Exponential Stability Analysis of Mixed Delayed Quaternion-Valued Neural Networks via Decomposed Approach

With the application of quaternion in technology, quaternion-valued neural networks (QVNNs) have attracted many scholars’ attention in recent years. For the existing results, dynamical behavior is an important studying side. In this paper, we mainly research the existence, uniqueness and exponential stability criteria of solutions for the QVNNs with discrete time-varying delays and distributed delays by means of generalized 2-norm. In order to avoid the noncommutativity of quaternion multiplication, the QVDNN system is ﬁrstly decomposed into four real-number systems by Hamilton rules. Then, we obtain the sufﬁcient criteria for the existence, uniqueness and exponential stability of solutions by special Lyapunov-type functional, Cauchy convergence principle and monotone function. Furthermore, several corollaries are derived from the main results. Finally, we give one numerical example and its simulated ﬁgures to illustrate the effectiveness of the obtained conclusion.


I. INTRODUCTION
After the models of various neural networks (NNs) (Hopfield NNs, Cohen-Grossberg NNs, memristive NNs, etc.) were built [1]- [3], they have been widely used to research pattern recognition, optimization problems, intelligent control, and so on. When NNs are utilized in the reality applications, dynamical characteristics of these systems are very important. Therefore, the study of dynamical behaviors has been one evergreen hot topic because of its significant influence on NNs [4]- [6], [12]. For instance, in [4], [9], the authors discussed the stability criteria of different NNs by means of diverse ways. Synchronization problems of NNs were investigated in [5], [7], [8], [11]. Passivity and Dissipativity of NNs were studied in [5], [6]. From these results, it can The associate editor coordinating the review of this manuscript and approving it for publication was Engang Tian . be seen that most of this references have studied the delayed NNs systems.
As we all know, time-delays are often the main cause of oscillation and instability, which are need to be considered when NNs models and neural circuits are constructed. Therefore, the dynamical characteristics of NNs with various time delays are necessary for the NNs researches [6], [10], [11], [13]- [21]. In particular, stability of delayed NNs (DNNs) is one of the most desirable dynamical properties when NNs models are used, which have drawn much attention of many scholars. There have been a large amount of related results in recent years [13]- [21]. For example, in [13], [15], [16], some important stability conclusions were obtained for those NNs with discrete and distributed delays. And discrete delays were considered in NNs to obtain the stability criteria in [14], [17]- [21]. Although there have been some significant stability results for the discrete delayed QVNNs (QVDNNs) systems [20]- [26], distributed delays have been rarely discussed. 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/ Although the signal propagation sometimes can be modeled by QVNNs with discrete delays, it can also be distributed in some certain periods. Hence, the distributed delays should also be considered simultaneously with discrete delays in a QVNNs system. In the past several decades, NNs have been mainly studied in the real number field [6], [10], [15] and in the complex number field [9], [17]- [19], and then, in the quaternion number field [20], [22], [26]. Quaternion was firstly given by W. R. Hamilton in 1843. Recently, it has been used in many areas, such as computer graphics, 3 or 4-D data modeling, array processing and so on. As one of the most important research contents of quaternion, QVNNs have drawn many researchers' attention [20]- [33], which is a natural continuation of complex-valued NNs (QVNNs) and real-valued NNs (RVNNs). Due to the noncommutativity of quaternion multiplication, decomposition and direct approaches are usually used to research the QVDNNs. In [21], [22], [29], [32], the authors used direct approach to investigate QVDNNs, while the authors in [20], [24], [26], [31], [33] studied QVDNNs by decomposition method.
Stability is one of the most fundamental important dynamical properties for QVDNNs [20]- [26], [31]- [33]. For example, in [21], [32], homeomorphic mapping was utilized for QVDNNs to study the existence, uniqueness and stability criteria of solutions by constructing a complex Lyapunov-Krasovskii functional. LMI-form sufficient conditions were derived in these papers. The existence and stability criteria of multiple equilibrium points were obtained for QVNNs by means of different ways in [22], [31]. And in [20], [25], {ξ, ∞}-norm, a generalized ∞-norm, was firstly used to study the existence and uniqueness criteria of solutions, and the µ-stable criteria for QVDNNs. It is worthy noting that generalized norm is an useful definition for QVDNNs to get their existence and stability criteria of solutions. Apart from generalized ∞-norm, there have been generalized 1-norm and 2-norm, which are named as {ξ, 1}-norm and {ξ, 2}-norm [4], [18], respectively. Although the method of ∞-norm can not be used to study some dynamical behaviors by 1-norm or 2-norm, the similar results can be derived by generalized 1-norm if some results of QVDNNs can be obtained by generalized 2-norm. Therefore, it is worthy studying the stability of QVNNs with discrete time-varying delays and distributed delays by {ξ, 2}-norm, which remains an open problem.
Based on the above analyses, this paper focuses on the existence and exponential stability of solutions for the QVNNs with discrete time-varying delays and distributed delays by generalized 2-norm ({ξ, 2}-norm). Because of the noncommutativity of quaternion multiplication, the QVDNN system is firstly decomposed into four real-number systems by Hamilton rules. Then, the novel stability definition of QVDNNs is introduced according to the definition of {ξ, 2}-norm. Meanwhile, some assumptions of discrete time-varying delays and distributed delays are given. In addition, by constructing {ξ, 2}-norm-type Lyapunov functional, the existence, uniqueness and exponential stability sufficient criteria of the discrete-distributed-delayed QVNNs are obtained by Cauchy convergence principle and monotone function. Several corollaries are derived from the main results. Finally, one numerical example about QVNNs with discrete time-varying delays and distributed delays is given to illustrate the effectiveness of the obtained conclusions.
The rest of this paper is organized as follows. In Section II, models and preliminaries are given. Then, the existence and stability sufficient criteria for discrete and distributed delayed QVNNs is obtained in Section III. In Section IV, a numerical simulation example is shown to illustrate the validity of obtained results. Finally, conclusions are given in Section V.

II. PRELIMINARIES
Notation: R and Q show the sets of real numbers and quaternion numbers, respectively. R n and Q n denote the n-dimensional Euclidean and quaternion spaces, respectively. R n×m and Q n×m are the sets of n × m real matrixes and n × m quaternion matrixes, respectively.
In this paper, we will consider the following QVNNs with discrete and distributed delays: where is the state vector, D = diag{d 1 , d 2 , · · · , d n } ∈ R n×n with d p > 0 is the self-inhibition matrix, f (x(t)) = (f 1 (x 1 (t)), f 2 (x 2 (t)), · · · , f n (x n (t))) T ∈ Q n and g(x(t)) = (g 1 (x 1 (t)), g 2 (x 2 (t)), · · · , g n (x n (t))) T ∈ Q n represent the quaternion-valued neuron vector-valued activation functions, which satisfy f q (0) = 0 and g q (0) = 0. A = [a pq ] n×n , B = [b pq ] n×n ∈ Q n×n and C = [c pq ] n×n ∈ Q n×n are the non-delayed, discrete-delayed and distributed-delayed connective weights matrixes, respectively. τ pq (t) > 0 is discrete time-varying delay. K (s) = [k pq (s)] n×n ∈ R n×n is the delayed kernel function matrix. u = (u 1 , u 2 , · · · , u n ) T ∈ Q n is an external input or bias vector. The initial condition is ϕ(s) = (ϕ 1 (s), ϕ 2 (s), . . . , ϕ n (s)) T ∈ C((−∞, 0], Q n ). Next, some basic definitions and properties of quaternion shows that the real quaternion field Q can be viewed as a 4-D vector space over R. According to Hamilton rules, its imaginary units ı,  , and κ obey the following rules: the addition h + s and product hs of h and s can be defined as Denote M = {R, I , J , K }, then the QVNN model with discrete and distributed delays (1) can be decomposed into real-valued systems with L ∈ M as follows: Definition 1 [4]: ≤ γ l q holds for any q = 1, 2, · · · , n, l ∈ M and x l q , y l q ∈ R, then g l q (x l q ) is said to belong to L 2 (γ R q , γ I q , γ J q , γ K q ). In order to study the existence and stability of the above delayed QVNN model (2), the following assumptions should be introduced: (H1) The activation functions f q (x q (t)) and g q (x q (t)) can be separated into one real and three imaginary parts as follows: where f l q (x l q (t)) f l q (t) : R → R belongs to class L 2 and g l q (x l q (t)) g l q (t) : R → R belongs to class L 2 for every l ∈ M . (H2) The discrete time-varying delays τ pq (t) : R → R + are continuously differential functions and satisfy τ pq (t) ≤ τ pq ≤ τ and |τ pq (t)| ≤ η pq < 1 for any p, q = 1, 2, · · · , n and t > 0, where τ pq , τ and η pq are real positive constants. (H3) For any p, q = 1, 2, · · · , n, the kernel k pq : [0, +∞) → [0, +∞) are real-valued nonnegative continuous functions and satisfy the following conditions: ∞ 0 k pq (s)ds = 1 and ∞ 0 e σ s k pq (s)ds < ∞, where σ is positive numbers. Remark 1: Different from assumption (H1), activation function f q (x q (t)) can be decomposed into in some ref- [20], [24], [25]. Actually, the decomposed form of assumption (H1) can simplifies research process and results of QVDNNs, which is used in this paper. Furthermore, when the exponential stable criteria of QVDNNs are studied, some special restrictions should be given, as in assumption (H2), to deal with discrete time-varying delays.
Based on (H1), the system (2) can be rewritten aṡ where , (J , I ), (K , R)} and ψ lw = 1 or −1 is the sign of a l pq f w q (·), b l pq g w q (·) and c l pq g w q (·). Then the concrete forms ofẋ R p (t)ẋ I p (t)ẋ J p (t) anḋ x K p (t) can be written as follows: Definition 2 [24]: holds for any p, q = 1, 2, · · · , n.

III. MAIN RESULTS
In this section, the existence and exponential stability criteria of the QVNNs (1) with time-varying discrete delays and distributed delays are studied by utilizing its decomposed form and the definition of {ξ, 2}-norm.
Remark 3: For this theorem, we discuss the existence and exponential stability criteria of QVNNs (1) with discrete time-varying delays and distributed delays by the definition of {ξ, 2}-norm and Cauchy convergence principle. It is clearly observed that discrete time-varying delays and distributed delays have important effects on the convergence conditions of QVNNs. If the discrete delays are time-invariant, i.e. τ pq (t) = τ pq , then the expression 1+η pq 1−η pq e ςτ pq |b l pq |γ w q of (5) will be changed into e ςτ pq |b l pq |γ w q . If the distributed delays are nonexistent, then the inequality (5) will be much simpler.
Remark 4: The difficulty of this theorem is how to deal with the discrete time-varying delays and distributed delays via {ξ, 2}-norm. By constructing special {ξ, 2}-norm-type Lyapunov functional, this problem is worked out. In addition, for the sake of discussion, we suppose that the activate functions f l q (x l q (t)) and g l q (x l q (t)) belong to class L 2 for any l ∈ M and q = 1, 2, · · · , n. If f l q (x l q (t)) belongs to class L 1 and g l q (x l q (t)) belongs to class L 2 for any l ∈ M and q = 1, 2, · · · , n, then the more complex results can be obtained, which can't be discussed in this paper. Furthermore, according to this theorem, we can give the exponential stability criteria of QVNNs (1) with time-invariant asynchronous delays as follows.
Corollary 1: Under assumptions (H1) and (H3), if there exist real constants ς > 0, and ξ l p > 0(p = 1, 2, · · · , n, l ∈ M ), such that 2ξ L q (−d q + ς) + holds for every L ∈ M and q = 1, 2, · · · , n. Then, the dynamical system (1) with time-invariant asynchronous discrete and distribute delays has an unique equilibrium point x * , which is globally exponentially stable. Proof: Let then we can prove this corollary by the above similar process, the details are omitted. From the above theorem and corollary, we can easily obtain the following two corollaries, respectively.
Corollary 2: Under assumptions (H1), (H2) and (H3), if there exist real constants ξ l p > 0(p = 1, 2, · · · , n, l ∈ M ), such that |b l qp |γ w p + |c l qp |γ w p + n p=1 (l,w)∈M L ξ L p |a l pq |λ w q + 1 + η pq 1 − η pq |b l pq |γ w q + |c l pq |γ w q < 0 holds for every L ∈ M and q = 1, 2, · · · , n. Then, the dynamical system (1) with time-varying discrete and distribute delays has an unique equilibrium point x * , which is globally exponentially stable. Corollary 3: Under assumptions (H1) and (H3), if there exist real constants ξ l p > 0(p = 1, 2, · · · , n, l ∈ M ), such that −2d q ξ L q + n p=1 (l,w)∈M L ξ L q |a l qp |λ w p + |b l qp |γ w p + |c l qp |γ w p + n p=1 (l,w)∈M L ξ L p |a l pq |λ w q + |b l pq |γ w q + |c l pq |γ w q < 0 holds for every L ∈ M and q = 1, 2, · · · , n. Then, the dynamical system (1) with time-invariant asynchronous discrete and distribute delays has an unique equilibrium point x * , which is globally exponentially stable. Remark 5: In this section, the existence and exponential stability sufficient criteria are derived by the definition of {ξ, 2}-norm and Cauchy convergence principle. If the known conditions are changed, more corollaries can be obtained, which are omitted. In the process of discussing, in order to deal with discrete and distributed delays, the construction of special Lyapunov-type functional is very important. It is worthy noting that the discrete time-varying τ pq (t) can be changed with p, q and t, which is asynchronous time-delay [25].
Obviously, for any p, q = 1, 2, f q (x q (t)) and g q (x q (t)) satisfy the assumption (H1), k pq (s) satisfies the assumption (H3), τ pq (t) ≤ 1 + 1 50π ≤ 1.01 is bounded, and |τ pq (t)| = | 1 100q cos( π 2 t)| ≤ 1 100 < 1. Therefore, assumptions (H1), (H2) and (H3) are all satisfied. Let ξ L p = 0.1 and ς = 0.53, it can be calculated that 2ξ L q (−d q + ς ) +      (6) with discrete time-varying delays and distributed delays has an unique global exponential stable equilibrium point, which can be shown by Figures 1-5. Remark 6: In this example, discrete time-varying delays and distributed delays are considered for QVNNs with external input u = 0. By calculation, the conditions of Theorem 1 are satisfied and the conclusions can be obtained. Figures 1-4 show the state trajectories of every real part and image parts,respectively. Figure 5 shows the state trajectories of all real part and image parts. From these figures, it is observed that discrete time-varying asynchronous delays and distributed delays have crucial effects on the convergence process of QVNNs.

V. CONCLUSIONS
In the past several decades, the existence and stability of solutions have been the evergreen important topics since various NN models were constructed. Recently, with the development of quaternion application in technology, QVNNs, as an important side of quaternion, have been presented and studied by many scholars. Similar to other NNs, the existence and stability of solutions is one of the most important research contents of QVNNs. Based on the facts, this paper has focused on the existence and exponential stability of solutions of the QVNNs with discrete time-varying delays and distributed delays by means of {ξ, 2}-norm. Due to the noncommutativity of quaternion multiplication, the delayed QVNNs system has been firstly decomposed into four real-number systems by Hamilton rules. Then the novel stability definition about delayed QVNNs has been introduced by the definition of {ξ, 2}-norm, and some assumptions of discrete time-varying delays and distributed delays have been given. By constructing special {ξ, 2}-norm-type Lyapunov functional and taking advantage of Cauchy convergence principle and monotone function, the existence, uniqueness and exponential stability sufficient criteria of solutions have been obtained. Several corollaries have been derived from the main theory. Although the stability of QVNNs with discrete time-varying delays and distributed delays has been studied by {ξ, 2}-norm, it can also be used to investigate the synchronization of NNs, which is our future research content. Finally, one numerical example and its simulated figures have been given to illustrate the effectiveness of the obtained conclusions in this paper.