin of coordinates.
This absolute-stability criterion for system (83) is similar to the asymptotic-stability criterion for the zero solution of
Wazewski systems given in Sec. 7. It should be noted, however, that the functions f x f xn n1 1
( ), , ( )� in (83) may not be
nondecreasing. Thus, system (83) may not be a Wazewski one.
9.3. Absolute Stability of Nonlinear Switched Systems. Suppose that the uncertainty of the right-hand side of the
system is not only because the only known fact about its nonlinearities is that they belong to some class of admissible functions,
but also because the parameters of the system can switch from one set of values to another. A switched system is a hybrid
dynamic system consisting of a family of subsystems and a switching law that defines which of the subsystems is active at every
instant [75]. For the zero solution of such a system to be asymptotically stable for any switching law, it is sufficient that for all
subsystems of the hybrid system, there exist a common Lyapunov function satisfying the conditions of some asymptotic-stability
theorem [75].
The switched system (83) was analyzed for absolute stability in [5].
Consider the system
� ( ) ( ) ( )( ) ( )
( ) ( )
x f x b f x f xi i i i ij n n
j
i
j
in
j
� �
�
�
& &
� �
1 1
1
1�
ki
�, i = 1, …, n. (87)
Let the scalar functions f xi i( ) satisfy the same conditions as in Sec. 9.2 and, additionally, be continuously
differentiable for | |xi� #, 0 � ��# , and 3 �f x
i i( ) 0for x
i� 0, i = 1, …, n. Let, as before, �
ip
j( )be nonnegative rational numbers
with odd denominators �
ip
j
p
n ( )
��
�
10, j = 1, …, k
i, i = 1, …, n. The piecewise-constant function & &� ( )t :
[ , ) { , , }0 1�� �Q N� defines the law of switching between various modes of operation; ai
s( )and b
ij
s( )are constant coefficients,
s = 1, …, N. Assume that for any values of the indices i and j, there exists at least one s such that bij
s( )� 0.
Thus, at every instant, the operation of the system is described by one of the subsystems
� ( ) ( ) ( )( ) ( )
( ) ( )
x f x b f x f xi i
s
i i ij
s
n n
j
i
j
in
j
� �
�
�
� �
1 1
1
1�
ki
�, i = 1, …, n, s = 1, …, N. (88)
Definition 20. System (87) is absolutely stable if its zero solution is asymptotically stable for any admissible functions
f x f xn n1 1
( ), , ( )� and for any switching law.
Let the following inequalities be valid:
bij
s( )� 0, j = 1, …, k
i, i = 1, …, n, s = 1, …, N. (89)
Since the functions f x f xn n1 1
( ), , ( )� are monotonic, subsystems (88) are Wazewski systems. The asymptotic-stability
criterion for such systems is given in Sec. 7. Note, however, that the asymptotic stability of the zero solution of each of
subsystems (88) is a necessary, but not sufficient condition for the asymptotic stability of the zero solution of the hybrid system
(87).
To establish the absolute-stability conditions for system (87), we will construct a common Lyapunov function for
subsystems (88):
507
V
f x
i i n
i i
i
i
( ) max
( )
, ,
x �
�
�
�
�
�
�
�
�
�
�
� �
� 1
, (90)
where � �
1, ,�
nare positive constants; � �
1, ,�
nare positive rational numbers with odd numerators and denominators.
Function (90) is positive definite.
The following statement was proved in [5].
Theorem 23. For system (87) to be absolutely stable, it is sufficient that for any . � 0 there exist numbers � �
1, ,�
n
satisfying the inequaltites
� � � �
� �
i
s
i ij
s
n
j
k
bi
j
in
ji
( ) ( )
( ) ( )
� �
�
�1
1
10� , i = 1, …, n, s = 1, …, N,
such that 0 � �� .
i, i = 1, …, n.
Corollary 1. Let system (87) have the form � ( )x P f x�
&
, where x x xn
� ( , , )1
�
T, f x f x f x
n n( ) ( ( ), , ( ))�
1 1�
T,
P PN1
, ,� are constant Metzler matrices. Then for this system to be absolutely stable, it is sufficient that there exists a positive
vector � � �� ( , , )1
�
n
Tsuch that P
s� � 0, s = 1, …, N.
Let us examine the case where inequalities (89) for the coefficients bij
s( )may hold. Suppose that
� max
, ,
( )a a
is N
i
s�
�1 �
,�
max | |
, ,
( )b b
ijs N
ij
s�
�1 �
, j = 1, …, ki, i = 1, …, n,
and consider the system
� � ( )�
( ) ( )
( ) ( )
x f x b f x f xi i i i ij n n
j
k
i
j
in
ji
� �
�
��
� �
1 1
1
1� , i = 1, …, n. (91)
Theorem 24. For system (87) to be absolutely stable, it is sufficient that system (91) be absolutely stable.
The absolute-stability conditions defined by Theorem 24 may appear too rough. They can be relaxed in some cases.
Consider a switched system:
� ( ) ( ) ( )( ) ( )
( ) ( )
x f x b f x f xi i i i ij n n
j
i
j
in
j
� �
�
�
& &
� �
1 1
1
1�
ki
�, i = 1, …, n, (92)
where a ai
s
i
s( ) ( )� , b b
ij
s
ij
s( ) ( )| |� , j = 1, …, k
i, i = 1, …, n, s = 1, …, N.
Let there be given a set of numbers w wn1
, ,� each of which is equal to +1 or –1. Let us consider a domain
G x x wn
i i� � �{ :� 0, i = 1, …, n}. The numbers w w
n1, ,� form the basis of the domain G.
Theorem 25. If the elements of some basis w wn1
, ,� satisfy the following relations for all s = 1, …, N:
bij
s
i n
i
j
in
j
( )
( ) ( )
! ! !
� �
1
10� � , j = 1, …, k
i, i = 1, …, n, (93)
then system (87) is absolutely stable if and only if so is system (92).
To prove this theorem, it is necessary to change variables x y wi i i� , i = 1, …, n, in (87).
Thus, if the conditions of Theorem 25 are satisfied, the analysis of system (87) for absolute stability is equivalent to the
analysis of system (92). To establish the absolute-stability conditions for system (92), it is possible to use Theorem 23.
Consider the subsystems
� ( ) ( ) ( )( ) ( )
( ) ( )
x f x b f x f xi i
s
i i ij
s
n n
j
i
j
in
j
� �
�
�
� �
1 1
1
1�
ki
�, i = 1, …, n, s = 1, …, N, (94)
508
of system (92).
It is easy to prove the following theorem.
Theorem 26. For subsystems (88) to have a common Lyapunov function of the form (90) whose derivative along the
solutions of each subsystem is negative definite, it is sufficient, and if for any s N�{ , , }1� , there exists a basis w wn1
, ,�
satisfying conditions (93), then it is also necessary that the Lyapunov function exist for subsystems (94).
Note that Theorem 26 assumes that there is an individual basis for each value s N�{ , , }1� , while Theorem 25 requires a
commin basis for subsystems (88).
Corollary 1. For system (87) to be absolutely stable, it is sufficient that for any . � 0 there exist numbers � �
1, ,�
n
satisfying the inequaltites
� � � �
� �
i
s
i ij
s
n
j
k
bi
j
in
ji
( ) ( )
( ) ( )
� �
�
�1
1
10� , i = 1, …, n, s = 1, …, N,
such that 0 � �� .
i, i = 1, …, n.
Example 13. Let system (87) have the form
� ( ) ( ) ( )( ) ( )
x a f x b f x f x1 1 1 1 1 2
2
2 3
3
3� �
& &
,
� ( ) ( )( ) ( ) /
x a f x b f x2 2 2 2 2 1
1 5
1� �
& &
,
� ( ) ( )( ) ( ) /
x a f x b f x3 3 3 3 3 1
1 5
1� �
& &
(95)
and consist of two subsystems (N � 2):
� ( ) ( ) ( )x f x f x f x1 1 1 2
2
2 3
3
3� � � , � ( ) ( ) ( )x f x f x f x
1 1 1 2
2
2 3
3
32 2� � � ,
� ( ) ( )/
x af x f x2 2 2 1
1 5
1� � � , � ( ) ( )
/x af x f x
2 2 2 1
1 5
12� � � ,
� ( ) ( )/
x f x f x3 3 3 1
1 5
12� � � , � ( ) ( )
/x f x f x
3 3 3 1
1 5
13� � � ,
where a is a positive parameter. Let us find the values of the parameter a at which system (95) is absolutely stable.
Theorem 24 suggests that for absolute stability of system (95) it is sufficient that a �1. Corollary 1 to Theorem 26 allows
relaxing this condition. Namely, system (95) will be asymptotically stable if a � 1/2.
Conditions (93) have the form w w1 3
0� , w w1 2
0� for the first mode of operation of system (95) and w w1 3
0� ,
w w1 2
0� for the second mode.
Thus, the required basis w1, w
2, w
3can be chosen for each mode. Indeed, let w w
1 3� � 1, w
2� –1 for s �1 and
w w1 2� � 1, w
3� –1 for s � 2. A common basis for both modes does not exist.
System (87) will be absolutely stable if a system of inequalities of special form has a positive solution in any indefinitely
small neighborhood of the origin of coordinates. Some special cases of such systems for which it was possible to find
constructive methods of testing the existence conditions for the solution were examined in [3, 5].
We will now consider a system consisting of two coupled nonlinear oscillators:
�� � ( , , � )( ) ( )
x a x b x q t1 1 1 1 1 1 1
1� � �
&2
&
� x x ,
�� � ( , , � )( ) ( )
x a x b x q t2 2 2 2 2 2 2
2� � �
&2
&
� x x , (96)
where x1, x
2��, x � ( , )x x
1 2
T; & &� ( )t : [ , ) { , , }0 1�� � N is, as before, the switching law; a
s
1
( ), a
s
2
( ), q
s
1
( ), q
s
2
( ), b
1, b
2are
constant coefficients, ai
s( )� 0, b
i� 0, i = 1, 2, s = 1, …, N; 2
11� , 2
21� are rational numbers with odd numerators and
denominators;�1
( , , � )t x x and�2
( , , � )t x x are functions that are continuous for t � 0, | | | |x � #, | | � | |x � #(# �const >0) and such that
509
| ( , , � )| ( � )/
� �
�
1 1 2
2
2
2 21t x xx x � , | ( , , � )| ( � )
/
� �
�
2 1 1
2
1
2 22t x xx x � ,
where �1, �
2, �
1, �
2are positive constants. Then system (96) has a zero solution.
System (96) describes the interaction of two nonlinear oscillators
�� �
( )x a x b x
i i i i i
i� � �
&2
0, i = 1, 2. (97)
Thus, the dissipation parameters of this system (which characterize the resistance forces) are switched. The functions
relating Eqs. (97) are switched too.
The zero solutions of the isolated subsystems
�� �
( )x a x b x
i i
s
i i i
i� � �
2
0, i = 1, 2, s = 1, …, N,
are known [7] to be asymptotically stable. Let us establish the conditions for the zero solution of the coupled system (96) to be
asymptotically stable for any switching law.
We will consider the subsystems of system (96):
�� � ( , , � )( ) ( )
x a x b x q ti i
s
i i i i
s
i
i� � �
2
� x x , i = 1, 2, s = 1, …, N. (98)
We choose the following Lyapunov functions:
v b
x x
x xi i
i i
i i i
i� � �
2 2
2 2
�
��
2
, �
i� const > 0, i = 1, 2.
Differentiating these functions along the solutions of the sth subsystem in (98), we obtain
� | � �
( )
( )v a x b x x x
i i
s
i i i i i i i i
i i i
98
1 1 12
� � � � �
� � �2 2 2
� � 2 �
i i
s
i ia x x
i i( )
�
2 2
� �(� ) ( , , � )( )
x x q ti i i i
s
i
i� �
2
x x , i = 1, 2, s = 1, …, N.
If�1
and�2
ar sufficiently small, there exist positive numbers ., ci1, c
i2, c
i3, c
i
s
4
( ), i = 1, 2, s = 1, …, N, such that
c x x b
x x
x x c x xi i i i
i i
i i i i i i
i
1
2 2
2 2
2
2 2
2 2
( � )
�
� ( � )� � � ��
2
,
| � | ( � )/
x x x xi i i i i
i� ��
22 2 1 2
,
� � � �
� � �
a x b x x x a xi
s
i
n
i i i i i i i i i
s
i
i i i( ) ( )
� �
1 1 12
� � 2 �
2 2 2
i i ix c x xi i
s
i i� ( � )
( ) ( )/2 2
� �
�
4
2 2 1 2
, i = 1, 2,
for | | | |x � .and | | � | |x � .whence follow the inequalities
�
( ) ( )/ ( ) //
v d v d v vs s
1 11 1
1 2
12 1
1 2
2
21 1
�
�2 �
,
�
( ) ( )/ ( ) //
v d v d v vs s
2 22 1
1 2
21 1
2
2
1 22 2 �
�2 �
,
where d c cs s
11 41 21
1 21
( ) ( ) ( )/
� �
� �2
, d q c c cs
12 1 1 31 11
1 2
12
21
�
��
| |( ) /
/
�
�
, d c cs s
22 42 22
1 22
( ) ( ) ( )/
� �
� �2
, d q c c cs s
21 2 2 32 12
1 2
11
22
( ) ( ) //
| |�
��
�
�
, s = 1,
…, N. The system
�
( ) ( )/ ( ) //
y d y d y y1 11 1
1 2
12 1
1 2
2
21 1
� �
�&
2&
�
,
510
�
( ) ( )/ ( ) //
y d y d y y2 22 2
1 2
21 1
2
2
1 22 2� �
�&
2&
�
, (99)
can be considered as a comparison system for Eqs. (96). Thus, the asymptotic stability of the zero solution of system (96) is
sufficient for the zero solution of system (99) to be asymptotically stable in the nonnegative orthant
��
� � �
2
1 2 1 20 0{( , ): , }y y y y .
System (99) is a system of the form (87), where f y yi i i
i( )
( )/
�
�2 1 2
, i = 1, 2. The exponents �ip
j( )in (87) are assumed to
be rational and to have odd denominators. Since system (99) can be analyzed only in the nonnegative orthant ��
2, it is possible to
use exponents with even denominators. For the same reason, it is not necessary that y f yi i i
( ) � 0 for yi� 0, i = 1, 2.
The above results suggest that for the zero solution of system (99) to be asymptotically stable in ��
2, it is sufficient that
one of the following conditions be satisfied:
(i) � � 2 2
1 2 1 2� ;
(ii) � � 2 2
1 2 1 2� and
max max
, ,
( )
( ), ,
( )
s N
s
ss N
sd
d
d
d� �
�
�
�
�
�
�
�
�
�
�
1
12
11
1
21
2
� �
2
1 2
1( )
/
s
�
�
�
�
�
�
�
�
�
� 2
.
9.4. Stability Conditions for Complex Systems. The results obtained in Sec. 7 can be used for the stability analysis of
complex systems. Consider the complex system
dx
dt
g t x h t xs
s s s� �( , ) ( , ), s = 1, …, n, (100)
where xs
ms
�� , x x xn
� ( , , )1
T T T� , the functions g t x
s s( , ) and h t x
s( , ) are continuous for t � 0, | | | |x H� (H � 0), and
g t h ts s( , ) ( , )0 0 0� � for all t � 0, i.å., system (100) has the equilibrium position x � 0. Let the zero solutions of isolated
( ( , )h t xs
- 0) subsystems be asymptotically stable. It is necessary to establish the conditions whereby the zero solution of system
(100) is asymptotically stable.
Theorem 27 [90]. Let functionsV t xs s( , ) and+
s st x( , ), s = 1, …, n, with the following properties exist in the domain
t � 0, | | | |x H� :
(a) the functionsV t xs s( , ) are continuously differentiable, positive definite, and decreasing, the functions+
s st x( , ) are
continuous and positive definite, and
dV
dt
b bs
ss s s sj j
j s( )100
2 �
�
�+ + + ,
where bss� 0, b
sj� 0, s j� ;
(b) there exists a positive solution ( , , )� �
1�
n
Tof the system of inequalities
b bss s sj j
j s
� �� �
�
�0, s = 1, …, n.
Then the zero solution of system (100) is asymptotically stable.
Remark 10. Some other aggregation methods for complex systems are addressed in the monographs [12, 73].
With the results of Sec. 7, it is possible to derive a generalization of the aggregation method for complex systems from
Theorem 27.
Theorem 28. Let functionsV t xs s( , ) and+
s st x( , ), s = 1, …, n, with the following properties exist in the domain t � 0,
| | | |x H� :
(a) the functionsV t xs s( , ) are continuously differentiable, positive definite, and decreasing, the functions+
s st x( , ) are
continuous and positive definite, and
511
dV
st
a bs
s s s sj n
j
k
s s s s
j
sn
js
( )
( )( )
100
1
1
1 �
�
�
+ + + +
� " ��
�
�
�, s = 1, …, n, (101)
where as� 0, b
sj� 0, and the parameters "
s� 0, �
s� 0, and �
si
j( )� 0are such that
�
" �
"
" �
si
j
i ii
n
s
s s
( )
�
�
�
�
�
1
, j = 1, …, ks, s = 1, …, n, (102)
(b) there exists a positive solution ( , , )� �
1�
n
Tof the system of inequalities
a bs s sj n
j
k
s s
j
sn
js
� � �
"�
�
� �
�
�1
1
10
( )( )
� , s = 1, …, n. (103)
Then the zero solution of system (100) is asymptotically stable.
To prove the theorem, it is sufficient to set up a Lyapunov functionV Vs ss
n
�
��
1and to show that if the conditions of
the theorem are satisfied, the positive constants s
can be selected so that the function
dV
dt( )100
is negative definite (see [45]).
Example 14 [45]. Consider the system
� ( ) ,
� ( )
x x x x x x
x x x x x a
1 1 1
2
2
2
1
2
2
2 1
3
2 1
2
2
2100 100
� � � �
� � � � x
x x b x
3
9
3 3
9 3
,
� | | �| | ,� � �
9
:
D
;
D
(104)
where a and b are some constants and � ( , )x x x�
1 2
T. Let the vector function V have the following components:
v
x x
1
1
2
2
2
100
2 2
� � , v x2 3�
�
, (105)
where � is some rational number with odd denominator and even numerator. With function (105), it is easy to obtain a
comparison system:
du
dt
u a u u1
1
2
1
1 2
2
9 80 04 2� � �. | | ( )
/ /,
du
dt
u b u u2
2
1 8
1
3 2
2
1 12� � �
� �
� �
� �/ / /| | ( ) .
Condition (c) of Theorem 21 means that the system of inequalities
� � �0 04 2 01
2
1
1 2
2
9. | | ( )
/ /� � �
�
a ,
� � �
� �
�� � � �
� �
2
1 8
1
3 2
2
1 12 0
/ / /| | ( )b
has a positive solution in some neighborhood of the point ( )u � �
�
02
� . If the parameters a, b are such that | |ab �0.01, then this
condition is satisfied and the zero solution of system (104) is uniformly asymptotically stable.
If � � 4 in (105), then
� ( , ) | | �| | | | | | �| | | |v x x x a x x1 1 2
4
3
9100� � � ,
512
� ( ) | | | | | | �| | | |v x x b x x2 3 3
12
3
34 4� � � ,
and we conclude, based on condition (b) of Theorem 28, that for its zero solution to be asymptotically stable, it is necessary and
sufficient that the system of inequalities
� � �100 01
4
1 2
9� � �| |a ,
� � �4 4 02
12
1
3
2
3� � �| |b , (106)
have a positive solution. It follows from system (106) that this condition is satisfied if | |ab �100.
Thus, selecting a candidate vector function and majorant, we can considerably extend the domain of parameters that
ensures the uniform asymptotic stability of the zero solution of the original system.
9.5. Stability of Complex Systems by Nonlinear Approximation. Let us address the class of complex systems that can
be analyzed for stability using the results of Sec. 9.4. Consider the system
dx
dt
F x Q t xs
s s sj
j
ks
� �
�
�( ) ( , )
1
, s = 1, …, n, (107)
where xs
ms
�� , x x xn
� ( , , )1
T T T� ; the elements of the vectors F x
s s( ) are continuous homogeneous functions of order "
s,
"
s�1; the vector functions Q t x
sj( , ) are continuous for t � 0, | | | |x H� (H � 0) and satisfy the conditions
| | ( , )| | | | | | | | | |
( ) ( )
Q t x c x xsj sj n
s
j
sn
j
1
1
��
� , csj� 0, �
si
j( )� 0. (108)
If �
si
j
i
n ( )
��
�
10, j = 1, …, k
s, s = 1, …, n, then system (107) has a zero solution. The functions F x
s s( ) describe the
dynamics of the subsystems, while the functions Q t xsj
( , ) characterize the interaction among the subsystems.
We assume that the zero solutions of the isolated subsystems
dx
dt
F xs
s s� ( ), s = 1, …, n, (109)
are asymptotically stable. It is necessary to establish the conditions whereby the zero solution of system (107) is asymptotically
stable. Thus, system (109) is considered a nonlinear approximation system for complex system (107).
The isolated subsystems (109) are known (see [17, 92]) to have Lyapunov functions V xs s( ) that are continuously
differentiable, positive definite, positive homogeneous of order �s�1, s = 1, …, n. These functions satisfy the inequalities
a x V x a xs s s s s s
s s
1
1
2
1
| | | | ( ) | | | |� �� �
,
>
>
�
V
x
a xs
s
s s
s
3| | | | ,
>
>
� "
V
x
F a xs
s
s s s
s s
�
�
�
�
�
�
�
�
�
�
T
4| | | | (110)
for all xs
ms
�� , where as1, a
s2, a
s3, a
s4are positive constants. For � �
1, ,�
n, we can take any positive values.
Then for t � 0, | | | |x H� , we have
dV
dt
a x a x c xs
s s s s sj
s s s s
( )
| | | | | | | | | | | |
107
4 3 1
1 � �
�� " ��
( ) ( )
| | | |
j
sn
js
xn
j
k
�
�
�
�
1
, s = 1, …, n.
Theorem 28 (where+s s
x� | | | |) suggests that if there exist nonnegative numbers � �
1, ,�
nsatisfying inequalities (102)
and positive numbers � �
1, ,�
nsatisfying the inequalities
513
� � �
�
�a a c
s s s sj n
j
k
s s
j
sn
js
4 3 1
1
10� � �
"�
�
( )( )
� , s = 1, …, n, (111)
then the zero solution of system (107) is asymptotically stable.
Remark 11. The necessary and sufficient conditions for the existance of nonnegative numbers � �
1, ,�
nsatisfying
inequalities (102) were established in [45]. It was proved that if these numbers are selected so that all inequalities (102) are strict,
then there exist positive numbers � �
1, ,�
nsatisfying inequalities (103).
An approach based on vector Lyapunov functions was proposed in [21] for the stability analysis of complex systems
with homogeneous subsystems. Applying this approach to system (100), we will construct a vector Lyapunov function
V V Vn
� ( , , )1
�
Tand a comparison system:
du
dt
p u u p u us
s s s sj n
s s
s
s
s
s
j
sn
� �
�
� ��
� "
�
�
�
�
�
�
1 1
1
1
1
1
( )(
�
j
n
s
j
k
)
� �
�
�
1
1
, s = 1, …, n,
where V xs s( ) are positive homogeneous Lyapunov functions of order �
s� �1 1 set up for the isolated subsystems (109) and
satisfying estimates (110), and the coefficients ps
and psj
are defined by
p a as s s
s s
s� �
�
�
�
4 2
1
� "
�
, p a c a a asj s sj s n
s
s
s
j
sn
j
n�
�
�
�
�
�
�
3 1
1
11
1
1
1
1
1
�
�
�
�
�
�
( )( )
� .
For this comparison system, the corresponding system of inequalities in the MO-condition has the form
p ps s sj n
j
k
s s
j
sn
js
� � �
"�
�
� �
�
�1
1
10
( )( )
� , s = 1, …, n. (112)
Remark 12. The coefficients in (111) and (112) depend, in the general case, on � �
1, ,�
n. Given � �
1, ,�
n, system (111)
yields better results than system (112) does, i.e., the approach proposed in [44, 45] allows improving the standard comparison
method in some cases. Unlike it, Kosov’s approach does not impose the additional constraints (102) on the parameters � �
1, ,�
n.
Remark 13. The case where the functions F xs s( ) in (107) are homogeneous in extended sense can be examined in a
similar way. In this case, it is convenient to use functions of the form+
s st x( , )(a norm in the space of functions homogeneous in
extended sense) for the functions+s s s
m
s
n
t x x s( , ) | |/
�
��
1
1in Theorem 28.
9.6. MO-Condition in the Dissipativity Analysis of Nonlinear Systems. We will show that the above ideas can also be
followed to establish the dissipativity conditions for nonlinear systems.
The behavior of such systems in the domain G x x Hn
1� � �{ : | | | | }� is of interest for stability problems and in the
domain G x x Hn
2� � �{ : | | | | }� for dissipation problems. Here H is a positive constant. In the former case, the number H � 0can
be chosen indefinitely small. All the components of the vector x are small as well. In the latter case, H should be indefinitely
large. Increase in | | | |x with H does not generally mean that all the components of the vector x increase. This is the main problem
encountered in applying the MO-condition in dissipativity analysis.
Let us first establish the dissipativity conditions for Wazewski’s autonomous systems of general form.
Consider the system
dx
dt
g x� ( ), (113)
where x x xn
� ( , , )1
�
Tand the vector function g x( ) is continuous for all x
n�� .
Let system (113) have the following properties:
(a) one solution x t t x( ; , )0 0
passes through each point ( , )t x0 0
, where t0� �� ��( , ), x
n�� ;
(b) there exists a number D � 0such that that system (113) has no equilibrium positions in the domain | | | |x D� ;
514
(c) system (113) is a Wazewski one.
Note that condition (b) is the necessary dissipativity condition.
Moreover, we will assume that the nonnegative cone ��
nis a positive invariant set for the system under consideration.
Definition 21. System (113) is dissipative in ��
nif there exists a positive number H such that for any Q � 0, there exists
T � 0for which the inequality | | ( ; , )| |x t t x H0 0
� holds for all t0� �� ��( , ) and t t T� �
0if only x
n
0�
�
� and | | | |x Q0
� .
Definition 22. System (113) satisfies the MO
~
-condition if for any number R � 0 there exists a point � � �� ( , )1
�
n
T
such that � � 0, | | | |� � R, and g Q( )� � .
Lemma 12 (Necessary Dissipativity Condition). If system (113) is dissipative in �m
�
, then it satisfies the
MO
~
-condition.
The validity of this statement follows from Lemmas 7 and 8.
The condition of dissipativity of system (113) in ��
nformulated in Lemma 12 is necessary, but not sufficient.
Example 15. Consider the system
dx
dt
x x x1
1 1
2
2� � � ,
dx
dt
x2
2� � . (114)
System (114) has properties (a)–(c) and its cone �m
�
is a positive invariant set. Let g ( )� � 0.
� � �� � �
1 1
2
20, � ��
20,
whence it follows that for any R � 0, there exists a point � � 0(for example, �1
1 2� / ( )R , �2� R) such that | | | |� � R and g ( )� � 0.
However, system (114) is not dissipative in �m
�
. Indeed, the solution of this system passing through the point
( , ) ( , )( ) ( )
x x1
0
2
02 1
T T� at t
00� has the form ( ( ), ( )) ( , )x t x t e e
t t
1 22
T T�
�
. As t ��, this solution increases without limit in
x1, which is indicative of nondissipativity.
Definition 23. System (113) satisfies the MO-condition if for any number R � 0 there exists a point � � �� ( , , )1
�
n
T
such that �s
R� , s = 1, …, n, and g ( )� � 0.
Lemma 13 (Sufficient Dissipativity Condition). If system (113) satisfies the MO-condition, then it is dissipative in ��
n.
The validity of this statement follows from Lemma 9.
If system (113) satisfies the MO-condition, then the number D in property (b) can be used for the number H in Definition
21.
If system (113) satisfies the MO-condition and has a unique equilibrium position x � 0 in ��
n, then this equilibrium
position is asymptotically stable in the large.
The condition of dissipativity of system (113) in ��
nformulated in Lemma 13 is sufficient, but not necessary.
Example 16. Consider the system
dx
dt
x x x1
1 1 2� � � ,
dx
dt
x2
2� � . (115)
All the assumptions made above are valid for system (115). It is easy to verify by direct integration that system (115) is
dissipative in �m
�
(moreover, it is dissipative in all the space �n
, and its zero solution is asymptotically stable in the large). At
515
the same time, if R �1, then for any point � � �� ( , )1 2
Tsuch that �
1� R, �
2� R, the inequalities g ( )� � 0that have the following
form in this example:
� � �� � �
1 1 20, � ��
20
do not hold.
Consider a system of the form (83). In addition to the constraints on the functions f x f xn n1 1
( ), , ( )� specified in Sec.
9.2, we assume that f xs s( )�� as x
s��, f x
s s( )�� as x
s��, s = 1, …, n.
Definition 24. System (83) is absolutely dissipative if it is dissipative for any admissible functions f x f xn n1 1
( ), , ( )� .
We again assume that the coefficients as
and bsj
of system (83) satisfy inequalities (85).
Theorem 29. For system (83) to be absolutely dissipative, it is necessary that it satisfy the MO
~
-condition.
Theorem 30. For system (83) to be absolutely dissipative, it is sufficient that it satisfy the MO-condition.
The proofs of Theorems 29 and 30 can be found in [46].
As with absolute stability, the conditions of absolute dissipativity of system (83) are similar to the dissipativity
conditions for Wazewski’s autonomous systems.
Let system (83) have the form
dx
dt
p f xs
sj j j
j
n
sj
�
�
�
�
( )
1
, s = 1, …, n, (116)
where �sj
are positive rational numbers with odd denominators, and �ss�1, p
sjare constant coefficients, p
ss� 0, p
sj� 0for
j s� , j, s = 1, …, n.
Theorem 31. For system (116) to be absolutely dissipative, it is necessary and sufficient that it satisfy the
MO-condition.
Theorem 31 states that for a system with additive constraints, the MO-condition is not only sufficient, but also necessary
condition of absolute dissipativity. At the same time, the MO
~
-condition is the necessary, but not sufficient condition of absolute
dissipativity of system (116).
Example 17. Let system (116) have the form
dx
dt
f x1
1 1� � ( ),
dx
dt
f x f x2
2 2 3
2
3� � �( ) ( ),
dx
dt
f x f x3
3 3 2
2
2� � �( ) ( ), (117)
It is easy to verify that system (117) satisfies the MO
~
-condition. However, if f x xs s s( ) � , s = 1, 2, 3, then this system is
not dissipative.
9.7. Dissipativity of Complex Systems by Nonlinear Approximation. To analyze complex systems for dissipativity, it is
possible to use an aggregation method similar to that in Theorem 28.
Let a complex system (100) be given, where the functions g t xs s( , )and h t x
s( , )are continuous for t � 0, x
m�� , where
m m mn
� � �
1� .
Theorem 32. Let functionsV t xs s( , )and+
s st x( , ), s = 1, …, n, be defined for all t � 0, x
s
ms
�� and have the following
properties:
(a) the functions V t xs s( , ) are continuously differentiable, and the functions +
s st x( , ) are continuous,
r x V t x r xs s s s s s1 2(| | | | ) ( , ) (| | | | ) , r x t x
s s s s3(| | | | ) ( , )+ , where r p
ks( ), k = 1, 2, 3, are functions given for p� ��[ , )0 ,
continuous, nonnegative, and monotonically increasing, r pks
( )�� as p ��;
516
(b) inequalities (101), where as� 0, b
sj� 0, hold in the domain t � 0, | | | |x H� (H = const > 0), and the parameters"
s� 0,
�
s� 0, and �
si
j( )� 0satisfy the conditions
�
" �
"
" �
si
j
i ii
n
s
s s
( )
�
�
�
�
1
, j = 1, …, ks, s = 1, …, n, (118)
(b) the system of inequalities (103) has a positive solution ( , , )� �
1�
n
T.
Then system (100) is uniformly dissipative.
As an application of Theorem 32, we will again consider the system of equations (107), where xs
are the state vectors of
subsystems of order ms, x x x
n� ( , , )
1
T T T� ; the elements of the vectors F x
s s( ) are continuous homogeneous functions of order
"
s, "
s� 0; the vector functions Q t x
sj( , ) are continuous for t � 0, x
m�� and satisfy inequalities (108) in the domain t � 0,
| | | |x H� (H is some constant).
Assume that the zero solutions of the isolated subsystems (109) are asymptotically stable, and a continuously
differentiable, positive homogeneous Lyapunov functionV xs s( )of order �
s� �1 1satisfying inequalities (110), s = 1, …, n, has
been constructed for each sth subsystem.
Let the nonlinear system (109) be a first approximation for (107). Let us establish the conditions for Q t xsj
( , )whereby
system (107) is uniformly dissipative.
Differentiating the functions V xs s( ) along the solutions of system (107) in the domain t � 0, | | | |x H� , we see that,
according to Theorem 32, if there exist nonnegative numbers � �
1, ,�
nsatisfying the system of inequalities (118) and positive
numbers � �
1, ,�
nsatisfying the system of inequalities (103) (where a a
s s� �
4, b a c
sj s sj�
3), then system (107) is uniformly
dissipative.
As in Sec. 9.5, if inequalities (118) are strict, it is not required to check whether the system of inequalities (103) is
solvable.
9.8. Homogeneous Cooperative Systems. One of the special cases of general monotonic systems (61) is homogeneous
cooperative systems. These systems of equations are used for the mathematical description of some processes in economy,
biology, and ecology (see [40, 70] and the references therein).
Definition 25. A continuous vector function gn n
:� � differentiable on �n
\{ }0 is cooperative if the Jackobian
>
>
g
x
a( )
is a Metzler matrix for any an
�
�
� \{ }0 .
Definition 26. A vector function gn n
:� � is positive homogeneous of the first order (or, simply, homogeneous) if
the relation g x g x( ) ( ) � holds for all xn
�� and for any � 0.
Next we need the following statements.
Lemma 14. Consider a dynamic system
dx
dt
g x t� ( ( )), (119)
where xn
�� , the vector function gn n
:� � is continuous on �n
and differentiable on �n
\{ }0 . If the function g is
homogeneous and cooperative, then for any solution x t x( , )0
, xn
0�
�
� , satisfying the condition x x x( , )00 0
� , the following
conditions are satisfied:
(a) x t x x t x( , ) ( , )
0 0� for any � 0for all t � 0;
(b) x t x x t x( , ) ( , )0 1
� for all t � 0as soon as x x0 1� , for any x x
n
0 1, �
�
� .
The proof of this lemma can be found in [78].
Denote bdn n n
( ) \ int( )� � �� � �
� , where int( )��
nis the interior of the cone �
�
n.
The vector function g is irreducible on ��
nif the matrix
>
>
g
x
a( ) is irreducible for any an
�
�
int( )� .
517
It is known [78] that for any a bdn
�
�
( ) \ { }� 0 , either the matrix
>
>
g
x
a( ) is irreducible or the inequality g ai( ) � 0follows
from ai� 0for all i = 1, 2, …, n.
Lemma 15. Let the vector function gn n
:� � in (119) be homogeneous and cooperative. If the equilibrium state x � 0
of system (119) is asymptotically stable in the large, then there exists a irreducible homogeneous cooperative function
gn n
1:� � satisfying the following conditions:
(a) g x g x1
( ) ( ) for all x � 0;
(b) the state x � 0of the system
dx
dt
g x t�
1( ( ))
is asymptotically stable in the large.
The proof of this statement can be found in [89].
Lemma 16. Let gn n
:� � be continuous and differentiable on �n
\{ }0 . If the vector function g is homogeneous,
cooperative, and irreducible, then there exist a vector � � 0 and a number ��
such that g ( )� � �
�
� . Moreover, the state x � 0 of
system (119) is asymptotically stable in the large if and only if ��
� 0.
The proof of this lemma can be found in [78].
The following corollary to Theorem 17 holds for cooperative and homogeneous systems.
Theorem 33 (see [89]). Let the vector function gn n
:� � be homogeneous and cooperative. The state x � 0of system
(119) is asymptotically stable in the large if and only if there exists a vector � � 0 such that q( )� � 0.
Proof. Sufficiency. Let there exist � � 0such that q( )� � 0. Let us consider the solution x t( , )� of system (119) as t ��.
Since q( )� � 0, it follows from (119) that
dx t
dt
( , )�
� 0 at t � 0.
There exists . � 0such that
x t x( , ) ( , )� . � �� � for all t � ( , ]0 . , (120)
whence it follows that x t( , )� �� and then it is possible to find 0 1� �� such that x( , ). � �� . According to Lemma 14, the
solutions of the cooperative system (119) are such that
x x x x( , ) ( , ( , )) ( , )2. � . . � . ��� .
Since system (119) is homogeneous, we have
x x x( , ) ( , ) ( , )22
. � . �� � . � � � � .
Moreover, since system (119) is cooperative and inequality (120) holds, we have x t( , )� ��� for all t � ( , ). .2 .
Therefore, for p = 2, 3, …, we get
x pp
( , ). � � � , …, x tp
( , )� � ��
�1for all t p p� �(( ) , )1 . . .
Hence, x t( , )� 0as t �.
Let xn
0�
�
� be given. Since � � 0, we can find a positive number k such that x k0� �. According to Lemma 14, the
solutions of system (119) are such that
x t x x k kx t( , ) ( , ) ( , )0
�. � � for all t � �.
In this case, as above, x t x( , )0
0 as t �, xn
0�
�
� . Then the solution x � 0of system (119) is asymptotically stable in
the large.
518
Necessity. Suppose that the solution x � 0of system (119) is asymptotically stable in the large. According to Lemma 15,
we choose an irreducible homogeneous cooperative vector function gn n
1:� � such that g x g x
1( ) ( )� for all x
n�
�
� and the
state x � 0of the system
dx
dt
g x t�
1( ( ))
is asymptotically stable. It follows from Lemma 16 that there exists a vector � � 0 such that g1
0( )� � . Since g g( ) ( )� �
1, we
have g ( )� � 0. This proves Theorem 33.
9.9. Stability of Cooperative Systems with Input. Consider a controllable nonlinear system
dx
dt
f x x ui
i n i� ( , , , )
1� , i = 1, 2, …, n, (121)
where xi
ni
�� , ui
mi
�� , n Nii
n
��
�
1, m M
ii
n
��
�
1, f
i
n m ni:� �
�
, fi( )% is locally Lipschitz and f
i( , )0 0 0� for all i = 1, 2,
…, n.
Definition 27 [93]. System (121) is globally input-to-state stable if there exist functions��KL-class and � �KR-class
such that for any initial conditions xN
0�� and for any input u
M�� , the following inequality holds:
| | ( , , )| | max{ (| | | | , ), (| | | | )}x t x u x t u0 0 0
� � for all t � 0,
where x t x u( , , )0
is a solution of system (121). Here | | | | sup | ( )|
[ , ]
u u s
s t t
0
0
�
�
ess , t0
0� .
If the conditions of Definition 27 are satisfied for x DN
0� �� and u U
M� �� , then system (121) is locally
input-to-state stable. Here D and U are compact sets in �N
and �M
, respectively.
Let for system (121) with ui� 0, we have a vector function
V x V x V xn n
( ) ( ( ), , ( ))�
1 1�
T, (122)
whose componentsV xi i( ) are such that
a x V x a xi i i i i i1 2(| | | | ) ( ) (| | | | ) , i = 1, 2, …, n, (123)
where a KRji
( )% � Hahn’s KR-class, i = 1, 2, …, n, j = 1, 2.
Let there exist locally Lipschitz functions �i, �
ij, �
iusuch that
dV
dt
x t V x t V x ti
i i i i ij j j i( ( )) ( ( ( ))) ( ( ( )))
( )121
� � �� � �
i
j i
u(| | | | )0
�
�. (124)
Denote
A v V xi i i
( ) ( ( ))� � , B v V xi ij j j
j i
( ) ( ( ))�
�
�� , C w w
i ii i( ) ( )� � , (125)
where wi
n�
�
� , and for inequalities (123), we will consider the following comparison system with input:
dy
dt
M y C w� �( ) ( ),
where M y A y B y( ) ( ) ( )� � � , y wn
, �
�
� .
Let the system
dy
dt
M y� ( ), yn
�
�
� , (126)
519
satisfy conditions (a)–(c) in Sec. 7. Applying Theorem 17 to system (126) leads to the following statement.
Theorem 34. For the zero solution of system (126) to be asymptotically stable in a nonnegative cone, it is necessary and
sufficient that this system satisfy the MO-condition.
Proof. Theorem 34 is a corollary of Theorem 17 for g y A y B y( ) ( ) ( )� � � .
Let ui
n mi i
� �
�
� � � , i.e., u u t xi i i� ( , ) and G u( ) is a vector function such that
G u u t x u t xu nu n n
( ) ( (| | ( , )| | ), , (| | ( , )| | ))� � �
1 1 1�
T.
Consider a comparison system with input
dy
dt
M y G u� �( ) ( ), yn
�
�
� , (127)
and formulate the following statement.
Theorem 35. Suppose that for system (121), we have a vector function (122) with components (123) and estimates
(124) along the solutions of system (121). Let the matrices M y( )and G u( )be defined by (125). Then the input-to-state stability
of system (121) follows from the input-to-state stability of system (127).
The proof of this theorem follows from the general comparison principle for continuous systems (see [23]). With the
function
L x V x( , ) ( )� ��
T, ��
�
�n
,
it is easy to obtain the estimate
>
>
� � � �
L
x
x t L x t u t( ( ), ) (| | ( ( ), )| | ) (| | ( )| | )
( )121
0 � � (128)
whence follows the input-to-state stability of system (121).
Theorem 36. If for system (126), the MO-condition is satisfied and there exists a continuous function Q:� �� �
,
Q s( ) � 0for all s � 0such that
Q s ds( )
0
�
�� �
and, moreover,
Q a y
L
y
( (| | | | ))
>
>
1 for all yn
�
�
� ,
then system (127) is input-to-state stable.
Proof. Since for system (126), the MO-condition is satisfied and there exists a positive definite and radially unbounded
function L y( ), we have
a y L y a y1 2
(| | | | ) ( ) (| | | | ) (129)
for all yn
�
�
� . Let w y L y( ) ( ( ))�B , where B( ( ))L y KR� -class and is defiend by
B( ) ( )r Q s ds
r
�
�
0
.
With (129), the function w y( ) can be estimated as B B( (| | | | ) ( ) ( (| | | | ))a y w y a y1 2
and
520
>
>
>
>
w
y
M y G y Q L y
L
y
M y
�
�
�
�
�
�
�
�
� �
�
�
�
�
�
�
�
�
T T
( ( ) ( )) ( ( )) ( ( )� G u( ))
� � �
>
>
(| | | | ) ( (| | | | ) ( (| | | | )) ( ) | | ( )| |y Q a y Q a y
L
y
y G u1 2
� �
~(| | | | ) | | ( )| |� y G u (130)
for all for all yn
�
�
� , where~
( ) ( ) ( ( ))� �s s Q a s�
1and �( )s is some KR-class function such that
>
>
�
L
y
y M y y( ) ( ) (| | | | )
�
�
�
�
�
�
�
�
�
T
for
all yn
�
�
� . If � �( ) max ( )s s
iiu
� (see (124)), then | | ( )| | (| | | | )G u y � for all yn
�
�
� and estimate (130) becomes
>
>
� �
w
y
M y G y y w
�
�
�
�
�
�
�
�
� � �
T
( ( ) ( ))~
(| | | | ) (| | | | )
whence follows Theorem 36.
Note that Theorem 36 remains valid if Q s c( ) /�1 , c = const > 0 or Q s
c c s
s c c s
( )
, ,
, ,
�
�
9
:
;
� �
� � �
1
1
2
1
1
1
1
2
1
1
1
where c1
0� , c2
1� .
9.10. Cooperative Hereditary Systems. Consider a linear hereditary system
dx
dt
Ax t Bx t� � �( ) ( )@ , (131)
where xn
�� , A and B are constant n n� -matrices.
The solution x t( , )� of Eqs. (131) with initial function � @� �
�
Cn
([ , ], )0 � exists and is unique for all t � � �[ , )@ . If
s� �[ , ]@ 0 , then x s s( , ) ( )� �� . The vector function xt
n:[ , ]� @ 0 � and x s x t s
t( ) ( )� � on the interval � @ s 0.
The system of equations (131) is positive if x t( , )� � 0 for all t � 0 follows from � � 0. System (131) is positive if the
matrix A is Metzler and B is nonnegative.
According to [89], the state x � 0 of positive system (131) is asymptotically stable for any @ � 0 if and only if the zero
solution of the following system is asymptotically stable:
dx
dt
A B x t� �( ) ( ).
Definition 28. The vector function gn n
:� � maintains the order of growth if for small x yn
, �
�
� , the inequality
g x g y( ) ( )� holds when x y� .
Consider a nonlinear hereditary system
dx
dt
f x t g x t� � �( ( )) ( ( ))@ , (132)
where the vector functions f gn n
, :� � are continuous on �n
and continuously differentiable on �n
\{ }0 .
Lemma 17. If the vector function f in (132) is cooperative and homogeneous, and the vector function g maintians the
order of growth and is homogeneous, then for initial functions � � @, ([ , ], )� �
�
Cn
0 � such that � � , the solutions x t( , )� and
x t( , )� are such that x t x t( , ) ( , )� � for all t � 0.
Theorem 37. Suppose that system (132) satisfies the conditions specified in Lemma 17. The equilibrium state x � 0of
system (132) is asymptotically stable in the large for any @ � 0if and only if the equilibrium state x � 0of the following system is
asymptotically stable in the large:
dx
dt
f x t g x t� �( ( )) ( ( )).
521
The proof of this theorem is based on Lemma 17 and can be found in [89].
10. Dissipativity and Stability in Some Mathematical Models of Population Dynamics. Dissipative systems play an
important role in mathematical biology. In analyzing the dynamics of population models describing the interaction of several
species, it is often necessary to find out whether the population size of some of the species increases without limit with time (see
[50, 70]). In biological systems, it is sometimes necessary that the population size of species remain within given limits during
evolution (see [40, 70]).
10.1. Dissipativity Conditions. We will consider generalized Volterra models of special form [4]. A generalized
MO-condition condition will be used to establish the conditions for the uniform dissipativity of the systems under consideration.
Consider a model
� ( ) ( ) ( )x g x b c f x a f xi i i i i i i ij j j
j
j i
n
iij
� � �
�
�
�
�
�
�
�
�
�
"�
1
�
�
�
�
�
�
, i = 1, …, n, (133)
where x ti( ) is the size of the ith population at time t; b
i, c
i, and �
ijare constant coefficients, c
i� 0; g x
i i( ) and f x
i i( ) are
functions defined for xi� ��[ , )0 and having special properties. The coefficients b
icharacterize the growth of the ith population
(specific birth rate minus specific death rate). The terms c f xi i i
i"
( ) describe the processes of population self-limitation. The
terms a f xij i j
( ) describe the interaction among populations. For example, if aij� 0, then the jth species has a beneficial
influence on the ith species, and if aij� 0, then the influence is adverse (the jth species is a competitior or a predator for the ith
species), and, finally, if aij� 0, then the jth species is said to be neutral to the ith one, "
i� 0, �
ij� 0, j i� , i, j = 1, 2, …, n.
If g x f x xi i i i i( ) ( )� � , i = 1, …, n, in (133), then we arrive at the well-known Gilpin–Ayala model widely used in many
applications (see, e.g., [50]). In [71], the Gilpin–Ayala model was used to describe the competitive processes in drosophila
populations and showed good agreement with experimental data.
Since system (133) is biological, it can be considered only in a nonnegative orthant ��
n. Let the functions g x
i i( ),
f xi i( ), i = 1, …, n, have the following properties:
(i) g xi i( ) and f x
i i( ) are continuous for x
i� 0;
(ii) for any t0� �� �( , ), x
( )0�
�
�n
, a unique solution x x( , , )( )
t t0
0of system (133) passes through the point ( , )
( )t0
0x ;
(iii) g fi i( ) ( )0 0 0� � and g x
i i( ) � 0, f x
i i( ) � 0 for x
i� 0;
(iv) f xi i( )�� as x
i��.
With these conditions, ��
nis an invariant set for the system under consideration. Let us additonally assume that there
exists a positive number � such that for any numbers � �
i� , i = 1, …, n, the following relations hold:
(v)
f
g
di
i
xii
�
@
@
@
( )
( )1
� �� as x
i��;
(vi)
f
g
di
i
i�
@
@
@
( )
( )1
1
�� �.
For example, the functions g x xi i i
mi
( ) � , f x xi i i
pi
( ) � , i = 1, …, n, have properties (i)–(vi) if the exponents mi
and pi
are such that mi�1, �
ij jp �1, j i� , i, j = 1, …, n.
Definition 29. System (133) is uniformly dissipative in �m
�
if there exists a number R � 0such that for any Q � 0, it is
possible to select T T Q� �( ) 0 so that the inequality | | ( , , )| |( )
x xt t R0
0 holds for all t
00� , x
( )0�K
Qand t T
0� .
Here K QQ m� �
�
{ : , | | | | }x x x� .
Let us show that Eqs. (133) are uniformly dissipative if two auxiliary systems of algebraic inequalities have positive
solutions.
522
Theorem 38. For system (133) to be uniformly dissipative in �m
�
, it is sufficient, and if all the coefficients bi, a
ijare
nonnegative and the functions f xi i( )are nondecreasing for x
i� ��[ , )0 , i, j = 1, …, n, j i� , it is also necessary that the following
conditions be satisfied:
(a) there exist positive numbers � �
1, ,�
nsuch that
�
�
�
�
"
� "
�
� "
i
i i
ij
j j
0 for j i� and aij� 0, i, j = 1, …, n, (134)
(b) there exist positive numbers � �
1, ,�
nsuch that
c ai i ij j
j
j i
n
ij
� �
"
�
� �
�
�
�
1
0, a aij ij� max{ ; }0 , i = 1, …, n. (135)
Proof. Sufficiency. Let inequalities (134) hold for positive numbers � �
1, ,�
n. Without loss of generality, we may
assume that � �
i� , i = 1, …, n.
Let us consider the following Lyapunov function on the set K x i ni
00 1� � �{ : , , , }x � :
V
f
g
di
i
n
i
i
xii
( )
( )
( )
x �
�
� �
@
@
@
�
1 1
,
where
1, ,�
nare positive constants. The functionV ( )x is continuously differentiable on K
0, andV ( )x �� as | | | |x ��.
Let us calculate its derivative along the solutions of Eqs. (133). For all x�K0
, we have
dV
dt
b f x c f x f xi
i
n
i i i i i i i i
i i i i
( )
( ) ( ) ( )
133 1
� � �
�
�
�
� � " �
a f xij j j
j
j i
n
ij�
( )
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
1
� �
�
�
�
� � " ��
i
i
n
i i i i i i i i ij jb f x c f x f x a f
i i i iij
1
( ) ( ) ( ) (xj
j
j i
n
)
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
1
.
To prove that system (133) is uniformly dissipative in K0
, it is sufficient to show that the positive coefficients
1, ,�
n
and number # � 0can be selected so that the function
W b y c y y a yi
i
n
i i i i i ij j
j
j i
n
i i i iij
( )y � � �
�
�
�
�
�
� �
� � " ��
1 1
�
�
�
�
�
�
�
�
�
�
�
satisfies the inequality W y( ) � 0 in the domain | | | |y � #.
Consider the auxiliary function
~
( )~
W c y y a yi
i
n
i i i ij j
j
j i
n
i i iij
y � �
�
�
�
�
�
�
�
�
�
�
� �
� " ��
1 1
�
�
�
�
�
�
,
where~a
ij� 0if the inequality in (134) corresponding to these values of the indices i and j is strict and
~a a
ij ij� if this inequality
becomes equality.
523
Note that
~
( )W y is a function homogeneous in extended sense [16]. According to [45], if system (135) has a positive
solution, then there exist positive coefficients
1, ,�
nat which the inequality
~
( )W y � 0holds for all y � 0. With the properties
of functions homogeneous in extended sense, the function W ( )y is negative in the domain | | | |y � # for the found values of
1, ,�
nand for sufficienly large # � 0.
Let some solution of system (133) emerge from a point x( )0
on the boundary of the set ��
n. Then this point has some
number of zero coordinates. For definiteness, we assume that x( ) ( ) ( )
( , , , , , )0
1
0 00 0� x x
m� �
T, where x
i
( )00� , i = 1, …, m,
1 �m n (the case x( )
( , , )0
0 0� �
Tis trivial). Considering the invariance of the set
K x i m x j m nm i j1
0 1 0 1, ,
{ : , , , , , , , }�
� �
�
� � � � � �x
and using the Lyapunov function
V
f
g
di
i
n
i
i
xii
( )
( )
( )
x �
�
� �
@
@
@
�
1 1
,
i� 0,
we see, as above, that conditions (a) and (b) of Theorem 38 ensure the uniform dissipativity of system (133) in Km1, ,�
�
.
Necessity. Assume that the coefficients aij
in (133) are nonnegative, and the functions f xi i( ) are nondecreasing for
xi� ��[ , )0 , i, j = 1, …, n, j i� . Then (133) is a Wazewski system. According to [32], if the system is uniformly dissipative in
�m
�
, then for any # � 0, positive numbers � �
1, ,�
ncan be selected so that the inequalities | | | |� � # hold and
b c ai i i ij j
j
j i
n
iij
� � �
�
�
�� �
"�
1
0, i = 1, …, n,
where � �7� �
1, , )�
n
T.
Let bi� 0, i = 1, …, n. In this case, these values of � �
1, ,�
nalso satisfy inequalities (135). In [46], it was proved that
then for any # � 0, a positive solution � �
1, ,�
nof system (135) can be selected so that the conditions �
i� #, i = 1, …, n, are
satisfied. This means that system (134) should also have a positive solution. The theorem is proved.
Corollary 1. If there exist positive numbers � �
1, ,�
nsuch that
�
�
�
�
�
"
� "
�
� "
i
i i
ij
j j
0 for j i� and aij� 0, i, j = 1, …, n, (136)
then system (133) is uniformly dissipative in �m
�
.
Proof. Let inequalities (136) hold for positive numbers � �
1, ,�
n. Then for sufficiently large @ � 0, the numbers
� @
� "
i
i i�
�1/( )
, i = 1, …, n, satisfy system (135).
Remark 14. The necessary and sufficient conditions for the existance of a positive solution of system (136) were
established in [37].
Example 18. Let Eqs. (133) have the form
� ( )x x b c x a x1 1 1 1 1 12 2
1� � �
"
,
� ( )x x b c x a x a x2 2 2 2 2 21 1 23 3
2� � � �
"
,
� ( )x x b c x a x3 3 3 3 3 32 2
3� � �
"
. (137)
524
Here n � 3, f x xi i i( ) � , � � � �
12 21 23 321� � � � . As before, we assume that c
i� 0, i = 1, 2, 3. Moreover, let the coefficients
b1, b
2, b
3, a
12, a
21, a
23, a
32be positive. Thus, the second species is in symbiotic relationship with the first and third species,
which are neutral to each other.
Consider the system of inequalities (134) corresponding to Eqs. (137):
� � "
1 1 20h h , � � "
2 2 10h h , � � "
2 2 30h h , � � "
3 3 20h h ,
where hi i i� �1/ ( )� " , i = 1, 2, 3. Rearranging these inequalities into the form h h h
2 1 1 1 2 2 " " " , h h h
3 2 2 2 3 3 " " " , we
see that for a positive solution to exist, it is necessary and sufficient that
" "
1 21� , " "
2 31� . (138)
Let us now set up the system of inequalities (135) corresponding to Eqs. (137):
c a1 1 12 2
10� �
"
� � , c a a2 2 21 1 23 3
20� � �
"
� � � , c a3 3 32 2
30� �
"
� � .
Applying the elimination method, we arrive at the sequence of inequalities
121
2
1 2
23
2
3 2
2 2� �
�
�
�
�
�
�
�
�
� �
�
�
�
�
�
�
�
�
�
� �a
c
a
c
� � � �
" "
� �
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
� �
�a
c c
a21
2
12
1
1
2
12
1
1 2 1
�
�
"
" " "
/
( )/3
2
32
3
1
2
1
3
3 2 3
c c
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
��
�
"
" " "
/
( )/
.
With (138), for positive numbers �1, �
2, �
3satisfying these inequalities to exist, it is necessary and sufficient that one
of the following conditions be satisfied:
(a) " "
1 21� , " "
2 31� ;
(b) " "
1 21� , " "
2 31� , ( / )( / )� � �a c a c
21 2 12 1
2 1"
;
(c) " "
1 21� , " "
2 31� , ( / )( / )� � �a c a c
23 2 32 3
2 1"
;
(d) " "
1 21� , " "
2 31� , ( / )( / ) ( / )( / )� � � � � �a c a c a c a c
21 2 12 1 23 2 32 3
2 2 1" "
.
Hence, according to Theorem 38, for system (137) to be uniformly dissipative in ��
3, it is necessary and sufficient that
one of conditions (a)–(d) be satisfied.
Remark 15. Of biological interest is the case where the solutions emerging from the positive orthant K0
arrive at a
bounded domain separated from the coordinate hyperplanes (populations do not die out) [40, 70]. Note that if bi� 0and a
ij� 0
for j i� , i.e., if the natural population increment is positive and all populations have beneficial influence on each other (they are
in symbiotic, commensal, or neutral relationship [40, 70]), then conditions (i) and (ii) of Theorem 38 ensure such a behavior of
the solutions of system (133). The functions g xi i( ), f x
i i( ), i = 1, …, n, may not satisfy the additional condition (138).
10.2. Interspecific Interaction Models with Switching. Let us consider the case where the parameters characterizing a
biological system can switch from one set of values to another. Such switchings may be trigerred by changes in the
environmental conditions. Consider the system
� ( ) ( ) ( )( ) ( ) ( )
x g x b c f x a f xi i i i i i i ij j j
j
j i
n
� � �
�
�
�
�
�
& & &
1
�
�
�
�
�
�
�
�
�
�
, i = 1, …, n, (139)
where g xi i( ) and f x
i i( ) are given functions continuous for x
i� ��[ , )0 ; & &� ( )t is a piecewise-constant function defining the
switching law, &( )t : [ , ) { , , }0 1�� �S N� ; bi
s( ), c
i
s( ), a
ij
s( )are constant coefficients such that c
i
s( )� 0, j i� , i, j = 1, …, n, s = 1,
…, N. Thus, at every instant, the system is described by one of the subsystems
525
� ( ) ( ) ( )( ) ( ) ( )
x g x b c f x a f xi i i i
s
i
s
i i ij
s
j j
j
j i
n
� � �
�
�
�
�
�
1
�
�
�
�
�
�
�
�
�
�
, i = 1, …, n, s = 1, …, N. (140)
System (139) fall into the class of so-called hybrid systems. Let us establish the conditions for the uniform dissipativity
of this system.
Note that the analysis and synthesis of switched systems frequently require that the system have prescribed properties
for any admissible switching law [75]. This is because the switching law may be either unknown or too complicated to be
explicitly described in the dynamic analysis of the system.
Let the functions g xi i( ) and f x
i i( ) satisfy conditions (i)–(iv) in Sec. 10.1 and f x
i i( ) be functions continuously
differentiable for xi� ��[ , )0 such that 3 �f x
i i( ) 0for x
i� 0, i = 1, …, n. Denote a a
ij
s
ij
s( ) ( )max{ , }� 0 , j i� , i, j = 1, …, n, s = 1, …, N.
Theorem 39. If there exist positive numbers � �
1, ,�
nsuch that
c ai
s
i ij
s
j
j
j i
n
( ) ( )� �� �
�
�
�
1
0, i = 1, …, n, s = 1, …, N, (141)
then system (139) is uniformly dissipative in �m
�
for any switching law.
Proof. Let � �
1, ,�
nbe a positive solution of the system of inequalities (141). Then, there exists a number . � 0 such
that
c ai
s
i ij
s
j
j
j i
n
( ) ( )� � .� � �
�
�
�
1
0, i = 1, …, n, s = 1, …, N.
Consider the Lyapunov function
V
f x
i n
i i
i
( ) max
( )
, ,
x �
�1 � �
and its upper right Dini derivative D V�
( )x along the solutions of the sth subsystem from (140).
Choose a point � ( � , � )x � �
�
x xn
n
1�
T� , �x � 0 and consider the solution x( ) ( ( ), , ( ))t x t x t
n�
1�
Tof the sth subsystem
energing from this point at t � 0. Let us find max ( � ) /
, ,i ni i i
f x
�1 �
� . Let this maximum be reached with indices i A n� � { , , }1� , i.e.,
V f xi i i
( � ) ( � ) /x � � for i A� ,V f xi i i
( � ) ( � ) /x � � for i A? . Then for each i A� we have
d
dt
f x t f x
g x bi i
it
i i
i
i i i
s( ( )) ( � )
( � )( )
� �
�
�
�
�
�
�
�
�
�
3
�
��0
c f x a f xi
s
i i ij
s
j j
j
j i
n
( ) ( )( � ) ( � )�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
1
3
� �
f x
g x b c f x
f x
ai i
i
i i i
s
i
s
i i
i i
i
ij
( � )
( � ) ( � )
( � )( ) ( ) (
� �
s
j
j
j i
n
i i
i
i i i
sf x
g x b V) ( )
( � )
( � )�
�
.
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
3
�
1
7 8( � )x
i.
Thus, the Dini derivative of the functionV ( )x along the solitions of any subsystem from (140) is such that D V�
�( )x 0
forV b( ) /x � ., where b is the maximum of bi
s( ), i = 1, …, n, s = 1, …, N.
Next, following a standard line of reasoning [4], we conclude that system (139) is uniformly dissipative in ��
nfor any
switching law. The theorem is proved.
526
Remark 16. It follows from the proof of the theorem that if b � 0, the zero solution of system (139) is asymptotically
stable in the large. In this case, all populations die out with time. If b � 0, then the inequality V b( ) /x . defines the domain
where system (139) is dissipative.
10.3. Reaction–Diffusion Equation as a Population Model. Consider, in an open domain + ��m
, the dynamics of
population density described by the following equation [97]:
du
dt
d u uf x u� H �
2( , ) for x�+,
u � 0 for x�>+, (142)
whereH2
is the Laplacian, >+ is the boundary of the domain+, d = const > 0 and f x u( , ) is an analytic function. Let E be the set
of equilibrium states of Eq. (142), ��
� � �{ : }u X u�
0 , X Lp
� ( )+ and ��(1/2, 1). The following statement is true.
Theorem 40. Let f be an analytic function and there exist �0
0� such that
f x u( , ) � 0 for u � �
0, x�+.
Then the set E is finite and one of its points is asymptotically stable in the cone ��
.
The proof of this statement can be found in [77].
Consider the equation
>
>
u
t
d u F x u� H �
2( , ) for x�+,
u � 0 for x�>+. (143)
For this model, the following result is well-known (see [55]).
Theorem 41. Let the function F x u( , ) be analytic and there exist real numbers a b 0 such that for all x�+, the
following inequalities hold:
F x u( , ) � 0 for u a� ,
F x u( , ) � 0 for u b� .
Then the set E of equilibrium states of Eq. (143) in the domain K u a u x b x� � �{ : ( ) , }� + is finite and there exists
an equilibrium state that is asymptotically stable in K.
In summary, let us consider the generalized Kolmogorov’s model in population dynamics:
>
>
u
t
d u uf x u v� H �
1
2
1( , , ) for x�+,
>
>
v
t
d v vf x u v� H �
2
2
2( , , ) for x�+,
u v� � 0 for x�>+, (144)
where fi
are analytic functions, di� 0, i = 1, 2, and
>
>
f
v
x u vi
( , , ) � 0,
>
>
f
u
x u vi
( , , ) � 0, i = 1, 2.
Following [55, 96], we assume that the equilibrium state E0� (0, 0) is repulsive and there exist unique equilibrium
states E u x1
0� ( ( ), ) and E v x2
0� ( , ( )) that attract all nontrivial solutions of Eqs. (144) emerging from the corresponding
coordinate axis.
Let us linearize system (144) about the equilibrium state E1
and we solve the eigenvalue problem for variational
equations in the neighborhood of E1:
527
>
>
u d u u f x u x u x
f
u
x u x u� H � �
I
J
K
L
M
N
�
1
2
1
10 0( , ( ), ) ( ) ( , ( ), ) ( ) ( , ( ), )x
f
v
x u x v
>
>
10 ,
v d v vf x u x� H �
2
2
20( , ( ), ), x�+,
u v� � 0 for x�>+. (145)
The second equation in (145) is not coupled with the first one, which allows us to consider the principal eigenvalue &12
of this equation. Accordign to Theorem 8.3.2 in [96],
(a) if &12
0� , then all the eigenvalues of problem (145) have negative real parts and the equilibrium state E1
is
asymptotically stable;
(b) if &12
0� , then E1
is unstable.
In cases (a) and (b), the equilibrium state E1
is isolated. The equilibrium state E2
can be analyzed in a similar fashion
(see [69]).
11. Stability of Two-Component Hybrid System. Mechanics problems related to complex systems with diverse
subsystems necessitate studying hybrid models of various phenomena. Examples of such systems are a pipeline conveying a
fluid, disturbed flight of a aircraft formation through a medium with uncertain parameters, oscillations of coupled pendulums
suspended from a flexible rod or an elastic thread, vibrations of beads on an elastic membrane, etc. Making certain assumptions
on the discrete and continuous components of such systems, we obtain complex systems consisting of two coupled subsystems,
one described by ordinary differential equations and the other by partial differential equations.
We will discuss the problem of stability in the context of the MO-condition of a two-component system consisting of a
system of ordinary differential equations and a system of partial differential equations.
Consider a system of perturbed equations of motion
dx
dt
X t x t g t z x t w t z� �( , ( )) ( , , ( ), ( , ))1
, x t x( )0 0
� ,
>
>
> >
w
t
L t z z w g t z x t w t z� �( , , / ) ( , , ( ), ( , ))2
, (146)
where w t z w z( , ) ( )0
0� , M t z z w w t s( , , / ) | ( , )> >
>+
�
1, s�>+,+ ��
k, X T D
n:
0� � , L B B:
1 2 , M B B:
1 3 , !
0
4�B , L
and M are some differential operators and B B1 4
, ,� are Banach spaces. The independent subsystems
dx
dt
X t x t� ( , ( )),
>
>
> >
w
t
L t z z w� ( , , / ) (147)
are related by the functions
g g t z x w T H Qn
1 1 0� � � � ( , , , ): + � ,
g g t z x w T H Qm
2 2 0� � � � ( , , , ): + � . (148)
Let for subsystems (147) and functions (148), there exist functions v C T H Qij� � �( , )
0� , i, j = 1, 2, v t x w
ij( , , ) that are
locally Lipschitz in x and w, comparison functions � �
i iK, � -class, i = 1, 2, positive constants �
ii, �
ii� 0, and arbitrary
constants �12
, �12
such that
� � � �11 1
2
11 11 2
2(| | | | ) ( , , ) (| | | | )x v t x w x ,
� � � �22 1
2
22 22 2
2(| | | | ) ( , , ) (| | | | )w v t x w w ,
528
� � � � � �12 1 1 12 12 2 2
(| | | | ) (| | | | ) ( , , ) (| | | | ) (| |x w v t x w x w| | ) (149)
for all for all x H� , w Q� , and t T�
0, | | | |x and | | | |w are vector norms in the respective spaces.
Lemma 18. WIth these conditions, the function
V t x w U t x w( , , ) ( , , )�� �
T, ��
�
�2
, (150)
is positive definite and decreasing if the 2�2-matrices
A1
11 12
21 22
�
�
�
�
�
�
�
�
�
� �
� �
, � �12 21
� ,
A2
11 12
21 22
�
�
�
�
�
�
�
�
�
� �
� �
, � �
12 21�
are positive definite.
The proof of this statement can be found in [85].
Consider a vector functionV t x w U t x w( , , ) ( , , )� � ,���
�2
with the following components:
V t x w v t x w v t x w1 11 1 12 2
( , , ) ( , , ) ( , , )� �� � ,
V t x w v t x w v t x w2 22 2 12 1
( , , ) ( , , ) ( , , )� �� � .
Given function U t x w( , , ) with elements (149), for the functions V t x w1
( , , ) and V t x w2
( , , ), there exist functions
G V Vi( , )
1 2, i = 1, 2, that are nondecreasing in both arguments and such that
G V V D V t x w1 1 2 1
0( , ) ( , , )� �
�
,
G V V D V t x w2 1 2 2
0( , ) ( , , )� �
�
(151)
for all x D� , w Q� , and t T�
0.
If inequalities (151) hold, system (146) has a nonlinear comparison system
du
dy
G u u1
1 1 2� ( , ),
du
dy
G u u1
1 1 2� ( , ). (152)
The following statement is true.
Theorem 42. Assume that for system (146), there exists a matrix-valued functionU t x w( , , ) with elements (149) such
that function (150) is positive definite and decreasing and the comparison system (152) satisfies conditions C1–C3 (Sec. 7).
Then, if system (152) satisfies the MO-condition, then the state ( , )x wT T T
� 0 of system (146) is uniformly asymptotically
stable.
The proof of this theorem is omitted as being similar to the proof of Theorem 21.
Remark 17. The problem of constructive construction of the nonlinear majorants G1
and G2
in (151) remains opened. If
the functionsG1
andG2
are sought in some parametric class of functions{ ( , , )}G u u p1 2
, where pl
� �� � , � is a compact set in
�l, then the search of G V V
i( , )
1 2, i = 1, 2, can be reduced to the solution of a mathematical programming problem [35]:
min { ( ( , , ), ) ( , , )}
p
G V t x w p D V t x w
�
�
�
�
5 ,
529
where 5 is some functional of the residual G V p D V t x w( , ) ( , , )�
�
in (151). In the general case, this problem remains unsolved
too.
12. Concluding Remarks. In fact, the general theorems of Lyapunov’s direct method [25] are based on elementary
differential inequalities. The study [20] initiated the regular application of differential inequalities in the theory of the stability of
motion.
Vector Lyapunov functions were proposed in the early 1960s. Together with the theorems of differential and integral
inequalities [1, 98], they formed the basis of a new method for the qualitative analysis of nonlinear complex systems. Theorems 1
and 2 illustrate the relationship between the dynamic properties of the comparison and original systems based on a scalar or
vector Lyapunov function. The Lyapunov function plays the role of a nonlinear transform that reduces the dimension of the
original system. The method of vector Lyapunov functions is developed in a great many articles (see [54, 84] and the references
therein) and monographs (see [2, 12, 23, 33–35, 99]).
The method of matrix Lyapunov functions as a development of Lyapunov’s direct method combines the advantages of
the method of vector Lyapunov functions (the possibility to construct a Lyapunov function from several components) and
Lyapunov’s direct method (which does not use the solutions of the original and/or comparison equations). Many results in this
field were obtained at the Department of Stability of Processes of the S. P. Timoshenko Institute of Mechanics, National
Academy of Sciences of Ukraine, and are reported in several monographs [83, 86, 87].
One of the ways to develop the method of matrix Lyapunov functions is to extend it to matrix systems of differential
equations (see [95]). Relevant results were obtained by setting up matrix Lyapunov functions [24] and by using the comparison
principle [29, 95]. The studies in this field have a great potential for further development.
Since the study [51], linear comparison systems have been intensively used in the method of matrix Lyapunov
functions. The supersufficiency of the associated stability conditions (see [36]) required improving the methods of deriving and
optimizing them in various areas. Many relevant results are summed up in the monographs [34, 99].
The uniform asymptotic stability criterion for a nonlinear comparison system was established in 1978 and published in
[31]. It was successfully used in [12, 88] to analyze the stability of motion in some problems of mechanics. It was further
developed and used in new areas in [3–5].
It is noted in [2] that testing the conditions that the equilibrium position y � 0is isolated may involve some difficulties.
A stability criterion for a comparison system was proposed in [19]. It is similar to the MO-condition and replaces the isolation
condition for the equilibrium state with the existence condition for a connected set Q located in a nonnegative cone and adjoining
the origin of coordinates such that g y( ) � 0 for all y Q� . Moreover, in this criterion does not require the solutions of the
comparison system to be unique.
In [89], this criterion is applied to cooperative homogeneous systems as a special case of monotonic systems. The
criterion was further developed in [45, 46] to be used in the analysis of the absolute stability and dissipativity of some classes of
nonlinear systems.
The theory of monotonic systems is now developed in many areas (see [27, 61–65, 67–69, 79–81, 96–97] and the
references therein). In some cases, real-world processes are adequately described by monotonic systems, i.e., a vector Lyapunov
function is not required to analyze dynamic monotonic systems, but what is needed are general stability criteria.
I would like to take this opportunity to express my gratitude to the professor Aleksandrov who drew my attention to
some papers reporting on results that are a special case of the MO-criterion (asymptotic stability of nonlinear monotonic
systems). It appears that the authors of these papers are not familiar with the results of the studies [31, 32] and their applications.
REFERENCES
1. N. V. Azbelev and Z. B. Tsalyuk, “On integral and differential inequalities,” in: Trans. 4th All-Union Math. Congr. [in
Russian], Vol. 2, Nauka, Leningrad (1964), pp. 384–390.
2. A. Yu. Aleksandrov and A. V. Platonov, Differential Inequalities and Stability of Motion [in Russian], Solo,
St. Petersburg (2006).
3. A. Yu. Aleksandrov and A. V. Platonov, “On absolute stability of one class of nonlinear switched systems,” Automat.
Remote Control, 69, No. 7, 1101–1116 (2008).
530
4. A. Yu. Aleksandrov, A. V. Platonov, and Y. Chen, “Dissipativity of some classes of dynamic population models,” Vestn.
SPbGU, Ser. 10, No. 2, 3–17 (2010).
5. A. Yu. Aleksandrov, A. V. Platonov, and Y. Chen, “A note on the absolute stability of nonlinear switched systems,”
Vestn. SPbGU, Ser. 10, No. 1, 122-136 (2008).
6. E. A. Barbashin, “Setting up a Lyapunov function for nonlinear systems,” in: Proc. 1st IFAC Congr. [in Russian],
Moscow (1961), pp. 742–751.
7. E. A. Barbashin, Lyapunov Functions [in Russian], Nauka, Moscow (1970).
8. R. Bellman and S. Dreyfus, Applied Dynamic Programming, Princeton Univ. Press, Princeton, N.J. (1962).
9. M. Bohner and A. A. Martynyuk, “Elements of Lyapunov stability theory for dynamic equations on time scale,” Int.
Appl. Mech., 43, No. 9, 949–970 (2007).
10. V. Volterra, Lectures on the Mathematical Theory of the Struggle for Existence [in French], Gauthier-Villars, Paris
(1931).
11. F. R. Gantmacher, Matrix Theory, Chelsea, New York (1959).
12. Lj. T. Grujic, A. A. Martynyuk, and M. Ribbens-Pavella, Large-Scale Systems Stability under Structural and Singular
Perturbations, Springer-Verlag, Berlin (1987).
13. B. P. Demidovich, Lectures on Mathematical Theory of Stability [in Russian], Nauka, Moscow (1967).
14. E. E. Dudnikov and M. V. Rybashov, “Neuron network with nonlinear feedbacks,” Avtom. Telemekh., 58, No. 6, 64–73
(1997).
15. V. I. Zubov, Methods of A. M. Lyapunov and Their Application, Noordhoff, Groningen (1964).
16. V. I. Zubov, Mathematical Methods for the Study of Automatic Control Systems. Macmillan, New York (1963).
17. V. I. Zubov, Stability of Motion [in Russian], Vysshaya Shkola, Moscow (1973).
18. M. A. Krasnosel’skij, J. A. Lifshits, and A. V. Sobolev, Positive Linear Systems. The Method of Positive Operators,
Heldermann Verlag, Berlin (1989).
19. R. I. Kozlov, “Estimates of solutions and stability of comparison systems,” in: V. D. Irtegov and V. M. Matrosov (eds.),
Stability Theory and Its Applications [in Russian], Nauka, Novosibirsk (1979), pp. 38–49.
20. C. Corduneanu, “The application of differential inequalities to the theory of stability,” Analde Stiint. Univ. Jasi. Seel., 7,
No. 2, 47–58 (1960).
21. A. A. Kosov, “Stability of complex systems by nonlinear approximation,” Diff. Uravn., 33, No. 10, 1432–1434 (1997).
22. G. Kron, Diakoptics: The Piecewise Solution of Large Scale Systems, MacDonald, London (1963).
23. V. Lakshmikantham, S. Leela, and A. A. Martynyuk, Stability of Motion: Comparison Method [in Russian], Naukova
Dumka, Kyiv (1991).
24. D. M. Lila and V. I. Slyn’ko, “Setting up a matrix-valued Lyapunov function for a linear system with quasiperiodic
coefficients,” Dop. NAN Ukrainy, No. 11, 60–65 (2007).
25. A. M. Lyapunov, The General Problem of the Stability of Motion, Taylor & Francis, London (1992).
26. N. N. Luzin, On Academician Chaplygin’s Method of Approximate Integration, GAAI, Moscow–Leningrad (1932).
27. A. G. Mazko, “Stability and comparison of states of dynamical systems with respect to a time-varying cone,” Ukr. Math.
J., 57, No. 2, 232–249 (2005).
28. A. A. Martynyuk, “On the theory of Lyapunov’s direct method,” Dokl. Math., 73, No. 3, 376–379 (2006).
29. A. A. Martynyuk, “The comparison principle for a system of matrix differential equations,” Dop. NAN Ukrainy, No. 12,
28–33 (2008).
30. Yu. A. Martynyuk-Chernienko, Uncertain Dynamical Systems: Stability and Motion Control [in Russian], Feniks, Kyiv
(2009).
31. A. A. Martynyuk and A. Yu. Obolenskii, Stability Analysis of Autonomous Comparison Systems [in Russian], Preprint
78.28, Inst. Mat. ANNU SSR, Kyiv (1978).
32. A. A. Martynyuk and A. Yu. Obolenskii, “Stability of solutions of Wazewski’s autonomous systems,”Diff. Uravn., 16,
No. 8, 1392–1407 (1980).
33. V. M. Matrosov, Method of Vector Lyapunov Functions: Analysis of the Dynamic Properties of Nonlinear Systems [in
Russian], Fizmatlit, Moscow (2001).
34. V. M. Matrosov, L. Yu. Anapol’skii, and S. N. Vasil’ev, The Comparison Method in Mathematical Systems Theory [in
Russian], Nauka, Novosibirsk (1980).
531
35. V. M. Matrosov and L. Yu. Anapol’skii (eds.), Setting up Vector Lyapunov Functions [in Russian], Nauka, Novosibirsk
(1980).
36. V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations, Dover, New York (1989).
37. A. A. Piontkovskii and L. D. Rutkovskaya, “Solving some stability problems with the method of vector Lyapunov
functions,” Avtomat. Telemekh., 28, No. 10, 23–31 (1967).
38. A. V. Platonov, “Stability of nonlinear complex systems,” Izv. RAN. Teor. Sist. Upravl., No. 4, 41–46 (2004).
39. A. Poincare, On Curves Defined by Differential Equations [Russian translation], GITTL, Moscow–Leningrad (1947).
40. Yu. A. Pykh, Equilibrium and Stability in Dynamic Population Models [in Russian], Nauka, Moscow (1983).
41. N. Rouche, P. Habets, and M. Laloy, Stability Theory by Liapunov’s Direct Method, Springer-Verlag, New
York–Heidelberg–Berlin (1977).
42. E. Spanier, Algebraic Topology, McGraw-Hill, New York (1966).
43. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge Univ. Press, Cambridge, MA (1985).
44. A. Yu. Aleksandrov and A. V. Platonov, “Aggregation and stability analysis of nonlinear complex systems,” J. Math.
Anal. Appl., 342, No. 2, 989–1002 (2008).
45. A. Yu. Aleksandrov and A. V. Platonov, “Construction of Lyapunov’s functions for a class of nonlinear systems,”
Nonlin. Dynam. Syst. Theor., 6, No. 1, 17–29 (2006).
46. A. Yu. Aleksandrov and A. V. Platonov, “Conditions of ultimate boundedness of solutions for a class of nonlinear
systems,” Nonlin. Dynam. Syst. Theor., 8, No. 2, 109–122 (2008).
47. D. Angeli and E. D. Sontag, “Monotone controlled systems,” IEEE Trans. Autom. Control, 48, 1684–1698 (2003).
48. D. Angeli and E. D. Sontag, “Multistability in monotone input/output systems,” Syst. Contr. Lett., 51, 185–202 (2004).
49. D. Angeli, J. E. Ferrell, and E. D. Sontag, “Detection of multi-stability, bifurcations, and hysteresis in a large class of
biological positive-feedback systems,” Proc. Nat. Acad. Sci. USA, 101, 1822–1827 (2004).
50. F. J. Ayala, M. E. Gilpin, and J. G. Eherenfeld, “Competition between species: theoretical models and experimental
tests,” Theor. Popul. Biol., No. 4, 331–356 (1973).
51. F. N. Bailey, “The application of Lyapunov’s second method to interconnected systems,” SIAM J. Control, A3, 443–462
(1965).
52. R. Bellman, “Vector Lyapunov function,” SIAM J. Control, A1, 32–34 (1965).
53. V. S. Bokharaie, O. Mason, and M. Verwoed, “Observations on the stability properties of cooperative systems,” Syst.
Contr. Lett., 58, 461–467 (2009).
54. P. Borne, M. Dambrine, W. Perruquetti, and J. P. Richard, “Vector Lyapunov functions: Nonlinear, time-varying,
ordinary and functional differential equations,” in: A. A. Martynyuk (ed.), Advances in Stability Theory at the End of the
20th Century, Taylor & Francis, London (2003), pp. 49–73.
55. R. Cantrell and C. Cosner, Spatial Ecology via Reaction–Diffusion Equations, Wiley, New York (2003).
56. F. Chen, “Some new results on the permanence and extinction of nonautonomous Gilpin–Ayalatype competition model
with delays,” Nonlin. Anal.: Real World Appl., No. 7, 1205–1222 (2006).
57. F. Chen, L. Wu, and Z. Li, “Permanence and global attractivity of the discrete Gilpin–Ayala type population model with
delays,” Comp. Math. Appl., 53, 1214–1227 (2007).
58. E. N. Dancer, “Some remarks on a boundedness assumption for monotone dynamical systems,” Proc. AMS, 126, No. 3,
801–807 (1998).
59. M. Z. Djordjeviæ, “Stability analysis of large scale systems whose subsystems may be unstable,” Large Scale Systems,
No. 5, 255–262 (1983).
60. M. Z. Djordjeviæ, Stability of Coupled Nonlinear Systems based on the Matrix Lyapunov Method [in German], No. 7690,
Diss. Techn.Wiss. ETH, Zurich (1984).
61. G. Enciso and W. Just, Analogues of the Smale and Hirsch theorems for cooperative Boolean and other discrete
systems,” arXiv:0711.0138v2[math.DS], November 18 (2007).
62. G. Enciso, “On a Smale theorem and nonhomogeneous equilibria in cooperative systems,” Proc. AMS, 136, No. 8,
2901–2909 (2008).
63. G. Enciso and E. Sontag, “Monotone bifurcation graphs,” J. Biolog. Dynam., 2, No. 2, 121–139 (2008).
64. G. Enciso, M. W. Hirsch, and H. L. Smith, “Prevalent behavior of strongly order preserving semi flows,” J. Dynam. Diff.
Eqns., 20, No. 1, 115–132 (2008).
532
65. G. Enciso and E. Sontag, “Monotone systems under positive feedback: multistability and a reduction theorem,” Syst.
Contr. Lett., 51, No. 2, 185–202 (2005).
66. W. Hahn, Stability of Motion, Springer, Berlin (1967).
67. M. W. Hirsch, “Systems ofdifferential equations which are competitive or cooperative II: Convergence almost
everywhere,” SIAMJ. Math. Anal., 16, No. 3, 423–439 (1985).
68. M. W. Hirsch, “Stability and convergence in strongly monotone dynamical systems,” Reine Angew. Math., 383, 1–53
(1988).
69. M. W. Hirsch and H. L. Smith, “Asymptotically stable equilibria for monotone semiflows,” Discrete Contin. Dyn. Syst.,
14, No. 3, 385–398 (2006).
70. J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge Univ. Press, Cambridge
(1998).
71. M. E. Gilpin and F. J. Ayala, “Global models of growth and competition,” Proc. Nat. Acad. Sci. USA, 73, 3590–3593
(1973).
72. M. E. Gilpin and F. J. Ayala, “Schoener’s model and Drosophila competition,” Theor. Popul. Biol., No. 9, 12–14 (1976).
73. L. T. Grujiæ, A. A. Martynyuk, and M. Ribbens-Pavella, Large Scale Systems Stability under Structural and Singular
Perturbations, Springer, Berlin (1987).
74. V. Lakshmikantham, S. Leela, and A. A. Martynyuk, Stability Analysis of Nonlinear Systems, Marcel Dekker, New York
(1998).
75. D. Liberzom and A. S. Morze, “Basic problems in stability and design in switched systems,” IEEE Contr. Syst.
Magazine, 19, No. 15, 59–70 (1999).
76. D. M. Lila and A. A. Martynyuk, “On the theory of stability of matrix differential equations,” Ukr. Math. J., 61, No. 4,
556–565 (2009).
77. P. L. Lions, “Structure of the set of steady states solutions and asymptotic behavior of semilinear heat equations,” J. Diff.
Eqns., 53, 362–386 (1984).
78. P. Leenher and D. Aeyels, “Extension of the Perron–Frobenius theorem to homogeneous systems,” SIAM J. Contr.
Optimiz., 41, 563–582 (2002).
79. P. Leenher, D. Angeli, and E. D. Sontag, “A tutorial on monotone systems with an applicationto chemical reaction
networks,” in: Proc. 16th Int. Symp. on Mathematical Theory of Networks and Systems (MTNS) (2004).
80. J. Mallet-Paret and G. Sell, “The Poincare–Bendixson theorem for monotone cyclic feedback systems with delay,”
J. Diff. Eqns., 125, 441–489 (1996).
81. J. Mallet-Paret and H. Smith, “The Poincare-Bendixon theorem for monotone cyclic feedback systems,” J. Dynam. Diff.
Eqns., No. 2, 367–421 (1990).
82. A. A. Martynyuk, “Matrix Lyapunov functions and stability analysis of dynamical systems,” in: A. A. Martynyuk (ed.),
Advances in Stability Theory at the End of 20th Century, Taylor & Francis, London–New York (2003), pp. 135–151.
83. A. A. Martynyuk, Qualitative Method in Nonlinear Dynamics. Novel Approaches to Liapunov’s Matrix Function,
Marcel Dekker, NewYork (2002).
84. A. A. Martynyuk, “Stability analysis by comparison technique,” Nonlin. Anal., 62, 629–641 (2005).
85. A. A. Martynyuk, “Stability of dynamical systems in metric space,” Nonlin. Dynam. Syst. Theor., No. 5, 157–167
(2005).
86. A. A. Martynyuk, Stability by Liapunov’s Matrix Function Method with Applications, Marcel Dekker, New York
(1998).
87. A. A. Martynyuk, Stability of Motion: The Role of Multicomponent Liapunov Functions, Cambridge Sci. Publ., London
(2007).
88. A. A. Martynyuk and A. Yu. Obolenskyi, “To the theory of one-side models in spaces with arbitrary cone,” in: Proc. 12th
IMACS Congr., 1, Paris (1988), pp. 348–351.
89. O. Mason and M. Verwoed, “Observations on the stability properties of cooperative systems,” Syst. Contr. Lett., 58,
461–467 (2009).
90. A. N. Michel, “Stability analysis of interconnected systems,” SIAM J. Control, 12, No. 3, 554–5790 (1974).
91. K. S. Navendra and J. Balakrishnan, “A common Lyapunov function for stable LTI systems with commuting
A-matrices,” IEEE Trans. Automat. Control, 39, No. 12, 2469–2471 (1994).
533
92. L. Rosier, “Homogeneous Lyapunov function for homogeneous continuous vector field,” Syst. Contr. Lett., 19, 467–473
(1992).
93. B. S. Ruffer, C. M. Kellett, and S. R. Weller, “Integral input-to-state stability of interconnected iISS systems by means of
a lower-dimensional comparison system,” in: Joint 48th IEEE Conf. on Decision and Control and 28th Chinese Control
Conf., Shanghai, P.R. China, December 16–18 (2009), pp. 638–643.
94. B. S. Ruffer, P. M. Dower, and H. Ito, Computational Comparison Principles for Large Scale System Stability Analysis,
Manuscript VIC 3010, University of Melbourn, Parkville, Australia (2009).
95. M. Shaw, “Generalized stability of motion and matrix Lyapunov functions,” J. Math. Anal. Appl., 189, 104–114 (1995).
96. H. L. Smith, Monotone Dynamical Systems, Math. Surveys and Monographs, Vol. 41, AMS, Providence, RI (1995).
97. H. L. Smith and H. Thieme, “Quasi convergence for strongly order preserving semiflows,” SIAM J. Math. Anal., 21,
673–692 (1990).
98. J. Szarski, Differential Inequalities, PWN, Warsaw (1967).
99. D. D. Siljak, Large-Scale Dynamic Systems: Stability and Structure, North-Holland, New York (1978).
100. T. Yoshizawa, Stability Theory by Liapunov’s Second Method, Math. Soc. Japan, Tokyo (1966).
534
![arXiv:1504.06964v6 [stat.ME] 5 Mar 2018 · 2018. 7. 14. · initial instantaneous drop followed by a monotonic rise towards an asymptotic level not exceeding the original function](https://img.pdfslide.us/doc/110x75/604a8cbdf4c4ea254c739f9c/arxiv150406964v6-statme-5-mar-2018-2018-7-14-initial-instantaneous-drop.jpg)
