A Delay-Dividing Approach to Robust Stability of Uncertain Stochastic Complex-Valued Hopﬁeld Delayed Neural Networks

: In scientiﬁc disciplines and other engineering applications, most of the systems refer to uncertainties, because when modeling physical systems the uncertain parameters are unavoidable. In view of this, it is important to investigate dynamical systems with uncertain parameters. In the present study, a delay-dividing approach is devised to study the robust stability issue of uncertain neural networks. Specifically, the uncertain stochastic complex-valued Hopfield neural network (USCVHNN) with time delay is investigated. Here, the uncertainties of the system parameters are norm-bounded. Based on the Lyapunov mathematical approach and homeomorphism principle, the sufﬁcient conditions for the global asymptotic stability of USCVHNN are derived. To perform this derivation, we divide a complex-valued neural network (CVNN) into two parts, namely real and imaginary, using the delay-dividing approach. All the criteria are expressed by exploiting the linear matrix inequalities (LMIs). Based on two examples, we obtain good theoretical results that ascertain the usefulness of the proposed delay-dividing approach for the USCVHNN model.

This article focuses on the robust stability issue of the USCVHNN model. Firstly, we formulate a general form of the system model by considering both aspects of parameter uncertainties and stochastic effects. Based on the studies in References [42,43], we divide the delay interval into N equivalent sub-intervals. Then, we exploit the homeomorphism principle along with the Lyapunov function and other analytical methods in our analysis. Specifically, we derive new sufficient conditions with respect to the robust stability issue of a USCVHNN model in terms of linear matrix inequalities (LMIs). We obtain the feasible solutions by using the MATLAB software package. Finally, the obtained theoretical results are validated by two examples. The remaining part of this paper is organized as follows. In Section 2, we formally define the proposed model. In Section 3, we explain the new stability criterion. In Section 4, we present the numerical examples and the associated simulation results. Concluding remarks are given in Section 5.
Notations: in this article, the Euclidean n-space and n × n real matrices are denoted by R n and R n×n , respectively. Note that i represents the imaginary unit, that is, i = √ −1. In addition, C n and C n×n denote the n dimensional complex vectors and n × the n complex matrices, respectively. The complex conjugate transpose and matrix transpositionhe superscripts * and T, respectively. Besides that, a matrix P > 0 (P < 0) indicates that P is a positive (negative) definite matrix. On the other hand, diag{·} denotes the diagonal of the block diagonal matrix, while I denotes an identity matrix, while indicates a symmetric term in a matrix. Finally, E{·} stands for mathematical expectation, 1, n = 1, 2, ..., n, and (Ω, F, P) indicates a complete probability space with a filtration {F t } t≥0 .

Assumption 1.
The activation function θ j (·), j = 1, n, is able to satisfy the Lipschitz condition for all σ 1 , σ 2 ∈ C, that is, where b j is a constant.
From Assumption 1, one can obtain where B = diag{b 1 , ..., b n }. Remark 1. Assumption 1 is a generalization of the real-valued function that is able to satisfy the Lipschitz condition on C. In addition, it is valid by dividing its real and imaginary parts.
It must be noticed that, in a realistic analysis, the uncertainties associated with the neuron weight coefficients are unavoidable. Besides that, the network model is susceptible to the stochastic effects. As a result, it is important to take the parameter uncertainties and stochastic disturbances into account when analysing the stability of NN models. Therefore, the model in (5) can be stated as where C, S ∈ R n×n are the known constant matrices, and ω(t) is the n dimensional Brownian motion defined on (Ω, F, {F t } t≥0 , P). It is assumed that the parameter uncertainties M(t), N(t) = N R (t) + i N I (t) in (10) are able to satisfy: where U and V 1 , V 2 , V 3 , are the known real matrices, while W(t) is the time-varying uncertain matrix that satisfies We define Now, the CVNN model in (10) can be separated into both the real and imaginary parts, that is, From (13), the system can be written in an equivalent form of which is equivalent to Let Then, the model in (15) can be equivalently re-written as From (11) and (12), the parameter uncertainties M , N , satisfy the following: From (9) one can obtain The model in (16) has the following initial condition: the model in (16) turns out to be the subsequent CVNN of Lemma 1. [41] Let H(σ) : R n → R n be a continuous map. It satisfies the following conditions: Lemma 2. [41] Given any vectors of the form M, N ∈ R n , along with a positive-definite matrix P ∈ R n as well as a scalar > 0, the following inequality is always true:

An Analysis of Uniqueness and Stability
In this sub-section, we present the sufficient conditions for the existence, uniqueness, as well as global asymptotic stability of the considered CVNN model.

Delay-Independent Stability Criteria
Theorem 1. Based on Assumption 1, we can divide the activation function into two parts: real and imaginary. The model in (20) has a unique equilibrium point. The model is deemed globally asymptotically stable if there exist matrix P > 0 and scalar > 0 that are able to satisfy the following LMI: Proof. A map associated with (20) is defined, that is, We prove that the map H(ω) is injective by contradiction. Given ω 1 and ω 2 with Multiplying both sides of (23) by (ω 1 − ω 2 ) T P yields Taking the transpose on (24), we have Adding (24) and (25) results in From Lemma (2) and (26), we have Since is a positive constant, we can obtain from (18) that From (27) and (28), we have If (21) holds, by Schur complement, we have From (29) and (30), note that ω 1 = ω 2 indicates a contradiction. As a result, the map H(ω) is injective. In addition we prove that H(ω) → ∞ as ω → ∞. Based on (30) for sufficiently small λ > 0. By using the same derivation method of (29), we have One can obtain from (32) that λ ω 2 ≤ 2 ω P ( H(ω) − H(0) ).
As a result, we have H(ω) → ∞ as ω → ∞. We can see from Lemma 1 that the map H(ω) is homeomorphic on R n . In addition, there exists a unique pointω such that H(ω) = 0. Therefore, the model in (20) has a unique equilibrium pointω. Now, we prove the global asymptotic stability of the equilibrium point possessed by the model in (20). Based on the transformationω = ω −ω, the equilibrium point of the model in (20) can be converted to the origin. As such, we have˙ω whereθ(ω(t − ν)) =θ(ω(t − ν) +ω) −θ(ω).
To analyse the global asymptotic stability with respect to the equilibrium point of the model in (34), the Lyapunov functional is utilized, that is, where P > 0 and Q > 0. As a result, the time derivative of V(t, ω t ) and the solution of the model in (35) yieldṡ From (18), we have for > 0. Then, together with (36) and (37), we havė where It is easy to observe that Ω < 0 if and only if Q > B and −PM −M T P + Q PN − I < 0.
If we choose Q > B , then Ω < 0 if the condition of (21) holds. This indicates thatV(t, ω t ) < 0; therefore the origin of the model in (34) is global asymptotically stable, which is equivalent to the case of the equilibrium point of the model in (20).

Delay-Dependent Stability Criteria
In this section, we address the delay-dependent stability criteria for the model in (16) using the delay-dividing method.
Theorem 2. Based on Assumption 1, we can divide the activation function into two parts: real and imaginary. For given scalars 0 < ν and 0 < µ, the model in (16) is robust as well as globally asymptotically stable in the mean square, subject to the existence of matrices P > 0, N) and positive scalars 1 , 2 , γ in a way that the following LMI is satisfied: Proof. The Lyapunov function candidate pertaining to the model in (16) is formed as follows: Based on the Itô's differential rule, we can obtain the stochastic derivative of V(t, ω t ) along with the trajectories of the model in (16) as follows: Based on Lemma (3), Moreover, from (18) it follows that for 1 , 2 > 0. Then, combining with (42)-(44), we have As a result, based on (45), we can have where andΞ 1 ,Ξ 2 ,Ξ 3 ,Ξ 4 ,Ξ 6 ,Ξ 7 ,Ξ 8 andΞ 9 are defined in (39). By using Schur complement lemma, Φ + 1 γ Π T Π + Λ T PΛ can be written as Θ. As a result, It can be easily deduced that for Θ < 0, there must exist a scalar δ > 0 such that By taking the mathematical expectation on both sides of the model in (47), The above result indicates that the model in (16) is robust. it is also globally asymptotically stable in the mean square. This completes the proof.

Remark 3.
The delay-dividing number N is a positive integer and, at the same time, N should not be lower than 2. This conservatism is reduced as N increases, and the time-delay tends to approach certain upper boundary. When N equals to 2 or 3, the conservatism is small enough.

Remark 4.
In the situation where no uncertainties exist, the model in (10) becomes
Corollary 1 is obtained by setting M = N = 0 in the proof of Theorem (2).

Corollary 1.
Based on Assumption 1, we can divide the activation function into two parts: real and imaginary. For given scalars 0 < ν and 0 < µ, the model in (51) is globally asymptotically stable in the mean square, subject to the existence of matrices P > 0, N) and positive scalars 1 , 2 in a way that the following LMI is satisfied:

Remark 5.
In the situation where no uncertainties and stochastic effects exist, the model in (10) becomes Simultaneously, the model in (16) becomes Corollary 2 is obtained by setting M = N =C =S = 0 in the proof of Theorem 2.

Corollary 2.
Based on Assumption 1, we can divide the activation function into two parts: real and imaginary. For given scalars 0 < ν and 0 < µ, the model in (54) is globally asymptotically stable, subject to the existence of matrices P > 0, .., N) and positive scalars 1 , 2 is a way that the following LMI is satisfied:

Remark 6.
It should be noticed that, due to the existence of modeling errors and measurement errors, we cannot solve some of the problems by using an exact mathematical model in realistic scenarios. The model described in this paper contains the following parameter uncertainties M(t), N(t). As a result, it is assumed that the effects of parameter uncertainties in our study are under the time-varying and norm-bounded condition.

Remark 7.
It is worth noting that several stability results have been published recently without taking into account the parameter uncertainties and stochastic effects on CVNNs [26,27]. In this paper, we take into account both the parameter uncertainties and stochastic effects. Therefore, this paper includes more general stability criteria than those in References [26,27].

Illustrative Examples
Here, two numerical examples are provided to ascertain the usefulness of the derived outcomes presented in the previous section. Example 1. The CVNN model defined in (10) that contains the following parameters is considered: Assume that θ j (σ) = θ R j (α, β) + iθ I j (α, β), j = 1, 2. We can obtain the following based on a simple calculation: When N = 3, ν(t) = 0.2sint + 0.3, which satisfies ν = 0.5 and µ = 0.3. By using the MATLAB software package, the LMI condition of (40) in Theorem (2) is true.
Based on the initial valuesω 1 (t) = 0.5 − 0.3i andω 2 (t) = −0.6 + 0.4i, Figures 1 and 2 depict the state trajectories pertaining to both the real and imaginary parts of the model in (10). On the other hand, Figure 3 illustrates the phase trajectories pertaining to both the real and imaginary parts of the model in (10). It is obvious form Figures 1-3 that the model converges to an equilibrium point, which means that the model in (10) or equivalently the model in (16) is robust as well as globally asymptotically stable.

Conclusions
In this paper, we have investigated the robust stability issue of the USCVHNN models along with the consideration with their time-varying delay and stochastic effects. Based on the delay-dividing approach, we have formulated a more general form of the model under scrutiny by incorporating the parameter uncertainties and stochastic effects. Following the studies in References [42,43], we have divided the delay interval into N equivalent sub-intervals. In addition, we have exploited the homeomorphism principle along with the Lyapunov function and other analytical methods in our analysis. As a results, we have derived new sufficient conditions in terms of LMIs, which allow us to analyse the robust stability issues pertaining to USCVHNN models. The feasible solutions have been obtained by using the MATLAB software package. Finally, the obtained theoretical results are ascertained through two numerical examples.
For further research, we will extend our proposed approach to analysing other relevant types of CVNN models. In this regard, we intend to undertake an investigation of the fuzzy stochastic CVNN models.