Some Stability and Boundedness Criteria for a Class of Volterra Integro-differential Systems

Using Lyapunov and Lyapunov-like functionals, we study the stability and boundedness of the solutions of a system of Volterra integrodieren tial equations. Our results, also extending some of the more well-known criteria, give new sucien t conditions for stability of the zero solution of the nonperturbed system, and prove that the same conditions for the perturbed system yield boundedness when the perturbation is L 2 .


Introduction
We consider the stability and boundedness of solutions of systems of Volterra integro-differential equations, with forcing functions, of the form in which A(t) is an n × n matrix function continuous on [0, ∞), B(t, s) is an n × n matrix continuous for 0 ≤ s ≤ t < ∞, f and g are n × 1 vector functions continuous on (−∞, ∞) and h is an n × 1 vector function defined almost everywhere on [0, ∞).Here, h(t) represent the forcing functions or external disturbances or inputs into system (1).
The qualitative behaviour of the solutions of systems of Volterra integrodifferential equations, especially the case where f (x) = g(x) = x and h(t) = 0, has been thoroughly analyzed by many researchers.Among the contributions in the 1980s, those of Burton are worthy of mention.His work ( [1], [2]) laid the foundation for a systematic treatment of the basic structure and stability properties of Volterra integro-differential equations, mainly, via the direct method of Lyapunov.This paper essentially looks into some of the many interesting results established by Burton and proposes ways of utilizing the form of the Lyapunov functionals proposed by Burton to construct new or similar ones for system (1).Now, if f (0) = g(0) = 0 and h(t) = 0, then system (1) reduces to so that x(t) ≡ 0 is a solution of (2) called the zero solution.The initial conditions for integral equations such as (1) or (2) involve continuous initial functions on an initial interval , say, x(t) = φ(t) for 0 ≤ t ≤ t 0 .Hence, x(t; t 0 , φ), t ≥ t 0 ≥ 0 denotes the solution of (1) or (2), with the initial function φ : [0, t 0 ] → R n assumed to be bounded and continuous on [0, t 0 ].The definitions of the stability and the boundedness of solutions of (1) are given in Burton [1].It is assumed that the functions in (1) are well-behaved, that continuous initial functions generate solutions, and that solutions which remain bounded can be continued.EJQTDE, 2002 No. 12, p. 2

Nonperturbed Case
Consider the scalar equation We suppose that For comparison sake, we first state Burton's theorem regarding the stability of the zero solution of (3).
We next state an extension of Theorem 1, which Burton proved via the Lyapunov functional the time-derivative along a trajectory of (3) of which is, EJQTDE, 2002 No. 12, p. 3 In the process, and motivated by the work of Miyagi et.al in the construction of generalized Lyapunov functions for power systems [4] and single-machine systems [5], we propose a new Lyapunov functional.As a simple example will show, the new stability criterion may be used in situations where Theorem 1 cannot be applied.
Theorem 2. Let ( 4)-(8) hold, with A(t) < 0, and suppose there are constants Suppose further there is some constant k > 0 such that and for t ≥ 0. Then the zero solution of (3) is stable.
Proof.Consider the functional We have, along a trajectory of (3), Recalling that A(t) < 0 for all t ≥ 0 and noting that the Schwarz inequality yields, The third and fourth terms of EJQTDE, 2002 No. 12, p. 5 and respectively.Thus, which will be nonpositive if equations ( 13) and ( 14) are satisfied.Finally, to prove the positive definiteness of V 2 , we see that if we define which is clearly positive definite given that ur(u) > 0 for u = 0.This completes the proof of Theorem 2.
Thus, we have proposed an alternate stability criterion for the scalar equation ( 3), and the criterion may be considered for cases where Burton's Theorem 1, though simpler, cannot be applied.

2
Example 2. Analysis via Theorem 2 shows that the zero solution of Then 0 < m < 1/3.Thus, we may choose 1/4 < m < 1/3 to satisfy all conditions of the theorem.Theorem 1 is not applicable.

Perturbed Case
The next two results, which extend Theorem 1 and Theorem 2, give a class of forcing functions that maintains the boundedness of the solutions of the equation where h : [0, ∞) → R is defined almost everywhere on [0, ∞).
then all solutions of (15) are bounded.
Proof.Let > 0 and consider the functional implying, therefore, the differentiability and hence the existence on [0, ∞) of the second term of the functional V 3 .Thus, we have This completes the proof of Theorem 3 since we can always find some > 0 small enough such that (ρ − ) > 0. Note that (16) ensures the radial unboundedness of V 3 .
In the same fashion, we prove the following extension of Theorem 2.
Proof.For > 0, the functional yields, given the definition, the time-derivative, This completes the proof of Theorem 4 since we can always find some > 0 small enough such that (ρ − τ 2 ) > 0. We note that EJQTDE, 2002 No. 12, p. 9 3 The Vector Equation

Nonperturbed Case
First, we consider the nonperturbed system (2).If we suppose that f , g ∈ C 1 [R n , R n ], then we can define which are defined for all x ∈ R n .Hence, assuming f (0) = g(0) = 0, system (2) can be written as the i-th component of which is noting that in the above equation d ij (x)x j and e ij (x)x j , for i, j = 1, . . ., n, are continuously differentiable with respect to x ∈ R n simply for the reason that D(x)x = f (x) and E(x The next result is new.
Putting n = 1 in Theorem 5 yields a new stability criterion for the scalar case (3), on the assumption that f, g ∈ C 1 [R, R] and f (0) = g(0) = 0, and on letting , Putting and b 11 (t, s) = B(t, s), we have the following result: If β(t, x) ≤ 0 for t ≥ 0 and x ∈ R, then the zero solution of (3) is stable.
Example 3.For the equation Corollary 1 is easier than either Theorem 1 or Theorem 2 to apply.Thus, we have, for all t ≥ 0 and x = 0, , with f (0) = g(0) = 0, and if for c 1 , c 2 > 0, t ≥ 0 and x ∈ R 2 , we have, using (18) and condition (b) of Theorem 5, The following simple, but illustrative, case is one such stable system: EJQTDE, 2002 No. 12, p. 16 or, in the form of (17), ds .
Now, for all t ≥ 0 and for all x ∈ R 2 , we have, Hence, we have shown that β i (t, x)x 2 i ≤ 0 for i = 1, 2, t ≥ 0 and x ∈ R 2 .The zero solution of system (20) is therefore stable by Theorem 5.
Proof.Let > 0 and consider the functional x 2 i .
We have thus proved the boundedness of solutions of (1), since we can always find > 0 such that (c − ) ≥ 0.
The following corollary follows directly from Theorem 6 by putting n = 1.
Corollary 2. Let the conditions of Corollary 1 hold, with the last condition replaced by the assumption that there is a constant c > 0 such that β(t, x) ≤ − c for t ≥ 0 and x ∈ R. If h(•) ∈ L 2 [0, ∞), then all solutions of the scalar equation (15) are bounded.