New Results of Global Asymptotical Stability for Impulsive Hopfield Neural Networks with Leakage Time-Varying Delay

In this paper, Hopfield neural networks with impulse and leakage time-varying delay are considered. New sufficient conditions for global asymptotical stability of the equilibrium point are derived by using Lyapunov-Kravsovskii functional, model transformation and some analysis techniques. The criterion of stability depends on the impulse and the bounds of the leakage time-varying delay and its derivative, and is presented in terms of a linear matrix inequality (LMI).


Introduction
As we know, time delay is a common phenomenon that describes the fact that the future state of a system depends not only on the present state but also on the past state, and often encountered in many fields such as automatic control, biological chemistry, physical engineer, neural networks, and so on [1] [2] [3] [4] [5].Moreover, the existence of time delay in a real system may lead to instability, oscillation, and bad dynamic performance [3] [4] [5].So, it is significant and necessary to consider the delay effects on stability of dynamical systems.In real numbers, +  the set of positive integers, n  the n-dimensional real space and n m ×  n m × -dimensional real space equipped with the Euclidean norm ⋅ , respectively.For E denotes the identity matrix with appropriate dimensions and { } , , n x t x t x t =  is the neuron state vector of the neural networks; ( )  is a diagonal matrix with > 0, i c i∈ Λ ; A and B are the connection weight matrix and the delayed weight matrix, respectively; J is an external input; f and g represent the neuron activation functions.Through-out this paper, we make the following assumptions: (H 1 ) ( ) ( ) , where , , , σ τ σ τ ρ ρ are some real constants; (H 2 )

( )
, : , [ ] ( ) The following Lemmas will be used to derive our main results.

S S =
, is equivalent to any one of the following conditions: (1) In the following, we assume that some normal conditions, such as Lipschitz continuity of f and g, etc, are satisfied so that the equilibrium point of system (1) does exist, see [13] [21] etc, in which the existence results of equilibrium point are established by employing contraction mapping theorem, Brouwer's fixed point theorem and some functional method.Note that these results are independent of time delays, so it is easy to extend the results in the literatures to an impulsive neural network with leakage time-varying delays and other delays, we omit the details and investigate the global asymptotic stability of the equilibrium point mainly in next section.As usual, we assume that ( )

Global Asymptotic Stability
In this section, we investigate the global asymptotic stability of the unique equilibrium point of system (1).For this purpose, the impulsive function k J which is viewed as a perturbation of the equilibrium point * x of model (1)   without impulses is defined by . Such a type of impulse describes the fact that the instantaneous perturbations encountered depend not only on the state of neurons at impulse times k t but also the state of neurons in recent history, which reflects a more realistic dynamics.Similar impulsive perturbations have also been investigated by some researchers recently [22] [23] [25].
For convenience, we let ( ) ( ) * y t x t x = − , then system (1) can be rewritten as is a solution of system (2).Therefore, to consider the stability of the equilibrium point of system (1), it is equal to consider the stability of zero solution of system (2).
In this paper, we assume that there exist constants which is a very important assumption for activation functions f and g.Using a model transformation, system (2) has an equivalent form as follows: In the following, we shall establish a theorem which provides sufficient conditions for global asymptotical stability of the zero solution of system (3).It implies that, if system (1) has an equilibrium point, then it is unique and globally attractive.

Q C PCQ C PC C PCQ C PC C PAQ A PC C PBQ B PC
By the well known Schur complements, we know that 0 Σ < if and only if the LMI (4) holds.Hence, one may derive that

D V t y y t y t t t t k
where , for some n + ∈  .Then integrating inequality (11)   at each interval [ ) In order to analyze (12), we need consider the change of V at impulse times.
Firstly, it follows from (5) that ( ) ( ) in which the last equivalent relation is obtained by Lemma 2.4.
Secondly, from system (3), it can be obtained that ∫ which together with (13) yields Obviously, we have .
Thus, we can deduce that Substituting the above inequality in ( 12) yields By simple calculation, it can be deduced that . 1  which implies that the equilibrium point of system (2) is locally stable, and uniformly bounded on [ ) 0, ∞ .Thus, considering the continuity of the activation function f and g, it can be deduced from system (2) that there exists some constant 0 R > such that  , where y  denotes the right-hand derivative of y at impulse times 1 , In the following, we shall prove that We first show that It is equivalent to prove that ( ) By (H 3 ), we define 1 min , 2 From (14), it can be obtained that Combining ( 16) and ( 18), one may deduce that, for any 0 >  , there exists a This completes the proof of (15 In particular, when the leakage delay and transmission delay are all constants, i.e., ( ) ( ) Q. Xi DOI: 10.4236/jamp.2017.5111732123 Journal of Applied Mathematics and Physics For system (22), we have the following result by Theorem 3.1.
Corollary 3.1.Assume that system (22) has one equilibrium and that assumptions (H 2 )-(H 5 ) hold.Then the equilibrium of system (22) is unique and is globally asymptotically stable if there exist n n × matrices 0, 0, 1, 2, ,5 such that the following LMI holds:  on our results, we would like to say that the stability of system (1) is more sensitive to leakage delay, leakage time-varying delay or leakage constant delay.
In other words, we should control not only the bound of leakage delay but also the bound of derivative of leakage delay, to obtain the stability of system (1), while the bound of transmission delay τ or ( ) t τ do not affect the stability of system in our results.Remark 3.3.So far, there are many papers to study the dynamics of time delay systems and impulsive systems, many effective methods and results have been developed [19]- [26].But, those results cannot be applied to systems with leakage time-varying delay and impulse which could affect the dynamics of system essentially.In this paper, we investigate the stability of impulsive Hopfield neural

Conclusion
We have studied the global asymptotic stability of the equilibrium point of impulsive Hopfield neural networks with leakage time-varying delay.Via an appropriate Lyapunov-Krasovskii functional and model transformation technique, a new stability criterion which depends on the impulse and the bounds of leakage time-varying delay and its derivative has been presented in matrix  is a symmetric and positive definite or negative definite matrix.The notation T  and −  denote the transpose and the inverse of  , respectively.If ,   are symmetric matrices, >   means that −   is positive definite matrix.the maximum eigenvalue and the minimum eigenvalue of matrix  , respectively.

Remark 3 . 2 .
The conditions in Corollary 3.1 are independent on transmission delay and dependent only on leakage delay as 0 τ ρ = in Theorem 3.1.So, based networks with leakage time-varying delay by model transformation technique and a certain Lyapunov-Krasovskii functional combined with LMI technique and construct a new criterion.How to improve the dynamics of systems with leakage time-varying delay and impulse may be an interesting problem and requires further research. . ). *,