International Applied Mechanics, Vol. 29, No. I0, 1993

M A T R I X M E T H O D O F C O M P A R I S O N IN T H E T H E O R Y O F

T H E S T A B I L I T Y O F M O T I O N

A. A. Mmrtynyuk UDC 531.36

The basic assertions of the principle of comparison with a Lyapunov matrix function are formulated for three classes

of functions (scalar, vector, and matrix). Analysis of the State. The method of Lyapunov matrix functions (MLMF) is an important tool for investigating complicated

processes and phenomena on the basis of simpler (sometimes unilateral) models. The correctness of the results obtained by this method is based on strict theorems of the method of comparison, which establish a relation between the functions characterizing the phenomenon investigated and its model. The following is a summary of the formulation of the MLMF:

- - discovery of double-index system of functions, as a structure suitable for constructing Lyapunov functions (A. A.

Martynyuk and R. Gutovskii [14]); - - formation of the basic concepts of the MLMF on the basis of double-index system of functions (see A. A. Martynyuk

[7, 8], M. Z. Djordjevic [25], Lj. T. arujic [28], and A. A. Martynyuk [34, 35]); - - formation of the principle of invariance and investigation of autonomous systems (see Yu. N. Krapivnyi and A. A.

Martynyuk [5], A. A. Martynyuk [8], A. A. Martynyuk and V. V. Shegai [20], Lj. T. Grujic, A. A. Martynyuk and M. Ribbens-

Pavella [29], and A. A. Martynyuk [331); - - development of methods for constructing matrix and hierarchical matrix Lyapunov functions (Yu. N. Krapivnyi and

A. A. Martynyuk [6], A. A. Martynyuk [7], A. A. Martynyuk [39], A. A. Martynyuk and V. G. Miladzhanov [17], V. V. Shegai [24], A. A. Martynyuk and K. Begmuratov [131, R. K. Azimov and A. A. Martynyuk [3], M. Z. Djordjevic [26], and

Lj. T. Grujic and H. Shaaban [30]); -- construction of sufficient conditions of stability for a) systems with lumped parameters (A. A. Martynyuk [7, 8], A. A. Martynyuk and Yu. N. Krapivnyi [15], and A.

A. Martynyuk and K. A. Begmuratov [t3]); b) systems with a small parameter multiplying a derivative (A. A. Martynyuk [9], A. A. Martynyuk [36], A. A. Martynyuk

and V. G. Miladzhanov [171, and V. G. Miladzhanov [21]); c) systems with random parameters (A. A. Martynyuk [40], A. A. Martynyuk [10], R. K. Azimov and A. A. Martynyuk

[3], and R. K. Azimov [2]); d) systems with pulsed action with structural perturbations (V. G. Miladzhanov [22]); e) discrete systems (A. A. Martynyuk and Yu. N. Krapivnyi [6,16], and M. Z. Djordjevic [26]); f) hybrid systems (A. A. Martynyuk [34, 37] and A. A. Martynyuk [8]); g) Lur'e-Postnikov automatic regulation systems (A. A. Martynyuk and V. G. Miladzhanov [18], and M. Z. Djordjevic

[261); h) large power generating systems (Lj. T. Grujic and H. Shaaban [30] and V. G. Miladzhanov [211); i) classes of systems a), b), and d) with structural perturbations (A. A. Martynyuk [11] and V. G. Miladzhanov [21]); j) systems modeling population dynamics (G. Fridman and A. A. Martynyuk [23]); extension of MLMF to the problem

of polystability of motion (A. A. Martynyuk [12]). The objective of the present paper is to formulate on the basis of the matrix Lyapunov function the basic theorems of

the comparison method for systems with a finite number of degrees of freedom.

1. Principle of Comparison for Scalar Functions of Class SL

We consider the system of differential equations modeling an arbitrary system with a finite number of degrees of freedom

S. P: Timoshenko Institute of Mechanics, Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 29, No. 10, pp. 116-122, October, 1993. Original article submitted June 14, 1993.

1063-7095/93/2910-0861512.50 © 1994 Plenum Publishing Corporation 861

d x / d t = [ ( t , x ) , x ( to ) = x o, t o ~ O, (1.1)

whe re f E C(R+ x R n, Rn).

We assume that the functionfis sufficiently smooth and satisfies conditions for which the system (1.1) describes continu-

ous motion. In addition, we assume that the solution x(t; t o, Xo) is continuous and unique for (t 0, Xo) E int (R+ × Rn).

For the system (1.1) we can construct a double-index system of functions

U ( t , x ) = [ % ( t , x ) ] , i , j = 1,2 . . . . . s, (1.2)

where

u u e C (g+ x R",R+) and U o e C (R. X R',R), i#.i. (1.3)

Definition. All scalar functions of the form

V ( t , x , y ) = y r u ( t , x ) Y ,

where the vector y can be specified as follows: a )y ~ RS, y ~ 0;

b)y = ~ E C(R'~,R~), 5(0) = o; A n $ c) y = 1/, e C ( R + x R ,R+), ~ ( t , 0 ) = 0;

for all t E R+; d) y =A s ~ER+, @>0

belong to the class SL.

For the function (1.4) we define the function

where

D ÷ V ( t , x , y ) = y r D + U ( t , x ) y , y E R ' ,

D * U ( t , x ) = h m s u p { l V ( t +O, x + O l ( t , x ) ) - U ( t , x ) I O - I : O ~ O ' } .

(1.4)

(1.5)

Applying the function (1.4) and the quasimonotonic function, nondecreasing as a function of u for each t, g:g E C(R+

× R n x R+, R), we formulate the following comparison theorem.

(1.6) THEOREM. Let the function U E C (R+ x R n, R TM) be locally Lipshitz as a function ofx. We assume that

the function D+V(t, x, y) satisfies the inequality

D ÷ v ( t , x , y ) ~ g ( t , x , V ( t , x , y ) ) ,

where

g E C ( R + × R ' × R + , R), g(t,O,O) = O.

Let w(t) = w(t, t 0, x 0, Uo) be a maximal solution of the scalar differential equation

d u / d t = g ( t , x ( t ) , u ) , U(to) = u o ~ O,

existing for t _> t o.

Then the inequality V(t o, x o, y) < u o implies the estimate

v ( t , x ( t ) , y ) = w ( t ) t ~ t o,

(1.7)

(1.8)

862

where x(t) is any solution of the system (1.1) existing for all t >_ t 0.

Proof. Let the vector y be chosen according to the specification a). Then the function (1.4) is a scalar Lyapunov function

and the proof of the estimate (1.8) is analogous to the proof of Theorem 3.1.1 of the monograph [31].

(1.9) COROLLARY. Let all conditions of the theorem (1.6) be satisfied and let the function g(t, x, V(t, x, y)) satisfy

one of the conditions

a) g(t, x, V(t, x, y)) = 0 for all t > to; b) g(t, x, V(t, x, y)) = prAp,

where q E C(R n, R+S), p(0) = 0, and A is an s x s constant matrix;

c) g(t, x, V(t, x, y)) = qTBq + R(t, w, y),

wherep C C(R n, R+S), q(O) = O, B is an s × s constant matrix, and R E C(R+ x R+ s x R s, R) is a polynomial of degree

higher than 2.

Then the estimate (1.8) holds, and the investigation of the comparison equation (1.7) simplifies.

2. Principle of Comparison for Vector Functions of Class VL

(2.1). Definition. All vector functions of the form

L ( t , x , y ) -- . 4u( t ,x )y , (2.2)

where A is an s x s constant matrix and the vector y is determined according to a)-d) from the definition (1.3), belong to the

class VL. For any function U E C(R+ x R n, RSXS), which is related to the system (1.1), we define the function

D* L ( t , x , y ) = a D ÷ V ( t , x ) y

for a l l ( t , x ) E R+ x R n, (y ~ O) E R s.

(2.4) T I t E O R E M . Let the following exist:

1) the matrix function U E C(R+ × R n, RS×S), where U(t, x) is locally Lipshitz in x for t >_ t o >_ 0;

2) a constant s x s matrix A, the vector y E R s and the vector n E R+ s such that

q r L ( t , x , y ) ~ a(llxll),

(2.3)

where a is a function of the Hahn class;

3) a vector function G E C(R+ x R n x R+ s, RS), where G(t, x, u) is quasimonotonic and nondecreasing as a function

of u for any t @ R+, x @ D C R n, and such that the estimate

D+ L ( t , x , y ) ,~ G ( t , x , L ( t , x , y ) ) (2.5)

holds;

4) the maximal solution W(t) = W(C t o, x o, Uo) of the system

exists for all t ~ t 0.

Then the inequality

d u / d t = G ( t , x , u ) , u(to) = u o ;~ 0

L(t,x(t),y) ,~ w(t)

holds as soon as

L(to,Xo,y) ,~ u o.

(2.6)

(2.7)

(2.8)

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Proof. The function (2.2), together with the function (2.3), satisfies all conditions of the standard theorems of the principle

of comparison with a vector Lyapunov function. For this reason, the proof of the estimate (2.7) from the theorem (2.4) does

not differ significantly from the existing proofs [1, 31, 32].

(2.9) COROLLARY. Let the conditions 1) and 2) of theorem 2.4 hold, and in the conditions (3) and (4) the function G E C(R+ × R n,Rs) .

Then the estimate (2.7) holds for the maximal solution if(t) = W(t; t o, u o) of the comparison system

d u / d t = G ( t , u ) , u ( t o ) = u o • O. (2.10)

(2.11) COROLLARY. Let the conditions 1) and 2) of the theorem 2.4 hold and let the function G have the form

G ( t , x , L ( t , x , y ) ) = P L ( t , x , y ) + m ( t , L ( t , x , y ) ,

where P = [p/j] is an s × s matrix with the elements p/j ___ 0 with i ;~ j , and m E C(R+ × R+ s, R s) is quasimonotonic on

L and l {tllm(/, LglI/IILIII: IILII - , o} = 0 uniformly for t _> t 0. Then the estimate (2.7) is correct for the maximal solution if(t) = if(t, t 0, u 0) of the comparison system

d u / d t = Pu + m ( t , u ) , u(to) = u o ;~ O.

3. Comparison Principle for the Class of Functions ML

(3.1) Definition. We assigned to the class ML all functions of the form (1.2), solving the problem of stability of the

state of equilibrium x = 0 of the system (1.1). In order to formulate the comparison theorem with a matrix Lyapunov function relative to an arbitrary cone K in the

space E, we require some auxiliary information. Let the structure in E be introduced with the help of a cone K thus:

Y~ ~ Ya =~ (Y2-Y~) E K;

0

Y~ < Y2 ~- ( Y 2 - Y , ) ~ K;

i~ is the interior of the cone K, and Y1 and Y2 are elements of the space E. Let K* be a cone conjugate to K and let OK be the

boundary of the cone. Let 0 E • C E, where • is an open convex subspace of the ordered space E.

Consider the system

d z / d t = I I ( t , z ) , H ( t , 0 ) = 0, (3.1)

where II E C(R+ × ~,l, E) and H(t, u) is locally Lipshitz as a function of u.

Let zx(t) = Zl(t," t o, Zlo), z2(t) = z2(t; to, Z2o), to --- 0 be the solutions of the system (3.1) with the initial conditions

(to, Zao) and (t o, Z2o), respectively. (3.2) Definition. It is said that the system (3,1) has a (strictly) monotonic solution if the inclusions

z2o - Zlo ~ K , Z~o ~ Z,o

for all t > t o, respectively, follow from

~ ( t ) - z , ( t ) 0

K , ( z 2 ( t ) - z , ( t ) ~ r ) .

(3.3) Definition. The system (3. I) is said to belong to class Wo(K) (Ws(K)) if the inequalities

(z - y) E O K , z ~ y

864

respectively, follow from

n ( t , z ) - rI( t ,y) ;, 0 (I-l(t,z) - II ( t ,y ) > O)

(3.4) Def'mition. The operator P(t, z) is positive on J x 9 , J C R+ if z E ,It implies the inequality P(t, z) - 0 for

all t @ J, relative to the cone K.

The theorem of the comparison principle in a space ordered by an arbitrary cone is formulated thus.

(3.5) T H E O R E M . Let

1) there exists a function II(t, z) E Wo(K), which is continuous in the open (t, z) set J1 × ~ and satisfies the conditions

of uniqueness of the solutions z(t; t o, Zo) of the comparison system

dz/dt = I I ( t , z ) , z(t,) ffi z , ; (3.6)

2) there exists a function ,b(t, y) that is continuous in the open (t, y) s e t J 2 × ,t, C Jz × 't,, J2 c J1 C R+ such that

II(t, y) - - cb(t, y) = P(t, y), where P(t, y) is a positive operator on the set J x ~ , where J = J1 C) J2-

Then, if Zo --Yo E K, then

z ( t ; t o , Z o ) - y ( t ; t o , y o ) ¢ K

for all t C J, where y (t; t o, Yo) is an arbitrary solution of the system

dy/dt = ~ ( t , y ) , Y( to) = Yo" (3.7)

Proof. Together with the system (3.6) we consider the weakly perturbed system

d z / d t ffi n ( t , z ) + e ( z , u ' ) u , z( to , , ) = Zo, (3.8)

where u E K, u* E K* and e E (0, e°], the solution z(t, ~) = z(t," t o, Zo, e) of which exists on J C R+,

lim{[z(t;to,zo,e)] :, -~ O} = z(t;to,Zo)

uniformly on J C R+.

For the function m(t, ~) = z(t; t o, Zo, e) - - y(t; t o, Yo) k is easy to obtain the equation

d m / d t = r l ( t , z ) - I I ( t , y ) + P ( t , y ) + e ( z , u ' ) u , (3.9)

where P(t, y) is a positive operator.

We construct with the set of points m from the boundary of the cone K the indicator function 6(- [ K), defined as

O, if m E K; ~ ( m l x ) ffi +®, if m~X.

For the indicator function ~i(m I K) we calculate the subgradient F(m) and form the scalar product with the right and left sides

of Eq. (3.9) and F(m). We obtain (I'(m), dm/dt) < --c~, c~ = const > 0 at the point t = t*. Therefore m(t, e) does not leave the cone K for t > t* and e --- 0, if for t = t*

and

This proves the theorem.

z ( c ; t o , Z o , ~ ) - y ( t ' ; t , , y o ) ~ o x

z ( t ' ; t o , z o ) ~- y ( t ' ; t o , y , ) .

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In conclusion we note that the theorems 1.6, 2.4, and 3.5 are the basis for using the comparison method, based on the matrix Lyapunov function, tn applications, as a rule, one of the estimates of the majorizing function which are presented in the corollaries is obtained.

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