Global Exponential Stability and Periodicity of Nonautonomous Impulsive Neural Networks with Time-Varying Delays and Reaction-Diffusion Terms

In this paper, we investigate the global exponential stability and periodicity of nonautonomous cellular neural networks with reaction-diﬀusion, impulses, and time-varying delays. By establishing a new diﬀerential inequality for nonautonomous systems, using the properties of M-matrix and inequality techniques, some new suﬃcient conditions for the global exponential stability of the system are obtained. Moreover, suﬃcient conditions for the periodic solutions of the system are obtained by using the Poincare mapping and the ﬁxed point theory. The validity and superiority of the main results are veriﬁed by numerical examples and simulations.


Introduction
Since Chua proposed the cellular neural networks (CNNs) in the 1980s [1], the neural network models have been widely studied and applied in the fields of signal recognition, image processing, pattern classification, and so on. Indeed, all these applications rely on the dynamic behaviors of neural networks.
e key to using neural networks to solve these problems is that the neural network must be globally dynamic stable; that is to say, each of its loci must converge to a unique balance. In the practical application of neural networks, time delay is inevitable because of the limited signal transmission and switching speed. As we all know, the time delay may make the system lose stability or produce nonlinear oscillation, which makes the dynamic behaviors of the system more complicated. Under the circumstances, the neural network model with time delay is introduced to better reflect the actual state. Hence, how to ensure the stability and good performance of the system is particularly important in the case of time delay. e dynamic analysis of time-delay neural networks has received considerable attention; for example, see [2][3][4][5][6][7].
In practical applications, neurons change over time and space. In order to better describe the state of neurons, a partial differential equation that considers diffusion is introduced in space. So, in 1995, Chua et al. [8] established the structure of reaction-diffusion CNNs. In [9,10], the authors have investigated the global exponential stability and the global exponential robust stability of reactiondiffusion neural networks with time-varying delays, by using partial differential equations to express the diffusion term of the CNNs, some interesting results were obtained. In [11], the authors used the properties and inequality analysis of the nonsingular M-matrix to study the exponential stability of CNNs with diffusion. In [12], considering the global exponential stability under the Neumann boundary conditions, using the properties of the M-matrix and algebraic inequality techniques, some sufficient conditions for the global exponential stability of the system are obtained. In [13], under Neumann boundary conditions, several sufficient conditions based on linear matrix inequalities for the global asymptotic stability of reactordiffusion CNNs with distributed delay are given. In [14,15], the Dirichlet boundary conditions of the rectangular region are introduced into the neural network with reactiondiffusion. In [16], under the Dirichlet boundary conditions, a diffusive-dependent Lyapunov functional related to the diffusion term is proposed to analyze the effect of the diffusion coefficient of neurons on the dynamics of the model, and the stabilization effect of the reaction-diffusion term is studied by using the extended Wirtinger inequality. In [17,18], the authors considered the exponential stability of reaction-diffusion neural networks with time delays in general spatial regions and obtained some new sufficient conditions for system stability by using Poincare's inequality technique.
As we all know, the phenomenon of nonautonomy arises in a lot of real systems frequently. In particular, when a system is in a dynamic behavior for a long time, the parameters of the system often change with time. Compared with autonomous CNNs, the study of nonautonomous CNNs is equally important. In fact, many authors have explored the dynamic behavior of different types of nonautonomous CNNs with time delays and produced some interesting results by constructing some Lyapunov-Krasovskii functionals [19][20][21][22][23][24][25][26]. In addition, some functions in these methods must be differentiable, while other features require additional constraints in terms of delay. In [27], the author improved some of the earlier Halanay-type inequalities and obtained the exponential stability of CNNs and the existence of periodic solutions through these new inequalities. In [28], sufficient conditions for the global exponential stability of a special class of nonautonomous differential systems with infinite time delay are given, without proving the existence of equilibrium point, periodic solution, or almost periodic solution. In addition, during the physical simulation of the neural network, the neural network will sometimes be interrupted due to the transient disturbance, which is called the impulse effect. Impulse may cause system instability or other more complex dynamic behavior. e influence of pulses on neural networks has been considered in many literatures [29][30][31][32][33]. erefore, it is of great significance to study the stability and periodicity of neural network models with dynamic impulses, time delays, and reaction-diffusion terms.
Here, we introduce a class of new nonautonomous impulsive neural networks with time-varying delays and reaction-diffusion terms as follows: where z p (t, x) is the state of the pth neuron in space x and time t.
. , m} is a bounded compact set with smooth boundary zΞ and mes x)) T ∈ R n . β p ∈ C(R + , R + ) denotes the rate at which the pth neuron resets its potential to the isolated resting state when disconnected from the network and external input; a pq ∈ C(R + , R) denotes the intensity of the pth element on the qth element at time t; b pq ∈ C(R + , R) represents the strength of the qth element due to delay at the pth element. τ pq (t) represents the transmission delay from the pth element to the qth element and satisfies 0 ≤ τ pq (t) ≤ τ pq ≤ τ(τ � max 1≤p,q≤n τ pq ); f q ∈ C(R, R) and g q ∈ C(R, R) denote the activation function and activation function caused by delay, respectively; J p (t) represents the external input of the pth neuron at time t. In (1b), t l > 0 satisfies t l < t l+1 , lim l⟶+∞ t l � +∞; z p (t − l , x) and z p (t + l , x) represent the left and right limits at t k , respectively; I pk shows impulsive perturbation of the pth neuron at time t k . Let z p (t + l , x) � z p (t l , x), l ∈ N. Equation (1c) denotes the Dirichlet boundary conditions and (1d) denotes the initial conditions.
e main contributions of this work are as follows: (I) We have formulated a class of new neural network models which assembles nonautonomous neural networks, reaction-diffusion cellular neural networks with time-varying delays, impulses, and the Dirichlet boundary conditions. (II) We have obtained several new criteria which guarantee the global exponential stability and periodicity of systems (1a)-(1d). ese criteria are shown in simple algebraic inequality forms, and they rely only on the system parameters. (III) An optimization method is proposed in order to estimate the exponential convergence rate of systems (1a)-(1d). e remainder of this paper is organized as follows. In Section 2, some assumptions and the necessary knowledge are provided. We give our main results in Section 3. In Section 4, two examples are given to show the effectiveness of the results obtained here. Finally, we give the conclusion in Section 5.

Preliminaries
In this section, we make some assumptions and introduce some notions.
(A2) ere exists a positive diagonal matrix I l � diag(I 1l , . . . , I nl ) such that For C, D ∈ R m×n , we define the Schur product by C ⊗ D � (c pq d pq ) m×n , and C ≥ D(C > D) means that the inequality c pq ≥ b pq (c pq > d pq ) holds. e � (1, 1, . . . , 1) T ∈ R n and E is a n-dimensional identity matrix.
x) satisfies the following conditions: (1) For t, z(t, x) is piecewise continuous with the first kind of discontinuity at the points t l , l ∈ N, and z(t, x) is right-continuous at every discontinuity point is a particular solution that satisfies the initial conditions for systems (1a)-(1d), and z(t, μ, x) represents the special solution of systems (1a)-(1d) under initial condition μ ∈ PC(Ξ).
Definition 3 (see [34]). A real matrix D � (d pq ) n×n is said to be a nonsingular M-matrix if d pq ≤ 0, p, q � 1, 2, . . . , n, p ≠ q, and all successive principal minors of D are positive.
From (17), we have that is, It is clear that this inequality and (16) contradict each other, and this implies that inequality (15) holds for all t ∈ [a, b).
Letting ε ⟶ 0, we obtain that e proof is completed.

Main Results
In this part, we establish a series of sufficient conditions to ensure the global exponential stability of systems (1a)-(1d). where where c l � max 1≤p≤n 1, I pl , l ∈ N; then, systems (1a)-(1d) are globally exponentially stable.
When we multiply both sides of (11) by Z p (t, ψ, x) and integrate them, we can get From the Dirichlet boundary conditions and Greens formula, we can obtain By Lemma 2, we have From (A1)-(A3) and Hoder inequality, we can get n q�1 a pq (t) In a similar way, we have
□ Complexity Remark 2. In eorem 1, the condition (C2) (c � sup l∈N lnc l /(t l − t l− 1 ) ) describes the influence of the impulsive intensity and the impulsive interval on the global exponential stability of systems (1a)-(1d). In the absence of impulses, the following optimization problem can be solved in order to estimate the exponential convergence rate of systems (1a)-(1d): Obviously, λ is associated with all time delays, the diffusion coefficients, the Dirichlet boundary conditions, and system parameters. When c ∈ [0, λ), eorem 1 shows that systems (1a)-(1d) are globally exponentially stable and its exponential convergence rate equals λ − c. About the exponential convergence rate λ, we have the following result.
Proof. By Lemma 3, λ * exists and λ * > 0. Let λ � λ * − ε for all ε ∈ (0, λ * ), then for all t ≥ 0. at is, for all t ≥ 0. erefore, max λ � λ * . e proof is completed. □ Remark 3. Some existing neural network models become special cases of systems (2a)-(2c) formulated in this paper; for example, see [15,16]. It is not hard to find out that all criteria in [15] are improved. Comparing with LMIs method in [16], our results are more concise, and these results do require the differentiability on time-varying delays, so our method has different advantages and disadvantages with the method in [16]. In addition, although those models in [37,38] are different from systems (2a)-(2c) because of the difference in reaction-diffusion terms and boundary conditions, all criteria in [37] are independent of diffusion effect. In [38], the authors had, respectively, obtained global exponential stability criteria of nonautonomous neural networks with time-varying delays and reaction-diffusion terms under the Neumann boundary conditions and the Dirichlet boundary conditions. However, our approach is similar to the one in [38] when the Dirichlet boundary conditions are considered.
Because of the uniform continuity, there exists a real number λ > 0 such that and this shows that condition (C1) holds. Reversely, let λ � 0 in (C1); it is easy to obtain that Δ(t) is a nonsingular M-matrix for t ≥ 0. Based on the above discussion, we have the following conclusions.
Proof. Note that ζ(λ) ≥ ζ(λ), it follows that is shows that (C1 ″ ) ensures (C1). In this part, we consider the case of the periodic solution of systems (1a)-(1d). Let us make the following assumptions: For p, q ∈ 1, 2, . . . , n { }, β p (t), a pq (t), b pq (t), τ pq (t), and J p (t) are periodic continuous functions and has a common period Ω > 0 for all t ≥ 0. (A5) For I l � diag(I 1l , . . . , I nl ) and the impulsive moments t l l∈N , there exists an integer r > 0 such that For the existence and the global exponential stability of periodic solutions for periodic systems (1a)-(1d), we have the following results.
Proof. On the basis of eorem 1, there exists a integer K > 0 such that Me − (λ− c)KΩ ≤ (1/2), and we define a Poincare mapping as follows: and it follows that Γ K (ω) � z KΩ (ω, x). Setting t � KΩ, we get Obviously, Γ K is a contraction mapping, and then there exists a unique fixed point ω * ∈ PC(Ξ) such that And then, we have and this implies that Γ(ω * ) ∈ PC(Ξ) is also a fixed point of Γ K . So, we have

Conclusion
We have formulated and investigated a class of new neural network models which assembles nonautonomous neural networks, reaction-diffusion cellular neural networks with time-varying delays, impulses, and the Dirichlet boundary conditions. Several new sufficient conditions have been obtained to ensure the global exponential stability and periodicity of systems (1a)-(1d), and these criteria are shown in simple algebraic inequality forms. In particular, an optimization method is proposed in order to estimate the exponential convergence rate of systems (1a)-(1d), and this method depends on the diffusion coefficient, the Dirichlet boundary conditions, the delays, system parameters, and impulses. Comparing with the method of Lyapunov functional in many previous publications, our method is simpler and more effective for stability and periodicity analysis of nonautonomous impulsive neural networks with timevarying delays and reaction-diffusion terms. Two examples have shown that our results improve and generalize previously known criteria. In near future, we will continue to study global exponential stability and periodicity of nonautonomous impulsive neural networks with distributed delays or leakage delays under the Dirichlet boundary conditions.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.