New Stabilization Method for Delayed Discrete-Time Cohen–Grossberg BAM Neural Networks

This paper deals with the state feedback stabilization problem of delayed discrete-time Cohen–Grossberg BAM neural networks. By the mathematical induction method, stabilizable conditions are derived to ensure that the resulting closed-loop system is globally exponentially stable, and thereby, the desired state feedback controller is designed. These stabilizable conditions are very simple, which can easily verified by using the standard toolbox software (for example, MATLAB). The proposed approach is directly based on the definition of global exponential stability, and does not involve the construction of any Lyapunov–Krasovskii functional. For a special case, it is theoretical proven that the proposed method is superior to an existing one. Moreover, several illustrative examples are given to validate the success of the derived theoretical results.


I. INTRODUCTION
In 1983, Cohen and Grossberg introduced a simplified model of neural networks, named as Cohen-Grossberg Neural Networks (CGNNs) [1], which is a single-layer auto-associative Hebbian correlator. Then the bidirectional associative memory neural networks (BAMNNs) were proposed by Kosko [2], [3], which is a type of recurrent neural networks. BAMNNs generalize the single-layer CGNNs to a two-layer pattern-matched heteroassociative circuits, and comes up with a complete and clear pattern stored in memory from an incomplete or fuzzy pattern.
In hardware implementation of neural networks, time delay is an unavoidable factor during the signal transmission between the neurons. Time delay may lead to some complex dynamical behaviors of the whole network, for example, instability, chaos, periodic, and poor performance [4]- [11]. Therefore, it is great important to determine sufficient conditions for the asymptotic or exponential stability of discrete-time delayed BAMNNs [12]- [23].
The associate editor coordinating the review of this manuscript and approving it for publication was Bo Shen .
A considerable number of outcomes have been investigated regarding the combination of CGNNs and BAMNNs (CGBAMNNs) which have been applied to many areas. The CGBAMNN will be a greater network system, which contains more neurons of interactions, since it considers the interactions between the two neural fields. CGBAMNNs include a number of models from neurobiology and population biology, such as Lotka-Votterra systems, CGNNs and BAMNNs as the special cases, thus they will have more functions in pattern recognition, signal processing, ecological system, parallel computing, associative memory, and combinatorial optimization [24]- [26]. Accordingly, many researchers paid more attention on CGBAMNNs. On the basis of the methods dealing with BAM and CGNNs, in recent years, some experts and scholars have proposed and discussed the stability of equilibrium for CGBAMNNs [26]- [37]. Here, we mention only those related to this paper closely. In [35], Cao and Song investigated several novel sufficient conditions ensuring the existence, uniqueness and global exponential stability of equilibrium by using the analysis method, inequality technique and the properties of M-matrix. Zhou et al. [36] analyzed global exponential stability of equilibrium for a class of CGBAMNNs with delays. VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ Under the assumptions that the activation functions only satisfy global Lipschitz conditions and the behaved functions only satisfy sign conditions, by applying the linear matrix inequality (LMI) method, degree theory and some inequality technique, a novel LMI-based sufficient condition is established for global exponential stability of the concerned neural networks. The assumptions on the activation functions and behaved functions are more general than ones in [29], [30], [32]. Ali et al. [37] studied the problem of asymptotic stability of neutral type CGBAMNNs with discrete and distributed time-varying delays. By constructing a suitable Lyapunov-Krasovskii functional and applying reciprocal convex technique and Jensen's inequality, delay-dependent sufficient conditions for the asymptotic stability are established. Due to the complex dynamic behaviors, the stabilization problems on the neural networks based on suitable control technique are important in the both theory and application sense. In some real applications, it is required that the dynamic behavior to converge to a stable equilibrium state. Accordingly, some researchers gave attention to study stabilization criteria under which the state trajectories of closed-loop system can be controlled to approach some periodic orbits or equilibriums and to keep them there then after. In [38], a continuous stabilization controller was designed for stabilizing the states of stochastic uncertain BAM neural networks in finite time. Aouiti et al. [39] dealt with the finite-time and fixed time stabilization problems for a class of high-order BAM neural networks with time varying delay. Chinnathambi et al. [40] designed a state feedback controller to stabilize CGBAMNNs with delays.
However, the stabilization problem for delayed discretetime CGBAMNNs is still an untreated topic in the existing literature. Motivated by the previous discussions, in this paper we study the stabilization issue for delayed discrete-time CGBAMNNs via state feedback controller. Based on the mathematical induction method proposed in [23], the state feedback controller is designed. We derive several novel global exponential stability criteria for the equilibrium of the resulting close-loop system, which have simple form, and hence they can be easily verified via the standard tool software (e.g., MATLAB). Compared with the existing results, the proposed approach does not require to construct any Lyapunov-Krasovskii functional. It is theoretically proven that the obtained global exponential stability criteria are less conservative than ones in [18].
The organization of this paper is as follows. Problem considered in this paper will be formulated in Section II. The stabilization method for delayed discrete-time CGBAMNNs via state feedback controller will be discussed in Section III. In Section IV, we will theoretically compare the methods proposed in this paper and [18]. Numerical examples are provided in Section V to illustrate the effectiveness of the proposed method. Finally, Section VI gives some concluding remarks.
Notations. Suppose Z, R and C are sets of all integers, real numbers and complex numbers, respectively. Let Z[a, b] be the subset of Z consisting of all integers between a and b, and let Z[a, ∞) = ∪ b>a Z[a, b]. For given positive integers p and q, let R p×q denote the set of all p × q matrices over R. Set For a matrix M ∈ R n×n , let λ(M ) = {z ∈ C : det(zI n − M ) = 0}. The spectral abscissa of M is defined by s(M ) := max{Reλ : λ ∈ λ(M )}, and the spectral radius of M is defined by ρ(M ) := max{|λ| : λ ∈ σ (M )}. We say that m is a Metzler matrix if all off-diagonal elements of M are nonnegative. 1 For

II. PROBLEM FORMULATION
Consider a class of discrete-time CGBAMNNs with time-varying delays and control inputs, which can be described as: where the subscripts i and j stand for the ith neuron from the neural field F X and the jth neuron from the neural field F Y , respectively, x i (k) and y j (k) are the states, U i (k) and V j (k) are the control inputs, α i (·) and β j (·) represent the amplification functions, a i (·) and b j (·) denote appropriately behaved functions, c ij , e ij , d ji and w ji are constants which denote the synaptic connection weights, f j (·), g j (·),f i (·) and g i (·) denote the activation functions, J i andJ j denote the external inputs, and h ij (k) and τ ji (k) denote the time-varying delays.
Assumption 3: The functions a i (·) and b j (·) satisfy and γ (2) i are known positive constants.
In the following we will always assume that (x * , y * ) ∈ R n × R n is the unique equilibrium of CGBAMNN (1) with U i (k) ≡ 0 and V j (k) ≡ 0, that is, for any i, j ∈ Z [1, n], where x * i and y * i are the ith components of x * and y * , respectively.
Set (1) and (2), we have where When the state feedback controller is applied to system (3), the resulting closed-loop system is obtained as follows: Here, η i and ζ j are control gains to be determined. The initial functions associated the closed-loop system (6) are given by for any ψ, ϕ ∈ C(Z[−σ, 0], R n ), where · 2 is the Euclidean norm on R n . Definition 1 [13]: The zero equilibrium of closed-loop system (6) is said to be globally exponentially stable, if there exist scalars K > 0 and γ > 0 such that every solution of (6), The main goal of this paper is to stabilize exponentially the delayed discrete-time system (3) via the state feedback controller (5), that is, find control gains η i and ζ j (i, j ∈ Z[1, n]) such that the zero equilibrium of closed-loop system (6) is globally exponentially stable.

III. EXPONENTIAL STABILIZATION
In this section, we will investigate sufficient conditions under which the zero equilibrium of closed-loop system (6) is globally exponentially stable. To describe conveniently our main conclusion, we define: 1 , γ Theorem 1: Under Assumptions 1-4, if there exist a scalar γ > 0 and vectorsũ,ṽ ∈ R n such that then the zero equilibrium of closed-loop system (6) is globally exponentially stable, that is, the delayed discrete-time system (3) can be stabilized exponentially via the state feedback controller Proof: Choose K 1 > 0 such that K 1ũ col (1, 1, . . . , 1), K 1ṽ col (1, 1, . . . , 1).
In summary, (12) holds. It follows from (10), (11) and (12) that Since ϕ, ψ ∈ C(Z[−σ, 0], R n ) is arbitrary, we obtain that the zero equilibrium of closed-loop system (6) is globally exponentially stable. This completes the proof. Remark 1: Theorem 1 gives global exponential stabilization conditions for the zero equilibrium of closed-loop system (6). From its proof, it is seen that the proposed method is applicable to the case that the numbers of neurons in the two neural fields are different.
To present more sufficient conditions, we introduce the following result.
Lemma 1 [41]: Let A 0 ∈ R n×n be a Metzler matrix and B 0 , C 0 , D 0 ∈ R n×n . Then the following statements (i)-(iii) are equivalent: Combining Theorem 1 and Lemma 1, one can easily derive the following conclusion.
Remark 2: Theorems 2 and 3 give the delay-dependent and -independent global exponential stabilization conditions for the closed-loop system (6), respectively.
Remark 3: Note that Proposition 3 does not require the conditionᾱ iλi ≤ 1 orβ jμj ≤ 1. This, together with Theorem 4, claims that Proposition 3 is less conservative than Proposition 1, and hence the method proposed in this paper is superior to one in [18]. Moreover, it can be observed from the proof of Theorem 3 that Proposition 2 is less conservative than Proposition 3.

V. ILLUSTRATIVE EXAMPLES
In this section we will present the effectiveness of the proposed method by several numerical examples. Now we design a state feedback controller (5) to stabilize the system under consideration. By employing the function eig of MATLAB, it is easy to verify that (ii) of Theorem 3 holds, and hence the considered system can be exponentially stabilized via the state-feedback controller: U 1 (k) = 0.5u 1 (k), V 1 (k) = 0.74v 1 (k), k ∈ Z[0, ∞). Notice that the condition (b) in Theorem 3 is delay-independent, the global exponential stability of the resulting closed-loop system is independent of the choice of delays h 11 (k) and τ 11 (k). Furthermore, the state responses of the closed-loop system are given in Figures 3 and 4 for different initial   functions, which demonstrates that the state trajectories of the resulting closed-loop system approach to zero and to keep them there. So, the resulting closed-loop system is stable due to the designed state-feedback controller. Now we design a state feedback controller (5) to stabilize the system under consideration. For γ = 0.022, by using the function eig of MATLAB, it is easy to verify that Theorem 2(b) holds, and hence the considered system can be exponentially stabilized via the state-feedback controller: U (k) = 0.5u(t), V (k) = 0.2v(t), k ∈ Z[0, ∞). Moreover, if we use p 11 = 5 instead of p 11 = 4 in this example, then Theorem 2(b) is not true, which implies that Theorem 2 is not available in this case. So, the global exponential stability criterion provided in Theorem 2 is delay-dependent. When h ij (k) and τ ji (k) (i, j ∈ Z [1,2]) are taken as above, the state responses of the closed-loop system are given in VOLUME 8, 2020    Figures 7 and 8, which demonstrates that the state trajectories of the resulting closed-loop system approach to zero and to keep them there. Hence, the resulting closed-loop system is stable due to the designed state-feedback controller.  Clearly, the inequalities in (4) are satisfied with α 1 = α 2 = 0.25,ᾱ 1 =ᾱ 2 = 0.75, β 1 = β 2 = 0.5,β 1 =β 2 = 5 6 , λ 1 = λ 2 =λ 1 =λ 2 = µ 1 = µ 2 =μ 1 =μ 2 = 2, γ (1) 2 = 0.5. By using MATLAB, it is obvious that the (i) of Proposition 1 (i.e., [18,Corollary 2]) is not satisfied, and hence Proposition 1 can not be used to check the global exponential stability of the considered CGBAMNN.
It is easily obtained thatǎ 1 =ǎ 2 = 0.5,b 1 =b 2 = 2 3 . By employing the function eig of MATLAB, it is easy to verify that (II) of Proposition 3 holds, and hence the zero equilibrium of CGBAMNN under consideration is globally exponentially stable. Furthermore, when u(s) ≡ col(−1.6, 4.1) and v(s) ≡ col(−1.3, −3.9) for any s ∈ Z[−6, 0], the state responses of the resulting closed-loop system under consideration is given in Figures 9 and 10.

VI. CONCLUSIONS
This paper addresses the stabilization problem for a class of delayed discrete-time CGBAMNNs via state feedback controllers. By using the mathematical induction method, delay-dependent and -independent stabilizable conditions are investigated to guarantee that the resulting closed-loop system is globally exponentially stable. These conditions involve only to solve several simple LMIs or compute the spectral abscissas and radiuses of constant matrices, which can be easily checked by employing the standard tool softwares (e.g., MATLAB). The stabilization method proposed in this paper is directly based on the definition of global exponential stability, and does not involve the construction of any Lyapunov-Krasovskii functional. Furthermore, the global exponential stability criteria for a special case of the considered CGBAMNN are derived, which has been theoretically proven to be less conservative than [18,Corollary 2]. Finally, the proposed method to design state feedback controllers is effective.