Non-fragile mixed H∞ and passive synchronization of Markov jump neural networks with mixed time-varying delays and randomly occurring controller gain fluctuation

This paper studies the non-fragile mixed H∞ and passive synchronization problem for Markov jump neural networks. The randomly occurring controller gain fluctuation phenomenon is investigated for non-fragile strategy. Moreover, the mixed time-varying delays composed of discrete and distributed delays are considered. By employing stochastic stability theory, synchronization criteria are developed for the Markov jump neural networks. On the basis of the derived criteria, the non-fragile synchronization controller is designed. Finally, an illustrative example is presented to demonstrate the validity of the control approach.


Introduction
There have been significant attentions on dynamic behaviors of neural networks, since they have various current and future potential applications, i.e., signal processing, optimization problems, pattern recognition and so forth. [1][2][3][4][5][6][7][8][9]. In particular, the study of Markov jump neural networks has been a significant topic during the past years, since this model can better describe the neural networks with different structures in real life. Generally speaking, the mode jumps displayed in the Markov jump neural networks are commonly considered to be governed by an ideal homogeneous Markov chain. With the help of analysis and synthesis for Markov jump systems, some remarkable results on Markov jump neural networks have been reported in the literature and the references therein [10][11][12][13][14][15][16][17][18][19].
On another research line, the synchronization problem has become a hot topic in the fields of neural networks [9,12]. When one neural network can synchronize the other neural network, they can display complicated dynamic behaviors, which can give an insight into the characteristics of neural network. As is well known, time delays exist in neural networks, such that there is a need for the synchronization problem with time delays [20,21]. Moreover, it is noted that another unavoidable factor affecting the synchronization in neural networks is the disturbance. As a result, several effective synchronization strategies for neural networks with disturbances have been proposed, especially for some finite-time cases [22][23][24][25][26][27][28]. PLOS  It is worth mentioning that the theory of passivity has provided a powerful tool in analyzing and synthesis of complex dynamic systems [29,30]. Note that some initial researches are on the mixed H 1 and passive filtering design, which can provide a more flexible design than common H 1 approach [31]. In addition, the non-fragile synchronization controller should be designed for controller gain fluctuation attenuation [32]. Furthermore, it can be found that the controller gain fluctuation can happen in a stochastic way [33]. Consequently, a natural question arises: how to solve the synchronization problem for Markov jump neural networks? Unfortunately, up to now, such a question has not been fully addressed and remains open.
This paper is to deal with the above question. In this paper, a stochastic variable is adopted for describing the controller gain fluctuation. Based on stochastic methods, synchronization criteria are first established, such that the drive and response Markov jump neural networks can be synchronized in the presence of mixed time-varying delays and disturbance. Base on the derived results, a design procedure is given for the synchronization controller.
The remainder of the paper is arranged as follows. The Markov jump neural network model is first introduced, and the non-fragile synchronization problem is formulated. The main results of the synchronization problem are then provided. Moreover, the simulation results are given and this paper is concluded in the end.
Notation: R n denotes the n dimensional Euclidean space, R mÂn denotes the set of m × n matrices. L 2 ½0; 1Þ denotes the space of square-integrable vector functions over [0, 1). ðO; F ; PÞ is a probability space, O is the sample space, F is the σ-algebra of subsets of the sample space and P is the probability measure on F . Pr{α} means the occurrence probability of the event α, and Pr{α|β} means the occurrence probability of α conditional on β. Efxg means the expectation of the stochastic variable x and Efxjyg means the expectation of the stochastic variable x conditional on the stochastic variable y. Ã denotes the ellipsis in symmetric block matrices and diagfÁ Á Ág denotes a block-diagonal matrix.
{r(t), t ! 0} is a right continuous continuous-time Markov process with Y ¼ fp ij g; 8i; j 2 I described as with Δt > 0, lim(o(Δt)/Δt) = 0 and π ij ! 0 (i; j 2 I ; j 6 ¼ i) is the transition rate from mode i at time t to mode j at time t + Δt, while p ii ¼ À P N j¼1; j6 ¼i p ij for 8i 2 I.

Assumption 1. The activation function f(x(t)) in Eq (1) is continuous and bounded, and satisfies
where f i (0) = 0, α, b 2 R, α 6 ¼ β and F À i and F þ i are known real constants. Denote Eq (1) as the drive neural network. For the sake of simplicity, we denote the Markov process r(t) by i indices. Moreover, it is assumed that the mode of the drive and the response neural networks can be identical all the time [34]. Then, the response neural network can be given by where u(t) denotes the control input and $ðtÞ 2 L 2 ½0; 1Þ is the disturbance. Define synchronization error as then it follows that We develop the following mode-dependent controller as: where K i 2 R nÂn is the mode-dependent controller gain and ΔK i is the controller gain fluctuation with where H i and E i are known constant matrices, Ç i ðtÞ 2 R kÂl satisfies with Pr where δ 2 [0, 1] is a known constant. Consequently, System (8) can be rewritten as The following definitions and lemmas are introduced. Definition 1. [31] System (15) is said to have mixed H 1 and passive performance γ, if there exists a constant γ > 0 such that for all t p > 0 and any non-zero $ðtÞ 2 L 2 ; where θ 2 [0, 1] represents the parameter that defines the trade-off between H 1 and passive performance. Definition 2. The mixed H 1 and passive synchronization of the Markov jump neural networks Eqs (1) and (6) is said to be achieved if System (15) can achieve the mixed H 1 and passive performance with the prescribed γ. Lemma 1. [35] For any positive symmetric constant matrix Q 2 R nÂn , scalars h 1 , h 2 satisfying h 1 < h 2 , a vector function 0 : ½h 1 ; h 2 ! R n , such that the integrations concerned are well defined, then

Results
In this section, delay-dependent synchronization conditions will be developed, based on which the non-fragile synchronization controller is designed. Theorem 1. For given upper bounds of mixed time-varying delays " t and " d, and given scalars θ and γ, the mixed H 1 and passive synchronization of the Markov jump neural networks Eqs (1) and (6) can be achieved in the sense of Definition 1 and 2, if there exist mode-dependent symmetric matrices P i > 0, symmetric matrices Q > 0, R > 0 and a constant ε > 0 such that hold for all i 2 I , respectively, wherê proof. Choose the Lyapunov-Krasovskii function as: where V 1 ðtÞ :¼ e T ðtÞP i eðtÞ; ð32Þ f ðeðtÞÞ :¼ f ðyðtÞÞ À f ðxðtÞÞ: The infinitesimal operator L of V(t) is defined by Then for each mode i, by taking the derivative of Eq (31) along the solution of System (15), one has p ij e T ðtÞP j eðtÞg; EfLV 2 ðtÞg ¼ Efe T ðtÞQeðtÞ À e T ðt À " tÞQeðt À " tÞg; ð39Þ By Lemma 1 and Lemma 2, it holds that: Non-fragile synchronization of Markov jump neural networks with mixed time-varying delays It follows from Assumption 1 that such that the following inequality holds eðtÞ f ðeðtÞÞ ½g À 1 ye T ðsÞeðsÞ À 2ð1 À yÞe T ðsÞ$ðsÞ À g$ T ðsÞ$ðsÞds: ð48Þ Noting that EfðdðtÞ À dÞ 2 g ¼ dð1 À dÞ, it can be deduced that where  ; ð52Þ Z 1 ðtÞ ¼ ½e T ðtÞ; e T ðt À tðtÞÞ; e T ðt À " tÞ; $ T ðtÞ T ; ð56Þ By Schur complement [38], it can be obtained that " P i < 0 is equivalent toP i < 0, wherê ; ð59Þ ; ð60Þ 1   2   6  6  6  6  6  6  4   3   7  7  7  7  7  7  5 : ð61Þ By performing congruence transformation of diagfI; I; I; I; I; I; I; P i ; P i g to Eq (58) and considering the inequality −P i R −1 P i R − 2P i ,P i can be further rewritten as where ; ð64Þ : Then, it can be derived by Lemma 3 that EfLVðtÞ þ g À 1 ye T ðsÞeðsÞ À 2ð1 À yÞe T ðsÞ$ðsÞ À g$ T ðsÞ$ðsÞg < 0 holds, if P i < 0. Therefore, under zero initial condition, it can be obtained by integrating both sides of Eq (48) that J 0 holds, which means that the mixed H 1 and passive synchronization of the Markov jump neural networks is well achieved according to Definition 2 and completes the proof.
Theorem 2. For given upper bounds of mixed time-varying delays " t and " d, and given scalars θ and γ, the mixed H 1 and passive synchronization of the Markov jump neural networks Eqs (1) and (6) can be achieved in the sense of Definition 1 and 2, if there exist mode-dependent symmetric matrices P i > 0, mode-dependent matrices V i , symmetric matrices Q > 0, R > 0 and a constant ε > 0 such that hold for all i 2 I , respectively, where ; ð69Þ 2   6  6  6  6  6  6  6  6  6  6  4   3   7  7  7  7  7  7  7  7  7  7  5 ; ð72Þ  (22) and the controller gain can be obtained as The rest of the proof can follow from the proof of Theorem 1 directly.

Discussion
In order to verify our designed synchronization scheme, the following simulation example is presented.
Consider the Markov jump neural networks with two modes, where ; and the neuron activation function is taken as which satisfies Assumption 1 with In the simulation, the transition probability matrix is given as where the time step is set as Δt = 0.01.
The controller gain fluctuation satisfies the condition Eq (9)  By solving C i < 0, i = 1, 2 in Theorem 2, the mode-dependent controller gains can be obtained as follows:

Conclusion
The non-fragile mixed H 1 and passive synchronization problem of Markov jump neural networks with mixed time-varying delays is addressed. By utilizing the stochastic stability theory, delay-dependent criteria are derived for ensuring that the desired synchronization is achieved and the non-fragile synchronization controller is designed. An interesting further research direction is extending the derived results to the uncertainty cases.