Novel Criteria of Stability for Delayed Memristive Quaternionic Neural Networks: Directly Quaternionic Method

: In this paper, we ﬁxate on the stability of varying-time delayed memristive quaternionic neural networks (MQNNs). With the help of the closure of the convex hull of a set the theory of differential inclusion, MQNN are transformed into variable coefﬁcient continuous quaternionic neural networks (QNNs). The existence and uniqueness of the equilibrium solution (ES) for MQNN are concluded by exploiting the ﬁxed-point theorem. Then a derivative formula of the quaternionic function’s norm is received. By utilizing the formula, the M -matrix theory, and the inequality techniques, some algebraic standards are gained to afﬁrm the global exponential stability (GES) of the ES for the MQNN. Notably, compared to the existing work on QNN, our direct quaternionic method operates QNN as a whole and markedly reduces computing complexity and the gained results are more apt to be veriﬁed. The two numerical simulation instances are provided to evidence the merits of the theoretical results.

have observable limitations: (i) they require the activation functions to be decomposable; yet, not all quaternion functions are decomposable; (ii) the decomposition methods often induce a bulky computing cost.
Given the foregoing discussion, this paper focuses on developing a new direct quaternionic method to research the GES of a class of multiple time-varying delayed MQNN. We establish a newly derivative formula of the quaternionic function. Then, based on the formula, and utilizing the inequality, the M-matrix theory, some novel stability outcomes for the considered MQNN are acquired. In this way, a new direct quaternionic approach for the analyzing stability of the MQNN is proposed. The main contributions of this paper are the following three aspects: (1) Memristors, time-varying delays, and quaternions are considered simultaneously in the neural network model, which extends some neural network models in previous papers.
(2) It is vital to estimate the norms of the quaternionic state variable through the given QNN for acquiring its stability. However, a quaternionic variable is a vector, while its norm is a real number. To deal with the challenge, a derivative formula of the quaternionic variable is established. With help of the formula, M-matrix theory and inequality techniques, a concise and efficient quaternionic method to study QNN has been established, which tackles the QNN as a whole without any decomposition and greatly reduces the computation burden.
(3) New GES criteria, coined in the form of M-matrix on quaternionic norm, are obtained. These criteria are easier to verify and have improved the some existing results. Besides, with these criteria, some restrictions on the MQNN have been removed.
In Section 2, we restate some quaternionic synopsis and define the considered model formally. We explain the new the uniqueness, existence as well as GES of the equilibrium solution (ES) in Section 3. The numerical examples and some comparisons with the previous results are given in Section 4. Concluding remarks are given in the last section.
Notations. R denotes the real number set in this paper. R m×n stands for the m × n the real matrix set. C([t 0 − τ, t 0 ]; S) represents the continuous mapping set from [t 0 − τ, t 0 ] to set S.

Mathematical Fundamentals and Model Statement
2.1. Quaternionic Synopsis q = r + I 1 + I 2 j + I 3 k is called as a quaternion, where r, I 1 , I 2 , I 3 ∈ R. Re(q) = r is known as the real part of p and Im(q) = I 1 i + I 2 j + I 3 k are known as the imaginary parts of q, where the imaginary unit i, j and k respect the following rules: Q denotes the quaternionic set and Q m×n represents the m × n quaternionic matrices. For any q = r + I 1 i + I 2 j + I 3 k , conjugateq and norm |q| of quaternion q are defined as q = r − I 1 i − I 2 j − I 3 k and |q| 2 = (r 2 + I 2 1 + I 2 2 + I 2 3 ). For q 1 = r 1 + iI 11 + I 12 j + I 13 k, q 2 = r 2 + I 21 i + I 22 j + I 23 k ∈ Q, their multiplication is defined as: q 1 q 2 =(r 1 r 2 − I 11 I 21 − I 12 I 22 − I 13 I 23 ) + (r 1 I 21 + r 2 I 11 + I 12 I 23 − I 13 I 22 )i + (r 1 I 22 + r 2 I 12 + I 13 I 21 − I 11 I 23 )j + (r 1 I 23 + r 2 I 13 + I 11 I 22 − I 12 I 21 )k.
Function q(t) = r(t) + I 1 (t)i + I 2 (t)j + I 3 (t)k is a quaternionic function on t, in which r(t), I 1 (t), I 2 (t) as well as I 3 (t) are all real-valued function on R → R. The quaternionic function q(t) is a differential iff r(t), I 1 (t), I 2 (t) and I 3 (t) are all differentiable, and d dt q(t) The more detailed property of quaternions can be restated as follows: Proposition 1. In [9] Set q 1 , q 2 ∈ Q, then the following relations hold,
In conformity with the memristor's feature and the current-voltage trait, the memristive coefficient a km (q k (t)) and b km (q k (t)) fulfill the following conditions: where k, m ∈ I, T k > 0 are said as the switching leaps,â km ,ȃ km ,b km ,b km ∈ Q are the known quaternionic constants. The initial conditions (IC) of the MQNN model (1) are given by q k (s) = φ k (s). Here φ k (s) is bounded function in ([−τ, 0], Q), for k ∈ I.

Remark 1.
In [19], the Lagrange stability of the MQNN is discussed. The following switching rules of a ij (q i (t)) are adopted as follows [19]: i (t))k. Clearly, the switching rule (4) requires that the quaternionic function a ij (q i (t)) can be decomposed into its four parts. However, not all quaternionic function can be decomposed in this way. Moreover, the switching rules (4) can be regraded as a special case of (3). The switching rule (3) can be applied in the cases whether a km (q k (t)) can be decomposed or not.
Due to the discontinuous function a km (q k (t)) and b km (q k (t)), the solutions of the MQNN model (1) are seen as the Filippov's sense. On account of the theory of differential inclusion, the MQNN model (1) can be reformulated into the form below: where co[S] denotes the closure of the convex hull of set S. For all k, m ∈ I, there exist the functionsã or the form of matrix, in The following hypothesis is necessary to gain the main results: (H) The continuous function f m (·) and g m (·) satisfy the following conditions:
(i) Each leading principal minors of matrix B is positive.

Main Results
Given the MQNN model in (1), we derive the new sufficient conditions to ensure the GES of its ES.

Theorem 1. Under hypothesis (H), supposing
Proof. We will first prove that the below matrix equation has a unique quaternionic solution p * to demonstrate that (7) holds an ES.
Next, we use the proof by contradiction to prove the uniqueness of solution of (8). Set q * to be another solution of (8), that is, Supposing This is a contradiction. Hence, q * = q * , that is, Equation (8) has a unique solution p * .
To explore the GES standard of system (1), we set up the below Theorem.
Theorem 2. Let u(t) : R → Q be differentiable, then the following equation is true,

Remark 2.
Let it be noted that u(t) is a quaternionic function while |u(t)| is a real-valued function. The significance of Theorem 2 is that it build a derivative relationship between a quaternionic function and its norm. The relationship lead ones to operate QNN as an entirety, which pave the way for researching the stability of QNN by utilizing direct quaternionic approaches. Proof. Set q * = (q * 1 , q * 2 , · · · , q * n ) to be the ES of the model (6). by translation q k (t) = q k (t) − q * k , k ∈ I, we can accept where Denote V(t) = ||q k (t)|| 2 , (k ∈ I). Computing dV(t) dt via (14), and by using Theorem 2, yields By using Proposition 1, hypothesis (H) as well asã km (t) ∈ co{â km ,ȃ km }, we can gain the following inequality, In the same way, we acquire that Combining (16) and (17) into (15), we can get That is equivalent to where [|q(t)|] = (|q 1 (t)|, |q 2 (t)|, · · · , |q n (t) Define the continuous function By (20), we see that H k (0) < 0. H k (ν) → +∞ as ν → +∞, for m ∈ I. We can acquire that there exists a constant δ > 0 meeting, according to the continuity of the function H k (ν), Let k (t) = e δt |q k (t)|, k ∈ I. By using (18), we receive Next, we will confirm that holds. Actually, if (23) does not hold, then there exists an unspecified positive integer k 0 and t * > 0 yielding However, in the light of (21)-(22) and (25), one can get, This is opposite to d dt k 0 (t * ) ≥ 0 in (23). Therefore (23) is verified, which shows By the initial valuesψ(s) = ψ(s) − q * , s ∈ [−τ, 0], is is easy to get where Π = max 1, max 1≤m≤n {η k }ν 0 min 1≤m≤n {sup −∞<s≤0 |ψ k (s)|} . Thus, we can receive

Remark 3.
In the proof of Theorem 2, Formula (13) plays a fundamental role. With the help of formula (13), the MQNN (6) can be analyzed as a whole without any decomposition. This concise method for analyzing MQNN can be applied to general QNN, regardless of whether the activity function of QNN can be decomposed, which greatly reducing the computational cost. Besides, the results obtained are easy to check in the practice. This is one of the distinguishing features and dedications of this paper.

Remark 4.
If a km (q k (t)) = a km , b km (q k (t)) = b km (k, m ∈ I), the MQNN model (1) reduces into the following QNN model By using Theorem 1 and 3, we can receive the following consequences.

Examples
We will give two instances to prove the obtained outcomes and make some comparisons with the previous works. (1) with Λ = diag{1.5, 3}, the activation function f m (·) = g m (·) = tanh(·), parameters a km (q k (t)) and b km (q k (t)) (k, m = 1, 2): is an M-matrix. Model (1) satisfies all conditions of Theorem 3, we know that model (1) is GES by mean of Theorem 3.

Example 1. Consider model
Taking   (1) can treated as a special case of system (2) in [18] when I(t) = constant vector. Theorem 3.1 in [18] should be able to be used to check the GES of the system (1). In fact, µ p (−D) + l Ã p + l B p = 0.3685 > 0 (p = 2), which does not meet Theorem 3.1 in [18]. Therefore, Theorem 3.1 in [18] can not be used for asserting the GES of the model (1). This shows that some improvements of Theorem 3.1 in [18] have been made.

Remark 6.
In [14], QNN with mixed delays were considered. Some sufficient conditions for the stability of the ES of the considered QNN system (1) were obtained by using the decomposing method (see Theorem 1 and 2 in [14]). The model (26) is a special case of the system (1) in [14]. Yet, Theorem 1 and 2 in [14] can not be used for checking the stability of the model (26), because Theorem 1 and 2 in [14] require the activity function to be decomposable, while there are not explicit real imaginary parts in the function tanh(q) q is a quaternion of system (26) and it is in-decomposable. This shows that the outcomes in [14] have been improved.

Conclusions
We have discussed the existence, uniqueness, and exponential stability of the equilibrium solution of MQNN with varying-time delays. Based on a new established derivative formula of the norm of quaternionic function, we have acquired some new GES criteria of the MQNN by employing the M-matrix theory and the inequality techniques. Conquering the shortcomings of the existing decomposition method, our direct quaternionic method can concisely analyze the MQNN. Compared with the existing decomposition method, our direct quaternionic method has a largely low computation cost. Moreover, the obtained algebraic criteria are formulated by the matrix of the quaternionic norm, which is easy to verify.
The direct quaternion method can contributed a new means to survey the dynamic behaviors for other types of QNN, such as QNN with impulses and stochastic QNN, which will be our further researches.