Stability and Practically Stability of Impulsive Integro-Differential Systems by Cone-Valued Lyapunov Functions

Impulsive integro-differential systems which are an important embranchment of nonlinear impulsive differential systems [1], arise from extensive applications in nature-science such as mathematic models of circuit simulation in physics and neuronal networks in biology. Consequently there are some results about stability of such systems by vector Lyapunov functions coupled with Razumikhin techniques [2,3]. However, it's difficult to choose a right vector Lyapunov function because of the restrict conditions. At the same time, the method of cone-valued Lyapunov functions is well known to be advantageous in applications [4]. Hence, the stability results for impulsive integro-differential systems could be improved via the method of cone-valued Lyapunov functions.


Introduction
Impulsive integro-differential systems which are an important embranchment of nonlinear impulsive differential systems [1], arise from extensive applications in nature-science such as mathematic models of circuit simulation in physics and neuronal networks in biology. Consequently there are some results about stability of such systems by vector Lyapunov functions coupled with Razumikhin techniques [2,3]. However, it's difficult to choose a right vector Lyapunov function because of the restrict conditions. At the same time, the method of cone-valued Lyapunov functions is well known to be advantageous in applications [4]. Hence, the stability results for impulsive integro-differential systems could be improved via the method of cone-valued Lyapunov functions.
Adeyeye [5] considered the comparison principle by cone-valued Lyapunov functions for a class of integro-differential system without impulses. But it was not proved and also cannot be applied to impulsive integrodifferential systems. In this paper we shall firstly prove the comparison principle. Then by employing cone-valued Lyapunov functions a new comparison principle for impulsive integro-differential systems with fixed moments of impulse effects is established, which is compared with impulsive differential systems whose stability is relatively easy to solve. Finally the relevant new comparison criteria of stability and practically stability [6] of impulsive integro-differential systems are obtained too.
The remainder of this paper is organized as follows. In section 2, we describe impulsive integro-differential systems and introduce some notions and concepts. In section 3, we get some comparison results of stability and practically stability of the impulsive integro-differential systems with fixed moments of impulse effects by using the method of cone-Lyapunov functions.

Preliminaries
Consider the following impulsive integro-differential systems of the form x t x (1) where N is the set of all positive integers, 0<t 1 <t 2 <…<t k <… and In addition we always assume that f, J k satisfy certain conditions such that the solution of system (1)exists on [ ] 0 , +∞ t and is unique. We x t x t t x the solution of system (1) with initial value , then x(t) = 0 is a solution of (1), which is called the trivial solution. Note that the solutions x(t) of (1) are right continuous, satisfying ( ) ( ) ( ) ( ) Let t 0 =0; the following sets are introduced: In addition, we introduce some definitions as follows: . For any , ∈ n x y R , we let ≤ x y if − ∈ y x Z and for any functions , : , for all x Z and , ∈ V t x V is locally Lipschitz in x relative to Z.
Derivative of ( ) 0 , ∈ V t x V along system(1) is defined:

Definition 4:
The trivial solution of (1) is said to be i. stable, if for any ii. uniformly stable, if δ in (i) is independent on t 0 ; iii. uniformly asymptotically stable, if it is uniformly stable, and there exists a such that for any and every 0

Definition 5:
The trivial solution of (1) is said to be i. practically stable, if for given number pair ( ) ii. uniformly practically stable, if(i)holds for every 0 iii. practically quasi-stable, if for given number ( ) iv. uniformly practically quasi-stable, if (iii) holds for every 0 v. strongly practically stable, if (i) and (iii) hold together; vi. strongly uniformly practically stable, if (ii) and (iv) hold together.
We also consider the comparison differential system: In addition we always assume that , Ψ k g satisfy certain conditions such that the solution of system(2) exists on [ ] 0 + ∞ t and is unique.
We denote by ( ) ( ) 0 0 , , = u t u t t u the solution of system(2) with initial value ( ) 0 0 , t u . Note that the solutions u(t) of (2) are right continuous,

Definition 6:
The trivial solution of (2) is said to be i.

Definition 7:
The trivial solution of (2) is said to be i.
ii. 0 φ − uniformly practically stable, if(i)holds for every 0 V t x is locally Lipschitz in x relative to the cone Z, therefore For small enough 0 ε ∀ > , consider system The solution of it is, then we have.
To prove the conclusion we only need to prove Therefore,

t t r t t u u t t t t r t t r r t t r t t t k
Then ( ) * u t is the solution of system (2), and V (t, x (t))≤ Z ( ) * u t ; since r(t; t 0 ; u 0 ) is the maximal solution of system(2) on Z, we get immediately quai-monotone nondecreasing in u for each fixed t on the cone Z and 0 0 ( ,0) 0, ( ) ( , , ) ≡ = g t r t r t t u is the maximal solution of system(2) on Z; iii) Then the ∅ stability properties of the trivial solution of system (2) imply the corresponding stability properties of the trivial solution of system (1).

Proof: For any
If it is not true, there exists a solution x(t) of system(1) such that a contradiction Then (3.2) holds, thus the trivial solution of system (1) is stable. If the trivial solution of system (2) is 0 ∅ -uniformly stable then it is clear that δ will be independent of t 0 and thus we get the uniform stability of the trivial solution of system (1). Assume that the trivial solution of system (2) is 0 ∅ -symptotically stable, consequently we get that the trivial solution of system(1) is stable, then for Of the trivial solution of system (2) we get that t t T so the trivial solution of system(1) is attractive.
Then the trivial solution of system (1) is asymptotically stable. If the trivial solution of system (2) is 0 ∅ -uniformly asymptotically stable, then it is clear that 0 δ T will be independent of t0, and thus we get the uniform asymptotically stability of the trivial solution of system (1).

Theorem 2:
Assume that theorem1 (i)-(v) hold and we have: Then the 0 ∅ -practical stability properties of the trivial solution of system (2) with respect to ( ( ), ( )) λ a b A imply the corresponding practical stability properties of the trivial solution of system (1) with respect to ( , ) λ A .
Proof: Set ( ) ( , ( )), suppose that the trivial solution of system (2) is 0 ∅ practically Stable with respect to ( ), ( )) λ a b A then exists 0 , For above 0 b A a contradiction thus (3.3) holds, so the trivial solution of system(1) is practically stable. Suppose that the trivial solution of system (2) is 0 ∅ -uniformly practically stable with respect to ( ( ), ( )) λ a b A then it is clear that t 0 will be independent of t 0 , and thus we get the trivial solution of system (1) is uniformly practically stable with respect to ( , ) λ A . Suppose that the trivial solution of system (2) is 0 ∅ -strongly practically stable with respect To ( ( ), ( ), ( ), ), λ a b A b B T consequently we get that the trivial solution of system (1) is practically stable with respect to ( , ) λ A then   (1) is practically quasi-stable with respect to ( , , ) λ B T .
Therefore the trivial solution of system (1) is strongly practically stable with respect to ( , , , ) If the trivial solution of system (2) is 0 ∅ -strongly uniformly practically stable with respect to ( , , , ) λ A B T then it is clear that the above proof establish for every t 0 , therefore we get the trivial solution of system (1) is strongly uniformly practically stable with respect to ( , , , ) λ A B T .