Global Exponential Stability of Antiperiodic Solutions for Discrete-Time Neural Networks with Mixed Delays and Impulses

The problem on global exponential stability of antiperiodic solution is investigated for a class of impulsive discrete-time neural networks with time-varying discrete delays and distributed delays. By constructing an appropriate Lyapunov-Krasovskii functional, and using the contraction mapping principle and the matrix inequality techniques, a new delay-dependent criterion for checking the existence, uniqueness, and global exponential stability of anti-periodic solution is derived in linear matrix inequalities LMIs . Two simulation examples are given to show the effectiveness of the proposed result.


Introduction
Over the past decades, delayed neural networks have found successful applications in many areas such as signal processing, pattern recognition, associative memories, and optimization solvers.In such applications, the qualitative analysis of the dynamical behaviors is a necessary step for the practical design of neural networks 1 .Many important results on the dynamical behaviors have been reported for delayed neural networks, see 1-5 and the references therein for some recent publications.Although neural networks are mostly studied in the continuous-time setting, they are often discretized for experimental or computational purposes.The dynamic characteristics of discrete-time neural networks have been extensively investigated, for example, see 6-10 and the references cited therein.
Impulsive differential equations are mathematical apparatus for simulation of process and phenomena observed in control theory, physics, chemistry, population dynamics, biotechnologies, industrial robotics, economics, and so forth 11, 12 .Consequently, many neural networks with impulses have been studied extensively, and a great deal of the norm in R n .If A is a matrix, denote by λ max A resp., λ min A the largest resp., smallest eigenvalue of A. For integers a, b, and a < b, N a, b denotes the discrete interval given by N a, b {a, a 1, . . ., b}.C N −τ, 0 , R n denotes the set of all functions φ : N −τ, 0 → R n .In symmetric block matrices, the symbol * is used as an ellipsis for terms induced by symmetry.Sometimes, the arguments of a function or a matrix will be omitted in the analysis when no confusion can arise.

Model Description and Preliminaries
In this paper, we consider the following impulsive discrete-time neural networks with timevarying discrete delays and distributed delays x i k r e ir x i k r , r 1, 2, . . .

2.1
or, in an equivalent vector form x k r e r x k r , r 1, 2, . . .

2.2
for k 1, 2, . .., where x k x 1 k , x 2 k , . . ., x n k T ∈ R n , x i k is the state of the ith neuron at time k; g x k g 1 x 1 k , g 2 x 2 k , . . ., g n x n k T ∈ R n , g j x j k denotes the activation function of the jth neuron at time k; I k I 1 k , I 2 k , . . ., I n k T ∈ R n , I i k represents the external input on the ith neuron at time k; the positive integer τ 1 k corresponds to the transmission delay and satisfies τ ≤ τ 1 k ≤ τ τ and τ are known integers such that τ > τ ≥ 0 ; the positive integer τ 2 describes the distributed time delay; A diag a 1 , a 2 , . . ., c n , a i 0 ≤ a i < 1 describes the rate with which the ith neuron will reset its potential to the resting state in isolation when disconnected from the networks and external inputs; B b ij n×n is the connection weight matrix, C c ij n×n and D d ij n×n are the delayed connection weight matrices; k r are the impulse instants satisfying 0 Remark 2.1.The delay term k τ 2 −1 m k g x m−τ 2 in the system 2.2 is the so-called distributed delay in the discrete-time setting, which can be regarded as the discretization of the integral form t t−τ 2 g x s ds for the continuous-time system.
The initial condition associated with system 2.2 is given by where τ max{ τ, τ 2 }.Throughout this paper, we make the following assumptions.
H1 g u and e r u are odd functions, τ 1 k and I k are ω-periodic function and ωantiperiodic function, respectively, that is,

2.4
And there exists a positive integer p such that k r p k r ω, e r p u e r u , u ∈ R n , r 1, 2, . . . .

2.5
H2 There exist constants ľj and l j j 1, 2, . . ., n such that H3 There exist constants ȟi and The proof of Lemma 2.3 can be carried out by following a similar line as in 10 , and hence it is omitted.

Main Result
The main objective of this section is to obtain sufficient conditions on the existence, uniqueness, and exponential stability of antiperiodic solution for system 2.2 .For presentation convenience, in the following, we denote hold, where then X is a Banach space with the topology of uniform convergence.For φ, ψ ∈ X, let x k, φ and x k, ψ be the solutions of system 2.2 with initial values φ and ψ, respectively.Define and then x k φ ∈ X for all k 1, 2, . ... It follows from system 2.2 that u r y k r e r x k r , φ − e r x k r , ψ , 3.9 system 3.8 can then be simplified as

3.10
Defining η k y k 1 − y k , we consider the following Lyapunov-Krasovskii functional candidate for system 3.10 as where

3.12
We proceed by considering two possible cases of k / k r and k k r .

3.20
From the definition of η k and 3.10 , we have

3.21
From Lemma 2.3, it can be shown that the following inequality holds: For positive diagonal matrices T 1 > 0 and T 2 > 0, we can get from assumption H2 that

3.27
it follows from 3.13 -3.26 that In the similitude of the proof of inequality 3.23 , we have

3.29
By using similar method in 3.28 , it follows from 3.13 to 3.22 , and 3.25 to 3.26 , and 3.29 that

3.30
Case 2 k k r r 1, 2, . . . .Note that the inequalities from 3.13 to 3.26 except 3.13 and 3.21 hold for k k r .Calculating ΔV 1 k r and η T k r Rη k r along the system 3.10 , we have

3.31
By using similar method in Case 1, we can obtain that Combining the above discussions in Case 1 and 2, we obtain from 3.2 , 3.3 , 3.28 , 3.30 , 3.32 , and 3.33 that where From the definition of V k , it is easy to verify that where

3.36
where For any scalar ρ ≥ 1, it follows from 3.34 and 3.35 that

3.37
Summing up both sides of 3.37 from 0 to k − 1 with respect to i, we have

3.39
From 3.35 , we have

3.40
It follows from 3.38 -3.40 that where

3.43
From the definition of V k , we have It follows form 3.43 and 3.44 that

3.46
We can choose a positive integer N such that

3.48
Then, we can derive from 3.46 and 3.47 that which shows that Γ N is a contraction mapping and therefore there exits a unique fixed point φ * ∈ X of Γ N , which is also the unique fixed point of Γ such that

3.51
Let x k, φ * be the solution of system 2.2 through 0, φ * .From assumption H1 we know that −x k ω, φ * is also a solution of system 2.2 .It follows from 3.51 that −x k ω, φ * is also through 0, φ * .By the uniqueness of solution we can know x k, φ * −x k ω, φ * , 3.52 for k 1, 2, . .., which indicates that x k, φ * is exactly one ω-antiperiodic solution of system 2.2 .To this end, it is easy to see that all other solutions converge exponentially to it as k → ∞.The proof is completed.
Remark 3.2.The conditions are dependent on both the lower bound and upper bound of delays.It has been shown that the delay-dependent stability conditions are generally less conservative than the delay-independent ones, especially when the size of the delay is small.
Remark 3.3.In this paper, the model includes both discrete and distributed delays simultaneously, and can be used to describe some well-known neural networks owing to its generality.
In 38 , only one kind of delay has been considered, which is a special case of neural networks with mixed delays.Furthermore, in 38 , the activations were assumed to be bounded functions, while the boundedness condition is removed in this paper.

Examples
In this section, some examples and numerical simulations are provided to illustrate our results. Example

4.2
Therefore, by Theorem 3.1, we know that system 2.2 with above given parameters has exactly one 8-antiperiodic solution and all other solutions of the system converge exponentially to it as k → ∞, which is further verified by the simulation given in Figure 1.

4.3
It can be verified that assumptions H1 , H2 , and H3 are satisfied with ľ1 ľ2 ľ3 −3, Therefore, by Theorem 3.1, we know that system 2.2 with the above-given parameters has exactly one 12-antiperiodic solution and all other solutions of the system converge exponentially to it as k → ∞, which is further verified by the simulation given in Figure 2.

Conclusions
In this paper, the discrete-time neural networks with mixed delays and impulses have been studied.A delay-dependent LMI criterion for the existence and global exponential stability of antiperiodic solutions has been established by constructing an appropriate Lyapunov-Krasovskii functional, and using the contraction mapping principle and the matrix inequality techniques.Moreover, two examples are given to illustrate the effectiveness of the results.

Example 4 . 2 .
Consider a discrete-time neural networks 2.2 with three neurons, where A