Stability Analysis of Pseudo-Almost Periodic Solution for a Class of Cellular Neural Network with D Operator and Time-Varying Delays

Cellular neural networks with D operator and time-varying delays are found to be effective in demonstrating complex dynamic behaviors. The stability analysis of the pseudo-almost periodic solution for a novel neural network of this kind is considered in this work. A generalized class neural networks model, combining cellular neural networks and the shunting inhibitory neural networks with D operator and time-varying delays is constructed. Based on the fixed-point theory and the exponential dichotomy of linear equations, the existence and uniqueness of pseudo-almost periodic solutions are investigated. Through a suitable variable transformation, the globally exponentially stable sufficient condition of the cellular neural network is examined. Compared with previous studies on the stability of periodic solutions, the global exponential stability analysis for this work avoids constructing the complex Lyapunov functional. Therefore, the stability criteria of the pseudo-almost periodic solution for cellular neural networks in this paper are more precise and less conservative. Finally, an example is presented to illustrate the feasibility and effectiveness of our obtained theoretical results.


Introduction
In recent years, the cellular neural networks (CNN), first proposed by Chua and Yang [1,2], have received significant attention because of their wide applications in science and engineering technology fields. Extensive research has been conducted on CNN in the past few decades, and one of the primary problems in designing CNN is to deal with the dynamic curves of existing solutions. Regarding the existence, uniqueness, and stability of the periodic, almost periodic solutions of neural networks, there have been many results in the fields of classification, signal processing [3], associative memory [4,5], optimal control [6], and filter problem [7][8][9]. The state estimation problem of delayed static neural networks has been considered by Wang and Xia et al. [10]. Donkers et al. showed stability analysis results of networked control systems employing a switched linear systems approach [11]. Liang and Wang et al. [12], mainly examined the robust synchronization issue for two-dimension discrete-time coupled dynamical neural networks. Due to the limited bandwidth speed and the constraints of physical property for circuit equipment, information propagation inevitably give birth to the emergence of time delays, meanwhile arising other problems of the CNN. The appearance of time lags usually causes turbulence, instability, and chaos [13,14]. The stability of discrete-time systems with time-varying delays via a novel summation inequality was discussed in [15,16]. For the reason that information transmission between neurons has time delays behavior, the neural network model with delay described by the time delays functional differential equation has been widely examined and implemented in various fields. supplement some of the previous research work. The r-neighborhood of a cell x ij defines as follows: where r ≥ 2 is a positive integer number. Since each cell of CNN is only connected to its neighboring areas, those cells that are not directly connected may be affected by continuous-time propagation effects and indirectly affect each other. Furthermore, due to the neural network's increasing distance, the influence between different neurons in CNN is correspondingly weakened. Several previous works are the motivation for us to propose the new r-neighborhood N 0 r (i, j), we examined the novel types of cellular neural networks with D operator and time-varying delays as follows: x ij (t) corresponds to the state of the ij − th cell (at the (i, j) position of the lattice) at time t, a ij (t) > 0 represents the rates with which the ij − th cell will reset its potential to the resting state in isolation when disconnected from the networks and external inputs at time t; f ij (·), g ij (·) denote the activation functions of signal transmission. p ij (t), b ij (t), c ij (t) denote the connection weights at time t, δ ij (t) ≥ 0, τ ij (t) ≥ 0 corresponding to the transmission delays, and L ij (t) are the external inputs on the ij − th cell at time t.
The initial conditions of the system (1) are assumed to be where φ ij (s) is a continuous function, The paper is organized as follows. In Section 2, we will introduce some definitions and lemmas, which will be used to obtain our results. In Section 3, we state and demonstrate the existence and global exponential stability of the pseudo-almost periodic solution. In Section 4, an example illustrates the feasibility and effectiveness of obtained theoretical results. In Section 5, a brief conclusion is given.

Preliminaries
In this section, we recall briefly some basic definitions and properties of pseudo-almost periodic functions and the exponential dichotomy. Definition 1. Let u(·) ∈ BC(R, R n ). u(·) is said to be (Bohr) almost periodic on R n , if for any is relatively dense, i.e., for any ε > 0, it is possible to find a real number l = l(ε) > 0 with the property that for any interval with length l(ε), there exists a number δ = δ(ε) in this interval such that (Bohr) almost periodic in t ∈ R uniformly in x ∈ K, where K is any bounded compact subset of R n , that is, if for each ε > 0, there exists l(ε) > 0 such that every interval of length l(ε) > 0 contains a number τ with the following property sup We denote by AP(t ∈ R × R n , R n ) the set of the almost periodic functions from R × R n to R n . Additionally, we define a class function as follows: which is a closed subspace of BC(R × R n ; R n ).

Definition 3.
A continuous function f ∈ BC(R × R n ; R n ) is called pseudo-almost periodic if it can be expressed as where f 1 ∈ AP(R × R n ; R n ) and f 2 ∈ PAP 0 (R × R n ; R n ). The collection of such functions is denoted PAP(R × R n ; R n ).
Definition 4. Let x ∈ R l and Q(t) be a l × l continuous matrix defined on R. The linear system is said to admit an exponential dichotomy on R l if exist positive constants k, λ > 0 and projection P and the fundamental solution matrix X(t) of (4) satisfying admits an exponential dichotomy on R l . Lemma 2. If the linear system x (t) = Q(t)x(t) has an exponential dichotomy, then almost periodic system has a unique pseudo-almost periodic solution x(t) which can be expressed as followings: Definition 5. Let x(t) = (x 11 , · · ·, x 1n , · · ·, x m1 , · · ·, x mn ) T be a continuous differentiable pseudoalmost periodic solution of system (1) with the initial value ψ(s) = (ψ 11 , · · ·, ψ 1n , · · ·, ψ m1 , · · ·, ψ mn ) T .

Main Results
In this section, we present some results on the existence and global exponential stability of pseudo-almost periodic solutions of the system (1).
We assume that the following conditions are adopted: and there exist positive constant numbers ρ, σ such that for all x ij , y ij ∈ R,
For any z ∈ PAP(R, R mn ), set Obviously, Ω is a closed convex subset of PAP(R, R mn ), and Firstly, let us prove that the mapping Γ is a self-mapping from Ω to Ω. In fact, for any , then the mapping Γ is a self-mapping from Ω to Ω.

We have
It is clear that Γ is a contraction mapping of Ω. Thus, by virtue of the Banach fixed point theorem, the mapping Γ has a unique fixed point, x(t) = (x 11 , · · ·, x 1n , · · ·, x m1 , · · ·, x mn ) T ∈ Ω which corresponds to the solution of the system (6) in Ω ⊂ PAP(R, R mn ), such that Then, combining with Equation (8), we get Hence, the system (1) has only one pseudo-almost periodic solution x(t). The proof is complete.
By the same way, for all 1 ≤ i ≤ m, 1 ≤ j ≤ n, according to (14), we have Therefore, the unique pseudo-almost periodic solution of the system (1.1.) is globally exponentially stable. The proof is complete.
Obviously, ρ ij = σ ij = 1 10 , a + ij = 1.5, a − ij = 1.0, p + ij = 1 10 , and This example is simulated through MATLAB according to the given parameters. Figure 1, Figure 2, and Figure 3 display the state trajectories x 11 (t), x 22 (t), and x 33 (t) of the pseudo-almost periodic solution for the neural network system (15) with three different initial values (2.5, 3.0, 3.5), (1.0, 1.5, 2.0), (1.0, 1.5, 1.8), respectively. Even with the change of initial points, the shapes of the trajectories are not changed. As can be seen that simulated the solution tends to be the pseudo-almost periodic solution of the neural network system (4.1). Figure 4 shows the dynamic behavior of the pseudo-almost solution x 11 (t) and x 22 (t) of the neural network system (15) with the same initial values x 11 (0) = x 22 (0) = 4. Similarly, Figure 5 exhibits the dynamic behavior of the pseudo-almost solution x 11 (t) and x 33 (t), and Figure 6 x 22 (t) and x 33 (t). The validity of the conclusions can be judged by comparing the two-state trajectories with each other.
Mathematics 2021, 9, 1951 14 theoretical results' sufficient conditions are effective for the neural network system Moreover, the phase response represents a bunch of pseudo-almost periodic trajecto which gives an idea of pseudo-almost periodic solutions for our described neural netw system (15). Considered the above relative parameters, all the conditions of Theorems 1 2 are satisfied. Therefore, the neural network system (15) has precisely one continuously ferential pseudo-almost periodic solution, which is also globally exponentially stable.   x , and initial values are 2.5,3.0,3.5, respectively.         Figure 7c describes the space behavior of the state variables x 11 (t), x 22 (t) for the neural network system (15). Similarly, Figure 8a,b shows the phase responses of state variables x 11 (t) and x 33 (t) for the neural network system (15) with different initial values (3.5, 1.0), (0.5, 1.5). Figure 8c space behavior of the state variables x 11 (t), x 33 (t) for the neural network system (15); and Figure 9a,b depict phase diagram x 22 (t) and x 33 (t) with (2.0, 1.5), (0.5, 1.0), Figure 9c reveals the space behavior of the state variables x 22 (t) and x 33 (t). Figure  7d, Figure 8d, and Figure 9d exhibitions the 3D space behavior of the state variables x 11 (t), x 22 (t) and x 33 (t) for the neural network system (15) with three different initial values (0.5, 1.0, 0.5), (0.5, 1.5, 1.5), (0.5, 2.5, 1.5). The time response confirms that our theoretical results' sufficient conditions are effective for the neural network system (15). Moreover, the phase response represents a bunch of pseudo-almost periodic trajectories, which gives an idea of pseudo-almost periodic solutions for our described neural network system (15). Considered the above relative parameters, all the conditions of Theorems 1 and 2 are satisfied. Therefore, the neural network system (15) has precisely one continuously differential pseudo-almost periodic solution, which is also globally exponentially stable.   Mathematics 2021, 9, 1951 20 (c) (d)

Conclusions
In this paper, the existence and stability criteria of the pseudo-almost periodic s tion for the novel type complex networks are examined. Based on the Banach fixed-p theorem and the exponential dichotomy of linear equations, the existence and uni ness of pseudo-almost periodic solutions are investigated. Through an integral var transformation, the global exponential stability condition of the CNN is evalu Compared with the previous work on the stability analysis of periodic solutions, the rived pseudo-almost periodic results are more precise and less conservative. The posed variable substitution can induce stability flexibility, overcome the bottle problem of constructing the complicated Lyapunov functional, and ensure the con gence results from more validity. The approach has a fast convergence speed, whi suitable for applications of complex systems. The obtained results in this work are v able in the design of neural network systems, which are used to solve efficiency and

Conclusions
In this paper, the existence and stability criteria of the pseudo-almost periodic solution for the novel type complex networks are examined. Based on the Banach fixed-point theorem and the exponential dichotomy of linear equations, the existence and uniqueness of pseudo-almost periodic solutions are investigated. Through an integral variable transformation, the global exponential stability condition of the CNN is evaluated. Compared with the previous work on the stability analysis of periodic solutions, the derived pseudoalmost periodic results are more precise and less conservative. The proposed variable substitution can induce stability flexibility, overcome the bottleneck problem of constructing the complicated Lyapunov functional, and ensure the convergence results from more validity. The approach has a fast convergence speed, which is suitable for applications of complex systems. The obtained results in this work are valuable in the design of neural network systems, which are used to solve efficiency and optimal control problems arising in practical engineering applications. The existence and stability conditions are expressed in simple algebraic form, and their verification is done.
In future work, the analysis method can also be applied to more complicated neural network systems such as fuzzy systems and fractional-order neural networks that arise in the various disciplines of engineering and scientific fields. Such as, Mittag-Leffer stability of the fractional-order neural networks with discontinuous activation functions and timevarying delays will also be explored. Moreover, synchronization and state estimation of the fractional-order memristor-based neural networks and stochastic delayed systems will also be examined in the future.