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Fixedtime synchronization of semiMarkovian jumping neural networks with timevarying delays
Advances in Difference Equations volumeÂ 2018, ArticleÂ number:Â 213 (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.
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 [1â€“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 [4â€“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 [10â€“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 [20â€“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 [34â€“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.
Motivated by the aforementioned discussions, we intend to realize the fixedtime synchronization goal for semiMarkovian jumping NNs with timevarying delays. By applying Lyapunov functional approach, the fixedtime synchronization conditions are presented in terms of LMIs. Therefore, the novelty of our contributions is in the following:

(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. âˆ— 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
Let \((\Omega,\mathscr{F},\{\mathscr{F}\}_{t\geq 0},\mathcal{P})\) be the complete probability space and the filtration \(\mathscr{F} _{t\geq 0}\) satisfies the usual conditions that it is right continuous and increasing while \(\mathscr{F}\) contains all \(\mathcal{P}\)null sets, where Î© is the sample space, \(\mathscr{F}\) is the algebra of events and \(\mathcal{P}\) is the probability measure defined on \(\mathscr{F}\). Let \(\{r(t),t\geq 0\}\) be a continuoustime semiMarkovian process taking values in a finite state space \(S=\{1,2,3,\ldots,N\}\). The evolution of the semiMarkovian process \(r(t)\) obeys the following probability transitions:
where \(h>0\), \(\lim_{h\rightarrow 0}\frac{o(h)}{h}=0\), \(\pi_{rk}(h)\geq 0 \) (\(r,k\in S\), \(r\neq k\)) is the transition rate from mode r to k and for any state or mode, it satisfies
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].
The model we consider in this paper is the neural networks model with semiMarkovian jumping parameters. The dynamical behavior of the drive system is described as the following stochastic differential equation:
and the corresponding response system is
where \(\{r(t),t\geq 0\}\) is the continuoustime semiMarkovian process and \(r(t)\) stands for the evolution of the mode at time t. \({x(t)}=(x_{1}(t),x_{2}(t),\ldots,x_{n}(t))^{T} \in R^{n}\), \({y(t)}=(y_{1}(t),y_{2}(t),\ldots, y_{n}(t))^{T} \in R^{n}\) denotes the state vector of the ith neuron at time t; \(D(r(t))\in R^{n}\) is a positivedefinite diagonal matrix; \(A(r(t)) \in R^{n\times n}\) and \(B(r(t))\in R^{n\times n}\) are matrices with real values in mode \(r(t)\); \({f(x(t))}=(f_{1}(x_{1}(t)),f _{2}(x_{2}(t)),\ldots,f_{n}(x_{n}(t)))^{T}\in R^{n}\) is the neuronal activation function; \(I=(I_{1},I_{2},\ldots,I _{n})^{T}\) denotes the external input on the ith neuron. \(u(t)=(u_{1}(t),u_{2}(t),\ldots,u_{n}(t))^{T}\) stands for the control input, which will be designed later. \(\psi (t)=(\psi _{1}(t),\psi _{2}(t),\ldots,\psi _{n}(t)\in \mathcal{C}([\tau,0];R^{n})\) and \(\phi (t)=(\phi _{1}(t),\phi _{2}(t),\ldots,\phi _{n}(t))\in \mathcal{C}([\tau,0];R^{n})\) are the initial conditions of system (1) and (2), respectively.
Variable \(\tau (t)\) denotes the timevarying delay function, and it is assumed to satisfy
where \(\tau >0\) and are known constants.
For notation simplicity, we replace \(D(r(t))\), \(A(r(t))\), \(B(r(t))\) with \(D_{r}\), \(A_{r}\) and \(B_{r}\), respectively, for \(r(t)=r\in S\). Then the neural networks models can be rewritten as follows:
For the purpose of this paper, we suppose that the activation function \(f_{i}(\cdot)\) satisfies the following assumption:
 (\(H_{1}\)):

For any \(x_{i}{(t)}\), \(y_{i} {(t)}\in R^{n} \), \({f_{i}}(\cdot)\) satisfies
$$\bigl\vert f_{i}\bigl(y_{i}(t)\bigr)f_{i} \bigl(x_{i}(t)\bigr) \bigr\vert \leq \mu _{i} \bigl\vert y_{i}(t)x_{i}(t) \bigr\vert \quad \mbox{and}\quad \bigl\vert f_{i}(\cdot) \bigr\vert \leq Q_{i}, $$where \(\mu_{i}>0\) and \(Q_{i}>0\) are both known constants.
Let \(e_{i}(t)=y_{i}(t)x_{i}(t)\) be the synchronization error, then the error dynamics system can be expressed as
where \({e(t)=(e_{1}(t),e_{2}(t),\ldots,e_{n}(t))^{T}}\), \(g_{i}(e_{i}(t))=f_{i}(y_{i}(t))f_{i}(x_{i}(t))\) and \(g_{i}(e_{i}(t \tau (t)))=f_{i}(y_{i}(t\tau (t)))f_{i}(x_{i}(t\tau (t)))\), and \({\varphi (t)}= \psi_{i}(t)\phi_{i}(t)\).
Remark 2.2
From assumption (\(H_{1}\)), we can conclude that \(g_{i}(\cdot)\) is also continuous and bounded, then
where \(H_{i}\) is a known positive constant.
Before proceeding our main results, some basic definitions and lemmas are introduced.
Definition 2.1
The neural network system (3) is said to be synchronized with the system (4) in finite time, if for any initial condition \({\varphi (t)}\), \(\tau \leq t\leq 0\), there exists a settling time function \(T_{\varphi }=T(\varphi)\), such that
Moreover, if there exists a constant \(T_{\max }>0\), such that \(T_{\varphi }\leq T_{\max }\), then the neural network system (3) is said to be synchronized onto system (4) in fixed time. \(T_{\max }\) is called the synchronization settling time.
Lemma 2.1
([40])
Given any scalar Îµ and matrix \(S\in R^{n\times n}\), the following inequality:
holds for any symmetric positivedefinite matrix \(W\in R^{n\times n}\).
Lemma 2.2
([41])
For any constant vector \(x\in R^{n}\) and \(0< c< l\), the following norm equivalence holds:
Lemma 2.3
Let \(U\in R^{n\times n}\) be a symmetric matrix, and let \(x\in R^{n}\), then the following inequality holds:
Lemma 2.4
([42])
Suppose there exists a continuous nonnegative function \(V(t): R^{n}\rightarrow R_{+}\cup {(0)}\), such that

(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:
$$\begin{aligned} \begin{aligned} D^{+}V\bigl(e(t)\bigr)\leq \alpha V^{\rho }\bigl(e(t)\bigr)\beta V^{\upsilon }\bigl(e(t)\bigr), \end{aligned} \end{aligned}$$
then the error system (5) is globally fixedtime stable for any initial conditions \(\varphi (t)\), and it satisfies
with the settling time estimated as
Lemma 2.5
([43])
Suppose there exists a continuous nonnegative function \(V(t): R^{n}\rightarrow R_{+}\cup {(0)}\), such that

(1)
\(V(e(t))>0\) for \(e(t)\neq 0\), \(V(e(t))=0\Leftrightarrow e(t)=0\);

(2)
for some Î±, \(\beta >0\), \(\rho =1\frac{1}{2p}\), \(\upsilon =1+\frac{1}{2p}\), \(p>1\), any solution \(e(t)\) satisfies
$$\begin{aligned} \begin{aligned} D^{+}V\bigl(e(t)\bigr)\leq \alpha V^{\rho }\bigl(e(t)\bigr)\beta V^{\upsilon }\bigl(e(t)\bigr), \end{aligned} \end{aligned}$$
then the error system (5) is globally fixedtime stable, and the settling time bounded by
Lemma 2.6
([44])
Suppose that there exists a positivedefinite, continuous differential function \(V(t)\) which satisfies
where \(\alpha >0\), \(0<\rho <1\) are two constants. Then we have \(\lim_{t\rightarrow {T^{*}}} V(t)=0\), and \(V(t)\equiv 0\), \(\forall t\geq {T^{*}}\), with the settling time \(T^{*}\) estimated as
3 Main results
In this subsection, the fixedtime synchronization conditions are developed between the system (3) and (4). For this purpose, we adopt the following discontinuous feedback controller:
where \(0<\rho <1\), \(\upsilon >1\), \(\lambda_{i1}\), \(\lambda_{i2}\), \(\lambda_{i3}\), \(\lambda_{i4}\), \(i=1,2\), are the parameters to be designed later.
Theorem 3.1
Under assumption (\(H_{1}\)), for given scalars \(0<\alpha <1\) and \(\beta >1\), if there exist symmetric positivedefinite matrices \(P_{r}\), \(W_{rk}\), such that
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
Consider the following Lyapunov functional:
For simplicity, here, we replace \(V(e(t),t,r)\), \(\mathscr{L}V(e(t),t,r)\) with \(V(t)\) and \(\mathscr{L}V(t)\), respectively. With regard to ItÃ´ formula, we have
where Î”t is a small positive number. Hence, for every \(r(t)={r}\in S\), it can be deduced that
Considering \(\pi_{rk}(h)=\pi_{rk}+\Delta \pi_{rk}\), \(\Delta \pi_{{rr}}=\sum_{k=1,k\neq r}^{N}\Delta \pi_{rk}\) and applying Lemma 2.1, we obtain
Then calculating the derivative of \(V(t)\) along the trajectory of (5), we have
Based on assumption (\(H_{1}\)), we get
Substituting the controller (6) into (10), it yields
By the condition (7), (11) can be rewritten as the following inequality:
In view of Lemmas 2.2 and 2.3, it derives that
According to (8), one obtains
Then, taking the expectation on both sides of (12), we can get
As is well known, for any \(t>0\), \(\mathscr{E}[(V(t))^{ \frac{1+\rho }{2}}]=(\mathscr{E}[V(t)])^{\frac{1+\rho }{2}}\) and \(\mathscr{E}[(V(t))^{\frac{1+\upsilon }{2}}]= (\mathscr{E}[V(t)])^{\frac{1+ \upsilon }{2}}\), then we have the following inequality:
By Lemma 2.4, we know that the error system (5) is globally fixedtime stable. And the settling time is estimated as
Hence, under the controller (6), the fixedtime synchronization conditions is derived. The proof is completed.â€ƒâ–¡
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.
In the following, the fixedtime synchronization conditions are addressed in terms of LMIs between the system (3) and (4). For this purpose, we adopt the following discontinuous feedback controller which includes the integral terms:
Theorem 3.2
Under assumption (\(H_{1}\)), for given scalars \(0<\alpha <1\) and \(\beta >1\), if there exist symmetric positivedefinite matrices \(P_{r}\), \(W_{rk}\), and \(K_{r}\), symmetric matrix \(K\geq 0\), such that
where \(\overline{\Omega }=K_{r}+\sum_{k=1}^{N}\pi_{rk}P_{k}+ \sum_{k=1,k\neq r}^{N} [\frac{\kappa_{rk}^{2}}{4}W_{rk}+(P_{k}P _{r})W_{rk}^{1}(P_{k}P_{r}) ]+\tau K\),
then the drive system (3) is synchronized onto the response system (4) in fixed time.
Proof
Consider the following Lyapunov functional:
For \(V_{1}(t)\), based on (8) and (9), we have
Calculating the derivatives of \(V_{2}(t)\) and \(V_{3}(t)\) along the trajectory of (5), it yields
and
Combining (16)â€“(18), we acquire
Based on assumption (\(H_{1}\)) and the error system (5), we have
Under the condition of the Theorem, we have the following inequality:
Substituting (13) into (19), we can obtain
By the conditions (14) and (15), then employing Lemma 2.3, we have
According to Lemma 2.2, we get
thus
Thus, (20) can be rewritten as
According to the conditions given in (15), then, based on Lemma 2.2, we get
where \(0<\rho <1\), \(\upsilon >1\), \(\lambda_{i3}\), \(\lambda_{i4}>0\).
Taking the expectation on both sides of (21), it yields
It is easily known that \(\mathscr{E}[(V(t))^{\frac{1+\rho }{2}}]=(\mathscr{E}[V(t)])^{\frac{1+\rho }{2}}\) and \(\mathscr{E}[(V(t))^{\frac{1+\upsilon }{2}}]=(\mathscr{E}[V(t)])^{\frac{1+\upsilon }{2}}\), then (22) can be rewritten as
Together with Lemma 2.4 and (23), we conclude that the error system (5) is globally fixedtime stable, and the settling time is estimated as
Hence, the fixedtime synchronization conditions are addressed in terms of LMIs. The proof is completed.â€ƒâ–¡
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 Î©Ì…. 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
Suppose the conditions in Theorem 3.2 admit. Under the controller (13), the drive system (3) is synchronized with the response system (4) in fixed time, and the settling time is estimated as
with \(p>1\).
Corollary 3.2
Under assumption (\(H_{1}\)), for given scalars \(0<\rho <1\), the drive system (3) is synchronized onto the response system (4) in a finitetime interval based on the following controller:
if there exist symmetric positivedefinite matrices \(P_{r}\), \(W_{rk}\), such that
where \(\widetilde{\Omega }=\sum_{k=1}^{N}\pi_{rk}P_{k}+\sum_{k=1,k \neq r}^{N} [\frac{\kappa_{rk}^{2}}{4}W_{rk}+(P_{k}P_{r})W_{rk} ^{1}(P_{k}P_{r}) ]\).
Meanwhile, the settling time is estimated as
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.
Consider the 2dimensional semiMarkovian jumping neural networks system. The parameters we choose as follows:
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:
For mode 1
For mode 2
Then we can get the parameters \(\pi_{rk}\), \(\kappa_{rk}\), where \(r,k \in S=\{1,2\}\).
Through simple computations, we have
Meanwhile, the parameters of the controller we choose as follows:
For mode 1
For mode 2
and the settling time \(T_{\max }\) can be calculated as 4.45.
Based on the values given above, then the first and second state trajectories of the systems (3) and (4) are displayed in Fig. 1 and Fig. 2, respectively. And the trajectories of the corresponding synchronization error system are depicted in Fig. 3. Hence, the correctness of Theorem 3.2 is proved.
Example 2
Consider the 3dimensional semiMarkovian jumping neural networks system. The parameters we choose as follows:
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.
For mode 1
For mode 2
Then we can get the parameters \(\pi_{rk}\), \(\kappa_{rk}\), where \(r, k \in S=\{1,2\}\),
Through simple computations, we have
For mode 1
For mode 2
and the settling time \(T_{\max }\) is evaluated as 4.45.
Under controller (13), the first, second and third state trajectories of the system (3) and (4) are plotted in Figs. 4, 5, and 6, respectively. Moreover, Fig. 7 shows the trajectories of the corresponding synchronization error system. The numerical simulation perfectly supports Theorem 3.2.
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.
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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.
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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).
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Zhao, W., Wu, H. Fixedtime synchronization of semiMarkovian jumping neural networks with timevarying delays. Adv Differ Equ 2018, 213 (2018). https://doi.org/10.1186/s136620181666z
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DOI: https://doi.org/10.1186/s136620181666z