On stability of nonlinear nonautonomous systems by Lyapunov's direct method

ON STABILITY OF NONLINEAR NONAUTONOMOUS SYSTEMS BY LYAPUNOV'S DIRECT METHOD

JONG YEOUL PARK*, Vu NGOC PHAT**, AND IL HYO JUNG*

ABSTRACT. The paper deals with asymptotic stability of nonlinear nonautonomous systems by Lyapunov's direct method. The pro- posed Lyapunov-like function V(t,x) needs not be continuous int and Lipschitz in x in a Banach space. The class of systems consid- ered is allowed to be nonautonomous and infinite-dimensional and we relax the boundedness, the Lipschitz assumption on the system and the definite decrescent condition on the Lyapunov function.

1. Introduction

Consider a nonlinear time-varying differential equation of the gen- eral form:

t 2: to ER,

(1) {

x(t) =f(t, x(t)), x(to) = xo,

where the states x(t) take values in X, f(t, x) : R x X ~ X is a given nonlinear function and f(t,O) = 0, for all t E R. We shall assmne that conditions are imposed on the system (1) such that existence of its solutions are guaranteed.

It is well-known that the Lyapunov direct or second method is one of the most useful and fruitful techniques in stability analysis of non- linear differential equations and has gained increasing significance in the development of stability theory of dynamical systems [6, 9, 20, 21]. The classical direct method of stability is based essentially on the existence of a positive definite Lyapunov function with a nega- tive derivative. There are a number of books and papers available

Received September 27, 1999. Revised January 3, 2000.

2000 Mathematics Subject Classification: 34D10, 34K20.

Key words and phrases: nonlinear system, nonautonomous, asymptotic stability, Lyapunov functions, Lipschitz condition, Banach space.

The work is supported by the Korea Scientific Engineering Foundation (KOSEF).

expouding the extensions and generalizations of Lyapunov functions, see, e.g., [1, 3, 8, 11, 12, 18] and references therein. It is recognized that the Lyapunov-like functions serve as a main tool to reduce a given complicated system into a relatively simpler system and provide useful applications to control systems [4, 16, 17, 19J. However, the problem of Lyapunov-like functions and their characterization have remained un- der active investigation and finding Lyapunov-like functions for general nonllnear systems is usually a difficult task.

In [10J, sufficient conditions for the asymptotic stability of system (1) were given with the boundedness assumption

(2) IIf(t,x)1I ::; M, V(t,x) ER x Rn,

(3)

and the Lyapunov function Vet,x) : R x D --+ R, is continuous in (t, x) E R x D and Lipschitz in x ED satisfying

DjV(t,x) ::; -'Y(lIxll) <0, V(t,x) ER x D \ {O},

where D ~ Rn isan open neighborhood of the origin, 'Y(.) :R+ --+ R+

is a given non-decreasing continuous function and

D+V( )=1' ·nf V(t+h,x+hJ)-V(t,x)

f t,x 1Ill 1 h .

h--+O+

The weaker stability conditions were proposed and presented in [8] for system (1), where the Lyapunov function Vet, x) satisfies the following conditions

a(llxll) ::; Vet, x) ::; b(llxlJ),

DjV(t, x) ::; -'Y(V(t, x)),

wherea(.), b(.),'Y(.) are continuous strictly increasing functions.

Thanks to a result of [16], the assumption on Lyapunov-like func- tions in the asymptotic stability for a class of autonomous systems has been considerably relaxed by the existence of two continuous positive definite functions V (x) : Rn ^--+R, 'Y(.) : Rn ^--+ R+ \ {O}, where V (x) is proper (i.e., Vex) --+ 00 as IIxll ^--+(0) satisfying condition

(4)

where

D V( )^{= li . f} V(x(t+h)) - V(t, x)

- x mm h .

h--+O+

Paper [2] proposed a nondifferentiable stepswise decreasing Lya- punov function V(t,x)for system (1), where X = Rn,and the function f(t, x) is assumed to be locally Lipschitzian.

Inspired by the results of [2,8] we consider a Lyapunov-like function V(t,x) for time-varying system (1), which needs not be continuous in

t and Lipschitz in x in a Banach space. ^Inthis general setup, the class of systems considered is allowed to be nonautonomous and infinite- dimensional and we relax the boundedness, the Lipschitz assumption on the system and the definite decrescent condition on the Lyapunov function.

The paper is organized as follows. In Section 2, we give main no- tations and definitions of Lyapunov-like functions needed later. Sec- tion 3 presents main theorems on asymptotic stability with proposed Lyapunov-like functions. The conclusion is drawn in Section 4.

2. Notations and definitions

We shall employ the following notations and definitions throughout:

X denotes an infinite-dimensional Banach space with the correspond- ing norm 11.11,

BE denotes the open unit ball with radius ^E,

R denotes the real line; R+ denotes the set of non-negative real num- bers,

Z+ denotes the set of non-negative integers, Rn denotes the n-dimensional Euclidean space.

DEFINITION 2.1. The zero solution of (1) is said to be stable iffor every ^E > 0,to E R, there exists a number [) > 0 (depending upon ^E and to) such that for any solution x(t) of (1) with Ilxoll < [) implies

Ilx(t)11 < ^E, for all t ~ to.

DEFINITION2.2. The zero solution of (1) is said to be asymptotically stable if it is stable and there isa number [)> 0 such that any solution x(t) with Ilxoll < [)satisfies limt--+oo Ilx(t)11 =o.

In the above definitions, if the number 0> 0 is independent on to, then the zero solution of the system is said to be uniformly (asymptot- ically) stable.

DEFINITION 2.3. Let H be a Hilbert space. A function f(t,x) : R x H -+ H is dissipative if there isa number L >0 such that for all (t,x) E R x H:

(5) <x,j(t,x) > ::; Lllx11^2, where < .,. > denotes the inner product in H.

Let V(t,x) :R x X -+ R be some given function, T > 0, andD <;X be some open neighborhood of zero.

DEFINITION2.4. A function V(t, x) : RxD -+X is a Lyapunov-like function for system (1) if it satisfies the following conditions:

(i) There exist a non-decreasing function a(t) : R -+ R+, and a non- increasing function b(t) :R-+ R+, and numbers a> 0,b> 0 such that for all (t,x) E R x ^{D :}

(6) a(t)llxll^{a ::;}V(t, x) ::; b(t)llxll^b•

(ii) For every f3 > 0, there are a number T > 0 and functions ,(.) : R+ -+ R+,c(.) :R-+ R+,strictly increasing, passing through zero and ,(.) isintegrabl~, such that, V(t,x) ER x D \ {O},

(7) ~TV(t,X):= V(t+T,x(t+T)) - V(t,x)

I

::; -c(t) ^t ,(llx(s)ll)ds< 0,

wherex(t)isa solution of (1) withx(t) =x.Ifthe numberT depending on f3 satisfies the condition 0 < T < {3, then we say that V(t,x) is a strict Lyapunov-like function for system (1).

3. Stability results

In the following theorem, we relax the boundedness condition (2) on the system and the conditions (3), (4) on V(t, x). We then provide a

sufficient condition for asymptotic stability of system (1) with the pro- posed Lyapunov-like functions, where V(t,x) needs not be continuous int and Lipschitz in x.

THEOREM 3.1. Assume that

(8) IIf(t,x)11 :S M(t), V(t,^{x) ER} x^X,

whereM (t) :R ~ R+ isanintegrable function satisfying the condition

(9) ^h--+Olim

I

Ifthe system (1) admits a strick Lyapunov-like function, then it is asymptotically stable.

Proof. a) The system is stable: We assume to the contrary that there are a number E> 0 and to E R, such that for every 8 > 0 and for some solution x(t) of system (1) with Ilxoll < 8, there is a number S> to such that Ilx(S)11 ~ Eo Let81 >0 be chosen such that B8₁ CD.

Let us take any 82 E (0,8d.

In view of condition (9), we can find a number{3> 0 such that

(10)

I

For this {3, from the condition (7) it follows that there is a number T E (0,h) such that for all (t,x) E R X B_{81 :}

(11) V(t+T,x(t+T)) - V(t,x):S o.

We now take a number 8 >0 such that

£ - . {£ _^{h [a(to)8}2J l/b [a(to)EUJl/b - ^h}

v <. mm VI 'b(to) 'b(to) .

For this 8 there is, by the contrary assumption, a number S > to such that IIx(S)11 ~ ^Eo Consider the solution x(t) with IIxoll < 8. Since

S > to, there are an integer k > 0 and number 7 E [0,T) such that S= to +kT+^7. Since

l

IIx(to+T)II ::; Ilxoll + M(s) ds

to

l

< 8+ M(s)ds

to

= 8+h < 81,

which gives x(to +T) E Bol ' Applying (6) and (11), we have a(to)llx(to+T)lI a ::; V(to +T, x(to+T))

::; V(to, xo) :Sb(to)lIxollb < b(to)8^b <_a(to)8~

and hence IIx(to +T)II < 82 . On the other hand,

l

IIx(to+2T)11 ::; IIx(to +T)II + M(s) ds

to+T

l

::; 82+ M(s)ds <82+h<81 , to+T

which impliesx(to+2T) E Bol .Thus, applying (6) and (11) again the following estimate holds

a(to)llx(to+2T)lla ::; V(to +2T, x(to +2T))

::; V(to+T, x (to +T)) ::; V(to, xo) ::; a(to)8~,

and hence Ilx(to+2T)II < 82 •Repeating the same arguments, we obtain IIx(t+kT)1I <82 , Vk E Z+.

Therefore, for everyk E Z+, we have

l

Ilx(S)1I = IIx(to +kT+7)11 ::; IIx(to +kT)1I + M(s) ds

to+kT

::; 82+h <81

which gives x(S) E B8₁^• Applying (6) and (11), we have a(tO)E^{U ::;} a(S)llx(S)IIU

::; V(S,x(S))

::; V (to+(k - l)T+r,x(to+(k - 1)T+r))

::; ... ::;

::; V(to +r,x(to+r)) ::;b(to)lI x (to +r)ll^b.

Since

l

Ilx(to +rll ::; Ilxoll + M(s) ds

to

l

::; 6+ M(s) ds

to

::;6+h

< [a(tO)E^U]lib

- b(to) ,

we obtain a(to)E^U < a(tO)E^U, which is a contradiction.

b) The sytem (1) is asymptotically stable: We remain to show that there is a number 6> 0, for every solutionx(t) of (1) with Ilx(to)11 < 6, for every ^E> 0, there exists a number N > 0 such that 11x(t)11 <^E for allt > to+N. For this, from the stability of the system it follows that for 61 > 0, where 61 is chosen so that B8₁ CD, we can find a number 62 > °such that any solution x(t) of the system with Ilx(to)11 < 6₂ implies

(12) IIx(t)11 < ⁶1 , "It> to.

Consider any solution x(t) of (1) with Ilxoll < 6 = min{61 ,82 }. We then have x(t) E D, for all t > to. Let E > °be an arbitrary given number, we define

~ ⁼ [a(to) u]^lib

u3 b(to)^{E •}

Let us take any 84 E (0, (_{3 ).} Due to the condition (9), there is a number (3 > 0 such that

For this (3 > 0 there is, by condition (7), a number T E (0, (3), and functions c(.), ,(.) such that for all (t,x) E R x D \ {O} :

I

(14) V(t+T, x(t+T)) - V(t, x) ~ -c(t) ^t ,(llx(s)ll)ds.

We shall show that there is an integerK > 0 such that

(15) IIx(to+^KT)11 < 8^{4 .}

Indeed, if (15) is not satisfied, then IIx(to+kT)11 284 for all k E Z+.

Taking (12) into account and applying (14), we have (16)

V(to+(k +l)T, x(to+(k+l)T))

l

~ V(to+kT, x(to+kT)) - c(to+kT) ,(llx(s)lI)ds

to+kT

1

~ V(to+kT,x(to+kT)) - c(to) _ ,(lIx(s)lI)ds,

to

where to ^:=to +kT. On the other hand, for every t E [to, to+(3], we have

Ilx(t)1I 2 IIx(to)II - (1/(s, x(s))jds

JEo

1

28_{4 -} _ M(s)ds.

to

Taking (13) into account, we have

1

Ilx(t)1I 284 - _ M(s)ds =TJ > 0,

to

and hence

1

to

Therefore, from (16), it follows that

V(to+(k+ I)T, x(to+(k+ I)T)) :s V(to +kT, x(to+kT)) - c(to)'(TJ)f3.

Repeating the same arguments we obtain

V(to +(k+I)T, x(to+(k +I)T)) :SV(to, xo) - (k +l)c(to)T(TJ)f3.

Since the Lyapunov-like function V(t,x) is non-negative, we have

o:s V(to, xo) - (k +l)c(to)T(TJ)f3, Vk E Z+, or equivalently

(k + l)c(to)'(TJ)f3 :S V(to, xo) :S b(to)//xoll^b < b(to)5^b < +00

which is a contradiction when letting k ----; 00. Thus, the condition (15) is proved. The proof is completed as follows. For every t ~ to+KT there are numbers ko> K,^T E [0,T) such that t - to = koT+^TO. We have

l

Ilx(to+KT+To)11 :S Ilx(to+KT)II + M(s) ds

t+KT

l

:S54 + M(s) ds < 53·

to+KT

Therefore

a(to)llx(t)l/a:s a(t)llx(t)lla:s V(t,x(t))

:S V(to +KT+T, x(to+KT+TO))

:S b(to)llx(to+ KT+T)ll^b :S b(to)5~ = a(to)fa, which gives Ilx(t)/1 < ^Eo The theorem is proved. D

REMARK 3.1. Note that since the functions aCt), bet) need not be continuous, the Lyapunov function Vet,x) is not continuous in t and it is, in general, not decreasing in t, we can not apply the techniqu.es used in [8J. Furthermore, if the functions aCt), bet) are independent on t, Le., are constant, then Theorem 3.1 holds for uniform asymptotic stability.

Theorem below asserts that if the boundedness off(t,x) is replaced by the dissipativity assumption (5)in Hilbert space, then we can derive asymptotic stability conditions for system(1)with a weaker Lyapunov- like function than the strict Lyapunov-like function.

THEOREM3.2. Assumethat X =H is a Hilbert space and the func- tionf(t, x) is dissipative in H. H the system(1)admits a Lyapunov-like function then it is asymptotically stable.

Proof. Let f(t, x) be a dissipative function satisfying condition (5) with the constant L > O. We first note that for every solution x(t) of (1) with x(to) = Xo the following property holds:

(17)

Indeed, since

Ilx(t)lI:::; ^IIxolle^{L(t-to), Vt}~ to.

d d

dt <x(t),x(t) > = dtllx(t)112 =2<x(t),f(t,x(t)) >,

and by integrating both sides of the above relation, we have IIx(t)1I 2= IIxoll2+² t < x(s), f(s, x(s)) > ds.

ito

Then

Applying the Gronwall's inequality, we obtain

which gives (17).

a) The system is stable. The proof is similar to that of Theorem 3.1, part a), where the positive numbers 02,0 are defined by

£^< £ ^-LT £^{< .} {£ ^-KL [a(to)02]1/b [a(to)€o]1/b ^-KL}

u2 u1 e , u mm U1 e 'b(to) 'b(to e .

b) The system is asymptotically stable. From the stability of the system it follows that there is a number 02 > 0 such that for every solution x(t) of (1) with IIxoll <02 implies IIx(t)11 <01, where 01 >0 is a number chosen so that B₀₁ cD. Consider any solution x(t) of(1) with Ilxoll < 0 = min{01,02}. We then have x(t) E BoI' ^{for all t} > to.

Let ^€ >0 be a given number. We take

[a(to) a]1/b -LT

03 = b(to)€ , 04 < 03^e .

Since V(t, x) is a Lyapunov-like function, take any number 0>0 such

that 0²

0< 2L~2.₁

We first prove that there is an integer K > 0 such that

(18) Ilx(to +KT)II < 04.

Indeed, if (18) is not satisfied, we have IIx(to + kT)1\ ~ 04, for all k E Z+. In view of (14) we have for all k E Z+ :

(19)

V(to +(k+I)T,x(to+(k+I)T))

l

::; V(to +kT, x(to +kT)) - c(to+kT) '"'((lIx(s)11)ds

to+kT

l

::; V(to +kT, x(to +kT)) - c(to) '"'((llx(s)11)ds.

to+kT

We estimate the value Ilx(t)11 on the interval 113 := [to +kT,to +

kT+,8]. On the other hand, since for allt E1_{13 :}

l

Ilx(t)11² _~ Ilx(to +kT)11²-12 ^< x(s), 1(s, x(s)) > dsl

to+kT

l

~ Ilx(to +kT)11^{2- 2L} Ilx(s)11²ds,

to+kT

and since Ilx(t)1I <81 for allt > to, we have

which gives

(20)

l

to+kT .

Therefore, combining (19) and (20), we obtain V(to +(k+I)T, x(to+(k+I)T))

~ V(to +kT, x(to +kT)) - c(to)[3,(T/)

~ V(to +kT, x(to +kT)) - M,

~ ... ~

~ V(to, xo) - (k +I)M,

whereM :=c(to)(3,(T/) which gives

(k+ I)M ~ V(to, xo) ~ b(to)8^b < +00.

The last inequality leads to a contradiction when lettingk - t 00.Thus, (18) is proved.

For everyt > to +KT, there are numbers ko> K,7 E [0,T), such that t - to =koT+^7. We have

a(to)lIx(t)lI^a ~ V(to +kaT+7, x(to+koT+7))

~ V(to +KT +7,X(to+KT+⁷⁾⁾

~ b(to)/Ix(to+KT+7)lI^b•

On the other hand, in view of (17) we have

and hence

which gives Ilx(t)11 < Eo The proof is complete. o

REMARK 3.2. The dissipative condition (2) can be replaced by the Holder-type condition

:3L> 0, :3aE (0,1] : Ilf(t, x)11 ~ LllxW\ Vet, x) ER x H.

In this case, the generalized Gronwall inequality [8, 22] is applied to obtain the estimate (17) of the form

{

IlxolleLCt-to) ifa =1

Ilx(t)1I ~ , ^{' 1}

Ilxoll[1+(1 - a)L(t - to)]2=a, ifa E (0,1),

and the positive numbers El,84 , and h in the proof of Theorem 3.2 are defined respectively by

where

{

eLT, ifa =1,

'TJ'-

.- [1 +^{(1 -} a)LT]²2"" ifa E (0,1).

REMARK 3.3. Theorem 3.2 remains true even if we replace the con- ditions (6), (7) by the following conditions:

aCt, r) ~ Vet, x) ~ bet, r)

(21) r^t+!3

~TV(t, x) ~ - it ,,(s, IIx(s)ll)ds < ^0,

wherea(t, r),b(t,r) :RXR+ -+R+ \{o} are continuous strictly increas- ing function inr,a(t,r) is non-decreasing int,b(t,r) is non-increasing in t, and ,,(t,h) : R ^X R+ -+ R+ is a strictly increasing in h E R+

function satisfying

I

lim inf /,(s, h)ds > 0,

h-+O+ t

Vt E R, h> 0.

EXAMPLE 3.1. To illustrate the stability result, we consider the asymptotic stability of a semilinear system in Hilbert space of the form (22) x(t) = { Ax(t)+get, x(t)), t 2:0,

x(to) = Xo, x(t) EH.

Let us assume that the linear operator A : H ^---tH is stable. Then, by a result of [5], there is a positive definite symmetric linear operator Q : H ^---t H such that

2< QAx, x > ~ -llxI1^2, Vx ^EH.

Assume that the nonlinear function9(t,x) satisfies the following growth condition

(23) /lg(t,x)/I _~K(t)/lx/l,. V(t,x) ER+ x H,

whereK(t) : R+ ^---t R+ is a bounded and integable function. Consider the Lyapunov function V (t, x) =< Qx, x > .The derivative along the trajectoriesx(t) of system (22) is given by

dtdV(t,x) = -llx(t)1I²+< Qx(t),g(t,x(t» > .

Thus, we can not apply the classical stability Lyapunov theorem since the derivative ofV (t,x) may take positive and negative values.

We willshow thatif

(24) ^{lim inf}

I

h--+O+ t

ViER,

then the system is asymptotically stable. Indeed, let us consider any solution x(t) of (22) with x(to) = XQ. By the assumption (24), there exist a sequence of positives numbers{t_{n }} going to zero and a number N >0 such that

(25)

I

Let j3 E (0,tn),n 2: N be an arbitrary number. We will show that there are a positive numberT >0, and a function1'(t,h) satisfying the

condition (21) and hence, by Theorem 3.2 and Remark 3.3, the system is asymptotically stable. For this, let Xo be an arbitrary initial state such that Ilxoll =^'TJ > O. Since the solution x(t) is continuous, we can find a number 8> 0 such that IIx(to+t)11 ~ 'TJ > 0 for allt E [to, to +8].

Let us take a number n > ^N such that t_n < 8 satisfying (25) and let T > tn > (3. From (23) it follows that for every to > 0, the following relation holds:

.6.TV(to, x) = V(to +T,x(to+T)) - V(to, xo)

l

= -dV(s, x(s)) ds

to t

l

(26) ^{= [}-llx(s)11²+ < Qx(s),g(s,x(s)) > ]ds

to

l

S - [1 -IIQIIK(s)]llx(s)11²ds.

to

On the other hand, setting,(t,h) = [1-IIQIIK(t)]h² and using (25) we have

l

l

= [1 - IIQIIK(s)]h²ds

to

l

~ h² [1 -IIQII]K(s) ds

to

= h²a > O.

Therefore

I

Hm inf ,(s, h)ds > 0,

h--+O+ t t E R,h > 0,

which together with (26) proves the assertion.

4. Conclusions

In this paper, asymptotic stability of nonlinear time-varying dif- ferential equations by Lyapunov direct method have been investigated.

Discontinuous Lyapunov-like function are proposed for sufficient stabil- ity conditions. Several stability results obtained earlier can be derived.

The stability results obtained in the paper can be considered as a fur- ther development of Lyapunov function characterization in stability analysis of nonlinear dynamical systems.

ACKNOWLEDGEMENTS. The paper was written during the first au- thor's visit to the Department of Mathematics, Pusan National Uni- versity, Korea.

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Jong Yeoul Park and Il Hyo Jung Department of Mathematics Pusan National University Pusan 609 - 735, Korea

E-mail: [email protected]

Vu Ngoc Phat

Institute of Mathematics

Box 631, Bo Ho, Hanoi, Vietnam