 Research
 Open Access
Fixedtime synchronization of semiMarkovian jumping neural networks with timevarying delays
 Wei Zhao^{1} and
 Huaiqin Wu^{1}Email author
https://doi.org/10.1186/s136620181666z
© The Author(s) 2018
 Received: 22 January 2018
 Accepted: 28 May 2018
 Published: 20 June 2018
Abstract
This paper is concerned with the global fixedtime synchronization issue for semiMarkovian jumping neural networks with timevarying delays. A novel statefeedback controller, which includes integral terms and timevarying delay terms, is designed to realize the fixedtime synchronization goal between the drive system and the response system. By applying the Lyapunov functional approach and matrix inequality analysis technique, the fixedtime synchronization conditions are addressed in terms of linear matrix inequalities (LMIs). Finally, two numerical examples are provided to illustrate the feasibility of the proposed control scheme and the validity of theoretical results.
Keywords
 Fixedtime synchronization
 SemiMarkovian jumping
 Neural networks
 Timevarying delay
1 Introduction
In the past decades, the neural networks (NNs) have been found extensive applications in many areas, such as pattern recognition, computer vision, speech synthesis, artificial intelligence and so on; see [1–3]. Such a wide range of applications attract considerable attention from many scholars to the dynamical behavior of the networks. Up to now, many significant works with respect to NNs have been reported; see [4–9], and the references therein.
Synchronization, which means that the dynamical behaviors of coupled systems achieve the same state, is a fundamental phenomenon in networks. At present, considerable attention has been devoted to the analysis of the synchronization of NNs and some effective synchronization criteria of NNs have been established in the literature [10–15]. Via the sliding mode control, the synchronization problem for complexvalued neural network was addressed in [12]. Reference [14] elaborates the impulsive stabilization and impulsive synchronization of discretetime delayed neural networks. By adopting the periodically intermittent control scheme, the exponential lag synchronization issue for neural networks with mixed delays was described in [15]. It should be pointed out that most of these synchronization criteria are based on the Lyapunov stability theory, which is defined over an infinitetime interval. However, from the practical perspective, we are inclined to realize the synchronization goal in a finitetime interval. Because in a finitetime interval the maximal synchronization time can be calculated through appropriate methods. Hence, it is significative to study the finitetime synchronization of NNs. In Ref. [16], the finitetime robust synchronization issue for memristive neural networks was discussed. By utilizing the discontinuous controllers, the finitetime synchronization issue for the coupled neural networks was addressed in [17]. And under the sampleddate control scheme, some finitetime synchronization criteria for inertial memristive neural networks were established in [18].
For the finitetime synchronization, the settling time heavily depends on the initial conditions, which may lead to different convergence times under different initial conditions. However, the initial conditions may be invalid in practice. In order to overcome these shortcomings, a new concept named fixedtime synchronization was firstly taken into account in [19]. Hints for future research on the fixedtime synchronization problem can be found in [20–25]. By designing a sliding mode controller, the fixedtime synchronization issue for complex dynamical networks was addressed in [21]. Robust fixedtime synchronization for uncertain complexvalued neural networks with discontinuous activation functions was introduced in [23]. Furthermore, the fixedtime synchronization issue for delayed memristorbased recurrent neural networks was investigated in [25].
As is well known, time delay is inevitable in the process of transitional information because of the finite velocity of the transmission signal. Time delays often cause the systems to be instable and oscillatory. Thus, considering the synchronization of NNs with delays is meaningful. Owing to the value of the delay not always being fixed, exploring the synchronization of NNs with timevarying delays has become the subject of great interests for many scholars. Finitetime and fixedtime synchronization analysis for inertial memristive neural networks with timevarying delays was addressed in [26]. Reference [27] also presents an intensive study of the fixedtime synchronization issue for the memristorbased BAM neural networks with timevarying discrete delays. In [28], the author elaborated the synchronization control problem for chaotic neural networks with timevarying and distributed delays. Moreover, the robust extended dissipativity criteria for discretetime uncertain neural networks with timevarying delays were investigated in [29].
By adding the Markovian process into the network systems of NNs, a new network model is developed. Up to now, the study concerning synchronization of Markovian jumping NNs, especially the global finitetime synchronization of Markovian jumping NNs have received wide attention from the scholars, and a number of results have been developed, such as finitetime synchronization [30], robust control [31], exponential synchronization [32], and state estimation [33]. However, the sojourntime of a Markovian process obeys an exponential distribution, which results in the transition rate to be a constant. That limits the application of Markovian process. Compared with Markovian process, semiMarkovian process can obey to some other probability distributions, such as Weibull distribution, Gaussian distribution, which makes the semiMarkovian process has a more extensive application prospect. Hence, the investigation for semiMarkovian jumping NNs is of great theoretical value and practical significance, which has been conducted in [34–38]. In [34], the finitetime \(H_{\infty }\) synchronization for complex networks with semiMarkov jump topology was investigated by adopting a suitable Lyapunov function and LMI approach. In [36], the exponential stability issue for the semiMarkovian jump generalized neural networks with interval timevarying delays was addressed. And in [38], the improved stability and stabilization results for stochastic synchronization of continuoustime semiMarkovian jump NNs with timevarying delays were also studied. However, to the best of our knowledge, little attention was paid to the synchronization issue for semiMarkovian jumping NNs. This motivates us to study the fixedtime synchronization of semiMarkovian jumping NNs with timevarying delays.
 (1)
A novel statefeedback controller, which includes doubleintegral terms, is designed to ensure the fixedtime synchronization, which can further improve the effectiveness of the convergence.
 (2)
A new formula for calculating the settling time for semiMarkovian jumping nonlinear system is developed; see Theorem 3.2.
 (3)
The timevarying delays and semiMarkovian processes are introduced in the construction of the NNs models.
 (4)
The fixedtime synchronization conditions are addressed in terms of LMIs.
The rest of this article is arranged as follows. Some preliminaries and model description are described in Sect. 2. In Sect. 3, we introduce the main results, the fixedtime synchronization conditions are derived under different nonlinear controllers. In Sect. 4, two examples are presented to show the correctness of our main results. Section 5, also the final part, the conclusion of this paper is shown.
Notation
R represents the set of real numbers. \(R^{n} \) denotes the ndimensional Euclidean space, and \(R^{n\times n}\) denotes the set of all \(n\times n \) matrices. Given column vectors \(x=(x_{1},x_{2}, \ldots,x_{n})^{T} \in R^{n}\), where the superscript T represents the transpose operator. \(X< Y\) (\(X>Y\)), which means that \(XY\) is negative (positive) definite. \(\mathscr{E}\) stand for mathematical expectation. \(\Gamma V(x(t),r(t),t)\) denotes the infinitesimal generator of \(V(x(t),r(t),t)\). For real matrix \(P=(p_{ij})_{n\times n}\), \(P=(p_{ij})_{n\times n}\), \(\lambda_{\min }(P)\) and \(\lambda_{ \max }(P)\) denote the minimum and maximum eigenvalues of P, respectively. ∗ stands for the symmetric terms in a symmetric block matrix. \(\x\\) stands for the Euclidean norm of the vector x, i.e., \(\x\=(x^{T}x)^{\frac{1}{2}}\). Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for the algebraic operation.
2 Preliminaries and model description
Remark 2.1
It is worth noting that in the continuoustime semiMarkovian process, the transition rate \(\pi _{rk}(h)\) is timevarying and depend on the sojourntime h. Meanwhile, the probability distribution of sojourntime h obeys the Weibull distribution, etc [39]. If the sojourntime h subjects to the exponential distribution, and the transition rate \(\pi _{rk}(h)=\pi _{rk}\), is a constant. Then the continuoustime semiMarkovian process recedes to the continuoustime Markovian process. On the other hand, the transition rate \(\pi_{rk}(h)\) is bounded, with \(\underline{\pi }_{rk}\leq \pi_{rk}(h)\leq \overline{ \pi }_{rk}\), \(\underline{\pi }_{rk}\) and \(\overline{\pi }_{rk}\) are known constant scalars, and \(\pi_{rk}(h)\) can be denoted as \(\pi_{rk}(h)=\pi_{rk}+\Delta \pi_{rk}\), where \(\pi_{rk}=\frac{1}{2}(\underline{ \pi }_{rk}+\overline{\pi }_{rk})\), and \( \Delta \pi_{rk} \leq \kappa_{rk}\) with \(\kappa_{rk}=\frac{1}{2}(\underline{\pi }_{rk}\overline{ \pi }_{rk})\), see [37].
 (\(H_{1}\)):

For any \(x_{i}{(t)}\), \(y_{i} {(t)}\in R^{n} \), \({f_{i}}(\cdot)\) satisfieswhere \(\mu_{i}>0\) and \(Q_{i}>0\) are both known constants.$$\bigl\vert f_{i}\bigl(y_{i}(t)\bigr)f_{i} \bigl(x_{i}(t)\bigr) \bigr\vert \leq \mu _{i} \bigl\vert y_{i}(t)x_{i}(t) \bigr\vert \quad \mbox{and}\quad \bigl\vert f_{i}(\cdot) \bigr\vert \leq Q_{i}, $$
Remark 2.2
Before proceeding our main results, some basic definitions and lemmas are introduced.
Definition 2.1
Lemma 2.1
([40])
Lemma 2.2
([41])
Lemma 2.3
Lemma 2.4
([42])
 (1)
\(V(e(t))>0\) for \(e(t)\neq 0\), \(V(e(t))=0\Leftrightarrow e(t)=0\);
 (2)for given constants \(\alpha >0\), \(\beta >0\), \(0<\rho <1\), and \(\upsilon >1\), any solution \(e(t)\) satisfies the following inequality:$$\begin{aligned} \begin{aligned} D^{+}V\bigl(e(t)\bigr)\leq \alpha V^{\rho }\bigl(e(t)\bigr)\beta V^{\upsilon }\bigl(e(t)\bigr), \end{aligned} \end{aligned}$$
Lemma 2.5
([43])
 (1)
\(V(e(t))>0\) for \(e(t)\neq 0\), \(V(e(t))=0\Leftrightarrow e(t)=0\);
 (2)for some α, \(\beta >0\), \(\rho =1\frac{1}{2p}\), \(\upsilon =1+\frac{1}{2p}\), \(p>1\), any solution \(e(t)\) satisfies$$\begin{aligned} \begin{aligned} D^{+}V\bigl(e(t)\bigr)\leq \alpha V^{\rho }\bigl(e(t)\bigr)\beta V^{\upsilon }\bigl(e(t)\bigr), \end{aligned} \end{aligned}$$
Lemma 2.6
([44])
3 Main results
Theorem 3.1
where \(\widetilde{\Omega }=\sum_{k=1}^{N}\pi_{rk}P_{k}+\sum_{k=1,k \neq r}^{N} [\frac{\kappa_{rk}^{2}}{4}W_{rk}+(P_{k}P_{r})W_{rk} ^{1}(P_{k}P_{r}) ]\), then the drive system (3) is synchronized onto the response system (4) in fixed time.
Proof
Remark 3.1
The function \(f_{i}(\cdot)\) we choose in this paper is continuous and bounded by a constant \(G_{i}\). It is a special condition for the function \(f_{i}(\cdot)\). The boundedness is not necessary in general conditions. In this paper, for estimating the parameter accurately, we choose the function bounded by \(G_{i}\). In other continuous cases, there only needs the condition \(f_{i}(y_{i}(t))f_{i}(x_{i}(t))\leq \mu _{i}y_{i}(t)x_{i}(t)\). Owing to \(\frac{f_{i}(y_{i}(t))f_{i}(x_{i}(t))}{y_{i}(t)x_{i}(t)}\leq \mu _{i}\), thus, for the constant \(\mu _{i}\), the value of \(\mu _{i}\) is determined by the selection of activation function \(f_{i}(\cdot)\).
Remark 3.2
To the best of our knowledge, of the current literature on the synchronization issue for NNs, only a part of the matrices in the network systems and Lyapunov functional are distinct for different system modes. Hence, the network systems and the Lyapunov functional in this paper are more general than the existing results (such as [24, 26]). Meanwhile, inspired by [33], the doubleintegral terms is introduced into the Lyapunov functional to deal with the adverse effect caused by the integral terms which include the semiMarkovian jumping parameters. The following theorem is established to show the advantage of this approach.
Theorem 3.2
Proof
Hence, the fixedtime synchronization conditions are addressed in terms of LMIs. The proof is completed. □
Remark 3.3
To the best of our knowledge, many existing works with respect to the fixedtime synchronization conditions for NNs, see [25, 27], address these in terms of algebraic inequalities. Compared with the approach used in [25], the fixedtime synchronization conditions obtained in Theorem 3.2 can be addressed in terms of LMIs, which can be solved by utilizing the LMI toolbox in Matlab. It should be mentioned that the condition (14) cannot be solved directly in terms of LMIs, because there exists a nonlinear term \(\sum_{r=1,k\neq r}^{N}(P_{k}P_{r})W_{rk}^{1}(P_{k}P_{r})\) in Ω̅. In order to overcome this difficulty, constructing a diagonal matrix \(\operatorname{diag}\{\sum_{r=1,k\neq r}^{N}(P_{k}P_{r})W_{rk}^{1}(P_{k}P_{r}), 0\}\) is necessary. Then, utilizing the condition of the transition rate \(\pi _{rk}(h)\) and Schur complement lemma which are mentioned in [37], the matrix inequalities is turned into the linear matrix inequalities, which can be solved in terms of LMIs.
Corollary 3.1
Corollary 3.2
Remark 3.4
Compared with the finitetime synchronization conditions obtained in [30], it needs more conditions to realize the fixedtime synchronization goal. For finitetime synchronization, there only needs such a term \(V^{\rho }(t)\), \(0<\rho <1\); whereas for fixedtime synchronization, it needs the two terms \(V^{\rho }(t)\) (\(0<\rho <1\)) and \(V^{\nu }(t)\) (\(\nu >1\)). Similar to the results in [30], the settling time \(T^{*}\) of the finitetime synchronization obtained in Corollary 3.2 depends on the initial condition \(V(0)\). When \(V(0)\) is so large that the \(T^{*}\) is not reasonable in practice application. However, the settling time \(T_{\varphi }\) of the fixedtime synchronization obtained in Theorem 3.2 is independent of any initial conditions. Thus, the settling time can be accurately evaluated by selecting appropriate control input parameters and semiMarkovian jumping parameters.
4 Numerical examples
Example 1
In this section, we perform two examples to demonstrate the correctness of Theorem 3.2.
The scalars we use in this paper are chosen as follows. The activation function is taken as \(f(t)=\tanh (t)\), thus \(\mu_{1}=\mu_{2}=1\), and \(G_{1}=G_{2}=1\). \(\rho =0.5\), \(\upsilon =2\). The timevarying delay function is assumed to be \(\tau (t)=0.5+0.5\cos (t)\), the initial value is \({x(t)}=(e^{3t},e^{3t})^{T}\), \({y(t)}=(\sin (3t),\tanh (3t))^{T}\), \(I=(0,0)^{T}\). We can easily see that its upper bound \(\tau =1\), .
The transition rates for each mode are given as follows:
Meanwhile, the parameters of the controller we choose as follows:
Example 2
It is assumed that the activation function and the timevarying delay function are taken as the same as Example 1. The initial conditions we choose as \({x(t)}=(3e^{2t},3e^{2t},3e^{2t})^{T}\), \({y(t)}=(3\sin (2t),3\sin (2t),3\tanh (2t))^{T}\), \(I=(0,0,0)^{T}\). And the relevant parameters are \(\mu_{1}=\mu_{2}=\mu _{3}=1\), \(G_{1}=G_{2}=G_{3}=1\), \(\rho =0.5\), \(\upsilon =2.0\).
The transition rates for each mode are given as follows.
5 Conclusion
In this paper, the fixedtime synchronization issue for semiMarkovian jumping neural networks with timevarying delays is discussed. A novel statefeedback controller is designed which includes doubleintegral terms and timevarying delay terms. Based on the linear matrix inequality (LMI) technique, the Lyapunov functional method, some effective conditions are established to guarantee the fixedtime synchronization of neural networks. Moreover, the upper bound of the settling time can be explicitly evaluated. To a certain extent, the results obtained in this paper have improved the previous works. More complex conditions, such as discontinuous functions, stochastic disturbances and fixedtime synchronization for complex dynamical networks will be taken into consideration in future research.
Declarations
Acknowledgements
The authors would like to thank the Editors and the Reviewers for their insightful and constructive comments, which have helped to enrich the content and improve the presentation of the results in this paper.
Availability of data and materials
Not applicable.
Funding
This work was jointly supported by the Natural Science Foundation of Hebei Province of China (A2018203288), the Postgraduate Innovation Project of Hebei Province of China (CXZZSS2018048) and High level talent project of Hebei Province of China (C2015003054).
Authors’ contributions
All authors contributed equally to the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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