On the theory of Lyapunov’s direct method

376

ISSN 1064–5624, Doklady Mathematics, 2006, Vol. 73, No. 3, pp. 376–379. © Pleiades Publishing, Inc., 2006.Original Russian Text © A.A. Martynyuk, 2006, published in Doklady Akademii Nauk, 2006, Vol. 408, No. 3, pp. 309–312.

The goal of this work is to describe two directions ofdeveloping the theory of Lyapunov’s direct method byconsidering a new class of auxiliary functions and thecomparison principle. The resulting conditions for theuniform asymptotic stability of nonlinear systems arenecessary and sufficient in certain special cases.

1. DEFINITIONS AND AUXILIARY RESULTS

Consider the nonlinear system

(1)

where

x

∈

R

n

,

t

∈

R

+

= [0,

∞

),

and

f

∈

C

(

R

+

×

R

n

,

R

n

)

.The system satisfies the conditions for the existence ofa solution

x

(

t

;

t

0

,

x

0

)

for any (

t

0

,

x

0

)

∈

int(

R

+

×

R

n

) and

x

(

t

0

;

t

0

,

x

0

) =

x

0

. It is assumed that

f

(

t

, 0)

= 0 for all

t

∈

R

+

.

To perform a dynamic analysis of system (1), we usethe two-index system of functions

(2)

where

v

ii

(

t

,

x

)

∈

C

(

R

+

×

R

n

,

R

+

),

v

ii

(

t

, 0) = 0,

v

ij

(

t

,

x

)

∈

C

(

R

+

×

R

n

,

R

)

for all

i

≠

j

, and

v

ij

(

t

, 0)

= 0. We usematrix-valued function (2) to construct the vector func-tion

(3)

and the scalar function

(4)

(see [1, 2]). We say that the two-index system of func-tions (2) is a suitable tool for constructing scalarLyapunov function (4) or vector Lyapunov function (3)if one of them, together with the total derivatives

dxdt------ f t x,( ), x t0( ) x0,= =

U t x,( ) v ij t x,( )[ ], i j, 1 2 … m,, , ,==

V t x η, ,( ) U t x,( )η, η R+m∈=

v t x η, ,( ) ηTU t x,( )η=

D

+

v

(

t

,

x

,

η

)

|

(1)

and

D

+

V

(t, x, η)|(1), solves the problem ofthe stability of the trivial solution to system (1). Here,

(5)

(6)

where

is calculated componentwise for all (t, x) ∈ R+ × Rn.

Definition 1. A function w is called angular if it iscontinuous and nondecreasing in its domain w: R+ → R+,w(0) = 0, and w(s) > 0 for s > 0.

In what follows, all functions with the propertiesindicated in Definition 1 are referred to as functions ofthe W class.

Assumption 1. The diagonal elements vii(t, x) andthe off-diagonal elements vij(t, x) of matrix-valuedfunction (2) satisfy the estimates

(7)

(8)

for all (t, xi, xj) ∈ R+ × × , xi ∈ , and xj ∈

, where = n; , > 0 for all i = 1, 2, …, m;

, and are arbitrary constants; and (ψ1i, ψ2i) ∈ Wclass.

Proposition 1. Under Assumption 1, function (4)satisfies the two-sided estimate

(9)

D+v t x η, ,( ) 1( ) ηT D+U t x,( )η,=

D+V t x η, ,( ) 1( ) D+U t x,( )η,=

D+U t x,( ) limsup U t θ+ x θf t x,( )+,( )[{=

– U t x,( ) ]θ 1– : θ 0+ }→

γiiψ1i

2 xi( ) v ii t x,( ) γ iiψ2i2 xi( ),≤ ≤

γijψ1i xi( )ψ1 j x j( ) v ij t x,( )≤

≤ γ ijψ2i xi( )ψ2 j x j( )

Rni R

n j Rni

Rn j ni

i 1=

m

∑ γii

γ ii

γij

γ ij

ψ1T x( )Aψ1 x( ) v t x η, ,( )≤

≤ ψ2T x( )Bψ2 x( )

MATHEMATICS

On the Theory of Lyapunov’s Direct MethodA. A. Martynyuk

Presented by Academician Yu.A. Mitropol’skii October 3, 2005

Received December 20, 2005

DOI: 10.1134/S1064562406030161

Institute of Mechanics, National Academy of Sciencesof Ukraine, ul. Nesterova 3, Kiev, 03057 Ukrainee-mail: [email protected]

DOKLADY MATHEMATICS Vol. 73 No. 3 2006

ON THE THEORY OF LYAPUNOV’S DIRECT METHOD 377

for all (t, x) ∈ R+ × Rn, where ψ1(||x||) = (ψ11(||x1||), …,ψ1m(||xm||))T, ψ2(||x||) = (ψ21(||x1||), …, ψ2m(||xm||))T,

A = YTGY, B = YT Y, Y = diag[η1, η2, …, ηm], G =

[ ], and = [ ] for i, j = 1, 2, …, m.

This result is proved by direct computation of upperand lower estimates for function (4) taking into accountinequalities (7) and (8).

2. THE FIRST DIRECTION OF DEVELOPMENT

It is well known that Lyapunov’s direct method in itsclassical version is based on scalar auxiliary functionswith special properties. We give some conditions for theuniform asymptotic stability of the trivial solution to sys-tem (1) based of scalar auxiliary function (4), which isconstructed by using matrix-valued function (2).

Theorem 1. Assume that the vector function f in sys-tem (1) is continuous on R+ × Rn and, additionally, thefollowing conditions are satisfied:

(i) There is a matrix-valued function (2) with ele-

ments (7) and (8) and a vector η ∈ such that function(4) is locally Lipschitz with respect to x for all t ∈ R+.

(ii) There are functions ψ1i, ψ2i, and ψ3i of the Wclass and m × m matrices A(η), B(η), and C(η) suchthat estimate (9) holds for all (t, x) ∈ R+ × N with N ⊆Rn and

(10)

for all (t, x) ∈ R+ × N, where m(t, u) satisfies the condi-tion

(11)

uniformly with respect to t ∈ R+.(iii) For δ > 0, the system of inequalities

has a solution η1, η2, …, ηm satisfying 0 < ηi < δ for alli = 1, 2, …, m.

Then the trivial solution x = 0 to system (1) is uni-formly asymptotically stable.

Proof. Under conditions (i) and (iii) in Theorem 1,function (4) satisfies estimates (9) and is positive def-inite and decreasing. Indeed, since λm(A) > 0 andλM(B) > 0, we have

for all (t, x) ∈ R+ × N. By condition (ii) of Theorem 1,ψ1i and ψ2i belong to the W class for i = 1, 2, …, m.

G

γij

G γ ij

R+m

D+v t x η, ,( ) 1( ) η3

T x( )C η( )η3 x( )≤+ m t η3 x( ),( )

m t u,( )u

--------------------u 0→lim 0=

A η( ) 0, B η( ) 0, C η( ) 0<> >

λm A( )ψ1T x( )ψ1 x( ) v t x η, ,( )≤

≤ λM B( )ψ2T x( )ψ2 x( )

Therefore, there are functions ω1 and ω2 that belong tothe W class such that

for all (t, x) ∈ R+ × N. It follows that

(12)

for all (t, x) ∈ R+ × N. As is known (see [3]), this con-dition is necessary and sufficient for function (4) to bepositive definite and decrease.

Under conditions (ii) and (iii) of Theorem 1, for any0 < µ < 1, we can choose δ(ε) > 0 such that

for all (t, x) ∈ R+ × S(δ). Therefore,

(13)

where ω3(||x||) ≥ (||x||)η3(||x||) and ω3 belongs to theW class. Since λM(C) < 0, we conclude that D+v(t, x,η)|(1) is a negative definite function for all (t, x) ∈ R+ ×S(δ), where S(δ) ⊂ N. Consequently, conditions (12)and (13) guarantee the uniform asymptotic stability ofthe trivial solution to system (1).

Corollary 1. Under the conditions of Theorem 1, ifλM(C) ≤ 0, then the trivial solution x = 0 to system (1) isuniformly stable, and if λM(C) ≤ 0 and B(η) = 0, thenx = 0 is stable.

In this direction, constructive results was obtainedconcerning the application of matrix-valued functionsto the stability theory of motion (see [4, 5]). Moreover,the concept of a matrix-valued function was justifiedand developed for a wide class of problems in the qual-itative theory of equations.

3. THE SECOND DIRECTIONOF DEVELOPMENT

This direction is related to the application of vectorfunction (3) and a comparison principle. Let us presentsome results from the theory of monotone systems ofdifferential equations.

In Rm, where m < n, we consider the cone K = {u ∈Rm: ui ≥ 0, i = 1, 2, …, m} with the interior K0.

A function G ∈ C( , Rm) is quasi-monotone non-

decreasing with respect to K if x y and ϕ(x – y) = 0for some ϕ ∈ implies ϕ(G(x) – G(y)) ≤ 0 where is the conjugate cone of K0 and ϕ(x) ≥ 0 for all x ∈ K.

Definition 2. An autonomous system

(14)

ω1 x( ) λm A( )ψ1T x( )ψ1 x( ),≤

ω2 x( ) λM B( )ψ2T x( )ψ2 x( ),≥

ω1 x( ) v t x η, ,( ) ω2 x( )≤ ≤

m t η3 x( ),( ) µλM C( )η3T x( )η3 x( )–≤

D+v t x η, ,( ) 1( ) 1 µ–( )λM C( )ω3 x( ),≤

η3T

R+m

≤K

K0* K0*

dudt------ G u1 u2 … um, , ,( ), u t0( ) u0= = 0≥

378


MARTYNYUK

is called a comparison system for system (1) if and onlyif its maximal uM(t, u0) and minimal um(t, u0) solutionsare related via functions (3) and (4) to the solutions ofthe original system (1) by the inequalities

(15)

(16)

for all t ∈ R+, where e = (1, …, 1)T ∈ . Inequality(15) is satisfied componentwise.

The vector function G in system (14) obeys the con-ditions stated below.

Assumption 2. Comparison system (14) is suchthat:

(i) the vector function G ∈ C( , Rm) is quasi-monotone nondecreasing with respect to K (satisfies theW0-condition);

(ii) the solution to Cauchy problem (14) is locallyunique;

(iii) there is a neighborhood D of the point u = 0 suchthat, for all u ∈ with u ≠ 0, G(u) ≠ 0 and G(0) = 0,

where is the closure of D.

Theorem 2. Assume that the vector function f in sys-tem (1) is continuous on R+ × Rn and the vector func-tion G in system (14) obeys Assumption 2. Additionally,let the following conditions be satisfied:


ments (7) and (8) and an vector η ∈ such that

for all (t, x) ∈ R+ × Rn.(ii) The matrices A and B in estimate (9) are positive

definite.(iii) For any δ > 0, the system of inequalities

has a solution θ1, θ2, …, θm such that 0 < θj < δ for allj = 1, 2, …, m.

Then the trivial solution to system (1) is uniformlyasymptotically stable.

Proof. Under condition (i) of Theorem 2, it is easyto derive comparison system (14). Condition (iii) inTheorem 2 is necessary and sufficient for the uniformasymptotic stability of an isolated trivial solution tosystem (14) (see [7]). Next, we use the comparisonprinciple (see [8]) and show the validity of Theorem 2.The uniform asymptotic stability of an isolated trivialsolution to system (14) implies that, for any δ1 with 0 <δ1 < r < +∞ and for any ε > 0, there is T = T(ε) > 0 such

um t u0,( ) V t x t; t0 x0,( ) η, ,( ) uM t u0,( ),≤ ≤

eTum t u0,( ) v t x t; t0 x0,( ) η, ,( ) eTuM t u0,( )≤ ≤

R+m

R+m

D

D

R+m

D+V t x η, ,( ) 1( ) G V1 t x η, ,( ) V2 t x η, ,( ) …,,,(≤

Vm t x η, ,( ) )

G j θ1 θ2 … θm, , ,( ) 0, j< 1 2 … m, , ,=

that, for any t1 ≥ t0 and initial values eTu(t1, u0) < δ1,

where e = (1, …, 1)T ∈ , we have the estimate

(17)

for all t ≥ t1 + T(ε), where 0 < λm(A) is the minimumeigenvalue of A in estimate (9) and a(||x||) ≤ϕT(||x||)ϕ(||x||) for all x ∈ N is a function of the W class.

Note that estimate (9) implies the sequence of ine-qualities

for all (t, x) ∈ R+ × N, where b(||x||) ≥ ψT(||x||)ψ(||x||) forx ∈ N and b(||x||) ∈ W class. Let δ be determined by thecondition λM(B)b(δ) ≤ δ1. If t1 ≥ t0 and ||x(t1)|| ≤ δ, thenu(t1) = v(t1, x, η) ≤ λM(B)b(||x(t1)||), where λM(B) > 0 byvirtue of conditions (ii) in Theorem 1.

Under inequality (17), for all t ≥ t1 + T(ε), accordingto the comparison principle, we have

Therefore, λm(A)a(||x(t)||) < λm(A)a(ε) for all t ≥ t1 + T(ε)and, consequently, ||x(t)|| < ε for all t ≥ t1 + T(ε) if t1 ≥ t0and ||x(t1)|| ≤ δ. This completes the proof of Theorem 1.

Corollary 2. Assume that system (1) is such that f ∈C(R+ × Rn, Rn), f(t, 0) = 0, and the following conditionsare satisfied:


ments (7) and (8), a vector η ∈ , and an m × m con-stant matrix P with nonnegative off-diagonal elementssuch that

for all (t, x) ∈ R+ × Rn.(ii) The m × m matrices A and B in estimate (9) are

positive definite.(iii) For any δ > 0, the system of inequalities

has a solution θ1, θ2, …, θm such that 0 < θj < δ for allj = 1, 2, …, m.

Then the trivial solution to system (1) is uniformlyasymptotically stable.

Corollary 3 (see [7]). An isolated trivial solution tosystem (14) is uniformly asymptotically stable if andonly if system (14) describes the perturbed motion of anactual system with a finite number of degree of free-doms, G is a vector function obeying Assumption 2, andcondition (iii) in Theorem 2 is satisfied.

Note that the vector function constructed from amatrix-valued function has been applied in the study ofthe polystability of motion (see [4] and the bibliogra-

R+m

eTu t u0,( ) λm A( )a ε( )≤

v t x η, ,( ) ψT x( )Bψ x( )≤

≤ λM B( )ψT x( )ψ x( ) λM B( )b x( )≤

v t x η, ,( ) ηTV t x η, ,( ) eTu t u,( ).≤=

R+m

D+V t x η, ,( ) 1( ) PV t x η, ,( )≤

pijθ j

j 1=

m

∑ 0, i< 1 2 … m, , ,=


ON THE THEORY OF LYAPUNOV’S DIRECT METHOD 379

phy therein). As applied to the analysis of the stabilityof large-scale systems, usual vector Lyapunov func-tions were used in numerous works (see [6, 10] and thebibliography therein).

4. DISCUSSIONIn fact, the conditions of Theorem 1 are necessary

and sufficient for the uniform (in t0, x0) asymptotic sta-bility of the trivial solution to system (1), since, underthese conditions, all the assumptions of Lyapunov’stheorem on asymptotic stability are satisfied withI.G. Malkin’s addition and conversion.

If system (1) is regarded as a large-scale one and itis analyzed by applying a decomposition–aggregationtechnique (see [4–6]), then linear or nonlinear compar-ison systems are derived in the context of Theorem 2 orits versions. Although condition (iii) in Theorem 2 isnecessary and sufficient for the uniform asymptotic sta-bility of an isolated trivial solution to comparison sys-tem (14), the sharpness of the estimate for the range ofparameters in system (1) that guarantee the uniformasymptotic stability of its trivial solution depends onthe “delicacy” of the majorants applied to the compo-nents of the total derivative of vector function (3) alongsolutions to system (1).

Thus, the range of parameter values guaranteeingthe uniform asymptotic stability of the trivial solutionto the original system can be considerably expanded bysuitably choosing the vector function and the majorant.

Note that there are other directions of developingthe theory of Lyapunov’s direct method based onmatrix-valued functions in addition to those described

in this paper. In particular, it is of interest to develop theproposed approach for dynamical systems in a metricspace [9] and for hybrid systems.

REFERENCES1. M. Z. Djordjevic, Large Scale Syst. 5, 255–262 (1983).2. A. A. Martynyuk, Dokl. Akad. Nauk SSSR 249, 59–63

(1985).3. W. Hahn, Stability of Motion (Springer-Verlag, Berlin,

1967).4. A. A. Martynyuk, Stability by Liapunov’s Matrix Func-

tion Method with Applications (Marcel Dekker, NewYork, 1998).

5. A. A. Martynyuk, Qualitative Methods in NonlinearDynamics: Novel Approaches to Liapunov’s MatrixFunctions (Marcel Dekker, New York, 2002).

6. Method of Lyapunov’s Vector Functions in Stability The-ory, Ed. by A. A. Voronov and V. M. Matrosov (Nauka,Moscow, 1987) [in Russian].

7. A. A. Martynyuk and A. Yu. Obolenskii, Differ. Uravn.16, 1392–1407 (1980).

8. V. Lakshmikantham, S. Leela, and A. A. Martynyuk, Sta-bility of Motion: The Comparison Method (NaukovaDumka, Kiev, 1991) [in Russian].

9. A. A. Martynyuk, Nonlin. Dyn. Syst. Theory 5, 157–167(2005).

10. Yongmao Wang, Yan Shi, and Hirofumi Sasaki, Nonlin.Dyn. Syst. Theory 4, 217–229 (2004).

11. V.I. Slyn’ko, Int. Appl. Mech. 40, 1171–1174 (2004).12. A. A. Martynyuk and V. G. Miladzhanov, Int. Appl.

Mech. 39, 875–894 (2003).13. A. A. Martynyuk, Int. Appl. Mech. 40, 322–327 (2004).

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