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. [1–9]. 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 [10–19].

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 [22–28].

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_∞ and passive filtering design, which can provide a more flexible design than common H_∞ 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: denotes the n dimensional Euclidean space, denotes the set of m × n matrices. denotes the space of square-integrable vector functions over [0, ∞). is a probability space, Ω is the sample space, is the σ-algebra of subsets of the sample space and is the probability measure on . Pr{α} means the occurrence probability of the event α, and Pr{α|β} means the occurrence probability of α conditional on β. means the expectation of the stochastic variable x and means the expectation of the stochastic variable x conditional on the stochastic variable y. * denotes the ellipsis in symmetric block matrices and denotes a block-diagonal matrix.

Methods

Consider the Markov jump neural networks with mixed time-varying delays in : (1) where x(t) = [x₁(t), x₂(t), …, x_n(t)]^T denotes the state of the neuron; f(x(t)) = [f₁(x₁(t)), f₂(x₂(t)), …, f_n(x_n(t))]^T is the neuron activation function; is a diagonal matrix with positive entries; Matrices A(r(t)) = (a_ij(r(t)))_n×n, B(r(t)) = (b_ij(r(t)))_n×n and D(r(t)) = (d_ij(r(t)))_n×n represent the connection weight matrix, the discretely delayed connection weight matrix and the distributively delayed connection weight matrix, respectively; τ(t) and d(t) denote the discrete delay and distributed delay, respectively, which satisfy (2) (3) where , μ, and are known positive constants. The initial condition of Eq (1) is given by x(s) = ϕ(s), .

{r(t), t ≥ 0} is a right continuous continuous-time Markov process with described as (4) with Δt > 0, lim(o(Δt)/Δt) = 0 and π_ij ≥ 0 () is the transition rate from mode i at time t to mode j at time t + Δt, while for .

Assumption 1. The activation function f(x(t)) in Eq (1) is continuous and bounded, and satisfies (5) where f_i(0) = 0, α, , α ≠ β and and 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 (6) where u(t) denotes the control input and is the disturbance.

Define synchronization error as (7) then it follows that (8)

We develop the following mode-dependent controller as: (9) where is the mode-dependent controller gain and ΔK_i is the controller gain fluctuation with (10) where H_i and E_i are known constant matrices, satisfies (11)

is defined by (12) with (13) (14) where δ ∈ [0, 1] is a known constant.

Consequently, System (8) can be rewritten as (15)

The following definitions and lemmas are introduced.

Definition 1. [31] System (15) is said to have mixed H_∞ and passive performance γ, if there exists a constant γ > 0 such that (16) for all t_p > 0 and any non-zero where θ ∈ [0, 1] represents the parameter that defines the trade-off between H_∞ and passive performance.

Definition 2. The mixed H_∞ 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_∞ and passive performance with the prescribed γ.

Lemma 1. [35] For any positive symmetric constant matrix , scalars h₁, h₂ satisfying h₁ < h₂, a vector function , such that the integrations concerned are well defined, then (17)

Lemma 2. [36] For any matrix M > 0, scalars τ > 0, τ(t) satisfying 0 ≤ τ(t)≤τ, vector function such that the concerned integrations are well defined, then (18) where (19) (20)

Lemma 3. [37] Let L^T = L, H and E be real matrices of appropriate dimensions with F(t) satisfying F^T(t)F(t) ≤ I. Then, L + HFE + E^T F^T H^T < 0, if and only if there exists a scalar ε > 0 such that L + ε⁻¹ HH^T + εE^T E < 0, or equivalently (21)

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 and , and given scalars θ and γ, the mixed H_∞ 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 (22) hold for all , respectively, where (23) (24) (25) (26) (27) (28) (29) (30)

proof. Choose the Lyapunov-Krasovskii function as: (31) where (32) (33) (34) (35) (36)

The infinitesimal operator of V(t) is defined by (37)

Then for each mode i, by taking the derivative of Eq (31) along the solution of System (15), one has (38) (39) (40) (41)

By Lemma 1 and Lemma 2, it holds that: (42) (43)

It follows from Assumption 1 that (44) (45) such that the following inequality holds (46) (47)

Define (48)

Noting that , it can be deduced that (49) where (50) (51) (52) (53) (54) (55) (56) (57)

By Schur complement [38], it can be obtained that is equivalent to , where (58) (59) (60) (61)

By performing congruence transformation of to Eq (58) and considering the inequality −P_i R⁻¹ P_i ≤ R − 2P_i, can be further rewritten as (62) where (63) (64) (65) (66)

Then, it can be derived by Lemma 3 that holds, if Π_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_∞ 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 and , and given scalars θ and γ, the mixed H_∞ 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 (67) hold for all , respectively, where (68) (69) (70) (71) (72) (73) , are defined in Eq (22) and the controller gain can be obtained as .

proof. Let V_i = P_i K_i. 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 and , such that

In the simulation, the transition probability matrix is given as where the time step is set as Δt = 0.01.

The time-varying delays are assumed to be τ(t) = 0.25 + 0.05 sin t and d(t) = 0.15 + 0.05 cos t, such that and . The disturbance is supposed to be The parameters δ, θ and γ are set by δ = 0.5, θ = 0.4 and γ = 0.2.

The controller gain fluctuation satisfies the condition Eq (9) with

By solving Ψ_i < 0, i = 1, 2 in Theorem 2, the mode-dependent controller gains can be obtained as follows:

Set the initial values as x(t) = [1, −1]^T and y(t) = [−5, 5]^T, respectively. Under the modes evolution shown in S1 Fig, it can be seen from S2 and S3 Figs that the synchronization can be achieved with the designed mode-dependent controllers, which demonstrates our control scheme.

Conclusion

The non-fragile mixed H_∞ 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.

Supporting information