On the Stability with Respect to H-Manifolds for Cohen–Grossberg-Type Bidirectional Associative Memory Neural Networks with Variable Impulsive Perturbations and Time-Varying Delays

The present paper is devoted to Bidirectional Associative Memory (BAM) Cohen–Grossberg-type impulsive neural networks with time-varying delays. Instead of impulsive discontinuities at fixed moments of time, we consider variable impulsive perturbations. The stability with respect to manifolds notion is introduced for the neural network model under consideration. By means of the Lyapunov function method sufficient conditions that guarantee the stability properties of solutions are established. Two examples are presented to show the validity of the proposed stability criteria.


Introduction
The Cohen-Grossberg-type neural network models were first proposed by Cohen and Grossberg [1] in 1983, and since then an extensive work on this subject has been done by numerous researchers due to the opportunities for applications of such models in key fields of science and engineering such as parallel computing, associative memory, pattern recognition, signal and image processing, etc. [2][3][4].
On the other side, it is well known that BAM neural networks were first proposed by Kosko [5][6][7] and this type of models also has been investigated intensively due to its extension of the single-layer auto-associative Hebbian correlation to two-layer hetero-associative circuits [8].
It is also well known that time delays naturally exist in neural network models, due mainly to the limited speed of signal transmissions and amplifiers switching. Time delays, also known as synaptic transmission delays, may affect the dynamical behaviors and synchronization control of neural networks. That is why numerous researchers considered delay effects on both Cohen-Grossberg and BAM neural networks, and excellent results have been reported in the literature. We will direct the reader to see [9][10][11] for some results on delayed Cohen-Grossberg neural networks, and [8,[12][13][14] for results on BAM neural networks with delays, including some very recent publications [15][16][17][18].
In addition, the hybrid Cohen-Grossberg-type BAM neural networks with delays are an important subject of research and hence, is very well studied by numerous researchers. See, for example, references [19][20][21][22][23] and the references therein. Most of the above cited authors considered time-varying delays in their investigations of such neural networks. Indeed, it is noted that in real-world applications, The remaining part of the paper is arranged as follows. In Section 2, the class of Cohen-Grossberg-type BAM impulsive neural networks with time-varying delays and variable impulsive perturbations is introduced. Some notations, assumptions and definitions are also given. Section 3 is devoted to our main h-global exponential stability result. The proof is performed by using the Lyapunov function technique and differential inequalities. In Section 4, two examples are provided to show the efficacy of the obtained criteria. Finally, some conclusions and open problems are presented in Section 5.

Preliminary Notes
Let R n denotes the n-dimensional Euclidean space endowed with the norm ||x|| = ∑ n i=1 |x i | and R + = [0, ∞). In the case when z = (x, y) T ∈ R n+m , ||z|| = ∑ n i=1 |x i | + ∑ m j=1 |y j |. The goal of this paper is to investigate the qualitative properties, in this case the global exponential stability of the solutions with respect to a manifold defined by a function, for Cohen-Grossberg-type BAM impulsive neural networks with time-varying delays and variable impulsive perturbations of the type: where the model parameters c ji , d ji , p ij , q ij , . . , n, j = 1, 2, . . . , m, System (1) is a generalization of the existing models of impulsive BAM Cohen-Grossberg neural networks with time-varying delays [39][40][41][42][43][44], where x(t) = (x 1 (t), x 2 (t), . . . , x n (t)) T , y(t) = (y 1 (t), y 2 (t), . . . , y m (t)) T , x i (t), i = 1, 2, . . . , n and y j (t), j = 1, 2, . . . , m, are the states of the ith unit and jth unit, respectively, at time t, the functions σ j (t) (0 ≤ σ j (t) ≤ σ j ),σ i (t) (0 ≤σ i (t) ≤σ i ) correspond to the transmission time delays at time t, c ji , p ij , d ji and q ij represent the connection weights, f j ,f i , g j andĝ i are the signal functions, I i and J j are the external inputs, a i andâ j represent the amplification functions, b i andb j represent the appropriately behaved functions, such that all solutions of (1) remain bounded.
Different from the existing models [39][40][41][42][43][44], we consider variable impulsive perturbations in (1) such that P ik and Q jk represent the abrupt changes of the states at the impulsive moments, are, respectively, the states of the ith unit from the first neural field and the jth unit from the second neural field before and after an impulsive perturbation at the moment t. Note that the abrupt changes P ik (x i (t)) = ∆x i (t) = x i (t + ) − x i (t) and Q jk (y j (t)) = ∆y j (t) = y j (t + ) − y j (t) can be considered as impulsive controls [31][32][33][34][35][36][37][38].
Let J be an interval, J ⊂ R + , and define the following class of piecewise continuous functions PC[J, R n ] = {s : J → R n : s(t) is piecewise continuous on J with points of discontinuity t k ∈ J at which s(t − k ) and s(t + k ) exist and s(t − k ) = s(t k )}.
For a t 0 ∈ R + we denote by z(t) = (x(t), y(t)) T = (x(t; t 0 , ϕ 0 ), y(t; t 0 , φ 0 )) T , x ∈ R n , y ∈ R m the solution of model (1) which satisfies initial conditions of the type: As usual, the solution z(t) = (x(t), y(t)) T of the problem (1), (2) is [29,[45][46][47][48][49] a function from the class PC[J, R n+m ], i.e., at the moments t l k when the integral curve of the solution (x(t), y(t)) meets the hypersurfaces we have: The points t l 1 , t l 2 , . . . (t 0 < t l 1 < t l 2 < . . . ) are called impulsive moments at which impulsive control techniques can be applied [34][35][36][37][38]. Note that, in general, the number k of the hypersurface θ k may not be equal to the number l k of the impulsive moment t l k . Furthermore, different solutions may have different impulsive moments.
Denote by ν = max{σ,σ}, To eliminate any opportunity of "beating" of solutions, and to assurance existence, uniqueness and continuability of the solution z(t) = z(t; t 0 , ψ 0 ) of the initial value problem (IVP) (1), (2), on the interval [t 0 , ∞) for ψ 0 ∈ P CB and t 0 ∈ R + we assume that: 1. τ 0 (x, y) ≡ t 0 for x ∈ R n , y ∈ R m , the functions τ k (x, y) are continuous, and the following relations hold: The next hypotheses will be very important in the proofs of our main results: Hypothesis 1. The functions a i ,â j , i = 1, 2, . . . , n, j = 1, 2, . . . , m are bounded and there exist positive constants a i ,â j such that a i ≤ a i (t) ≤ a i andâ j ≤ a j (t) ≤â j for t ∈ R. Hypothesis 2. For the functions b i andb j there exist positive constants B i ,B j respectively, such that for any χ 1 , χ 2 ∈ R, χ 1 = χ 2 and i = 1, 2, . . . , n, j = 1, 2, . . . , m.
Introduce the following sets In our investigations, we will use the Lyapunov-Razumikhin approach, which for impulsive systems requires a definition of Lyapunov's like piecewise continuous functions [29]. Definition 2. We will say that a function V : R + × R n+m → R + , V = V(t, x, y) = V(t, z), belongs to the class V 0 if: 1. The function V is continuous in G and locally Lipschitz continuous with respect to (x, y) on each of the sets G k , k = 1, 2, . . . .

H-Stability Results
We will now derive our main h−stability results for the equilibrium state of the model (1).
Then the equilibrium z * of the Cohen-Grosberg-type BAM impulsive delayed neural network system (1) is globally exponentially stable with respect to the function h.
The last estimate implies the global exponential stability of the equilibrium state z * of (1) with respect to the function h.

Remark 1.
The concept of stability with respect to manifolds defined by a particular function h generalizes numerous stability notions. Hence, Theorem 1 can be applied to a number of concrete situations depending on the choice of the norm ||z|| and the Lyapunov function V(t, z). Some of the most applicable cases are when the function h is where z * is an arbitrary nontrivial solution of (1) (an equilibrium, periodic solution, almost periodic solution, etc.); where A ⊂ R n+m and d is the distance function. Therefore, the proposed stability result extends and generalizes the existing stability results for Cohen-Grossberg-type BAM impulsive neural networks with time-varying delays.
Remark 2. The stability criteria provided by Theorem 1 also generalize the results in [39][40][41][42][43][44] considering variable impulsive perturbations and h−manifolds. It is worth noting that, in the case when h(t, z) = z − z * , the impulsive moments of both solutions z(t) and z * (t) can be different which is not considered in [39][40][41][42][43][44]. However, considering impulsive perturbations at variable time in impulsive neural network models is more natural and realistic, and, therefore, the new results offer an extended horizon for applications. Observe also that, if the impulsive events are realized at fixed times or when τ k (x, y) = t k , k = 1, 2, . . . , and the function h(t, z) = z, then the exponential stability criteria in [39][40][41][42][43][44] can be obtained as corollaries from our result.

Illustrative Examples
In this section, we will demonstrate the validity of the obtained in Theorem 1 criteria for global exponential stability with respect to manifolds. Example 1. Consider the following Cohen-Grossberg-type BAM impulsive neural networks with time-varying delays with impulsive perturbations of the type x(t), t = τ k (x(t), y(t)), k = 1, 2, . . . , where t > 0, We have that all assumptions of Theorem 1 are satisfied for We can verify that condition 2 of Theorem 1 is satisfied for 0 < µ ≤ 0.2.

Conclusions
In this paper, the important notion of global exponential stability of single solutions of Cohen-Grossberg BAM impulsive neural networks with time-varying delays is extended and generalized. We introduce the concept of stability with respect to a manifold defined by a function h with specific properties. Thus, our research generalize some existing results in the literature on global exponential stability of solutions of impulsive BAM Cohen-Grossberg neural networks. In addition, instead of impulsive effects at fixed moments of time, we consider variable impulsive perturbations. The proposed notion and the results obtained in the paper can be extended to various other types of impulsive neural network models.