Perturbing Lyapunov functions and stability criteria for initial time difference

Applied Mathematics and Computation 117 (2001) 313±320 www.elsevier.com/locate/amc

Perturbing Lyapunov functions and stability criteria for initial time di€erence F.A. McRae Department of Mathematics, Catholic University of America, Washington, DC 20064, USA

Abstract Stability criteria for di€erential equations where the initial time for each solution is di€erent is developed using the method of perturbing Lyapunov functions. Ó 2001 Elsevier Science Inc. All rights reserved. Keywords: Initial value problems; Variable initial times; Perturbing Lyapunov functions; Stability criteria

1. Introduction In dealing with real world phenomenon it is impossible not to make errors in the starting time. Hence it is important to study the variation in initial times. This creates problems in comparing any two solutions which di€er in initial starting time. An investigation of initial value problems of di€erential equations where the initial time changes with each solution in addition to the change of space has been initiated in [3]. In those situations where the Lyapunov function does not satisfy all the desired conditions, it is fruitful to perturb the Lyapunov functions, rather than discard it [1,2]. Thus, the notion of perturbing Lyapunov functions enables us to discuss nonuniform properties of solutions of di€erential systems under weaker assumptions. In this paper we develop stability criteria for initial value problems of differential equations with initial time di€erences using the method of perturbing Lyapunov function so that the usual results on stability criteria result as a consequence. E-mail address: [email protected] (F.A. McRae). 0096-3003/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 9 9 ) 0 0 2 0 6 - 4

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2. Preliminaries Consider the di€erential system x0 ˆ f …t; x†;

…2:1†

where f 2 C‰R‡  Rn ; Rn Š. Let x…t; t0 ; x0 † and x…t; s0 ; y0 † be the solutions of (2.1) through …t0 ; x0 † and …s0 ; y0 †, respectively. Suppose that x…t† ˆ x…t; t0 ; x0 † is the given solution relative to which we shall study stability criteria. Let g ˆ s0 ÿ t0 . We shall ®rst de®ne the notion of stability in the present framework. De®nition 2.1. The solution x…t† of (2.1) is said to be (i) equistable if given  > 0 and t0 2 R‡ , there exist d ˆ d…t0 ; † > 0 and r ˆ r…t0 ; † > 0 such that kx0 ÿ y0 k < d, jgj < r implies kx…t ‡ g; s0 ; y0 †ÿ x…t; t0 ; x0 †k < , t P t0 ; (ii) uniformly stable if d and r in (1) is independent of t; (iii) equi-asymptotically stable if (1) holds and given  > 0, t0 2 R‡ , there exist d0 …t0 † > 0, r0 …t0 † > 0 and a T ˆ T …t0 ; † > 0 such that kx0 ÿ y0 k < d0 , jgj < r0 implies kx…t ‡ g; s0 ; y0 † ÿ x…t; t0 ; x0 †k < , t P t0 ‡ T ; (iv) uniform asymptotically stable if (2) holds and d0 , r0 and T in (3) are independent of t0 . We need the following comparison theorem [4] before we can proceed. Theorem 2.1. Assume that V 2 C‰R‡  Rn ; R‡ Š, V …t; u† is locally Lipschitzian in u and D‡ V …t; u; g† 6 g…t; V …t; u†; jgj†, where u ˆ u…t† ˆ x…t ‡ g; s0 ; y0 † ÿ x…t; t0 ; x0 †, g 2 C‰R3‡ ; RŠ and r…t; t0 ; x0 ; jgj† is the maximal solution of x0 ˆ g…t; x; jgj†, w…t0 † ÿ w0 P 0, existing the t P t0 . Then V …t0 ; x0 ÿ y0 † 6 x0 implies V …t; x…t ‡ g; s0 ; y0 † ÿ x…t; t0 ; x0 †† 6 r…t; t0 ; x0 ; jgj†;

t P 0:

3. Main results Theorem 3.1. Assume that (i) V1 2 C‰R‡  S…q†; R‡ Š, V1 …t; x† is locally Lipschitzian in x and V1 …t; 0†  0 and D‡ V1 …t; x† 6 g1 …t; V1 …t; x†; jgj†;

…t; x† 2 R‡  S…q†;

where g1 2 C‰R3‡ ; RŠ and S…q† ˆ ‰x 2 Rn : kxk < qŠ; (ii) for every c > 0, there exists a V2 2 ‰R‡  S…q† \ S c …c†; R‡ Š, V2 …t; x† is locally Lipschitzian in x, b…kxk† 6 V2 …t; x† 6 a…kxk†;

…t; x† 2 R‡  S…q† \ S c …c†;

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where a; b 2 K ˆ f/ 2 C‰…0; q†; R‡ Š : /…u† is increasing in u and /…u† ! 0 as u ! 0g and D‡ V1 …t; x† ‡ D‡ V2 …t; x† 6 g2 …t; V1 …t; x† ‡ V2 …t; x†; jgj†; …t; x† 2 R‡  S…q† \ S c …c†; where g2 2 C‰R3‡ ; RŠ; (iii) the scalar differential equation w01 ˆ g1 …t; w1 ; jgj†;

w1 …t0 † ˆ w10 P 0;

…3:1†

is equistable and the scalar differential equation w02 ˆ g2 …t; w2 ; jgj†;

w2 …t0 † ˆ w20 P 0;

…3:2†

is uniformly stable. Then the solution x…t† ˆ x…t; t0 ; x0 † of the system (2.1) is equistable. Proof. Let 0 <  < q and t0 2 R‡ be given. The uniform stability of (3.2) implies that, given b…† > 0 and t0 2 R‡ , there exist a d0 ˆ d0 …† > 0 and r1 ˆ r1 …† > 0 such that 0 6 w20 < d0 and jgj < r1 implies w2 …t; t0 ; w20 ; jgj† < b…†;

t P t0 ;

…3:3†

where w2 …t; t0 ; w20 ; jgj† is any solution of (3.2). Because of the hypothesis on a…u†, there exists d2 ˆ d2 …† > 0 such that a…d2 † <

d0 : 2

…3:4†

Also, since (3.1) is equistable, given …d0 =2† > 0, t0 2 R‡ , there exists d ˆ d …t0 ; † and r2 ˆ r2 …t0 ; † > 0 such that 0 6 w10 < d and jgj < r2 implies w1 …t; t0 ; w10 ; jgj† <

d0 ; 2

t P t0 ;

…3:5†

where w1 …t; t0 ; w10 ; jgj† is any solution of (3.1). Choose w10 ˆ V1 …t0 ; x0 ÿ y0 †. Since V1 …t; x† is continuous and V1 …t; 0†  0, there exists d1 > 0 such that kx0 ÿ y0 k < d1

and

V1 …t0 ; x0 ÿ y0 † < d

…3:6†

hold simultaneously. Set d ˆ min…d1 ; d2 †, r ˆ min…r1 ; r2 †. We claim that kx0 ÿ y0 k < d and jgj < r implies that kx…t ‡ g; s0 ; y0 † ÿ x…t; t0 ; x0 †k < , t P t0 . If not, there exists a solution x…t; s0 ; y0 † of (2.1) and t1 ; t2 > t0 such that kx…t1 ‡ g; s0 ; y0 † ÿ x…t1 ; t0 ; x0 †k ˆ d2 ;

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kx…t2 ‡ g; s0 ; y0 † ÿ x…t2 ; t0 ; x0 †k ˆ 

…3:7†

and d2 6 kx…t ‡ g; s0 ; y0 † ÿ x…t; t0 ; x0 †k 6 ;

t 2 ‰t1 ; t2 Š:

Set d2 ˆ c so that the existence of V2 satisfying (ii) is assured. Now, setting m…t† ˆ V1 …t; x…t ‡ g; s0 ; y0 † ÿ x…t; t0 ; x0 †† ‡ V2 …t; x…t ‡ g; s0 ; y0 † ÿ x…t; t0 ; x0 ††, t 2 ‰t1 ; t2 Š we get, using standard arguments, D‡ m…t† 6 g2 …t; m…t†; jgj†;

t 2 ‰t1 ; t2 Š;

which yields, using Theorem 2.1 and (ii), V1 …t2 ; x…t2 ‡ g; s0 ; y0 † ÿ x…t2 ; t0 ; x0 †† ‡ V2 …t2 ; x…t2 ‡ g; s0 ; y0 † ÿ x…t2 ; t0 ; x0 †† 6 r2 …t2 ; t1 ; V1 …t1 ; x…t1 ‡ g; s0 ; y0 † ÿ x…t1 ; t0 ; x0 †† ‡ V2 …t1 ; x…t1 ‡ g; s0 ; y0 † ÿ x…t1 ; t0 ; x0 †; jgj†; where r2 …t; t1 ; w20 ; jgj† is the maximal solution of (3.2) r2 …t1 ; t1 ; w10 ; jgj† ˆ w20 . We also have using Theorem 2.1 and (i) V1 …t1 ; x…t1 ‡ g; s0 ; y0 † ÿ x…t1 ; t0 ; x0 †† 6 r1 …t1 ; t0 ; V …t0 ; x0 ÿ y0 †; jgj†;

…3:8† such

that …3:9†

where r1 …t; t0 ; w10 ; jgj† is the maximal solution of (3.1). By (3.5) and (3.6), we get V1 …t1 ; x…t1 ‡ g; s0 ; y0 † ÿ x…t1 ; t0 ; x0 †† < d0 =2:

…3:10†

Also, by (3.4), (3.7) and (ii), we have V2 …t1 ; x…t1 ‡ g; s0 ; y0 † ÿ x…t1 ; t0 ; x0 †† 6 a…kx…t1 ‡ g; s0 ; y0 † ÿ x…t1 ; t0 ; x0 †k† ˆ a…d2 † < d0 =2:

…3:11†

Therefore, using (3.8), (3.10), (3.11) and (3.3), we get V1 …t2 ; x…t2 ‡ g; s0 ; y0 † ÿ x…t2 ; t0 ; x0 †† ‡ V2 …t2 ; x…t2 ‡ g; s0 ; y0 † ÿ x…t2 ; t0 ; x0 †† < b…†:

…3:12†

From (3.7), (ii) and the fact that V1 …t; x† P 0, we get b…† ˆ b…kx…t2 ‡ g; s0 ; y0 † ÿ x…t2 ; t0 ; x0 †k† 6 V2 …t2 ; x…t2 ‡ g; s0 ; y0 † ÿ x…t2 ; t0 ; x0 †† 6 V1 …t2 ; x…t2 ‡ g; s0 ; y0 † ÿ x…t2 ; t0 ; x0 †† ‡ V2 …t2 ; x…t2 ‡ g; s0 ; y0 † ÿ x…t2 ; t0 ; x0 ††:

…3:13†

Hence (3.12) and (3.13) lead to the contradiction b…† < b…†, proving the equistability of (2.1).  Next, we consider stability criteria for equi-asymptotic stability of (2.1).

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Theorem 3.2. Assume that (i) V1 2 C‰R‡  S…q†; R‡ Š, V …t; x† is locally Lipschitzian in x, V1 …t; 0†  0 and D‡ V1 …t; x† 6 ÿ c…W …t; x†† ‡ g1 …t; V1 …t; x†; jgj†;

…t; x† 6 R‡  S…q†;

where g1 2 C‰R3‡ ; RŠ, g10 …t; u; jgj† is increasing in u, W 2 C‰R‡  S…q†; R‡ Š, W …t; x† is locally Lipschitzian in x, D‡ W …t; x† is bounded above or below and W …t; x† P b0 …kxk† where b0 ; C 2 K. (ii) Assumptions (ii) and (iii) in Theorem 3.1 hold. Then, the solution x…t† ˆ x…t; t0 ; x0 † of (2.1) is equi-asymptotically stable. Proof. The system (2.1) is equistable by Theorem 3.1. For  ˆ q; let d0 ˆ d0 …q; t0 † > 0, r0 ˆ r0 …q; t0 † > 0 so that kx0 ÿ y0 k < d0 and jgj < r0 imply kx…t ‡ g; s0 ; y0 †ÿ x…t; t0 ; x0 †k < q, t P t0 . We want to show that kx…t ‡ g; s0 ; y0 † ÿ x…t; t0 ; x0 †k ! 0

…3:14†

0

as t ! 1 when kx0 ÿ y0 k < d and jgj < r0 . Since W …t; x† is positive de®nite, to prove (3.14) it is enough to show that limt!1 W …t; x…t ‡ g; s0 ; y0 †† ÿx…t; t0 ; x0 † ˆ 0. We claim that limt!1 W …t; x…t ‡ g; s0 ; y0 † ÿ x…t; t0 ; x0 †† ˆ 0 when kx0 ÿ y0 k < d0 and jgj < r0 . If not, then for any a > 0, there exist two divergent sequences ftn g, ftn0 g such that W …ti ; x…ti ‡ g; s0 ; y0 † ÿ x…ti ; t0 ; x0 †† ˆ a=2;

…3:15†

W …ti0 ; x…ti0 ‡ g; s0 ; y0 † ÿ x…ti0 ; t0 ; x0 †† ˆ a and a 6W 2

…t; x…t ‡ g; s0 ; y0 † ÿ x…t; t0 ; x0 †† 6 a;

t 2 ‰ti ; ti0 Š;

i ˆ 1; 2; . . .

We could also have instead of (3.15), W …ti ; x…ti ‡ g; s0 ; y0 † ÿ x…ti ; t0 ; x0 †† ˆ a; W …ti0 ; x…ti ‡ g; s0 ; y0 † ÿ x…ti0 ; t0 ; x0 †† ˆ a=2; a=2 6 W …t; x…t ‡ g; s0 ; y0 † ÿ c…t; t0 ; x0 †† 6 a; t 2 ‰ti ; ti0 Š; i ˆ 1; 2; . . . R t0 Suppose that D‡ W …t; x† 6 M. Then tii D‡ W …s; x…s†† ds 6 M…ti0 ÿ ti † which implies ti0 ÿ ti P a=2M

for each i:

Set

Z L…t; x† ˆ V1 …t; x† ‡

t0

t

c…W …s†; x…s†† ds:

…3:16†

…3:17†

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We get, by using standard arguments and (i), D‡ L…t; x† 6 D‡ V1 …t; x† ‡ C…W …t; x…t††† 6 g1 …t; V …t; x††

6 g1 …t; L…t; x†† …3:18†

Hence, by Theorem 2.1 and (i), we obtain L…t; x…t ‡ g; s0 ; y0 † ÿ x…t; t0 ; x0 †† 6 r1 …t; t0 ; V1 …t0 ; x0 †; jgj†;

t P t0 ;

…3:19†

where r1 …t; t0 ; w10 ; jgj† is the maximal solution of (3.1). Now, using (3.15)± (3.18), we get the contradiction 0 6 V1 …t; x…t ‡ g; s0 ; y0 † ÿ x…t; t0 ; x0 †† Z t c…W …s; x…s ‡ g; s0 ; y0 † ÿ x…s; t0 ; x0 ††† ds 6 r1 …t; t0 ; V1 …t0 ; x0 ÿ y0 †; jgj† ÿ t0

n aX  a  na d0 d0 <0 …ti0 ÿ ti † < ÿ c 6 ÿc 2 2 2 iˆ1 2 2M

for suciently large n. Therefore, W …t; x…t ‡ g; s0 ; y0 † ÿ x…t; t0 ; x0 †† ! 0 as t ! 1 and since, by assumption, W …t; x† P b…kxk† it follows that kx…t ‡ g; s0 ; y0 † ÿ x…t; t0 ; x0 †k ! 0 as t ! 1 and the proof is complete.  Remark 3.1. In Theorem 3.1, 1. if V1 …t; x†  0, g2 …t; u; jgj†  0 and V1 …t; x† is positive de®nite, the system (2.1) is equistable, 2. if V1 …t; x†  0, g1 …t; u; jgj†  0, the system (2.1) is uniformly stable. Remark 3.2. In Theorem 3.2, if V2 …t; x†  0, g2 …t; u; jgj†  0 and the V1 …t; x† is positive de®nite, the system (2.1) is equi-asymptotically stable. Remark 3.3. In Theorem 3.1, let V1 …t; x†  0, g1 …t; u; jgj†  0. Then, for uniform asymptotic stability of (2.1), we need to strengthen the assumption on D‡ V2 …t; x†, which we state as a corollary. Corollary 3.1. In Theorem 3.1, assume V1 …t; x†  0 and g1 …t; u; jgj†  0. Suppose the condition on D‡ V2 …t; x† is strengthened to D‡ V2 …t; x† 6 ÿ c…W …t; x†† ‡ g2 …t; V2 …t; x†; jgj†;

…t; x† 2 R‡  S…q† \ S c …c†;

where W 2 C‰R‡  S…q†; R‡ Š, W …t; x† P b…kxk†, c; b 2 K and g2 …t; u; jgj† is nondecreasing in u. Then, the solution x…t† ˆ x…t; t0 ; x0 † of (2.1) is uniformly asymptotically stable.

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Proof. Let 0 <  < q. By uniform stability of the solution x…t† of (2.1), we have kx0 ÿ y0 k < d…†; implies

jgj < r…†

kx…t ‡ g; s0 ; y0 † ÿ x…t; t0 ; x0 †k < ;

t P t0 :

Choose  ˆ q and let d0 ˆ d…q†, r0 ˆ r…q†, then kx0 ÿ y0 k < d0 implies kx…t ‡ g; s0 ; y0 †ÿ c…t; t0 ; x0 †k < q, t P t0 . To prove uniform attractivity, we shall show that there exists a t in ‰t0 ; t0 ‡ T Š, where T ˆ

b…q† ‡1 c…b…d††

…3:20†

such that W …t ; x…t †† < b…d…q†† for any solution of (2.1) with kx0 ÿ y0 k < d0 . If not, for all t 2 ‰t0 ; t0 ‡ T Š, W …t; x…t†† > b…d…††;

t 2 ‰t0 ; t0 ‡ T Š:

…3:21†

Using (3.20) and (3.21), the fact that r2 …t; t0 ; w20 † < b…q† the assumption on D‡ V2 …t; x† and Theorem 2.1, we get the contradiction 0 6 V2 …t0 ‡ T ; x…t0 ‡ T ; g; s0 ; y0 † ÿ x…t0 ‡ T ; t0 ; c0 †† 6 r2 …t0 ‡ T ; t0 ; V2 …t0 ; x0 ÿ y0 †; jgj† Z t0 ‡T c…W …s; x…s†††ds < b…q† ÿ c…b…d††T ÿ t0   b…q† ‡ 1 < 0: ˆ b…q† ÿ c…b…d†† c…b…s†† Hence W …t ; x…t †† < b…d…††. Now, since W …t; x† is positive de®nite, we get b…kx…t †k† 6 w…t ; x…t †† < b…d…†† or kx…t †k < d…†. Therefore, by uniform stability of (2.1), we get kx…t†k < , t P t0 ‡ T , and hence the solution of x…t† of (2.1) is uniform asymptotically stable.  R1 Remark 3.4. The function g2 …t; u; jgj† ˆ jgjk…t†, with 0 k…s† ds ˆ N is admissible in Remark 3.1 (2) to yield uniform stability of (2.1) where d0 ˆ

b…† 2

and

r1 ˆ

b…† : 2N

Remark 3.5. The function g2 …t; l; jgj† ˆ jgjk…t† as in Remark 3.4 is admissible in Remark 3.3 to yield uniform asymptotic stability of (2.1) where d0 ˆ d…q†, r0 ˆ b…q†=2N , T ˆ T …q† ˆ b…q†=c…b…d†† ‡ 1.

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References [1] V. Lakshmikantham, S. Leela, On perturbing Lyapunov functions, Math. Systems Theory 10 (1976) 85±90. [2] V. Lakshmikantham, S. Leela, A.A. Martynyuk, Stability Analysis of Nonlinear Systems, Marcel Dekker, New York, 1989. [3] V. Lakshmikantham, A.S. Vatsala, Di€erential inequalities with initial time di€erence and applications, J. Inequalities Appl. (to appear). [4] V. Lakshmikantham, S. Leela, J. Vasundhara Devi, Stability criteria for solutions of di€erential equations relative to initial time di€erence (to appear).