Robust dissipativity and passivity of stochastic Markovian switching CVNNs with partly unknown transition rates and probabilistic time-varying delay

: This article addresses the robust dissipativity and passivity problems for a class of Markovian switching complex-valued neural networks with probabilistic time-varying delay and parameter uncertainties. The main objective of this article is to study the proposed problem from a new perspective, in which the relevant transition rate information is partially unknown and the considered delay is characterized by a series of random variables obeying bernoulli distribution. Moreover, the involved parameter uncertainties are considered to be mode-dependent and norm-bounded. Utilizing the generalized Itˆ o ’s formula under the complex version, the stochastic analysis techniques and the robust analysis approach, the ( M , N , W )-dissipativity and passivity are ensured by means of complex matrix inequalities, which are mode-delay-dependent. Finally, two simulation examples are provided to verify the e ﬀ ectiveness of the proposed results.


Introduction
Over the past several decades, dynamical performances of complex-valued neural networks (CVNNs) have drawn a lot of sensitive attention owing to their broad application prospect, such as signal processing, associative memory, pattern recognition, engineering optimization [1][2][3] and the references therein. CVNNs can effectively solve not only the real-valued information problems but also the complex-valued ones under complex plane condition. In addition to this, CVNNs have the strong advantage in comparison with the real-valued neural networks (RVNNs), rates with uncertainties. For instance, the stability issue of delayed Markovian networks has been investigated [25], where transition rate information is partly known. The exponential stability issue of mixed delayed impulsive Markovian jump networks with general incomplete transition rates has been studied in [26]. Regardless of these recent developments, up till now, when simultaneously consider all factors, including the stochastic disturbances, Markovian switching with partly known transition rates, probabilistic time-varying delay and uncertain parameters, there is no relevant results on dissipativity/passivity problem for complex-valued networks in the complex domain, which becomes the most important motivation to investigate this research.
In response to the statements given above, the main goal is to study the robust dissipativity and passivity issues of Markovian switching CVNNs, which involve stochastic disturbance, probabilistic time-varying delay and partly unknown transition rates. Here, the main novelties are primarily summarized as follows. (1) Partly unknown transition rates are considered for the first attempt to address robust passivity and dissipativity problems for stochastic Markovian switching CVNNs with probabilistic time-varying delay and norm-bounded uncertainties. (2) A stochastic variable in time-varying delay is introduced to analyse dissipativity and passivity issues for considered delayed complex-valued neural networks, which satisfies the Bernoulli random binary distribution. (3) By taking advantage of the robust analysis technique, stochastic analysis approach, Lyapunov stability theory and the generalised Itô's formula, sufficient criteria on (M, N, W)-dissipativity/passivity are obtained with the intuitionistic form of complex matrix inequalities, which are delay-mode-dependent, (4) Simulation results are given, which could clearly show that the stochastic factors, i.e., the Markovian process and the Brownian motion, have significant effect on the dissipativity/passivity performance index.
In this paper, the remainder is outlined as follows. Section 2 shows the considered model description and some necessary preliminaries. Section 3 derives the robust dissipativity and passivity criteria for the stochastic delayed Markovian switching CVNNs with probabilistic time-varying delay and partly known transition rates through utilizing the general Lyapunov functional method in the complex domain. Section 4 gives two illustrative numerical simulations to verify the viability of the presented results. In the end, the conclusion is given in Section 5.
Notations: R n and C n denote, respectively, n-dimensional real vectors and n-dimensional complex vectors. R m×n and C m×n are m × n real and complex matrices. I denotes the identity matrix with appropriate dimensions. The (Ω, X, {X t } t≥0 , W) is the complete probability space, in which X t is monotonically right continuity, and X 0 includes whole W-null sets. The superscript 'T ' stands for the matrix transposition. The superscript 'H' denotes the matrix complex conjugate transpose. i denotes the imaginary unit. ' * ' denotes the elements involved by symmetry in a matrix. col E{·} means the mathematical expectation.

Problem formulation and preliminaries
Consider the following stochastic Markovian switching CVNNs with probabilistic time-varying delay and uncertain parameters: where x(t) = col(x ι ) n ι=1 ∈ C n means the state vector of the network at time t, which involves n nodes. C(s(t)) = diag{c ι (s(t))} n ι=1 ∈ R n×n stands for the self-feedback weight matrix with every entry c ι (s(t)) > 0, A(s(t)) = (a ιm (s(t))) n×n and B(s(t)) = (b ιm (s(t))) n×n denote, respectively, the connection weight matrix and the delayed connection weight matrix and they belong to C n×n . f (x(t)) = col( f ι (x ι (t))) n ι=1 : C n → R n and g(x(t)) = col(g ι (x ι (t))) n ι=1 : C n → C n stand for, respectively, the neuron activation function without and with time delay. (t) = col( ι (t))) T ∈ R n and y(t) = col(y ι (t))) T ∈ R n stand for, respectively, the external input vector and the output vector. h(t, x(t), x(t−τ(t))) : R×C n ×C n → C n×n is the noise density function. ω(t) stands for the n-dimensional Brownian motion, which is defined on (Ω, X, {X t } t>0 , W). τ(t) is called as time-varying probabilistic delay, which often satisfies the following equation: (2.2) in which τ 1 (t) ∈ [τ 1 ,τ] and τ 2 (t) ∈ (τ, τ 2 ] with τ 1 ≤τ ≤ τ 2 being known positive numbers. Moreover, The stochastic process {s(t), t ≥ 0}, taking valid values in a set S {1, 2, . . . , N}, denotes a continuous-time Markov process, where the transition rate matrix Π [ ab ] N×N is defined in the form of probability type as follows: where θ > 0, when a b, ab ≥ 0 refers to the transition rate which jumps mode a at time t to mode b at time t + θ, lim θ→0 (o(θ)/θ) = 0, and aa = − N b=1,b a ab . Obviously, the well-known fact is that transition rates under the Markov process can directly influence the behavior of the Markovian switching systems, it is further assumed that some elements of the transition rates are partly available. Next, for every a ∈ S, let S S a uk ∪ S a uc with S a uk {b : ab is unknwon} and S a uc {b : ab is uncertain}. Moreover, if S a 2 ∅, S a 1 can be expressed as in which m is a positive integer belonging to {1, . . . , N − 2}. In transition rate matrix Π, K a s (s ∈ {1, 2, . . . , m}) denotes the sth foreknown element in the ath row. For further facilitate analysis, when s(t) = a, the presented matrices C(s(t)), A(s(t)), B(s(t)), ∆C(s(t)), ∆A(s(t)), and ∆B(s(t)) are, respectively, simplified as C a , A a , B a , ∆C a , ∆A a , and ∆B a .
The mode-dependent parameter uncertainties ∆C a ∈ R n×n , ∆A a ∈ C n×n , and ∆B a ∈ C n×n are assumed to satisfy To simplify further analysis, η(t), a Bernoulli distributed white sequence, is introduced as follows: Combined with the above analysis, we rewrite the network (2.1) as Remark 2.1. It is worth noticing that random variable η(t) owns the corresponding statistical properties with is independent with ω(t) and s(t).
For further discussion, the given nonlinear activation functions satisfy the following conditions which will be used later.
Assumption 2.1. The considered activation functions f ι (·), g ι (·) (ι = 1, 2, . . . , n) satisfy the Lipschitz condition and f ι (0) = g ι (0) = 0, i.e., there exist positive constants σ ι , ρ ι such that Assumption 2.2. There exist positive semi-definite Hermitian matrices V 1 and V 2 of appropriate dimensions satisfying the inequality below: Remark 2.2. It's worth noting that, the nonlinear functions in Assumption 2.1 are usually looked upon as the expansion of the real-valued ones with the Lipschitz condition. Moreover, the existing literatures concerning CVNNs are adopted to decompose the considered CVNNs into two real-valued networks, which make the achieved matrix dimension will be twice as large and increase the computational complexity [27,28]. In view of these points, it is urgent to further consider the dynamic behaviors of CVNNs in complex domain.
In this article, the robust dissipativity/passivity criteria will be established for system (2.1) by utilizing the mode-dependent Lyapunov-Krasovskii functional. Before stating the main results, we present some useful definitions and lemmas.
For CVNN (2.1), set energy input-output function H as H( , y, t) 2 y, N t + , W t + y, My t , ∀t ≥ 0, (2.7) in which N is a real matrix, M and W are Hermitian matrices, and y, N t stands for t 0 y H (s)N (s)ds.
Definition 2.1. when the initial constraint is zero, if there owns a scalar γ > 0 that makes the inequality below Definition 2.2. when the initial constraint is zero, if there owns a scalar γ > 0 that makes the inequality below hold, from the input (·) to the output y(·), system (2.1) is robustly passive in the sense of expectation.
Definition 2.3. [29,30] Consider a n-dimensional stochastic Markovian switching complex-valued differential equation: , and the conjugate R derivative of Ψ [31] as in which the conjugate vector of φ isφ. All functions Ψ(t, φ, a) : R + × C n × S → R + are C 1,2 (R + × C n × S, R + ), which means twice differentiable in φ andφ and once continuously differentiable in t. Then for all Ψ(t, φ, a), the complex version of the generalized Itô's formula could be given as the form below: , a) for simplicity, In addition, the operator L on Ψ(t, φ, a) is defined as (2.9) Lemma 2.1.

Remark 3.2.
It is an apparent fact that the presented matrix inequalities (3.1)- (3.4) in Theorem 3.1 are all complex-valued, which cannot be directly solved via Matlab Toolbox. In view of this, we can utilize the method firstly proposed in [39] to solve a complex Hermitian matrix P satisfies P < 0 if and only if Re(P) Im(P) −Im(P) Re(P) < 0, where Re(P) and Im(P) refer to, respectively, the real and imaginary part of matrix P. In this case, the obtained complex-valued matrix inequalities can be transformed into real-valued matrix inequalities, which can be solved by adopting the standard Matlab Toolbox.
After acquiring the analysis in Theorem 3.1, set N = I, M = 0 and W = 2γI, it can directly obtain the robust passivity criterion of system (2.1), which can be presented in the theorem below. Hermitian matrix U a with appropriate dimensions, diagonal matrices Λ ς > 0 (ς = 1, 2, 3, 4, 5, 6), constants ϑ > 0, λ > 0, δ > 0 and ν ς > 0 such that matrix inequalities (3.1), (3.3) and (3.4) and the inequality below uniformly are valid: inaccessible, which leads to two common cases: S a uk = ∅, S a k = S or S a k = ∅, S a uk = S. Therefore, it takes great limitations or restrictions to practical applications. In reality, it is very difficult to be measure and require the transition rate information due to random factors. Hence, taking into partly unknown transition rates account, it is urgent to research Markovian switching systems and some relevant results have been reported [42,43]. Based on these considerations, analysis of this paper are more meaningful.
Remark 3.4. It should be pointed out that when dealing with the stochastic CVNNs, the considered system will be decomposed into real and imaginary parts, which means that the dimensions will be doubled and the computational complexity will increase [36,37]. Besides, the adopted method is the general real Itô's formula. However, in this paper, compared to [36,37], the main advantages of the results are three parts. The first one is that replacing the real-imaginary separation technique, we discuss the system performance in the complex domain; the second one is that in virtue of the generalised Itô's formula in the complex domain and stochastic analysis method, mode-delay-dependnt criteria are obtained; the third one is that the considered transition rate information is partly unknown, which further reflect realistic significance.

Numerical examples
This section provides two examples to show the effectiveness and validity of the obtained results.

Conclusions
The robust dissipativity/passivity problem for stochastic Markovian switching CVNNs with probabilistic time-varying delay is probed in this work. The considered probabilistic delay is characterized by a series of random variables obeying bernoulli distribution. Moreover, the concerned parameter uncertainties are not only norm-bounded but also mode-dependent. For the aim of reflecting more realistic dynamics of the presented model, transition rate information is partly acquainted. Combined robust analysis tools, stochastic analysis methods with generalized complex Itô's formula, some sufficient mode-delay-dependent criteria on the (M, N, W)-dissipativity/passivity have been derived by means of complex linear matrix inequalities. In the end of paper, two effective examples are presented to support and clarify the validity and correctness of our proposed research results.