Novel Criteria of ISS Analysis for Delayed Memristive Simplified Cohen–Grossberg BAM Neural Networks

.e memristor as the fourth circuit element, it can capture some key aspects of biological synaptic plasticity. So, it is significant that the characteristic of memristors is considered in neural networks. .is paper investigates input-to-state stability (ISS) of a class of memristive simplified Cohen–Grossberg bidirectional associative memory (BAM) neural networks with variable time delays. In the sense of Filippov solution, some novel sufficient criteria for ISS are obtained based on differential inclusions and differential inequalities; when the input is zero, the stability of the total system is state stable. Furthermore, numerical simulations are illustrated to show the feasibility of our results.


Introduction
In 1988, Kosko proposed a class of bidirectional associative memory (BAM) neural networks [1]. Because of potential applications of associate memory and pattern recognition, many research studies are increasingly concerned about dynamics behaviors of such neural networks.
ere have been many results on the stability for BAM neural networks with or without delays [2][3][4][5][6][7][8]. At the same time, in 1983, Cohen and Grossberg discussed a class of competitive neural networks, which is called Cohen-Grossberg neural networks. A lot of models can be described by this model, such as Hopfield-type neural networks, population biology. Meanwhile, this model has also been applied in some areas such as associate memory, pattern recognition, and optimization problems.
erefore, stability problems have attracted many researchers'attentions [9][10][11][12][13][14][15][16]. Based on nonsmooth analysis and stability theory, we obtained sufficient conditions of the globally asymptotical stability of the mix-delayed Cohen-Grossberg neural networks with nonlinear impulse [13]. Considering characteristics of BAM neural networks and Cohen-Grossberg neural networks, many researchers investigated qualitative analysis of Cohen-Grossberg BAM neural networks [16][17][18]. Bao [16] investigated the existence and exponential stability of periodic solutions for fuzzy Cohen-Grossberg BAM neural networks with mixed delays by M-matrix theory and inequality techniques. Rao et al. [17] considered stability results for p-Laplace dynamical equations including Cohen-Grossberg BAM neural networks by constructing suitable Lyapunov functional, M-matrix technique, and Yang inequality. Chinnathambi et al. [18] explored finitetime stabilization of delayed Cohen-Grossberg BAM neural networks under suitable control schemes by using Lyapunov function and some algebraic conditions.
In 1971, Chua predicted memristors as the fourth circuit element [19]. Strukov et al. found the first practical memristor device in 2008 [20]. It can remember its past dynamic history, similar to biological synapse. Because of the characteristic of its memory, it attracted some theoretical researchers' interests and notices [21][22][23][24][25]. e stability of neural networks with memristors has become a hot topic [24][25][26][27][28][29]. Considering the characteristic of memristors is significant in neural networks, it can give a potential hope to build brain-like computer and emulate human brain.
In practice, information transmission unavoidably exists in delays. In neural networks, finite propagation velocity often results in delays. Delayed neural network reduces to more complex dynamical behaviors, such as periodic, instability, and chaos. us, considering time delays is significant in neural networks [2-14, 16, 18, 24-34]. Recently, dynamics analysis for memristive neural networks with delays has attracted considerable attention [21][22][23][24][25][26][27][28][29][30][31][32][33][34]. Wu and Zeng [27] investigated the Lagrange stability of memristive neural networks by the nonsmooth analysis and control theory, which is considered to discuss the stability of the total system and not only the stability of the equilibria. Zhao et al. [28] considered the input-to-state stability of a class of memristive Cohen-Grossberg-type neural networks with variable time delays, which include some known results as particular cases. Zhong et al. [29] obtained some sufficient conditions for the input-to-state stability of a class of memristive neural networks with time-varying delays. Furthermore, Zhao et al. [30] discussed dynamics of memristive BAM neural networks with variable time delays and obtained some novel sufficient conditions of the inputto-state stability.
From these papers, we know the input-to-state stability is general stability. When the inputs are bounded, it is also called bounded-input bounded-output (BIBO) stable. When the inputs are zero, the state of the system is asymptotically stable.
erefore, considering input-to-state stability of system (1) is necessary and significant. us, inspired by the above discussions, we will here investigate stability problems of memristive simplified Cohen-Grossberg BAM neural networks with variable time delays: where c i (·) and d i (·) are the amplification functions, f j (j � 1, 2, . . . , m) and g i (i � 1, 2, . . . , n) denote the signal transmission functions, and I i (t)(i � 1, 2, . . . , n) and , c ij (y j (t)), and d ij (y j (t)) represent memristorbased weights, and According to the feature of a memristor and its currentvoltage characteristic, we have is system includes some well-known systems as particular cases. e solution of system (1) is represented by (x(t), y(t)) T � (x 1 (t), . . . , x n (t), y 1 (t), . . . , y m (t)) T ∈ R n+m . System (1) is supplemented with the initial values 2 Complexity where φ i (s), ψ j (s) denotes a real-valued continuous function defined on (− τ, 0]. To obtain our solutions of system (1), furthermore, we assume as follows: (H3)f j (·) and g i (·) satisfy globally Lipschitz conditions with positive constants k j > 0 and l i > 0, respectively, such that for any x(t), y(t) ∈ R. e rest of this paper is organized as follows. In Section 2, we introduce some notations, definitions, and some preliminary results. In Section 3, we present sufficient conditions for the input-to-state stability of system (1). Finally, in Section 4, an example illustrates the feasibility of our results.

Preliminaries
In this section, we give some notations, definitions, and some preliminaries, which will be necessary for our main results.
In this paper, the solutions of all systems are considered in Filippov's sense [35]. co a, � a { } denotes the closure of the convex hull generated by the real numbers a and � a.
In what follows, we also need to introduce some definitions: Definition 2. For the system dx/dt � g(x), x ∈ R n , with discontinous right-hand sides, a set-valued map is defined as where co[E] is the closure of the convex hull of the set E, B(x, δ) � y: y − x ≤ δ , and μ(N) is a Lebesgue measure of the set N. A solution in Filippov's sense of the Cauchy problem for this system with initial condition Definition 3 (see [29,37]). A scalar continuous function α(r) defined for r ∈ [0, a) is said to belong to the class κ if it is strictly increasing, and α(0) � 0. It is said to belong to the class κ ∞ for all r ≥ 0 and also α(r) ⟶ +∞ as r ⟶ +∞.
Definition 4 (see [29,37]). A function β(r, s), defined for r ∈ [0, a) and s ∈ [0, ∞) is said to belong to the class κL if for each fixed s ≥ 0, the mapping β(r, s) belongs to the class K with respect to r and for each fixed r, and the mapping β(r, s) is decreasing with respect to s and β(r, s) ⟶ 0 as s ⟶ ∞.
Definition 5 (see [29,37]). System (1) is said to be input-tostate stable if there exists a κL function β and a κ ∞ function α such that for any x 0 ∈ R n , y 0 ∈ R m , I(t), J(t) ∈ L n ∞ .
Remark 1. When the inputs I(t) and J(t) are bounded, note that β is a κ ∞ function and is also bounded. When the inputs I(t) and J(t) are zero, system (1) is asymptotically stable. From (8), ‖x(t; x 0 , I(t))‖ + ‖y(t; y 0 , J(t))‖ is bounded. erefore, system (1) is input-to-state stable and also called bounded-input bounded-output (BIBO) stable.

Complexity
Proof. Taking (14), we choose a sufficiently small positive constant ε > 0 such that and consider Lyapunov function where Combining (17) and (18), then 6 Complexity us, it follows that and from the definition, system (1) is input-to-state stable.
Corollary 2. Assume that (H1)-(H3) hold. en, system (24) is input-to-state stable if Complexity Remark 2. It is worth noting that the result of eorem 1 can be improved and simplified. Next, we give related results, which are less conservative than eorem 1 and are more useful.
□ Similar to eorem 1, we have related corollaries for special cases as follows.
Based on the calculation, it is easy to know that the conditions of eorem 1 and eorem 2 are satisfied. us, system (39) is input-to-state stable. When initial values x(0) � [0.06; 0.05] T , Figure 1 shows that time evolutions of system (39) with periodic inputs are bounded. erefore, it is input-to-state stable. When inputs are zeros, Figure 2 shows that the state of system (39) with identical initial values is exponentially stable.

Conclusions
Memristors can show synaptic dynamic characteristic of the human brain, so it can become ideal implementation of the artificial synapse. Based on the memristors, some theoretical researchers look forward to propose some better model for understanding and simulating the characteristic of the human brain. In this paper, based on nonsmooth analysis, set-valued maps, and differential inequality analysis, we investigate input-tostate stability of a class of memristive simplified Cohen-Grossberg bidirectional associative memory (BAM) neural networks with variable time delays. Some sufficient conditions guarantee the input-to-state stability of such networks. Furthermore, when the input is zero, the stability of the total system is state stable. Our results extended to some known results, which can be applied in wider situations [6,8,[24][25][26]. At the same time, an illustrative example shows the feasibility and effectiveness of our results.

Data Availability
No data were used to analyse this theoretical study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.