Sampled-Data Exponential Synchronization of Markovian Jump Chaotic Lur'e Systems with Multiple Time Delays

This paper studies the mean square exponential synchronization control of the Markovian jump chaotic Lur’e systems (MJCLSs) with multiple time delays via aperiodic sampling. Firstly, by introducing the mode-dependent single/double integral terms and considering the system state at the sampling instant, a new mode-dependent Lyapunov functional based on the looped-functional is constructed. On the basis of the constructed functional and by estimating the expectation of the solution of MJCLSs with multiple time delays, an exponential synchronization criterion in the mean square sense is presented. Then, based on the obtained stability conditions, a design method for aperiodic sampled-data controller (ASDC) to ensure the exponential synchronization is proposed. Finally, the effectiveness of the developed method is demonstrated by the Chua’s circuit model and a neural network model. Compared with the existing literature, an admissible larger upper bound of sampling period can be obtained.


Introduction
The Lur'e system is a special kind of nonlinear systems proposed by Lur'e in the 1940s when studying the automatic pilot control of aircraft. It consists of a linear term and a sector-bounded nonlinear term. In [1][2][3][4], a large number of actual nonlinear systems such as positive rolling attractors, hyperchaotic systems, Chua's circuits and neural networks can all be simulated as chaotic Lur'e systems. However, the structure of an actual control system is often changed due to sudden external environment disturbances, changes in the coupling part of the subsystem and the random component failures and maintenance. In this case, we can use lems. Considering the unknown transfer rate, and by constructing a mode-dependent Lyapunov functional, sampled-data synchronization of MJNNs was reported in [32,33]. Based on constructing mode-dependent one-sided and two-sided looped-functionals, the random sampled-data synchronization problem of MJNNs was studied in [34] and Ref. [34] concluded that two-sided looped-functionals has lower conservative. The H∞ synchronization problems of MJNNs were studied by constructing a composite disturbance rejection controller and a mode-dependent synchronous controller in [35,36], respectively.
Unlike such a wealth of research on sampled-data synchronization of MJNNs, the sampled-data synchronization of MJCLSs is less concerned, therefore, it needs further Consideration. Considering the event-trigger mechanism and the quantization effect, Ref. [37] studied the asynchronous control problem of MJCLSs. By considering partial accessibility and using hidden Markov model, Ref. [38] discussed the problem of asynchronous control of MJCLSs with limited time interval. By fully considering the probability distribution characteristics of packet loss, the random synchronization problem of a semi-MJCLSs with packet loss and multi-period sampling was studied in [39]. Through the above discussion, we can find that most of the chaotic Lur'e systems studied in the existing published papers are with single time-delay, and rarely literature considers the situation of multiple time delays. In addition, for MJCLSs, compared with the mode-independent Lyapunov functional, the mode-dependent Lyapunov functional is less conservative in many cases. However, in most studies of MJCLSs under aperiodic sampling, only the mode-dependent of the partial Lyapunov matrices is considered. However, when the single integral and double integral terms are introduced to construct the functional, the introduction of the mode-dependent matrices will inevitably bring about less conservative results. Furthermore, if one considers the use of sampled states to construct functionals, is it possible to construct mode-dependent functionals as well? Considering the above problems, the ASDC is further designed to guarantee exponential synchronization of the error system by constructing a lower conservative functional.
The following contributions have been presented: 1. A lemma to estimate the expectation of the solution of the MJCLSs with multiple time delays is proposed.
2. By introducing mode-dependent single/double integral terms to the Lyapunov functional, and considering mode-dependent information of the sampled-state, a innovative Lyapunov functional is constructed.
3. On the basis of the constructed Lyapunov functional and the improved integral inequality, the exponential synchronization criterion in the mean square sense is presented, and the ASDC is designed.
Notations: The following are the significant symbols used in this article. R: real numbers set; R n : n-
The objective of this research is to develop a new design method for ASDC via a novel Lyapunov functional to make the master system M and the slave system N mean square exponential synchronize. Before giving the results of stability analysis, we supply a stability definition and two useful lemmas. there exists a decay coefficient β > 0 and a decay rate α > 0 such that where Lemma 1. Concerning system (3), the following inequality holds: where Proof. Integrating system (3) from t k to t leads to In addition, from Cauchy-Schwarz inequality, we can easily obtain that It can be found from (4) that ξ(Dς(t), y(t)) 2 ≤ S D 2 ς(t) 2 , and 5h M E

Stability analysis and controller design
In this section, a mode-dependent two-sided looped-functional was constructed. Using the novel functional, we discuss the exponentially synchronization in the mean square sense for MJCLSs with multiple time delays.
The ASDC design scheme for error system is proposed. For simplicity, we use nomenclature for vectors and matrices in the following.
Theorem 1. Given scalars α > 0, δ and γ, the slave system (2) is exponentially synchronization in the mean square sense to the master system (1) if there exist positive definite matrices . . , m: Furthermore, the ASDC gain matrix in (3) is Proof. Constructing mode-dependent two-sided looped-functional for the system (3) with Let L be the weak infinitesimal operator of the random process Then, for any i ∈ N , calculating LV(ς(t), i, t) of (3) gives and It follows from (11) and that is, from (11), there is the following inequality Therefore, Similarly, the inequality (12) implies the following inequality: We further apply Lemma 2 and get that is, The inequality (13) implies the following inequality: so it is easy to get .
In addition, for any a free weighting matrix G i , and scalars γ and δ, we always have Furthermore, on the basis of (5), the following inequality is satisfied: , y(t)) − ξ(Dς(t), y(t)) T V 2,i ξ(Dς(t), y(t)) , We insert the right side of (19) and (20) to LV and let L i = G i K i , then LV(ς(t), i, t) can be expressed as: It is noted thatΞ From (9) and (13), we can find thatΞ Thus, we can further get that It suggests that, for t ∈ [t k , t k+1 ), one has On the basis of Lemma 1 and (23), we can conclude that It can be computed that where ̺ = max{̺ 1 + ̺ 2 + ̺ 3 , ̺ 4 + ̺ 5 + ̺ 6 }, Combining (24) with (25), we can get that Hence, by Definition 1, the master-slave systems (1) and (2) are exponentially synchronous in the mean square.

Remark 2.
In the proof of Theorem 1, we introduced mode-dependent single/double integral terms and considered the mode-dependent sampled-state information to construct functionals. For example, since dif-ferent U i is selected for distinct system mode, t t k e 2αsς T (s) N j=1 r ij U j ς(s)ds is consequentially involved in LV 4 (ς(t), i, t). To achieve the stability result, we need to cope with it. As seen clearly from (18), Similarly, it can be seen from (16) and (17) Remark 3. In Theorem 1, one of the innovation of Lyapunov functionals V(ς(t), i, t) in (15) lies in that some mode-dependent two-sided functions i.e., V 5 , V 6 , V 7 which not only fully utilize the interval message from ς(t k ) to ς(t) but also consider the information from ς(t) to ς(t k+1 ). The looped-functional approach has been used to analyze system stability in [42]. However, Ref. [42] did not consider the influence of Markovian parameters on the construction of functionals. To bridge this gap, this paper constructs mode-dependent looped-functional.
To make comparison, on the basis of Theorem 1, a simplified delayed system is proposed in which Markovian parameters is not considered.
Consider the following master system: and slave system N : where u(t) = K (q(t k ) − p(t k )), t k ≤ t < t k+1 , d is the constant time-delay. Hence, the error system (3) reduces toς For simplicity, we use nomenclature for vectors and matrices in the following.
Proof. Introduce simplified two-sided looped-functional, with V 1 (t) = e 2αt ς T (t)P ς(t), According to the derivative of V (t) along the trajectory of the error system (29), we can obtaiṅ In addition, for any free weighting matrix G, and scalars γ and δ, we always have: and similarly the following inequality is satisfied: Furthermore, referring to the proof of Theorem 1, the ASDC gain matrix in (29) is

Numerical Examples
In this section, two examples are given to demonstrate the reduced conservatism of the synchronization criterion and the effectiveness of the ASDC design approach.
Furthermore, consider the Chua's circuit with a single time-delay and without Markovian jump parameters, that is, let A = A 1 , B = B 1,1 , W = W 1 , C = C 1 , d = 1, and keep η σ (x σ (t)), δ, γ and h m unchanged. Based on Corollary 1, the specific numerical comparison is shown in Table 2. From comparison with [19] and [20], it can be seen that our method can enlarge the upper bound of the sampling period h M . Therefore, the method developed in this paper significantly reduces the conservativeness.
Take it one step further, choosing d = 0, α = 0, Table 3 shows the comparison of our result with some previous results when considering the Lur'e system without time-delay. It is clear that the results obtained in this paper are better than the existing results.   (1) and (2) with the following parameters: Based on above parameters, the system is reduced to a three neurons neural network with two mode.
Moreover, the activation functions is η σ (x σ (t)) = 1 2 (|x σ (t) + 1| − |x σ (t) − 1|), σ = 1, 2, 3, and thus s 1 = s 2 = s 3 = 1. Denote δ = 0.5, γ = 2, h m = 10 −5 . Using Theorem 1, the relationship between the maximum values of h M and α is displayed in Table 4. Choosing α = 0.2 and h M = 0.3288, by solving the LMIs (9)-(13), the   Figure 1: Open-loop state trajectory of the error system, closed-loop state trajectory of the error system, sampled-data control input and aperiodic sampled-data interval, chaotic behaviors of master system and slave system gain matrix K 1 , K 2 can be computed as:   Figure. 2(a) when there is no ASDC. We can get that the error system (3) itself is unstable. On the basis of the obtained ASDC, by Theorem 1, Figure. 2(b) displays the Markovian jump mode and the corresponding state trajectories of the error system (3). Furthermore, Figure. 2(c) displays the control input under the action of ASDC and aperiodic sampling interval, from which one can get that the closed-loop system is stable. Figure. 2(d) shows the displayed chaotic behavior. Therefore, the ASDC   Figure 2: Open-loop state trajectory of the error system, closed-loop state trajectory of the error system, sampled-data control input and aperiodic sampled-data interval, chaotic behaviors of master system and slave system can successfully stabilize the unstable error system. The master-slave systems exponential synchronization is achieved.
Furthermore, consider the neurons neural network without Markovian jump parameters, that is, let A = A 1 , W = W 1 , C = C 1 , and keep D, η σ (x σ (t)), δ, γ and h m unchanged. Baed on Corollary 1, Table 5 shows the comparison of our result with some previous results, our method can enlarge the upper bound of the sampling period h M . Therefore, the approach obtained in this article is effective and can ensure a lower conservativeness.

Conclusion
The presented paper has proposed the mean square exponential synchronization criterion and the ASDC design method for the MJCLSs with multiple time delays. A novel construction of Lyapunov functionals is given by introducing mode-dependent single/double integral terms and mode-dependent sampled-state information. The estimation of the expectation of the solution plays a vital role in obtaining the exponential stability criterion in the mean square sense. On the basis of the stability criterion, a mode-dependent ASDC has been presented. Finally, a Chua's circuit and a neural network have been provided as numerical simulation examples to demonstrate the effectiveness of the raised methods.