ON THE ASYMPTOTIC STABILITY IN TERMS OF TWO MEASURES FOR FUNCTIONAL DIFFERENTIAL EQUATIONS G. Makay Student in Mathematics, University of Szeged, Szeged, H-6726, Hungary Key words and phrases: Lyapunov functional, Lyapunov's direct method, process, nonautonomous dierential systems, partial stability. 1. INTRODUCTION Denote by C the space of the continuous functions : [?r; 0]! R n, where r > 0 is a constant. In the function space C we use the norms kk = maxfj(s)j : s 2 [?r; 0]g; jjjjjj = s Z 0 j(s)j 2 ds ;?r where j j denotes an arbitrary norm in R n. If x : [t 0? r; T )! R n (0 t 0 < T 1) is continuous and t 2 [t 0 ; T ), we dene x t 2 C by x t (s) := x(t + s) for s 2 [?r; 0]. Consider the functional dierential equation _x = f(t; x t ); (1:1) where f : R + C! R n is continuous and f(t; 0) = 0 for all t 0. We suppose, that for each t 0 2 R + and each 2 C there is a unique solution x(; t 0 ; ) dened on an interval [t 0 ; t 0 + ); > 0. We assume, that the function x continuously depends on the initial data t 0 ;. Denote by K the set of strictly increasing continuous functions w : R +! R + with w(0) = 0. If V is a continuous functional from R + C into R, then _ V denotes the derivative of functional V with respect to equation (1:1) dened by _V (t; ) = _ V (1:1) (t; ) = lim sup!0+ V (t + ; x t+ (; t; ))? V (t; ) : In this paper we study the asymptotic stability of the zero solution of equation (1:1) using Lyapunov functionals. In this topic the following two theorems are basic : 1
Theorem A (see [5, Theorem 4.1]). Suppose, that for every M > 0 there is an L(M) > 0 such that [t 2 R + ; 2 C M := f 2 C : kk Mg] =) jf(t; )j L(M): (1:2) If there are M > 0, a continuous functional V : R + C M! R and functions w 1 ; w 2 ; w 3 2 K such that w 1 (j(0)j) V (t; ) w 2 (kk) ; (1:3) _V (t; )?w 3 (jjjjjj) (1:4) for all t 2 R +, 2 C M, then the zero solution of equation (1.1) is uniformly asymptotically stable. Theorem B ([3, Theorem 5.2.1]). Suppose that conditions (1.2) and (1.3) of Theorem A hold, and, instead of condition (1.4), the inequality _V (t; )?w 3 (j(0)j) (1:5) holds. Then the zero solution of equation (1.1) is uniformly asymptotically stable. It is an old problem (see [1, p. 252]), whether the boundedness condition can be dropped from Theorems A and B. Note, that - as is usual in the stability theory - the proofs of these theorems need conditions (1.4) or (1.5) only along the solutions of (1.1). We will show by examples that if conditions (1.4) and (1.5) are required only along the solutions of (1.1), then the boundedness condition cannot be dropped. At rst we replace the norms jj; kk and jjjjjj in Theorems A and B by abstract "measures" (see [6],[7] and [8]). It will be pointed out that, under these general circumstances, to the boundedness condition there corresponds an estimate between the measures. Our examples will show that these estimates are essential. 2. THEOREMS AND COROLLARIES Let X be a Banach-space. A continuous function h : R X! R + is called a measure in X, if h(t; 0) = 0. The continuous function u : R X R +! X is said to be a process (see e.g. [3, Chapter 4.1]), if u(t 0 ; x; 0) = x and u(t 0 + t 1 ; u(t 0 ; x; t 1 ); t) = u(t 0 ; x; t 1 + t) for all t 0 2 R; t 1 ; t 2 R +. For example, let U(t 0 ; ; t? t 0 ) := x t (; t 0 ; ), where x(t; t 0 ; ) (t t 0 ) is the solution of equation (1.1) with x(t 0 ; t 0 ; ) =. It is easy to see, that U is a process. 2
Let h 0 and h be measures. If u is a process and t 0 2 R, x 0 2 X, then the function u(t 0 ; x 0 ; ) is said to be a motion. Let x denote the state of this motion at t t 0, i.e. x := u(t 0 ; x; t? t 0 ). The motion is (h 0 ; h)-stable or stable in measures (h 0 ; h), if for each t t 0 and > 0 there is a (x 0 ; t 0 ; t ; ) > 0 such that h 0 (t ; x? x ) < implies h(t + t; u(t ; x; t)? u(t ; x ; t)) < for all t 0. The stability is uniform, if can be chosen independently of t. The motion u(t 0 ; x 0 ; ) is said to be (h 0 ; h)-attractive, if there is a (x 0 ; t 0 ; t ) > 0 such that, if h 0 (t ; x? x ) <, then h(t + t; u(t ; x; t)? u(t ; x ; t))! 0 (t! 1). The attractivity is uniform, if is independent of t and the convergence is uniform in t. The motion is said to be (uniformly) asymptotically stable, if it is (uniformly) stable and (uniformly) attractive. For a V : R X! R we dene the derivative _ V with respect to the process u by _V (t; x) := lim sup!0+ V (t + ; u(t; x; ))? V (t; x) : Examples: The zero solution of equation (1.1) is stable in the usual sense (see e.g. [3, Chapter 5.1]), if and only if the zero motion (U(t 0 ; 0; )) is stable in measures h 0 (t; ) = kk and h(t; ) = j(0)j. The solution x(; t 0 ; 0 ) of equation (1.1) is stable if and only if the motion U(t 0 ; 0 ; ) is stable in the previous measures. The zero solution of the ordinary dierential equation _x = F (t; x) (x 2 R n ) is partially stable (see [4]), if the zero motion of this equation is stable in the measures h 0 (t; x) = p x 2 1 + x2 2 + ::: + x2 n and h(t; x) = p x 2 1 + ::: + x2 s, where 0 < s n and x = (x 1 ; x 2 ; :::; x n ). The stability of an invariant set A with respect to this equation is equivalent to the stability in measures h 0 (t; x) = h(t; x) = d(x; A), where d(x; A) means the distance between x and A in R n. (For further examples for processes, among them partial dierential equations, see e.g. [9].) Proposition 1. Let the measures h 0 ; h be given. Suppose that there are a continuous functional V : R X! R, functions w 1 ; w 2 ; w 3 ; w 4 2 K and a measure h 1 satisfying the following conditions: w 1 (h(t; x)) V (t; x) w 2 (h 0 (t; x)) (i) _V (t; x)?w 3 (h 1 (t; x)) h 0 (t; x) w 4 (h 1 (t; x)) (ii) (iii) for all t 2 R and x 2 X. Then the zero motion of the process u is uniformly asymptotically (h 0 ; h)-stable. Proof: At rst we prove the uniform stability. Let an > 0 be given and dene () := w?1 2 (w 1()). Now if h 0 (t ; x) <, then V (t ; x) w 1 () by (i). Condition (ii) and the continuity of function V (t + 3
t; u(t ; x; t)) implies, that this function is nonincreasing in t, so w 1 (h(t + t; u(t ; x; t))) V (t + t; u(t ; x; t)) V (t ; u(t ; x; t? t )) = V (t ; x) w 1 (): Consequently, h(t + t; u(t ; x; t)) which proves the uniform stability. The conditions imply _V (t + t; u(t ; x; t))?w 3 (h 1 (t + t; u(t ; x; t)))?w 3 (w 4 (h 0 (t + t; u(t ; x; t))))?w 3 (w 4 (w?1 2 (V (t + t; u(t ; x; t))))); which is a dierential inequality for V (t + t; u(t ; x; t)). By [6, Theorem 3.1.1] we get lim V (t + t; u(t ; x; t)) = 0 t!1 uniformly in t. Obviously, inequality (i) proves the uniform asymptotic stability. Note, that inequalities (ii) and (iii) could be replaced by _V (t; x)?w 5 (h 0 (t; x)); with an appropriate w 5 2 K. In spite of this fact we separated them because it is inequality (iii) that corresponds to the boundedness condition in Theorem A. In order to formulate the problem corresponding to that of omitting the boundedness condition from Theorem A, we weaken some conditions of Proposition 1. It can be seen that conditions (ii) and (iii) may be asked only along the motions, even we can assume on the motions in condition (iii) that h(t + t; u(t ; x; t)) B for all t 0, where B is a constant independent of t ; x. Conditions (ii) and (iii) can be further weakened if we need only (nonuniform) asymptotic stability. The process u is said to be h 0 -continuous (with respect to x) if for every > 0, t 0 2 R, t 1 0 and x 0 2 X there is = (t 0 ; t 1 ; x 0 ; ) > 0 such that h 0 (t 0 ; x?x 0 ) < implies h 0 (t 0 +t 1 ; u(t 0 ; x; t 1 )?u(t 0 ; x 0 ; t 1 )) <. If the process u is h 0 -continuous, then conditions (ii) and (iii) can be asked only for suciently large values of t. This modications result in the following. 4
Theorem 1. Let measures h 0, h be given. Assume that the process u is h 0 -continuous with respect to x. Suppose that there are continuous functional V : R X! R, functions w 1 ; w 2 ; w 3 ; w 4 2 K, measure h 1, and constants T; B > 0 satisfying the following conditions: w 1 (h(t; x)) V (t; x) w 2 (h 0 (t; x)) (i) for all t t 0 and x 2 X, _V (t + t; u(t ; x; t))?w 3 (h 1 (t + t; u(t ; x; t))) (ii) for all t t 0, t T and x 2 X, and h 0 (t + t; u(t ; x; t)) w 4 (h 1 (t + t; u(t ; x; t))) (iii) for each t t 0, t T and x 2 X such that h(t + s; u(t ; x; s)) < B for all s T. Then the zero motion of the process u is asymptotically (h 0 ; h)-stable. Corollary 1 (Theorem A revisited). Suppose that all but condition (1.4) of Theorem A are satised. Suppose, in addition, that there is a T = T (t 0 ) > 0 such that the inequality _V (t; x t )?w 3 (jjjx t jjj) (1:4 0 ) holds for all t t 0 + T and for every solution x : [t 0? r; 1)! R n of (1.1). Then the zero solution of equation (1.1) is asymptotically stable. Proof: Let h 0 (t; ) := kk, h(t; ) := j(0)j and h 1 (t; ) := jjjjjj in Theorem 1. It is enough to prove, that condition (iii) in Theorem 1 follows from the boundedness condition of Theorem A. Inequalities (i) and (ii) imply the stability, so we can assume, that the solutions are bounded above in the measure h with an arbitrary B. Suppose that T r. We have from the boundedness (see condition (1.2)), that the absolute value of the derivative of function x(t; t 0 ; ) less or equal than L(B). For every t t 0 there exists t 1 2 [t? r; t] with jx(t 1 )j = kx t k. So if t t 0 + r, then s 2 [t 1? kx t k=2l(b); t 1 + kx t k=2l(b)] implies jx(s)j kx t k=2 and we have the inequality jjjx t jjj s kx t k 3 8L(B) : This means that condition (iii) is satised and the proof is complete. Theorem B cannot be deduced from Theorem 1, because one cannot estimate below the measure j(0)j by the measure kk. So we must replace inequality (iii) with a more general condition. 5
Theorem 2. Assume that all but condition (iii) of Theorem 1 are satised. Suppose, in addition that if w 2 (h 0 (t + t; u(t ; x; t))) > > 0 for all t T, then Z 1 w 3 (h 1 (t + t; u(t ; x; t)))dt = 1: T Then the zero motion of the process u is asymptotically (h 0 ; h)-stable. Proof: By h 0 -continuity of the process u, for every > 0 there exists a (t 0 ; x 0 ; t ; T; ) > 0 such that h 0 (t ; x? x ) < implies h 0 (t + T; u(t ; x; T )? u(t ; x ; T )) <. From the proof of Proposition 1, for each > 0 we get an (t 0 ; x 0 ; t ; T; ) > 0 such that if h 0 (t + T; u(t ; x; T )? u(t ; x ; T )) <, then h(t + t; u(t ; x; t)? u(t ; x ; t)) < for all t T. This proves the stability of the zero motion. To complete the proof it is enough to show that lim t!1 V (t + t; u(t ; x; t)) =: v 0 = 0: Suppose v 0 > 0. Then v 0 w 2 (h 0 (t + t; u(t ; x; t))) for all t T and the last condition of the theorem gives Z 1 w 3 (h 1 (t + t; u(t ; x; t)))dt = 1: By condition (ii) we have 0 V (t + t; u(t ; x; t)) (t! 1), which is a contradiction. V (t + T; u(t ; x; T ))? T Z t T w 3 (h 1 (t + t; u(t ; x; t)))dt!?1 Corollary 2 (Theorem B revisited). Suppose that all but condition (1.5) of Theorem B are satised. Suppose, in addition, that there is a T = T (t 0 ) > 0 such that the inequality _V (t; x t )?w 3 (jx(t)j) (1:5 0 ) holds for all t t 0 + T and for every solution x : [t 0? r; 1)! R n of (1.1). Then the zero solution of equation (1.1) is asymptotically stable. Proof: Let h 0 (t; ) := kk and h(t; ) = h 1 (t; ) := j(0)j in Theorem 2. We use the boundedness condition to prove the new condition in Theorem 2. As we saw in the proof of Corollary 1, for every t t 0 + r we have t 1 2 [t? r; t] such that jx(s)j kx t k=2 for all s 2 [t 1? kx t k=2l(b); t 1 + kx t k=2l(b)]. If kx t k for all t t 0, then it follows from the last property that Z 1 t 0 w 3 (jx(t)j)dt = 1; which proves the new condition. 6
3. EXAMPLES The following two examples show that the boundedness condition cannot be dropped from the Corollary 1 and 2. Consider the ordinary scalar dierential equation _x = _ (t) x; (x 2 R) (3:1) (t) where : [?1; 1)! (0; 1) is continuously dierentiable (r = 1; t 0 = 0). Obviously, the functions x(t) = c (t) (c 2 R) are the solutions of equation (3.1). In both examples we choose the functional V (t; ) = kk + jjjjjj. We have to construct a function which is bounded on [?1; 1) and satises an inequality _V (t; c t )?w 3 (jjjc t jjj) (Cor: 1) (3:2) respectively _V (t; c t )?w 3 (jc (t)j) (Cor: 2) (3:3) for all t 1; c 2 R and, at the same time, (t) 6! 0 as t! 1. Let a function : [?1; 1)! (0; 1) be such that (a) (n) = 1 + 1=2 n, (b) (t? 1)? (t) 1=2 n+1 for all t 2 [n; n + 1), (c) jjj jjj 1=2 n?1 for all t 2 [n; n + 1) for n =?1; 0; 1; :::. (Continuously dierentiable functions with these properties can be constructed from pieces of lines and parabolas.) Condition (b) implies that k t k is monotone nonincreasing. So _V (t; c t ) jjjc t jjj?jcjw( 1 2 n)?w 3(jjjc t jjj) for all t 0; c 2 R, i.e. inequality (3.2) holds. Condition (a) implies, that (t) 6! 0 as t! 1, so the zero solution of equation (3.1) is not asymptotically stable. We now construct a function satisfying (3.3). Let a sequence of intervals f[a n ; b n ]g 1 n=0 be given such that [a n+1?1; b n+1?1] [a n ; b n ], a 0 = 0, b 0 = 1=4. Consider a function having the following properties: (a) (t) = 1=2 n+1 for t 2 [b n ; a n+1 ], (b) (t? 1)? (t) 1=2 n+1 for all t 2 [a n ; a n+1 ), (c) _ (t)?1 and (t) 1 + 1=2 n+2 for t 2 [a n+1? 1; b n+1? 1], 7
(d) (t) monotone decreasing on interval [b n+1? 1; b n ], (e) maxf (t) : t 2 [a n ; a n+1? 1]g = 1 + 1=2 n+1 and (f) (t) 4 for all t?1. Since k t k is nonincreasing, properties (a) and (b) implies _V (t; c t ) jcj jjj t jjj?jcjw( 1 2 n )?w 3(jc (t)j) for all t 2 [b n ; a n+1 ] (n = 0; 1; :::) and c 2 R with appropriate functions w; w 3 2 K. If t 2 [a n ; b n ], then _V (t; c t ) jcj k t k?jcj?jcj (t) 4 =? 1 jc (t)j: 4 Consequently, (3.3) is satised for all t 1. On the other hand, condition (e) guarantees (t) 6! 0 (t! 1), i.e. the zero solution of equation (3.1) is not asymptotically stable. REFERENCES 1. Burton T. A., Volterra Integral and Dierential Equations, Academic Press (1983). 2. Burton T. & Hatvani L., Stability theorems for nonautonomous functional dierential equations by Lyapunov functionals, Tohoku Math. J. 41, 65-104 (1989). 3. Hale J., Theory of functional dierential equations, Springer-Verlag New York-Heidelberg-Berlin (1977). 4. Hatvani L., On partial asymptotic stability and instability, Acta Sci. Math. 45, 219-231 (1983). 5. Hatvani L., On the asymptotic stability of the solutions of functional dierential equations, Coll. Math. Soc. J. Bolyai 53, 227-238 (1988). 6. Lakshmikantham V., Leela S. & Martynyuk A. A., Stability analysis of nonlinear systems, Marcel Dekker, Inc. New York and Basel (1989). 7. Lakshmikantham V. & Xin Zhi Liu, Perturbing families of Lyapunov functions and stability in terms of two measures, J. Math. Anal. Appl. 140, 107-114 (1989). 8. Movchan A. A., Stability of processes with respect to two metrics, J. Appl. Math. Mech. 24, 1506-1524 (1961). 9. Stephen H. Saperstone, Semidynamical Systems in Innite Dimensional Spaces, Springer-Verlag New York- Heidelberg-Berlin (1981). 8