LMI-Based Stability Criterion for Impulsive Delays Markovian Jumping Time-Delays Reaction-Diffusion BAM Neural Networks via Gronwall-Bellman-Type Impulsive Integral Inequality

Lyapunov stability theory, variational methods, Gronwall-Bellman-type inequalities theorem, and linear matrices inequality (LMI) technique are synthetically employed to obtain the LMI-based global stochastic exponential stability criterion for a class of timedelays Laplace diffusion stochastic equations with large impulsive range under Dirichlet boundary value, whose backgrounds of physics and engineering are the impulsive Markovian jumping time-delays reaction-diffusion BAM neural networks. As far as the authors know, it is the first time to derive the LMI-based criterion byway ofGronwall-Bellman-type inequalities, which can be easily and efficiently computed by computer Matlab LMI toolbox. And the obtained criterion improves the allowable upper bounds of impulse against those of some previous related literature. Moreover, a numerical example is presented to illustrate the effectiveness of the proposed methods.


Introduction
In this paper, we consider the stability of a class of time-delays Laplace diffusion stochastic equations with large impulsive range under Dirichlet boundary value, whose backgrounds of physics and engineering are the impulsive Markovian jumping time-delays reaction-diffusion bidirectional associative memory (BAM) neural networks.In 1987, Kosko [1] introduced originally the BAM neural networks model.Owing to its generalization of the single-layer autoassociative Hebbian correlation to two-layer pattern-matched heteroassociative circuits, the BAM neural networks have been proved to have widespread applications in many areas, such as pattern recognition, automatic control, signal and image processing, artificial intelligence, and parallel computation and optimization problems.Generally, an important precondition of the applications mentioned above is that the equilibrium of the BAM neural networks should be stable to some extent.So the stability analysis for neural networks has been attracting wide publicity ( [2][3][4][5][6] and their references therein).

Mathematical Problems in Engineering
Recently, Gronwall inequalities, Gronwall-Bellman-type inequalities, and their applications have attracted abundant interests [16,[20][21][22][23].In this paper, we will synthetically employ Lyapunov stability theory, variational methods, Gronwall-Bellman-type inequalities theorem, and linear matrices inequality (LMI) technique to derive the LMI-based global stochastic exponential stability criterion for Markovian jumping time-delays reaction-diffusion BAM neural networks with large impulsive range allowable.The main purpose of this paper is to improve the allowable upper bounds of impulse.In our new stability criterion, the harsh condition (1) is unnecessary.This paper is organized as follows.In Section 2, the new BAM neural network model is formulated, and some necessary preparation knowledge is provided.In Section 3, we firstly employ variational methods to obtain an inequalities lemma and then use the Lyapunov functional method and Schur Complement technique to deduce a LMI-based exponential stability criterion.In Section 4, an example is provided to illustrate the effectiveness of the proposed methods.In the end, Section 5 contains some conclusions of this paper.
Remark 1.In [24,25], a class of delay differential inequalities ([24, Lemma 2.2] and [25, Lemma 3]) were employed to obtain the stability criteria for deterministic systems.However, a stability criterion of Markovian jumping stochastic system can be obtained via Gronwall-Bellman-type impulsive integral inequality in this paper.To some extent, the restrictive conditions of Gronwall-Bellman-type impulsive integral inequality lemma are simpler than those of the delay differential inequality lemmas ([24, Lemma 2.2] and [25, Lemma 3]), for there are some advantages of the utilization of Gronwall-Bellman-type inequalities (e.g., the allowable upper bounds of time-delays), which will be illustrated in Numerical Example.

Model Description and Preliminaries
In 1987, the bidirectional associative memory (BAM) neural networks were introduced by Kosko (see [1,26]).He set up the following mathematical model [1]: We can know from the above model that it has generalized the single-layer autoassociative Hebbian correlator to twolayer pattern-matched heteroassociative circuits.Neurons are placed in the two layers, and neurons of the same layer are not connected while the neurons of the different layers are connected.There exists the bidirectional information transfer between the two layers of neurons.Such class of networks has wide applications in many fields such as pattern recognition, associative memory, and artificial intelligence.But the important precondition of these applications is that the system should be stable.So Kosko proved the stability of the above model in [1].Since then, the stability analysis of the BAM neural network becomes the most active area of research ( [2][3][4][5][6] and their references therein).For a power system of signal transmission, there inevitably is a time lag problem.And the existence of time-delay often results in unstable phenomenon of a network system.Besides, in the real world, impulsive phenomena exist in the process of changing dynamic behaviors.In order to give an exact description of these process, adoption of delay impulsive differential equations for BAM neural networks is a more effective method (see [3,4,[27][28][29] and their references therein).In addition, the BAM neural networks are often disturbed by environmental noise.The noise may influence the stability of the equilibrium and vary some structure parameters, which usually satisfies the Markov process ( [30,31] and their references therein).And diffusion effect exists really in the BAM neural networks when electrons are moving in asymmetric electromagnetic fields ( [2][3][4][5][6] and their references therein).
Proof.Since  and V is a solution of problem (3)-( 5), it follows by Gauss formula and the Dirichlet boundary condition that Similarly, we can prove that ∫ Ω V    ∇ ⋅ (D ∘ ∇V) ⩽ − 1  d‖V‖ 2 .Thus, the proof is completed.
Proof.Consider the following Lyapunov-Krasovskii functional: Mathematical Problems in Engineering 5 Let  be the weak infinitesimal operator such that In view of we have It follows by the Lipschitz assumption (7) and Lemma 5 that Similarly, we have From the above analysis, Lemma 7, ( 19), ( 20), ( 23), (24), and Schur Complement theorem, we can deduce where Then we can derive that, for  ∈ (  ,  +1 ), where  =  − Hence,

Numerical Example
where From the above data, we know that  = 0.005 and d = 0.005.Consider two modes for the Markovian jumping impulsive system (51).For mode 1, Moreover, the above data into (25) results in which implies that all the assumptions of Theorem 8 are satisfied.And then the equilibrium point  = 0 and V = 0 of system (51) is globally stochastically exponential stability.On the whole, Example 1 illustrates more effectiveness and less conservativeness of our Theorem 8 than other related literatures (see Table 1 for details).
Remark 11.In Example 1, to verify the condition (25), we need to employ comprehensively the computer Matlab LMI toolbox and the method of trial and error.To compare the upper bounds of time-delay and impulse in various related literature, we need to compute and compare the ratios between the maximum of allowable delay (or impulsive) and maximum of parameters in various related literature, because the maximum of allowable delay may rise as parameters of numerical examples become bigger.From Table 1, the ratios of our Example 1 are bigger than those of [24,25] to some extent.And hence, there are some advantages of the utilization of Gronwall-Bellman-type inequality compared to that of other inequalities in [24,25].Moreover, LMIbased criterion in our Theorem 8 is more effective than handcomputation based criteria of [16,36].

Table 1 :
Ratios between allowable upper bounds of time-delay  (or impulse C) and maximum of parameters P.
this paper, Schur Complement technique and Gronwall-Bellman-type impulsive integral inequality are synthetically applied to stability analysis of impulsive delayed Markovian jumping reaction-diffusion bidirectional associative memory (BAM) neural networks, and LMI-based exponential stability criterion is obtained.It is the first time to obtain the LMIbased stability criterion for BAM neural networks via Gronwall-Bellman-type impulsive integral inequality.The main result of this paper improves the allowable upper bounds of impulse in [10, Theorem 3.1], [9, Theorem 1], [15, Theorem 1], [3, Theorem 4.1], [4, Theorem 3.2], and [19, Theorem 3.2].Finally, a numerical example in Section 4 is presented to illustrate the effectiveness of the proposed method (see Remarks 10 and 11).