Global Stability of Integral Manifolds for Reaction–Diffusion Delayed Neural Networks of Cohen–Grossberg-Type under Variable Impulsive Perturbations

: The present paper introduces the concept of integral manifolds for a class of delayed impulsive neural networks of Cohen–Grossberg-type with reaction–diffusion terms. We establish new existence and boundedness results for general types of integral manifolds with respect to the system under consideration. Based on the Lyapunov functions technique and Poincar`e-type inequality some new global stability criteria are also proposed in our research. In addition, we consider the case when the impulsive jumps are not realized at ﬁxed instants. Instead, we investigate a system under variable impulsive perturbations. Finally, examples are given to demonstrate the efﬁciency and applicability of the obtained results.


Introduction
The neural network models of Cohen-Grossberg type have been initially introduced in 1983 [1]. Since the above pioneered publication, the theory and applications of the Cohen-Grossberg neural networks (CGNNs) have been developed in numerous research papers [2][3][4]. In addition, delayed CGNNs have been massively investigated due to their enormous opportunities of applications in diverse areas of science and engineering [5][6][7][8][9][10].
Moreover, many researchers considered the effect of reaction-diffusion terms on the dynamic behavior of neural networks [11][12][13][14][15]. Indeed, diffusion effects are essential in modelling and scientific understanding of natural and artificial neural networks. The properties of the hybrid type of CGNNs with reaction-diffusion terms have been also widely investigated in the existing literature [16][17][18][19].
In addition, the effect of various types of impulsive perturbations such as those at fixed moments of time, at variable times or delayed impulsive perturbations has been found to be remarkably important in the behavior and control of numerous systems. That is why impulsive differential equations and impulsive control systems are intensively used as tools in modelling of processes studied in widespread areas of the mathematical, physical, chemical, engineering, and statistical sciences [20][21][22][23][24][25][26][27][28][29].
In this paper we will introduce the integral manifolds approach to the following CGNN model with reaction-diffusion terms and time-varying delays under variable impulsive perturbations x)), i = 1, 2, . . . , m, k = 1, 2, . . . , ∆u i (t, x) = J ik (u i (t, x)), t = τ k (u i (t, x)), i = 1, 2, . . . , m, k = 1, 2, . . . , where: a m, m ≥ 2 is the number of neurons in the GCNN model, t > 0, x = (x 1 , x 2 , . . . , x n ) T ∈ Θ, u i (t, x) denotes the state of the i-th neuron at time t and in space x, u(t, x) = (u 1 (t, x), u 2 (t, x), . . . , u m (t, x)) T ∈ R m ; b s j (t) denote the time-varying delays of the i-th neural unit, the functions s j are continuous and t > s j , j = 1, . . . , m, 0 ≤ s j (t) ≤ ν, ds j (t) dt < δ j (ν > 0, δ j < 1); c a i (u i (t, x)) ≥ 0, i = 1, 2, . . . , m, are the amplification functions and are continuous on their domains; d b i (u i (t, x)) are appropriately behaved continuous functions with real values, i = 1, 2, . . . , m; e c ij (t) and w ij (t) are the connection weight and time-varying delay connection weight of j-th neural unit on the i-th neural unit at time t, respectively, and are continuous real-valued functions; f f j (u j (t, x)) and g j u j (t − s j (t), x) , j = 1, . . . , m are the activation function and time-varying delay activation function of the j-th neuron, respectively, and for any j = 1, 2, . . . , m the functions f j , g j are continuous with real values; g D iq = D iq (t, x) ≥ 0, q = 1, 2, . . . , n, i = 1, 2, . . . , m are the diffusion coefficients along the q-th coordinate for i-th neural unit and are continuous functions for any i = 1, 2, . . . , m and q = 1, 2, . . . , n. h J ik (u i (t, x)), i = 1, 2, . . . , m, k = 1, 2, . . . , are the real-valued functions that characterize the weights of the impulsive perturbations on the i-th nodes at the at the variable times for is the state of the i-th neuron before the jump perturbation at t = τ k (u i (t, x)) and u i (t + , x) is the impulsively controlled output of the i-th unit.
Denote by u(t, x) = u(t, x; ϕ 0 ) the solution of the delayed impulsive reaction-diffusion CGNN model (1) under the following initial and boundary conditions: According to the theory [27,31,49], the solution (1) with variable impulsive perturbations is such that at the moments t l k when the integral surface of u(t, x) meets the hypersurfaces the following is true The above points t l 1 , t l 2 , . . . are the impulsive moments. It is also well known [27,31,49] that, in general, k = l k . In addition, due to the nature of the variable impulsive perturbations, different nodes u i (t, x), i = 1, 2, . . . , m may have different impulsive moments. In this paper, we will investigate such nodes the motion along with which is established by a suitable choice of the impulsive forces. That is why we will assume that, τ 0 (u i ) ≡ t 0 = 0 for u i ∈ R, i = 1, 2, . . . , m, all functions τ k (u i ), i = 1, 2, . . . , m, k = 1, 2, . . . are continuous and Further, we will use the following classes of functions: The class of all nonnegative continuous functions defined on R + that are strictly increasing and are zeros at zero will be denoted by K; The class of all functionsσ : R × Θ → R m that are continuous everywhere on their domains except at points of the type (t l k , P C will denote the class of all piecewise continuous functions ϕ = (ϕ 1 , ϕ 2 , . . . , for all points (s, x) ∈ [−ν, 0] × Θ which must be finite number; (iv) PCB will denote the class of all functions ϕ ∈ P C that are bounded on [−ν, 0] × Θ.
In this paper we will apply the method of integral manifolds for the delayed reaction-diffusion impulsive CGNN model (1). To this end, we will adopt the following definition for an integral manifold related to (1) [41][42][43][44][45][46][47][48]. Definition 1. We will say that a manifold M in the extended phase space For a manifold M ⊂ [−ν, ∞) × Θ × R m we introduce the following sets and distances: The set of all u ∈ R m such that (t, An ε-neighborhood of M(t, x) is denoted by M(t, x, ε) and is defined by The distance between a function ϕ ∈ P C and M 0 (s, x) is defined as We will also use the closures Throughout this paper, we will assume that: A1. For the nonnegative continuous functions a i there exist constants a i and a i such that The continuous functions f j and g j are bounded, there exist positive constants L j , for all χ 1 , χ 2 ∈ R, χ 1 = χ 2 , and f j (0) = g j (0) = 0 for any j = 1, 2, . . . , m. A4. For the continuous functions D iq there exist constants d iq ≥ 0 such that for any i = 1, 2, . . . , m and q = 1, 2, . . . , n.
Next, the following boundedness and stability definitions for an integral manifold M ⊂ [−ν, ∞) × Θ × R m are introduced. They generalize and extend the known boundedness and stability definitions for impulsive delayed CGNNs with reaction-diffusion terms used in [34,35,37,39,40] to the integral manifolds case. Definition 2. We will say that an integral manifold M of system (1) is: Definition 3. We will say that an integral manifold M of system (1) is said to be: (c) uniformly globally asymptotically stable, if it is a uniformly stable, uniformly bounded, and It is clear that, in particular cases, Definitions 2 and 3 can be used to investigate the behavior of single solutions, such as, zero states, equilibria, periodic solutions, etc. In such cases, the integral manifolds contain only the corresponding solutions. Note that such a generalization of the concepts for systems with variable impulsive perturbations is not trivial, since for such systems [31,[49][50][51][52], different states may have different impulsive moments.
In addition, the introduced definitions extend the opportunities for applications of the integral manifolds methods [41][42][43][44][45][46][47][48] to specific systems studied in numerous areas of science and technologies, such as impulsive CGNNs with reaction diffusion terms and time-varying delays.
Next Poincarè-type integral inequality [54] for the set Θ = ∏ n q=1 [a q , b q ], a q = const ∈ R, b q = const ∈ R, q = 1, 2, . . . , n will be applied in the proofs of our main results. Lemma 1. [53,54] For any real-valued function v(x) that belongs to C 1 (Θ) the following relation is valid Some generalizations of Lemma 1 also exist in the literature. See, for example, [16]. Finally, some basic notations and results from the method of piecewise continuous Lyapunov-type functions [6,11,13,15,27,28,31,35] are in order.
Define the sets We will use Lyapunov-type functions from the class (1), (2), (3). In fact, each of the points t k is a solution of some of the equations t = τ k (u(t, x)), t ≥ 0, x ∈ Θ, k = 1, 2, . . . , i.e., t k are the impulsive points at which the integral curve (t, u(t, x; ϕ 0 )) of the initial value boundary problem (1), (2) (3) meets each of the hypersurfaces σ k , k = 1, 2, . . . . Now, for a given function V ∈ V M , t ∈ R + , t = t k , k = 1, 2, . . . andφ ∈ P C define the following derivative with respect to system (1) We will use the following key lemma.

Existence and Boundedness Results
In this section, existence and boundedness of an integral manifold with respect to system (1) will be investigated.
Next is the boundedness result.

Theorem 2. Under the conditions of Theorem 1 the integral manifold M of model (1) is uniformly bounded.
Proof. For the integral manifold M, consider the Lyapunov-type function (5). We have that V ∈ V M and there exist v 1 , v 2 ∈ K such that Let η > 0 be chosen. It follows from v 1 , v 2 ∈ K that the number b = b(η) > 0 can be chosen so Now, we suppose that u(t, x) = u(t, x; ϕ 0 ) is the solution of the problem (1), (2) and (3) with initial function ϕ 0 ∈ S α (PC 0 ) ∩ M 0 (s, x, η) for α > 0. From (18) and (16) Therefore, u(t, x; ϕ 0 ) ∈ M(t, x, b) for t ∈ R + and the proof is completed.

Integral Manifolds Stability Analysis
In the next we will use measurable functions of the type λ : for t = τ k (u), k = 1, 2, . . . , then the integral manifold M of the impulsive reaction-diffusion delayed CGNN (1) is uniformly globally asymptotically stable.
Proof. The uniform boundedness of the integral manifold M follows from Theorem 2. Now, we will prove that M is uniformly stable and globally attractive. Consider the uniform stability. For the function V ∈ V M from Theorem 1, the inequalities (18) are true for functions v 1 , v 2 ∈ K.
If (21) is not true for at least one t * ∈ [0, T], then for any (t, x) ∈ [0, T] × Θ. In this case, by (6), (20) and (22), for any t ∈ [0, T] it follows that Furthermore, from the integral positivity of the function λ(t) we can choose the number T so that Now, for t = T, from (23) and the uniform boundedness of the integral manifold M, we have which is a contradiction. Therefore, there exists a t * ∈ [0, T], such that the inequality (21) is satisfied. Now, (6), (20) and the fact that V is nonincreasing along the solution u(t, x; ϕ)) of (1) imply that for t ≥ t * we have The above estimates are true for t ≥ T as well, and hence, u(t, x) ∈ M(t, x, ), t ≥ T, which shows that the integral manifold M of (1) is globally attractive.

Theorem 4.
If the conditions of Theorem 3 are met and there exists a constant λ 1 such that λ(t) ≥ λ 1 > 0, t ∈ R + , then the integral manifold M of (1) is globally exponentially stable.
Proof. Let ϕ 0 ∈ PCB and u(t, x; ϕ 0 ) be the solution of the initial value boundary problem (1), (2), (3). By Theorem 3, for the Lyapunov function V ∈ V M , we get (6) and (20). Then, by Lemma 2,and λ(t) Next, for the Lyapunov function V ∈ V M defined by (5), we have for a constant k 1 > 1 2 > 0. Then from the choice of V ∈ V M , conditions of Theorem 4, (24) and (25), we obtain which proves the global exponential stability of the integral manifold M.
The results in theorems 3 and 4 extend and generalize the existing stability results for single solutions for impulsive reaction-diffusion CGNNs [34,35,[37][38][39][40] to the integral manifolds case. The new stability results are obtained for the set Θ = ∏ n q=1 [a q , b q ], where a q , b q ∈ R, 0 ∈ Θ, and can be easily applied to the most studied particular case, when the set Θ of points x, x = (x 1 , x 2 , ..., x n ) T is such that |x q | < l q , l q > 0, q = 1, 2 . . . , n.
where u C = (u C 1 , u C 2 ) T is a constant solution of the model (26). The existence of an equilibrium u C of the impulsive delayed reaction-diffusion CGNN model (26) is guaranteed by conditions A1-A7 and the assumptions on the impulsive functions and hypersurfaces.
Finally, we have that condition (4) of Theorem 1 holds for Therefore, according to Theorem 1, the manifold M defined by (27) is an integral manifold of (26), and by Theorem 2, we conclude that it is uniformly bounded. The global exponential stability of the integral manifold M follows from Theorem 4 for λ 1 such that 0 < λ 1 ≤ 1.3.
In the above example, since by means of the impulsive control the stability properties of the system without impulsive perturbations are preserved. If the impulsive functions do not satisfy condition A7, then due to the impulsive jumps the stable neuronal behavior can be changed momentarily. Thus, our results offer an insight on the effects of impulsive stability and control strategies on the interactions of neurons.

Remark 1.
In the last example we extend the image encryption scheme proposed by [55] considering Cohen-Grossberg type reaction-diffusion delayed neural network and variable impulsive perturbations. The functions b i play the role of feedback gains~in the synchronization mechanism. Thus we again demonstrate the great opportunities for applications of our results.

Conclusions
In this paper, the integral manifolds technique is applied to propose boundedness and stability criteria for a class of impulsive delayed reaction-diffusion CGNNs. The proposed results complement and extend some existing qualitative results for such models [34,35,[37][38][39][40]. The consideration of variable impulsive perturbations, as well as, the use of a Poincarè-type integral inequality additionally increase the degree of generality. Two examples are provided to illustrate the proposed integral manifold method. The demonstrated integral manifold approach can be extended in the investigation of different classes of neural network and related systems. Our future studies will be focused on the consideration of systems with distributed delays and non-instantaneous impulses based on this study. Considering the case of anti-diffusion is also an important and interesting topic for future investigations.