synchronization of chaotic Hopfield networks with time-varying delay: a resilient DOF control approa
本站小编 Free考研考试/2022-01-02
Xin Huang(黄鑫)1, Youmei Zhou(周幼美)1, Qingkai Kong(孔庆凯)2,3, Jianping Zhou(周建平),1,4, Muyun Fang(方木云)11School of Computer Science & Technology, Anhui University of Technology, Ma’anshan, 243032, China 2Department of Electrical Engineering, Yeungnam University, 280 Daehak-Ro, Kyongsan 38541, Republic of Korea 3School of Electrical & Information Engineering, Anhui University of Technology, Ma’anshan 243032, China 4Research Institute of Information Technology, Anhui University of Technology, Ma’anshan, 243000, China
*Natural Science Foundation of the Anhui Higher Education Institutions.KJ2017A064 Natural Science Foundation of the Anhui Higher Education Institutions.KJ2018ZD007 Excellent Youth Talent Support Program of Universities in Anhui Province.GXYQZD2019021 National Natural Science Foundation of China.61503002
Abstract This paper focuses on the issue of resilient dynamic output-feedback (DOF) control for ${{ \mathcal H }}_{\infty }$ synchronization of chaotic Hopfield networks with time-varying delay. The aim is to determine a DOF controller with gain perturbations ensuring that the ${{ \mathcal H }}_{\infty }$ norm from the external disturbances to the synchronization error is less than or equal to a prescribed bound. A delay-dependent criterion for the ${{ \mathcal H }}_{\infty }$ synchronization is derived by employing the Lyapunov functional method together with some recent inequalities. Then, with the help of some decoupling techniques, sufficient conditions on the existence of the resilient DOF controller are developed for both the time-varying and constant time-delay cases. Lastly, an example is used to illustrate the applicability of the results obtained. Keywords：Hopfield network;Chaos synchronization;Time delay;Dynamic output feedback
PDF (783KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Xin Huang(黄鑫), Youmei Zhou(周幼美), Qingkai Kong(孔庆凯), Jianping Zhou(周建平), Muyun Fang(方木云). ${{ \mathcal H }}_{\infty }$ synchronization of chaotic Hopfield networks with time-varying delay: a resilient DOF control approach*. Communications in Theoretical Physics, 2020, 72(1): 015003- doi:10.1088/1572-9494/ab5452
1. Introduction
Hopfield network, a type of interconnected artificial neural network introduced by John J. Hopfield in 1982 [1], provides an innovative model for understanding and imitating memory in the human brain. Over the last two decades, there is a growing interest in the study of dynamic behaviors of Hopfield networks owing to the fact that they are closely related to many applications such as combinatorial optimization [2], content-addressable memory [3], and blind signal detection [4]. In [5], a Hopfield network with symmetric connection weight matrix was considered. It was proved therein that the network is absolute stable if and only if the weight matrix is negative semi-definite. When one is faced with the problem of asymmetric weight matrix, some sufficient conditions for global asymptotic stability were reported in [6]. A qualitative Hopf bifurcation analysis of a Hopfield network consisting of four neurons was given in [7], where the features of the limit cycles related to the bifurcations were acquired with the aid of normal forms theory. Periodic oscillation of discrete Hopfield networks was discussed [8], where several criteria for the uniqueness and exponential stability of oscillatory solutions were derived by utilizing coincidence degree theory. In [9], the chaotic regime of a six-neuron Hopfield network was analyzed, and it was demonstrated that many attractors can be obtained under appropriate choices of the network parameters.
The results in the above references rely on the assumption that the neurons can communicate and respond to each other instantaneously, which is questionable in hardware implementation due to the limited switching speed of amplifiers. In [10], a constant time delay was first introduced into the continuous-time Hopfield networks. It was shown that, for a certain type of connection topology, even a small delay may affect the system performance seriously. Later, multiple constant time delays were introduced in [11] to reflect different communication channels. And when the time delay was allowed to be time-varying, the existing criteria of almost periodic solutions were proposed in [12], while sufficient conditions for global asymptotic or even global exponential stability were given in [13]. It is now well recognized that the introduction of time delays into a Hopfield network can greatly enrich the dynamic behavior of the network. In particular, it has been shown that time-delay Hopfield networks are able to generate a large variety of strange chaotic attractors including single-scroll chaotic attractors [14], double-scroll chaotic attractors [15], three-chaos and periodic four-torus attractors [16], etc. Correspondingly, much attention has been paid to synchronization design of chaotic time-delay Hopfield networks in recent years in view of its potential application in secure communication, and some valid control schemes have been adopted; see, e.g., [17] for the adaptive control scheme and [18] for the memory control scheme.
One of the most important issues in control theory is how to construct controllers with acceptable performance in the presence of exogenous disturbances. Generally, one has no prior knowledge of the statistics of the disturbances. This situation has led to the development of ${{ \mathcal H }}_{\infty }$ control in the past several decades as such a control strategy offers many advantages, especially the insensitive property to the disturbance statistics [19–21]. For chaotic neural networks with time delay, the problem of exponential ${{ \mathcal H }}_{\infty }$ synchronization control was considered in [22] and [23] in continuous-time and discrete-time respectively, where some sufficient conditions on the existence of delayed state-feedback controllers were presented. In [24], a delay-dependent ${{ \mathcal H }}_{\infty }$ synchronization control scheme was developed in terms of linear matrix inequalities (LMIs). In the context of packet dropouts, sampled-data ${{ \mathcal H }}_{\infty }$ synchronization control strategies were designed in [25] by employing the Lyapunov functional theory. It should be noted that, although there are a number of studies on ${{ \mathcal H }}_{\infty }$ synchronization control of chaotic time-delay neural networks available in the literature, most of them are founded on full state feedback, which requires that all neuron states should be available. In practice, a complete measurement of the neuron states may be hard, costly, and even infeasible to implement. In this situation, the output-feedback control strategy is more preferable for engineering applications as the outputs of a system are generally easier to obtain than the states [26–29].
Motivated by the above analysis, in the present study, we focus on the design of resilient dynamic output-feedback (DOF) controller for ${{ \mathcal H }}_{\infty }$ synchronization of a class of chaotic Hopfield networks with time-varying delay. The time-delay function is supposed to be continuous and bounded, but not necessarily differentiable. Our purpose is to determine a DOF controller with gain perturbations ensuring the ${{ \mathcal H }}_{\infty }$ norm from the external disturbances to the synchronization error is less than or equal to a prescribed bound (called the ${{ \mathcal H }}_{\infty }$ disturbance attenuation bound). The issue of resilient DOF controller design is non-convex in nature and remains challenging. By selecting a proper Lyapunov functional together with using the Wirtinger inequality and the reciprocally convex combination lemma, we derive a delay-dependent criterion for ${{ \mathcal H }}_{\infty }$ synchronization. For the constant time delay case, a delay-dependent ${{ \mathcal H }}_{\infty }$ synchronization condition is also proposed. On the basis of the criteria and with the help of some decoupling techniques, we develop sufficient conditions on the existence of the resilient DOF controller for both the time-varying and constant time-delay cases. It is worth mentioning that the desired controller gain can be obtained through the feasible solution of a few LMIs, which is easy to solve by the well-known mathematical software Matlab. Lastly, we give an example to illustrate the applicability of the results obtained.
Let ${{\mathbb{R}}}^{p}$ be the p-dimensional Euclidean space, ${{\mathbb{R}}}^{p\times q}$ be the group of p×q real matrices, and ${{\mathbb{S}}}^{p}$ (respectively, ${{\mathbb{S}}}_{+}^{p}$) be the group of real symmetric square matrices (respectively, real symmetric positive-definite square matrices) of order p. In a symmetric matrix, the blocks that can be obviously inferred by symmetry are represented by $\ast $. Let Ip and 0p be the p-order unit matrix and zero matrix, respectively. The notation $\parallel \cdot \parallel $ refers to the Euclidean vector norm, $\mathrm{diag}\{\cdot \cdot \cdot \}$ denotes a diagonal matrix, MT denotes the transpose of matrix M, and He(M) stands for M+MT.
2. Preliminaries
Let us consider a neural network with dynamic model:$ \begin{eqnarray}\begin{array}{rcl}\dot{x}(t) & = & {Ax}(t)+{A}_{\tau }x(t-\tau (t))+{Bf}(x(t))\\ & & +{B}_{\tau }g(x(t-\tau (t)))+J,\end{array}\end{eqnarray}$$ \begin{eqnarray}y(t)={Cx}(t),\end{eqnarray}$where $x(t)\in {{\mathbb{R}}}^{{n}_{1}}\ $ is the neuron state; $y(t)\in {{\mathbb{R}}}^{m}$ represents the output; $A,{A}_{\tau },B,{B}_{\tau },C$ are constant real matrices; $J\in {{\mathbb{R}}}^{{n}_{1}}$ denotes a constant input; $f(\cdot )\in {{\mathbb{R}}}^{{n}_{1}}$ and $g(\cdot )\in {{\mathbb{R}}}^{{n}_{1}}$ are two activation functions that both are global Lipschitz continuous [30–32]; i.e., there are Lipschitz constants Lf and Lg such that$ \begin{eqnarray}\parallel f({\delta }_{1})-f({\delta }_{2})\parallel \leqslant {L}_{f}\parallel {\delta }_{1}-{\delta }_{2}\parallel ,\end{eqnarray}$$ \begin{eqnarray}\parallel g({\delta }_{1})-g({\delta }_{2})\parallel \leqslant {L}_{g}\parallel {\delta }_{1}-{\delta }_{2}\parallel \end{eqnarray}$for any ${\delta }_{1},{\delta }_{2}\in {{\mathbb{R}}}^{{n}_{1}}$. One can refer to [33, 34] for more relaxed activation functions. τ (t) stands for the time delay that is continuous and belonging to a given interval as [35]:$ \begin{eqnarray}0\leqslant {\tau }_{1}\leqslant \tau (t)\leqslant {\tau }_{2},\ {\tau }_{12}\mathop{=}\limits^{{\rm{\Delta }}}{\tau }_{2}-{\tau }_{1},\end{eqnarray}$where τ1 and τ2 are known constants. Note that the time delay is time-varying, which is more general than the constant time delay discussed in [36, 37]. It is also worth mentioning that, unlike that commonly used in the literature (see, e.g., [38–42]), the time-varying delay considered herein is not required to be differentiable with respect to t.
The response system under consideration is given as follows:$ \begin{eqnarray}\begin{array}{rcl}\dot{\hat{x}}(t) & = & A\hat{x}(t)+{A}_{\tau }\hat{x}(t-\tau (t))+{Bf}(\hat{x}(t))\\ & & +{B}_{\tau }g(\hat{x}(t-\tau (t)))+J+{Hu}(t)+D\omega (t),\end{array}\end{eqnarray}$$ \begin{eqnarray}\hat{y}(t)=C\hat{x}(t),\end{eqnarray}$where $\hat{x}(t)\in {{\mathbb{R}}}^{{n}_{1}}$ and $\hat{y}(t)\in {{\mathbb{R}}}^{{n}_{1}}$ are, respectively, the state and the output; $u(t)\in {{\mathbb{R}}}^{{n}_{1}}$ and $\omega (t)\in {{\mathbb{R}}}^{k}$ denote, respectively, the control input and the external disturbance belonging to ${{ \mathcal L }}_{2}[0,\infty );$$D\in {{\mathbb{R}}}^{{n}_{1}\times k}$ is a constant real matrix. Define $e(\cdot )=\hat{x}(\cdot )-x(\cdot )$. Then one is able to establish the following error system$ \begin{eqnarray}\begin{array}{rcl}\dot{e}(t) & = & {Ae}(t)+{A}_{\tau }e(t-\tau (t))+B\left(f\left(\hat{x}(t,)\right)-f\left(x(t\right),)\right)\\ & & +{B}_{\tau }\left(g\left(\hat{x}(t-\tau (t),)\right)-g\left(x(t-\tau (t),)\right)\right)\\ & & +{Hu}(t)+D\omega (t).\end{array}\end{eqnarray}$
In this paper, we are interested in the design of a resilient DOF controller as$ \begin{eqnarray}{\dot{e}}_{{}_{K}}(t)={\tilde{A}}_{K}{e}_{{}_{K}}(t)+{\tilde{B}}_{K}\left(\hat{y}(t)-y(t,)\right),\end{eqnarray}$$ \begin{eqnarray}u(t)={\tilde{C}}_{K}{e}_{{}_{K}}(t)+{\tilde{D}}_{K}\left(\hat{y}(t)-y(t,)\right),\end{eqnarray}$where$ \begin{eqnarray}\left[\begin{array}{cc}{\tilde{A}}_{K} & {\tilde{B}}_{K}\\ {\tilde{C}}_{K} & {\tilde{D}}_{K}\end{array}\right]=\left[\begin{array}{cc}{A}_{K} & {B}_{K}\\ {C}_{K} & {D}_{K}\end{array}\right]+\left[\begin{array}{cc}{\rm{\Delta }}{A}_{K} & {\rm{\Delta }}{B}_{K}\\ {\rm{\Delta }}{C}_{K} & {\rm{\Delta }}{D}_{K}\end{array}\right].\end{eqnarray}$In (8), ${e}_{K}(t)\in {{\mathbb{R}}}^{{n}_{c}}$ denotes the state of the controller, ${A}_{K},{B}_{K},{C}_{K},{D}_{K}$ are the controller gains to be determined later, and ${\rm{\Delta }}{A}_{K},{\rm{\Delta }}{B}_{K}$, ${\rm{\Delta }}{C}_{K},{\rm{\Delta }}{D}_{K}$ stand for the gain perturbations, which are assumed to be norm-bounded as [43, 44]:$ \begin{eqnarray}\left[\begin{array}{cc}{\rm{\Delta }}{A}_{K} & {\rm{\Delta }}{B}_{K}\\ {\rm{\Delta }}{C}_{K} & {\rm{\Delta }}{D}_{K}\end{array}\right]={MFN},\end{eqnarray}$where M, N are constant real matrices, and F is an uncertain matrix satisfying ${F}^{{\rm{T}}}F\leqslant I$. It is worth mentioning that the resilient DOF controller to be designed is of full-order when nc=n1 and of reduced-order when nc<n1. Particulary, when nc=0, the controller reduces to the usual static output-feedback controller.
Although there are many results on ${{ \mathcal H }}_{\infty }$ synchronization control of chaotic time-delay neural networks available in the literature, only a few of them consider the robustness of controllers. Due to the rounding error of digital computing, the restriction of process memory, and the noise of analog-digital conversion, it is often the case that the designed controllers possess parameter inaccuracies to some degree, and even a tiny perturbation to the controller gain may greatly undermine the controller’s performance [45–48]. Thus the design of resilient controller is of practical importance.
Combining (6) with (7), we get$ \begin{eqnarray}\begin{array}{rcl}\dot{\xi }(t) & = & {{ \mathcal A }}_{K}\xi (t)+{{ \mathcal A }}_{\tau }\xi \left(t-\tau (t,)\right)+{ \mathcal B }\bar{f}\left(\xi (t,)\right)\\ & & +{{ \mathcal B }}_{\tau }\bar{g}\left(\xi \left(t-\tau (t\right),)\right)+{ \mathcal D }\omega (t),\end{array}\end{eqnarray}$ where $ \begin{eqnarray*}\begin{array}{rcl}\xi (t) & = & \left[\begin{array}{c}e(t)\\ {e}_{{}_{K}}(t)\end{array}\right]\in {{\mathbb{R}}}^{n},\,{{ \mathcal A }}_{K}=\tilde{A}+\tilde{B}\left(K+{\rm{\Delta }}K\right)\tilde{C},\\ {{ \mathcal A }}_{\tau } & = & \left[\begin{array}{cc}{A}_{\tau } & 0\\ 0 & 0\end{array}\right],\,{ \mathcal B }=\left[\begin{array}{cc}B & 0\\ 0 & 0\end{array}\right],\\ {{ \mathcal B }}_{\tau } & = & \left[\begin{array}{cc}{B}_{\tau } & 0\\ 0 & 0\end{array}\right],\,{ \mathcal D }=\left[\begin{array}{c}D\\ 0\end{array}\right],\\ \bar{f}(\xi (t)) & = & \left[\begin{array}{c}f\left(e(t)+x(t,)\right)-f\left(x(t,)\right)\\ 0\end{array}\right],\\ & & \bar{g}(\xi (t-\tau (t)))\\ & = & \left[\begin{array}{c}g\left(e(t-\tau (t))+x(t-\tau (t),)\right)-g\left(x(t-\tau (t),)\right)\\ 0\end{array}\right]\end{array}\end{eqnarray*}$ with $ \begin{eqnarray*}\begin{array}{rcl}\tilde{A} & = & \left[\begin{array}{cc}A & 0\\ 0 & 0\end{array}\right],\ \tilde{B}=\left[\begin{array}{cc}0 & H\\ I & 0\end{array}\right],\ \tilde{C}=\left[\begin{array}{cc}0 & I\\ C & 0\end{array}\right],\\ K & = & \left[\begin{array}{cc}{A}_{K} & {B}_{K}\\ {C}_{K} & {D}_{K}\end{array}\right],\ {\rm{\Delta }}K={MFN},n={n}_{1}+{n}_{c}.\end{array}\end{eqnarray*}$
Before ending the section, we recall the definition of ${{ \mathcal H }}_{\infty }$ synchronization and prepare some lemmas that are needed for establishing our main results.
([49] ${{ \mathcal H }}_{\infty }$ synchronization).Error system (6) is called ${{ \mathcal H }}_{\infty }$ synchronized, if, under the zero initial condition, $ \begin{eqnarray*}{\int }_{0}^{t}{e}^{{\rm{T}}}(\sigma ){Q}_{1}e(\sigma ){\rm{d}}\sigma \leqslant {\int }_{0}^{t}{\gamma }^{2}{\omega }^{{\rm{T}}}(\sigma )\omega (\sigma ){\rm{d}}\sigma \end{eqnarray*}$ holds for a pre-defined level $\gamma \gt 0$ (called the ${{ \mathcal H }}_{\infty }$ disturbance attenuation bound), where ${Q}_{1}$ is a positive symmetric matrix.
([50] Improved Wirtinger’s inequality).For a specific matrix $R\in {{\mathbb{S}}}_{+}^{n}$ and any function $\eta $ that is differentiable in $\left[p,q\right]\to {{\mathbb{R}}}^{n}$, one has$ \begin{eqnarray}{\int }_{p}^{q}{\dot{\eta }}^{{\rm{T}}}(u)R\dot{\eta }(u){\rm{d}}{u}\geqslant \displaystyle \frac{1}{q-p}{\left[\begin{array}{c}{\eta }_{0}\\ {\eta }_{1}\end{array}\right]}^{{\rm{T}}}\tilde{R}\left[\begin{array}{c}{\eta }_{0}\\ {\eta }_{1}\end{array}\right],\end{eqnarray}$in which $ \begin{eqnarray*}\begin{array}{rcl}{\eta }_{0} & = & \eta (q)-\eta (p),\\ {\eta }_{1} & = & \eta (q)+\eta (p)-\displaystyle \frac{2}{q-p}{\displaystyle \int }_{p}^{q}\eta (u){\rm{d}}{u},\\ \tilde{R} & = & \mathrm{diag}(R,3R).\end{array}\end{eqnarray*}$
[51] Suppose that the following condition $ \begin{eqnarray*}\left[\begin{array}{cc}{\rm{\Lambda }} & {U}_{1}+\rho {V}_{1}\ \cdots \ {U}_{N}+\rho {V}_{N}\\ * & \mathrm{diag}\left\{-\rho {W}_{1}-\rho {W}_{1}^{{\rm{T}}}\ \cdots \ -\rho {W}_{N}-\rho {W}_{N}^{{\rm{T}}}\right\}\end{array}\right]\lt \ 0\end{eqnarray*}$ regarding a real scalar $\rho \gt 0$ and real matrices ${\rm{\Lambda }},{U}_{i},{V}_{i}$ and ${W}_{i}(i=1,\,\cdots ,\,N)$ is satisfied. Then one can write $ \begin{eqnarray*}{\rm{\Lambda }}+\displaystyle \sum _{i=1}^{N}{He}({U}_{i}{W}_{i}^{-1}{V}_{i}^{{\rm{T}}})\lt 0.\end{eqnarray*}$
Assume that ${\rm{\Phi }}(\beta )$ is a parameter dependent matrix ensuring $ \begin{eqnarray*}{\rm{\Phi }}(\beta )\leqslant (1-\beta ){\rm{\Phi }}(0)+\beta {\rm{\Phi }}(1)\end{eqnarray*}$ for any $\beta $ in $\left[0,1\right]$. If there are matrices $R\in {S}_{+}^{n}$ and ${N}_{1},{N}_{2}\in {R}^{m\times n}$ such that $ \begin{eqnarray}\left[\begin{array}{cc}{\rm{\Phi }}(\beta )-{{\rm{\Gamma }}}^{{\rm{T}}}{ \mathcal R }(\beta ){\rm{\Gamma }}-{He}\left({{\rm{\Gamma }}}^{{\rm{T}}}\left[\begin{array}{c}(1-\beta ){N}_{1}^{{\rm{T}}}\\ \beta {N}_{2}^{{\rm{T}}}\end{array}\right]\right) & * \\ \beta {N}_{1}^{{\rm{T}}}+(1-\beta ){N}_{2}^{{\rm{T}}} & -R\end{array}\right]\lt 0\end{eqnarray}$ for $\beta =\left\{0,1\right\}$, in which $ \begin{eqnarray*}{ \mathcal R }(\beta )=\left[\begin{array}{cc}(2-\beta )R & 0\\ 0 & (1+\beta )R\end{array}\right].\end{eqnarray*}$ Then, for any $\beta \in (0,1)$ one can write $ \begin{eqnarray*}{\rm{\Phi }}(\beta )-{\rm{\Theta }}(\beta )\lt 0,\ \end{eqnarray*}$ where $ \begin{eqnarray*}{\rm{\Theta }}(\beta )={{\rm{\Gamma }}}^{{\rm{T}}}\left[\begin{array}{cc}\tfrac{1}{\beta }R & 0\\ 0 & \tfrac{1}{1-\beta }R\end{array}\right]{\rm{\Gamma }}.\end{eqnarray*}$
3.${{ \mathcal H }}_{\infty }$ synchronization analysis
This section focuses on the ${{ \mathcal H }}_{\infty }$ synchronization analysis for the error system. A sufficient condition is given by the following theorem:For given scalar $\gamma \gt 0$ and ${Q}_{1}$ in ${{\mathbb{S}}}_{+}^{{n}_{1}}$, system (6) is ${{ \mathcal H }}_{\infty }$ synchronized, if there are scalars ${\varepsilon }_{f}\gt 0,{\varepsilon }_{g}\gt 0$ and matrices $P$ in ${{\mathbb{S}}}_{+}^{3n},{S}_{1},{S}_{2},{R}_{1},{R}_{2}$ in ${{\mathbb{S}}}_{+}^{n},{N}_{1},$${N}_{2}$ in ${R}^{9n\times 2n}$ such that $ \begin{eqnarray}\left[\begin{array}{ccc}{{\rm{\Phi }}}_{0}(\alpha )-{{\rm{\Gamma }}}^{{\rm{T}}}{{ \mathcal R }}^{0}(\alpha ){\rm{\Gamma }}-{He}\left({{\rm{\Gamma }}}^{{\rm{T}}}\left[\begin{array}{c}(1-\alpha ){N}_{1}^{{\rm{T}}}\\ \alpha {N}_{2}^{{\rm{T}}}\end{array}\right]\right) & * & * \\ \alpha {N}_{1}^{{\rm{T}}}+(1-\alpha ){N}_{2}^{{\rm{T}}} & -{\tilde{R}}_{2} & * \\ {\left({G}_{1}^{{\rm{T}}}(\alpha )P\tilde{D}+{g}_{0}^{{\rm{T}}}({\tau }_{1}^{2}{R}_{1}+{\tau }_{12}^{2}{R}_{2}){ \mathcal D }\right)}^{{\rm{T}}} & 0 & {\rm{\Pi }}\end{array}\right]\lt 0\end{eqnarray}$ holds for all $\alpha =\left\{0,1\right\}$, where $ \begin{eqnarray*}\begin{array}{rcl}{{\rm{\Phi }}}_{0}(\alpha ) & = & {He}\left({G}_{1}^{{\rm{T}}}(\alpha ){{PG}}_{0}\right)+{\tilde{S}}_{{fg}}\\ & & -{G}_{2}^{{\rm{T}}}{\tilde{R}}_{1}{G}_{2}+{g}_{{}_{0}}^{{\rm{T}}}\left({\tau }_{1}^{2}{R}_{1}+{\tau }_{12}^{2}{R}_{2}\right){g}_{{}_{0}},\\ {{ \mathcal R }}^{0}(\alpha ) & = & \left[\begin{array}{cc}(2-\alpha ){\tilde{R}}_{2} & 0\\ 0 & (1+\alpha ){\tilde{R}}_{2}\end{array}\right],\ {\rm{\Gamma }}=\left[\begin{array}{c}{G}_{3}\\ {G}_{4}\end{array}\right],\ \tilde{D}=\left[\begin{array}{c}{ \mathcal D }\\ 0\\ 0\end{array}\right],\\ {G}_{1}(\alpha ) & = & \left[\begin{array}{ccccccccc}I & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & {\tau }_{1}I & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \alpha {\tau }_{12}I & (1-\alpha ){\tau }_{12}I & 0 & 0\end{array}\right],\\ {\rm{\Pi }} & = & {{ \mathcal D }}^{{\rm{T}}}({\tau }_{1}^{2}{R}_{1}+{\tau }_{12}^{2}{R}_{2}){ \mathcal D }-{\gamma }^{2}I\end{array}\end{eqnarray*}$ with $ \begin{eqnarray*}\begin{array}{rcl}{\tilde{S}}_{{fg}} & = & \mathrm{diag}\{{S}_{1}+Q+{\varepsilon }_{f}{L}_{f}^{2}\bar{I},\\ & & -{S}_{1}+{S}_{2},{\varepsilon }_{g}{L}_{g}^{2}\bar{I},-{S}_{2},{0}_{3n},-{\varepsilon }_{f}I,-{\varepsilon }_{g}I\},\\ {g}_{0} & = & \left[\begin{array}{ccccccccc}{{ \mathcal A }}_{K} & 0 & {{ \mathcal A }}_{\tau } & 0 & 0 & 0 & 0 & { \mathcal B } & {{ \mathcal B }}_{\tau }\end{array}\right],\\ \bar{I} & = & \left[\begin{array}{cc}I & 0\\ 0 & 0\end{array}\right],\,Q=\left[\begin{array}{cc}{Q}_{1} & 0\\ 0 & 0\end{array}\right],\\ {G}_{0} & = & \left[\begin{array}{ccccccccc}{{ \mathcal A }}_{K} & 0 & {{ \mathcal A }}_{\tau } & 0 & 0 & 0 & 0 & { \mathcal B } & {{ \mathcal B }}_{\tau }\\ I & -I & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & I & 0 & -I & 0 & 0 & 0 & 0 & 0\end{array}\right],\\ {G}_{2} & = & \left[\begin{array}{ccccccccc}I & -I & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ I & I & 0 & 0 & -2I & 0 & 0 & 0 & 0\end{array}\right],\\ {G}_{3} & = & \left[\begin{array}{ccccccccc}0 & I & -I & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & I & I & 0 & 0 & -2I & 0 & 0 & 0\end{array}\right],\\ {G}_{4} & = & \left[\begin{array}{ccccccccc}0 & 0 & I & -I & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & I & I & 0 & 0 & -2I & 0 & 0\end{array}\right],\\ {\tilde{R}}_{i} & = & \mathrm{diag}\{{R}_{i},3{R}_{i}\},\ i\in \{1,2\}.\end{array}\end{eqnarray*}$
Let us consider an augmented Lyapunov-Krasovskii functional (LKF) given by$ \begin{eqnarray}V({\xi }_{t},{\dot{\xi }}_{t})={V}_{1}({\xi }_{t})+{V}_{2}({\xi }_{t})+{V}_{3}({\xi }_{t},{\dot{\xi }}_{t}),\end{eqnarray}$where $ \begin{eqnarray*}\begin{array}{rcl}{V}_{1}({\xi }_{t}) & = & {\left[{G}_{1}(\beta )\zeta (t)\right]}^{{\rm{T}}}P\left[{G}_{1}(\beta )\zeta (t)\right],\\ {V}_{2}({\xi }_{t}) & = & {\displaystyle \int }_{t-{\tau }_{1}}^{t}{\xi }^{{\rm{T}}}(s){S}_{1}\xi (s){\rm{d}}s\\ & & +{\displaystyle \int }_{t-{\tau }_{2}}^{t-{\tau }_{1}}{\xi }^{{\rm{T}}}(s){S}_{2}\xi (s){\rm{d}}s,\\ {V}_{3}({\xi }_{t},{\dot{\xi }}_{t}) & = & {\tau }_{1}{\displaystyle \int }_{-{\tau }_{1}}^{0}{\displaystyle \int }_{t+\theta }^{t}{\dot{\xi }}^{{\rm{T}}}(s){R}_{1}\dot{\xi }(s){\rm{d}}s{\rm{d}}\theta \\ & & +{\tau }_{12}{\displaystyle \int }_{-{\tau }_{2}}^{-{\tau }_{1}}{\displaystyle \int }_{t+\theta }^{t}{\dot{\xi }}^{{\rm{T}}}(s){R}_{2}\dot{\xi }(s){\rm{d}}s{\rm{d}}\theta ,\end{array}\end{eqnarray*}$ with $ \begin{eqnarray*}\begin{array}{rcl}{G}_{1}(\beta ) & = & \left[\begin{array}{ccccccccc}I & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & {\tau }_{1}I & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \beta {\tau }_{12}I & (1-\beta ){\tau }_{12}I & 0 & 0\end{array}\right],\\ \zeta (t) & = & {\left[{\zeta }_{1}(t){\zeta }_{2}(t){\zeta }_{3}(t)\right]}^{{\rm{T}}},\\ {\zeta }_{1}(t) & = & \left[{\xi }^{{\rm{T}}}(t)\ \ \ {\xi }^{{\rm{T}}}(t-{\tau }_{1})\ \ \ {\xi }^{{\rm{T}}}(t-\tau (t))\ \ \ {\xi }^{{\rm{T}}}(t-{\tau }_{2})\right],\\ \beta (t) & = & \displaystyle \frac{\tau (t)-{\tau }_{1}}{{\tau }_{12}},\\ {\zeta }_{2}(t) & = & \left[\displaystyle \frac{1}{{\tau }_{1}}{\int }_{t-{\tau }_{1}}^{t}{\xi }^{{\rm{T}}}(s){\rm{d}}s\ \ \ \displaystyle \frac{1}{\tau (t)-{\tau }_{1}}{\int }_{t-\tau (t)}^{t-{\tau }_{1}}{\xi }^{{\rm{T}}}(s){\rm{d}}s\right],\\ {\zeta }_{3}(t) & = & \left[\displaystyle \frac{1}{{\tau }_{2}-\tau (t)}{\int }_{t-{\tau }_{2}}^{t-\tau (t)}{\xi }^{{\rm{T}}}(s){\rm{d}}s\right.\\ & & \left.{\bar{f}}^{{\rm{T}}}(\xi (t))\ {\bar{g}}^{{\rm{T}}}(\xi (t-\tau (t)))\Space{0ex}{3.25ex}{0ex}\right].\end{array}\end{eqnarray*}$ Taking the derivative of ${V}_{1},{V}_{2}$ and V3 along the trajectories of system (10) results in$ \begin{eqnarray}\begin{array}{rcl}{\dot{V}}_{1}({\xi }_{t}) & = & {\zeta }^{{\rm{T}}}(t)\left({He}\left({G}_{1}^{{\rm{T}}}(\beta ){{PG}}_{0}\right)\right)\zeta (t)\\ & & +2{\zeta }^{{\rm{T}}}(t){G}_{1}^{{\rm{T}}}(\beta )P\tilde{D}\omega (t)\end{array}\end{eqnarray}$$ \begin{eqnarray}{\dot{V}}_{2}({\xi }_{t})={\zeta }^{{\rm{T}}}(t)\tilde{S}\zeta (t),\end{eqnarray}$$ \begin{eqnarray}\begin{array}{rcl}{\dot{V}}_{3}({\xi }_{t},{\dot{\xi }}_{t}) & = & {\left[{g}_{0}\zeta (t)+D\omega (t)\right]}^{{\rm{T}}}({\tau }_{1}^{2}{R}_{1}+{\tau }_{12}^{2}{R}_{2})\\ & & \times \left[{g}_{0}\zeta (t)+D\omega (t)\right]\\ & & -{\tau }_{1}{\displaystyle \int }_{t-{\tau }_{1}}^{t}{\dot{\xi }}^{{\rm{T}}}(s){R}_{1}\dot{\xi }(s){\rm{d}}s\\ & & -{\tau }_{12}{\displaystyle \int }_{t-{\tau }_{2}}^{t-{\tau }_{1}}{\dot{\xi }}^{{\rm{T}}}(s){R}_{2}\dot{\xi }(s){\rm{d}}s,\end{array}\end{eqnarray}$respectively, where $ \begin{eqnarray*}\tilde{S}=\mathrm{diag}\{{S}_{1},-{S}_{1}+{S}_{2},{0}_{n},-{S}_{2},{0}_{5n}\}.\end{eqnarray*}$ Using lemma 1 to deal with the first integral of (17), one obtain $ \begin{eqnarray*}\begin{array}{l}-{\tau }_{1}{\displaystyle \int }_{t-{\tau }_{1}}^{t}{\dot{\xi }}^{{\rm{T}}}(s){R}_{1}\dot{\xi }(s){\rm{d}}s\\ \quad \leqslant \,-{\left[\begin{array}{c}\xi (t)-\xi (t-{\tau }_{1})\\ \xi (t)+\xi (t-{\tau }_{1})-\tfrac{2}{{\tau }_{1}}{\displaystyle \int }_{t-{\tau }_{1}}^{t}\xi (s){\rm{d}}s\end{array}\right]}^{{\rm{T}}}{\tilde{R}}_{1}\\ \qquad \times \,\left[\begin{array}{c}\xi (t)-\xi (t-{\tau }_{1})\\ \xi (t)+\xi (t-{\tau }_{1})-\tfrac{2}{{\tau }_{1}}{\displaystyle \int }_{t-{\tau }_{1}}^{t}\xi (s){\rm{d}}s\end{array}\right],\end{array}\end{eqnarray*}$ which can be formulated as$ \begin{eqnarray}-{\tau }_{1}{\int }_{t-{\tau }_{1}}^{t}{\dot{\xi }}^{{\rm{T}}}(s){R}_{1}\dot{\xi }(s){\rm{d}}s\leqslant -{\zeta }^{{\rm{T}}}(t){G}_{2}^{{\rm{T}}}{\tilde{R}}_{1}{G}_{2}\zeta (t).\end{eqnarray}$For the second integral, it follows from lemma 1 that$ \begin{eqnarray}-{\tau }_{12}{\int }_{t-{\tau }_{2}}^{t-{\tau }_{1}}{\dot{\xi }}^{{\rm{T}}}(s){R}_{2}\dot{\xi }(s){\rm{d}}s\leqslant -{\zeta }^{{\rm{T}}}(t){\rm{\Theta }}(\beta )\zeta (t),\end{eqnarray}$in which $ \begin{eqnarray*}\begin{array}{rcl}{\rm{\Theta }}(\beta ) & = & {{\rm{\Gamma }}}^{{\rm{T}}}\left[\begin{array}{cc}\tfrac{1}{\beta }{\tilde{R}}_{2} & 0\\ 0 & \tfrac{1}{1-\beta }{\tilde{R}}_{2}\end{array}\right]{\rm{\Gamma }},\ \forall \beta \in (0,1),\\ {\tilde{R}}_{2} & = & \mathrm{diag}\{{R}_{2},3{R}_{2}\}.\end{array}\end{eqnarray*}$ Moreover, by (2) and (3), one can write$ \begin{eqnarray}{\varepsilon }_{f}{\bar{f}}^{{\rm{T}}}\left(\xi (t,)\right)\bar{f}\left(\xi (t,)\right)\leqslant {\varepsilon }_{f}{L}_{f}^{2}{\xi }^{{\rm{T}}}(t)\bar{I}\xi (t),\end{eqnarray}$$ \begin{eqnarray}\begin{array}{l}{\varepsilon }_{g}{\bar{g}}^{{\rm{T}}}\left(\xi \left(t-\tau (t\right),)\right)\bar{g}\left(\xi \left(t-\tau (t\right),)\right)\\ \quad \leqslant \,{\varepsilon }_{g}{L}_{g}^{2}{\xi }^{{\rm{T}}}\left(t-\tau (t,)\right)\bar{I}\xi \left(t-\tau (t,)\right).\end{array}\end{eqnarray}$Combining (15)–(20) with (21), one has$ \begin{eqnarray}\dot{V}({\xi }_{t},{\dot{\xi }}_{t})\leqslant {\left[\begin{array}{c}\zeta (t)\\ \omega (t)\end{array}\right]}^{{\rm{T}}}\,{{\rm{\Xi }}}_{1}\,\left[\begin{array}{c}\zeta (t)\\ \omega (t)\end{array}\right],\end{eqnarray}$ where $ \begin{eqnarray*}\begin{array}{rcl}{{\rm{\Xi }}}_{1} & = & \left[\begin{array}{cc}{{\rm{\Phi }}}_{0}(\beta )-{\rm{\Theta }}(\beta )-\mathrm{diag}(Q,{0}_{8n}) & {G}_{1}^{{\rm{T}}}(\beta )P\tilde{D}+{g}_{0}^{{\rm{T}}}({\tau }_{1}^{2}{R}_{1}+{\tau }_{12}^{2}{R}_{2}){ \mathcal D }\\ * & {{ \mathcal D }}^{{\rm{T}}}({\tau }_{1}^{2}{R}_{1}+{\tau }_{12}^{2}{R}_{2}){ \mathcal D }\end{array}\right],\\ {{\rm{\Phi }}}_{0}(\beta ) & = & {He}\left({G}_{1}^{{\rm{T}}}(\beta ){{PG}}_{0}\right)+{\tilde{S}}_{{fg}}-{G}_{2}^{{\rm{T}}}{\tilde{R}}_{1}{G}_{2}+{g}_{{}_{0}}^{{\rm{T}}}\left({\tau }_{1}^{2}{R}_{1}+{\tau }_{12}^{2}{R}_{2}\right){g}_{{}_{0}}.\end{array}\end{eqnarray*}$ Define: $ \begin{eqnarray*}{J}_{t}\mathop{=}\limits^{{\rm{\Delta }}}{\int }_{0}^{t}\left({\xi }^{{\rm{T}}}(\sigma )Q\xi (\sigma )-{\gamma }^{2}{\omega }^{{\rm{T}}}(\sigma )\omega (\sigma ,)\right){\rm{d}}\sigma ,\end{eqnarray*}$ where $Q=\mathrm{diag}\{{Q}_{1},0\}$. Then, under the zero initial condition, by $V(t)\geqslant 0$ and (22) one can write $ \begin{eqnarray*}\begin{array}{rcl}{J}_{t} & = & {J}_{t}+{\displaystyle \int }_{0}^{t}\dot{V}(\sigma ){\rm{d}}\sigma -(V(t)-V(0))\\ & \leqslant & {J}_{t}+{\displaystyle \int }_{0}^{t}\dot{V}(\sigma ){\rm{d}}\sigma \\ & = & {\displaystyle \int }_{0}^{t}{\left[\begin{array}{c}\zeta (\sigma )\\ \omega (\sigma )\end{array}\right]}^{{\rm{T}}}\,{\rm{\Xi }}\,\left[\begin{array}{c}\zeta (\sigma )\\ \omega (\sigma )\end{array}\right]{\rm{d}}\sigma ,\end{array}\end{eqnarray*}$ where $ \begin{eqnarray*}{\rm{\Xi }}=\left[\begin{array}{cc}{{\rm{\Phi }}}_{0}(\beta )-{\rm{\Theta }}(\beta ) & {G}_{1}^{{\rm{T}}}(\beta )P\tilde{D}+{g}_{0}^{{\rm{T}}}({\tau }_{1}^{2}{R}_{1}+{\tau }_{12}^{2}{R}_{2}){ \mathcal D }\\ * & {{ \mathcal D }}^{{\rm{T}}}({\tau }_{1}^{2}{R}_{1}+{\tau }_{12}^{2}{R}_{2}){ \mathcal D }-{\gamma }^{2}I\end{array}\right].\end{eqnarray*}$ If ${\rm{\Xi }}\lt 0$ holds for $\beta =\tfrac{\tau (t)-{\tau }_{1}}{{\tau }_{12}}$, then one can get ${J}_{t}\lt 0$ for any ${\left[\begin{array}{cc}{\zeta }^{{\rm{T}}}(t) & {\omega }^{{\rm{T}}}(t)\end{array}\right]}^{{\rm{T}}}\ne 0$, which means $ \begin{eqnarray*}{\int }_{0}^{t}{e}^{{\rm{T}}}(\sigma ){Q}_{1}e(\sigma ){\rm{d}}\sigma \leqslant {\int }_{0}^{t}{\gamma }^{2}{\omega }^{{\rm{T}}}(\sigma )\omega (\sigma ){\rm{d}}\sigma \end{eqnarray*}$ and, thus system (6) is ${{ \mathcal H }}_{\infty }$ synchronized in the sense of definition 1. From the above analysis, now one just need to show that inequality ${\rm{\Xi }}\lt 0$ can be guaranteed by the condition of this Theorem. In fact, by Schur’s complement one can rewrite ${\rm{\Xi }}\lt 0$ as$ \begin{eqnarray}{\rm{\Phi }}(\beta )-{\rm{\Theta }}(\beta )\lt 0,\end{eqnarray}$ where $ \begin{eqnarray*}{\rm{\Phi }}(\beta )={{\rm{\Phi }}}_{0}(\beta )-\phi {\left({{ \mathcal D }}^{{\rm{T}}}({\tau }_{1}^{2}{R}_{1}+{\tau }_{12}^{2}{R}_{2}){ \mathcal D }-{\gamma }^{2}I\right)}^{-1}{\phi }^{{\rm{T}}}\end{eqnarray*}$ with $ \begin{eqnarray*}\phi ={G}_{1}^{{\rm{T}}}(\beta )P\tilde{D}+{g}_{0}^{{\rm{T}}}({\tau }_{1}^{2}{R}_{1}+{\tau }_{12}^{2}{R}_{2}){ \mathcal D }.\end{eqnarray*}$Since Φ (β) is quadratic with respect to β, one can see that Φ (β) is convex function. Then, using lemma 3, (23) holds for any $\beta =\tfrac{\tau (t)-{\tau }_{1}}{{\tau }_{12}}\in (0,\ 1)$ if$ \begin{eqnarray}\left[\begin{array}{cc}{\rm{\Phi }}(\beta )-{{\rm{\Gamma }}}^{{\rm{T}}}{{ \mathcal R }}^{0}(\beta ){\rm{\Gamma }}-{He}\left({{\rm{\Gamma }}}^{{\rm{T}}}\left[\begin{array}{c}(1-\beta ){N}_{1}^{{\rm{T}}}\\ \beta {N}_{2}^{{\rm{T}}}\end{array}\right]\right) & * \\ \beta {N}_{1}^{{\rm{T}}}+(1-\beta ){N}_{2}^{{\rm{T}}} & -{\tilde{R}}_{2}\end{array}\right]\lt 0\end{eqnarray}$is satisfied for all β$=\,\left\{0,1\right\}$. Using Schur’s complement again, (24) can be rewritten as (13) with α replaced by β. Thus, the proof is completed.□
By utilizing augmented LKF (14) together with the improved Wirtinger’s inequality and the generalized reciprocally convex combination lemma, a delay-dependent condition for the ${{ \mathcal H }}_{\infty }$ synchronization of error system (6) is derived without the requirement of differentiability of the delay function. From the proof, it is not difficult to see that the condition also ensures the asymptotical stability of error system (6).
When the time delay is constant (i.e., τ(t)=τ), augmented system (10) turns to$ \begin{eqnarray}\begin{array}{rcl}\dot{\xi }(t) & = & {{ \mathcal A }}_{K}\xi (t)+{{ \mathcal A }}_{\tau }\xi \left(t-\tau \right)+{ \mathcal B }\bar{f}\left(\xi (t,)\right)\\ & & +{{ \mathcal B }}_{\tau }\bar{g}\left(\xi \left(t-\tau \right)\right)+{ \mathcal D }\omega (t).\end{array}\end{eqnarray}$
In this situation, one can establish the following theorem:
For given scalar $\gamma \gt 0,Q$ in ${{\mathbb{S}}}_{+}^{n}$, system (6) with constant time delay is ${{ \mathcal H }}_{\infty }$ synchronized, if there are scalars ${\varepsilon }_{f}\gt 0,{\varepsilon }_{g}\gt 0$ and matrices $P$ in ${{\mathbb{S}}}_{+}^{2n},{S}_{1},{R}_{1}$ in ${{\mathbb{S}}}_{+}^{n}$ such that$ \begin{eqnarray}\left[\begin{array}{cc}{{\rm{\Phi }}}_{0} & {G}_{1}^{{\rm{T}}}P\tilde{D}+{\tau }^{2}{g}_{0}^{{\rm{T}}}{R}_{1}{ \mathcal D }\\ * & {\tau }^{2}{{ \mathcal D }}^{{\rm{T}}}{R}_{1}{ \mathcal D }-{\gamma }^{2}I\end{array}\right]\lt 0,\end{eqnarray}$ where $ \begin{eqnarray*}\begin{array}{rcl}{{\rm{\Phi }}}_{0} & = & {He}({G}_{1}^{{\rm{T}}}{{PG}}_{0})+{\tilde{S}}_{{fg}}+{\tau }^{2}{g}_{0}^{{\rm{T}}}{R}_{1}{g}_{0}-{G}_{2}^{{\rm{T}}}{\tilde{R}}_{1}{G}_{2},\\ {G}_{1} & = & \left[\begin{array}{ccccc}I & 0 & 0 & 0 & 0\\ 0 & 0 & \tau I & 0 & 0\end{array}\right],\ \tilde{D}=\left[\begin{array}{c}{ \mathcal D }\\ 0\end{array}\right],\\ {g}_{0} & = & \left[\begin{array}{ccccc}{{ \mathcal A }}_{K} & {{ \mathcal A }}_{\tau } & 0 & { \mathcal B } & {{ \mathcal B }}_{\tau }\end{array}\right]\end{array}\end{eqnarray*}$ with $ \begin{eqnarray*}\begin{array}{rcl}{G}_{0} & = & \left[\begin{array}{ccccc}{{ \mathcal A }}_{K} & {{ \mathcal A }}_{\tau } & 0 & { \mathcal B } & {{ \mathcal B }}_{\tau }\\ I & -I & 0 & 0 & 0\end{array}\right],\ \bar{I}=\left[\begin{array}{cc}I & 0\\ 0 & 0\end{array}\right],\\ {\tilde{S}}_{{fg}} & = & \mathrm{diag}\{{S}_{1}+Q+{\varepsilon }_{f}{L}_{f}^{2}\bar{I},\\ & & -{S}_{1}+{\varepsilon }_{g}{L}_{g}^{2}\bar{I},{0}_{n},-{\varepsilon }_{f}I,-{\varepsilon }_{g}I\},\\ {G}_{2} & = & \left[\begin{array}{ccccc}I & -I & 0 & 0 & 0\\ I & I & -2I & 0 & 0\end{array}\right],\ {\tilde{R}}_{1}=\mathrm{diag}\{{R}_{1},3{R}_{1}\}.\end{array}\end{eqnarray*}$
Let us choose a LKF $V({\xi }_{t},{\dot{\xi }}_{t})$ as follows $ \begin{eqnarray*}V({\xi }_{t},{\dot{\xi }}_{t})={V}_{1}({\xi }_{t})+{V}_{2}({\xi }_{t},{\dot{\xi }}_{t}),\end{eqnarray*}$ where $ \begin{eqnarray*}\begin{array}{rcl}{V}_{1}({\xi }_{t}) & = & {\zeta }^{{\rm{T}}}(t){G}_{1}^{{\rm{T}}}{{PG}}_{1}\zeta (t),\\ {V}_{2}({\xi }_{t},{\dot{\xi }}_{t}) & = & {\displaystyle \int }_{t-\tau }^{t}{\xi }^{{\rm{T}}}(s){S}_{1}\xi (s){\rm{d}}s+\tau \\ & & \times {\displaystyle \int }_{-\tau }^{0}{\displaystyle \int }_{t+\theta }^{t}{\dot{\xi }}^{{\rm{T}}}(s){R}_{1}\dot{\xi }(s){\rm{d}}s{\rm{d}}\theta \end{array}\end{eqnarray*}$ with $ \begin{eqnarray*}\begin{array}{rcl}\zeta (t) & = & \left[{\xi }^{{\rm{T}}}(t)\ \ {\xi }^{{\rm{T}}}(t-\tau )\ \ \displaystyle \frac{1}{\tau }{\displaystyle \int }_{t-\tau }^{t}{\xi }^{{\rm{T}}}(s){\rm{d}}s\right.\\ & & {\times \left.{\bar{f}}^{{\rm{T}}}(\xi (t)){\bar{g}}^{{\rm{T}}}(\xi (t-\tau ))\right]}^{{\rm{T}}}.\end{array}\end{eqnarray*}$ Thus taking the derivatives of V1 and V2 along the trajectories of augmented system (25) results in$ \begin{eqnarray}{\dot{V}}_{1}({\xi }_{t})=2{\zeta }^{{\rm{T}}}(t){G}_{1}^{{\rm{T}}}{{PG}}_{0}\zeta (t)+2{\zeta }^{{\rm{T}}}(t){G}_{1}^{{\rm{T}}}P\tilde{D}\omega (t),\end{eqnarray}$$ \begin{eqnarray}\begin{array}{rcl}{\dot{V}}_{2}({\xi }_{t},{\dot{\xi }}_{t}) & = & {\zeta }^{{\rm{T}}}(t)\tilde{S}\zeta (t)+{\tau }^{2}{\dot{\xi }}^{{\rm{T}}}(t){R}_{1}\dot{\xi }(t)\\ & & -\tau {\displaystyle \int }_{t-\tau }^{t}{\dot{\xi }}^{{\rm{T}}}(s){R}_{1}\dot{\xi }(s){\rm{d}}s,\end{array}\end{eqnarray}$respectively, where $ \begin{eqnarray*}\tilde{S}=\mathrm{diag}\{{S}_{1},-{S}_{1},{0}_{3n}\}.\end{eqnarray*}$ In view of the definition of the matrices g0 and $\zeta (t)$, the second term of (28) can be formulated as follows$ \begin{eqnarray}\begin{array}{l}{\tau }^{2}{\dot{\xi }}^{{\rm{T}}}(t){R}_{1}\dot{\xi }(t)={\tau }^{2}{\zeta }^{{\rm{T}}}(t){g}_{0}^{{\rm{T}}}{R}_{1}{g}_{0}\zeta (t)\\ \quad +\,2{\tau }^{2}{\zeta }^{{\rm{T}}}(t){g}_{0}^{{\rm{T}}}{R}_{1}{ \mathcal D }\omega (t)+{\tau }^{2}{\omega }^{{\rm{T}}}(t){{ \mathcal D }}^{{\rm{T}}}{R}_{1}{ \mathcal D }\omega (t).\end{array}\end{eqnarray}$Applying lemma 1 to the third term of (28), the following upper bound is obtained$ \begin{eqnarray}-\tau {\int }_{t-\tau }^{t}{\dot{\xi }}^{{\rm{T}}}(s){R}_{1}\dot{\xi }(s){\rm{d}}s\leqslant -{\left[{G}_{2}\zeta (t)\right]}^{{\rm{T}}}{\tilde{R}}_{1}\left[{G}_{2}\zeta (t)\right].\end{eqnarray}$Moreover, by the Lipschitz conditions given in (2) and (3), one can write$ \begin{eqnarray}{\varepsilon }_{f}{\bar{f}}^{{\rm{T}}}\left(\xi (t,)\right)\bar{f}\left(\xi (t,)\right)\leqslant {\varepsilon }_{f}{L}_{f}^{2}{\xi }^{{\rm{T}}}(t)\bar{I}\xi (t),\end{eqnarray}$$ \begin{eqnarray}{\varepsilon }_{g}{\bar{g}}^{{\rm{T}}}\left(\xi \left(t-\tau \right)\right)\bar{g}\left(\xi \left(t-\tau \right)\right)\leqslant {\varepsilon }_{g}{L}_{g}^{2}{\xi }^{{\rm{T}}}\left(t-\tau \right)\bar{I}\xi \left(t-\tau \right).\end{eqnarray}$Now, one can get from (27)–(32) that $ \begin{eqnarray*}\begin{array}{l}\dot{V}({\xi }_{t},{\dot{\xi }}_{t})={\left[\begin{array}{c}\zeta (t)\\ \omega (t)\end{array}\right]}^{{\rm{T}}}\\ \times \,\left[\begin{array}{cc}{{\rm{\Phi }}}_{0}-\mathrm{diag}(Q,{0}_{4n}) & {G}_{1}^{{\rm{T}}}P\tilde{D}+{\tau }^{2}{g}_{0}^{{\rm{T}}}{R}_{1}{ \mathcal D }\\ * & {\tau }^{2}{{ \mathcal D }}^{{\rm{T}}}{R}_{1}{ \mathcal D }\end{array}\right]\left[\begin{array}{c}\zeta (t)\\ \omega (t)\end{array}\right].\end{array}\end{eqnarray*}$ Then, along similar lines as those in the proof of theorem 1, one can easily show that system (6) with constant time delay is ${{ \mathcal H }}_{\infty }$ synchronized. The reminder is omitted for brevity.□
4.${{ \mathcal H }}_{\infty }$ synchronization synthesis
This section focuses on the design of resilient DOF controller to ensure the ${{ \mathcal H }}_{\infty }$ synchronization of the error system. A constructive approach is presented by the following theorem:For given scalars $\gamma \gt 0$ , $\rho \gt 0$, resilient DOF controller (7) ensures the ${{ \mathcal H }}_{\infty }$ synchronization of drive system (1) and response system (5), if there exist scalars $\varepsilon \gt 0,{\varepsilon }_{f}\gt 0,{\varepsilon }_{g}\gt 0$, and matrices $P={({P}_{{ij}})}_{3\times 3}$ in ${{\mathbb{S}}}_{+}^{3n},{S}_{1},{S}_{2},{R}_{1},{R}_{2}$ in ${{\mathbb{S}}}_{+}^{n},{N}_{1},{N}_{2}$ in ${R}^{9n\times 2n},X,Y$ such that the following LMI holds true$ \begin{eqnarray}\left[\begin{array}{ccc}{\rm{\Lambda }} & {{\rm{\Delta }}}_{B}-{{\rm{\Delta }}}_{J}X+\rho {{\rm{\Delta }}}_{C}^{{\rm{T}}}{Y}^{{\rm{T}}} & {{\rm{\Delta }}}_{B}M\\ * & -\rho X-\rho {X}^{{\rm{T}}} & 0\\ * & * & -\varepsilon I\end{array}\right]\lt 0\end{eqnarray}$for all $\alpha =\left\{0,1\right\},$ where $ \begin{eqnarray*}\begin{array}{rcl}{{\rm{\Delta }}}_{B} & = & {\left[\begin{array}{ccccccccccccc}{{\rm{\Delta }}}_{1}^{{\rm{T}}} & 0 & 0 & 0 & {{\rm{\Delta }}}_{5}^{{\rm{T}}} & {{\rm{\Delta }}}_{6}^{{\rm{T}}} & {{\rm{\Delta }}}_{7}^{{\rm{T}}} & 0 & 0 & 0 & 0 & {{\rm{\Delta }}}_{12}^{{\rm{T}}} & {{\rm{\Delta }}}_{13}^{{\rm{T}}}\end{array}\right]}^{{\rm{T}}},\\ {{\rm{\Delta }}}_{C} & = & \left[\begin{array}{ccccccccccccc}\tilde{C} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right],\\ {{\rm{\Delta }}}_{J} & = & {\left[\begin{array}{ccccccccccccc}{\tilde{B}}^{{\rm{T}}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]}^{{\rm{T}}},\end{array}\end{eqnarray*}$ $ \begin{eqnarray*}\begin{array}{rcl}{\rm{\Lambda }} & = & {He}({{\rm{\Delta }}}_{J}Y{{\rm{\Delta }}}_{C})+{{\rm{\Omega }}}_{1}+{{\rm{\Omega }}}_{2}+\varepsilon \,{{\rm{\Delta }}}_{C}^{{\rm{T}}}{N}^{{\rm{T}}}N{{\rm{\Delta }}}_{C},\\ {{\rm{\Omega }}}_{1} & = & \left[\begin{array}{cccc}\unicode{x003DD} & * & * & * \\ \alpha {N}_{1}^{{\rm{T}}}+(1-\alpha ){N}_{2}^{{\rm{T}}} & -{\tilde{R}}_{2} & * & * \\ {\left({G}_{1}^{{\rm{T}}}(\alpha )P\tilde{D}\right)}^{{\rm{T}}} & 0 & {\rm{\Pi }} & * \\ 0 & 0 & 0 & -\left({\tau }_{1}^{2}{R}_{1}+{\tau }_{12}^{2}{R}_{2}\right)\end{array}\right],\\ {{\rm{\Omega }}}_{2} & = & \left[\begin{array}{cccc}{He}\left({G}_{1}^{{\rm{T}}}(\alpha ){{PG}}_{01}\right) & * & * & * \\ 0 & 0 & * & * \\ {\left({g}_{01}^{{\rm{T}}}({\tau }_{1}^{2}{R}_{1}+{\tau }_{12}^{2}{R}_{2}){ \mathcal D }\right)}^{{\rm{T}}} & 0 & 0 & * \\ \left({\tau }_{1}^{2}{R}_{1}+{\tau }_{12}^{2}{R}_{2}\right){g}_{01} & 0 & 0 & 0\end{array}\right],\\ {G}_{01} & = & \left[\begin{array}{ccccccccc}\tilde{A} & 0 & {{ \mathcal A }}_{\tau } & 0 & 0 & 0 & 0 & { \mathcal B } & {{B}}_{\tau }\\ I & -I & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & I & 0 & -I & 0 & 0 & 0 & 0 & 0\end{array}\right],\\ \unicode{x003DD} & = & {\tilde{S}}_{{fg}}-{G}_{2}^{{\rm{T}}}{\tilde{R}}_{1}{G}_{2}-{{\rm{\Gamma }}}^{{\rm{T}}}{{ \mathcal R }}^{0}(\alpha ){\rm{\Gamma }}\\ & & -{He}\left({{\rm{\Gamma }}}^{{\rm{T}}}\left[\begin{array}{c}(1-\alpha ){N}_{1}^{{\rm{T}}}\\ \alpha {N}_{2}^{{\rm{T}}}\end{array}\right]\right),\\ {g}_{01} & = & \left[\begin{array}{ccccccccc}\tilde{A} & 0 & {{ \mathcal A }}_{\tau } & 0 & 0 & 0 & 0 & { \mathcal B } & {{B}}_{\tau }\end{array}\right],\ \\ {{\rm{\Delta }}}_{1} & = & {P}_{11}\tilde{B},\ {{\rm{\Delta }}}_{5}={\tau }_{1}{P}_{12}^{{\rm{T}}}\tilde{B},\ {{\rm{\Delta }}}_{6}=\alpha {\tau }_{12}{P}_{13}^{{\rm{T}}}\tilde{B},\ \\ {{\rm{\Delta }}}_{7} & = & (1-\alpha ){\tau }_{12}{P}_{13}^{{\rm{T}}}\tilde{B},\ {{\rm{\Delta }}}_{12}={{ \mathcal D }}^{{\rm{T}}}\left({\tau }_{1}^{2}{R}_{1}+{\tau }_{12}^{2}{R}_{2}\right)\tilde{B},\\ {{\rm{\Delta }}}_{13} & = & \left({\tau }_{1}^{2}{R}_{1}+{\tau }_{12}^{2}{R}_{2}\right)\tilde{B},\end{array}\end{eqnarray*}$ where ${\tilde{S}}_{{fg}},{\rm{\Gamma }},{G}_{2}$, and ${\rm{\Pi }}$ are the same as those in theorem 1. In this case, the desired gain is given by$ \begin{eqnarray}K={X}^{-1}Y.\end{eqnarray}$
Using Schur’s complement, inequality (13) is equivalent to$ \begin{eqnarray}{{\rm{\Omega }}}_{1}+{{\rm{\Omega }}}_{2}+{He}({{\rm{\Delta }}}_{B}(K+{\rm{\Delta }}K){{\rm{\Delta }}}_{C})\lt 0.\end{eqnarray}$In view of ${F}^{{\rm{T}}}F\leqslant I$, it is not difficult to write$ \begin{eqnarray}{He}({{\rm{\Delta }}}_{B}{\rm{\Delta }}K{{\rm{\Delta }}}_{C})\leqslant {{\rm{\Delta }}}_{\varepsilon },\end{eqnarray}$where $ \begin{eqnarray*}{{\rm{\Delta }}}_{\varepsilon }=\displaystyle \frac{1}{\varepsilon }{{\rm{\Delta }}}_{B}{{MM}}^{{\rm{T}}}{{\rm{\Delta }}}_{B}^{{\rm{T}}}+\varepsilon {{\rm{\Delta }}}_{C}^{{\rm{T}}}{N}^{{\rm{T}}}N{{\rm{\Delta }}}_{C}.\end{eqnarray*}$ Then, using (36) and noting $K={X}^{-1}Y$ , one can see that (35) is guaranteed by$ \begin{eqnarray}\begin{array}{l}{{\rm{\Omega }}}_{1}+{{\rm{\Omega }}}_{2}+{He}\left({{\rm{\Delta }}}_{J}Y{{\rm{\Delta }}}_{C}\right.\\ \quad \left.+\left({{\rm{\Delta }}}_{B}-{{\rm{\Delta }}}_{J}X\right){X}^{-1}Y{{\rm{\Delta }}}_{C}\right)+{{\rm{\Delta }}}_{\varepsilon }\lt 0.\end{array}\end{eqnarray}$ On the other hand, one can obtain from (33) that $ \begin{eqnarray*}\left[\begin{array}{cc}{{\rm{\Omega }}}_{1}+{{\rm{\Omega }}}_{2}+{He}({{\rm{\Delta }}}_{J}Y{{\rm{\Delta }}}_{C})+{{\rm{\Delta }}}_{\varepsilon } & {{\rm{\Delta }}}_{B}-{{\rm{\Delta }}}_{J}X+\rho {{\rm{\Delta }}}_{C}^{{\rm{T}}}{Y}^{{\rm{T}}}\\ * & -\rho X-\rho {X}^{{\rm{T}}}\end{array}\right]\lt 0,\end{eqnarray*}$ which, together with lemma 2, implies (37). In this way, the proof is completed.□
For the situation of constant time delay, the following result can be established via following similar lines as those in the derivation of theorem 3:For given scalars $\gamma \gt 0$ , $\rho \gt 0$, resilient DOF controller (7) ensures the ${{ \mathcal H }}_{\infty }$ synchronization of drive system (1) and response system (5) with constant time delay, if there exist scalars $\varepsilon \gt 0,{\varepsilon }_{f}\gt 0,{\varepsilon }_{g}\gt 0$, and matrices $P={({P}_{{ij}})}_{2\times 2}$ in ${{\mathbb{S}}}_{+}^{2n},{S}_{1},{R}_{1}$ in ${{\mathbb{S}}}_{+}^{n},X,Y$ such that the following LMI $ \begin{eqnarray*}\left[\begin{array}{ccc}{\rm{\Lambda }} & {{\rm{\Delta }}}_{B}-{{\rm{\Delta }}}_{J}X+\rho {{\rm{\Delta }}}_{C}^{{\rm{T}}}{Y}^{{\rm{T}}} & {{\rm{\Delta }}}_{B}M\\ * & -\rho X-\rho {X}^{{\rm{T}}} & 0\\ * & * & -\varepsilon I\end{array}\right]\lt 0\end{eqnarray*}$ holds, where $ \begin{eqnarray*}\begin{array}{rcl}{\rm{\Lambda }} & = & {He}({{\rm{\Delta }}}_{J}Y{{\rm{\Delta }}}_{C})+{{\rm{\Omega }}}_{1}+{{\rm{\Omega }}}_{2}+\varepsilon {{\rm{\Delta }}}_{C}^{{\rm{T}}}{N}^{{\rm{T}}}N{{\rm{\Delta }}}_{C},\\ {{\rm{\Delta }}}_{B} & = & {\left[\begin{array}{ccccccc}{{\rm{\Delta }}}_{1}^{{\rm{T}}} & 0 & {{\rm{\Delta }}}_{3}^{{\rm{T}}} & 0 & 0 & {{\rm{\Delta }}}_{6}^{{\rm{T}}} & {{\rm{\Delta }}}_{7}^{{\rm{T}}}\end{array}\right]}^{{\rm{T}}},\\ {{\rm{\Delta }}}_{C} & = & \left[\begin{array}{ccccccc}\tilde{C} & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right],\\ {{\rm{\Delta }}}_{J} & = & {\left[\begin{array}{ccccccc}{\tilde{B}}^{{\rm{T}}} & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]}^{{\rm{T}}}\end{array}\end{eqnarray*}$ with $ \begin{eqnarray*}\begin{array}{rcl}{{\rm{\Omega }}}_{1} & = & \left[\begin{array}{ccc}{\tilde{S}}_{{fg}}-{G}_{2}^{{\rm{T}}}{\tilde{R}}_{1}{G}_{2} & {G}_{1}^{{\rm{T}}}P\tilde{D} & 0\\ * & {\tau }^{2}{D}^{{\rm{T}}}{R}_{1}D-{\gamma }^{2}I & 0\\ * & * & -{\tau }^{2}{R}_{1}\end{array}\right],\\ {{\rm{\Omega }}}_{2} & = & \left[\begin{array}{cccc}{He}\left({G}_{1}^{{\rm{T}}}{{PG}}_{01}\right) & * & * & * \\ 0 & 0 & * & * \\ {\tau }^{2}{{ \mathcal D }}^{{\rm{T}}}{R}_{1}{g}_{01} & 0 & 0 & * \\ {\tau }^{2}{R}_{1}{g}_{01} & 0 & 0 & 0\end{array}\right],\end{array}\end{eqnarray*}$ $ \begin{eqnarray*}\begin{array}{rcl}{G}_{01} & = & \left[\begin{array}{ccccc}\tilde{A} & {{ \mathcal A }}_{\tau } & 0 & { \mathcal B } & {{ \mathcal B }}_{\tau }\\ I & -I & 0 & 0 & 0\end{array}\right],\ {g}_{01}=\left[\begin{array}{ccccc}\tilde{A} & {{ \mathcal A }}_{\tau } & 0 & { \mathcal B } & {{ \mathcal B }}_{\tau }\end{array}\right],\\ {{\rm{\Delta }}}_{1} & = & {P}_{11}\tilde{B},\ {{\rm{\Delta }}}_{3}=\tau {P}_{12}^{{\rm{T}}}\tilde{B},\ {{\rm{\Delta }}}_{6}={\tau }^{2}{{ \mathcal D }}^{{\rm{T}}}{R}_{1}\tilde{B},\ {{\rm{\Delta }}}_{7}={\tau }^{2}{R}_{1}\tilde{B},\end{array}\end{eqnarray*}$ where ${\tilde{S}}_{{fg}}$ and ${G}_{2}$ are the same as those in theorem 2. In this case, the desired gain is given by (34).
With the help of lemma 2 and Schur’s complement, theorems 3 and 4 give sufficient conditions on the existence of resilient DOF controller (7) for ensuring the ${{ \mathcal H }}_{\infty }$ synchronization of drive system (1) and response system (5) with time-varying delay and constant delay, respectively. These conditions possess the form of LMIs and, hence, are able to be easily checked via the well-known mathematical software Matlab. Once the LMI conditions are feasible, the desired control gain can be obtained by (34).
5. Simulation examples
In the section, an example is applied to show the applicability of the presented DOF ${{ \mathcal H }}_{\infty }$ synchronization control approaches.
Consider a time-delay Hopfield neural network as follows: $ \begin{eqnarray*}\begin{array}{rcl}\left[\begin{array}{c}{\dot{x}}_{1}(t)\\ {\dot{x}}_{2}(t)\end{array}\right] & = & A\left[\begin{array}{c}{x}_{1}(t)\\ {x}_{2}(t)\end{array}\right]+B\left[\begin{array}{c}\tanh ({x}_{1}(t))\\ \tanh ({x}_{2}(t))\end{array}\right]\\ & & +{B}_{\tau }\left[\begin{array}{c}\tanh ({x}_{1}(t-\tau (t)))\\ \tanh ({x}_{2}(t-\tau (t)))\end{array}\right],\end{array}\end{eqnarray*}$ where system parameters $A,B$, and ${B}_{\tau }$ are given in [15]: $ \begin{eqnarray*}\begin{array}{rcl}A & = & -\left[\begin{array}{cc}1 & 0\\ 0 & 1\end{array}\right],\ B=\left[\begin{array}{cc}2 & -0.1\\ -5 & 2.0\end{array}\right],\\ {B}_{\tau } & = & \left[\begin{array}{cc}-1.5 & -0.1\\ -0.2 & -1.5\end{array}\right],\end{array}\end{eqnarray*}$ and the time delay is taken as $\tau (t)={\tau }_{2}-{\tau }_{12}{e}^{-t}$. It can be seen that (2)–(4) hold true with ${L}_{f}={L}_{g}=1$.
Let us set $ \begin{eqnarray*}\begin{array}{rcl}C & = & \left[\begin{array}{cc}1 & -1\\ 0 & 0.5\end{array}\right],\ D=\left[\begin{array}{cc}1.4 & 1\\ -1.1 & 1\end{array}\right],\\ H & = & \left[\begin{array}{cc}-1.2 & 0\\ 0 & -1.7\end{array}\right],\ {Q}_{1}=\left[\begin{array}{cc}0.1 & 0\\ 0 & 0.1\end{array}\right],\\ M & = & {[\begin{array}{ccc}{M}_{1} & \cdots & {M}_{{n}_{1}+{n}_{c}}\end{array}]}^{{\rm{T}}},\\ F & = & 0.5,\ N=[\begin{array}{ccc}{N}_{1} & \cdots & {N}_{{n}_{1}+{n}_{c}}\end{array}],\end{array}\end{eqnarray*}$ where Mi and Ni ($i=1,\,\cdots ,\,{n}_{1}+{n}_{c}$) are random numbers belonging to $(-3,3)$. Then, theorem 3 can be utilized to seek the resilient DOF controller (7) ensuring the ${{ \mathcal H }}_{\infty }$ synchronization of the drive-response systems. Especially, when taking ρ=0.1 and τ1=0.5, the minimum allowed ${{ \mathcal H }}_{\infty }$ disturbance attenuation bound for different (nc, τ2) is given in table 1, from which one can see the designed resilient DOF controller of different orders yields different results.
In the following, let us further set that τ2=1.2, nc=1, and γ=0.108 4. Then, solving the LMIs in theorem 3 results in $ \begin{eqnarray*}\begin{array}{rcl}X & = & \left[\begin{array}{ccc}62 & 46683 & -34886\\ -46683 & 51 & 3693\\ 34886 & -3716 & 41\end{array}\right],\\ Y & = & \left[\begin{array}{ccc}-620 & 560200 & -65730\\ 466830 & 620 & 126790\\ -348860 & -44590 & -87780\end{array}\right].\end{array}\end{eqnarray*}$ Thus, according to theorem 3, the ${{ \mathcal H }}_{\infty }$ synchronization of the drive-response systems can be ensured by the designed resilient DOF controller with gain $K={X}^{-1}Y$.
Suppose that $\omega (t)={\left[5{e}^{-2t},3{e}^{-2t}\right]}^{{\rm{T}}}$ and $ \begin{eqnarray*}\left[{x}_{1}(s),{x}_{2}(s)\right]=\left[{\hat{x}}_{1}(s),{\hat{x}}_{2}(s)\right]=[1.7,2.5],\ s\in [-1,0].\end{eqnarray*}$ Then, the phase-plane trajectories of the drive system and unforced response system are shown in figures 1 and 2, respectively. The state trajectories are depicted in figure 3. One can observe that both the drive and unforced response systems have chaotic behaviors, and they are not synchronized even under the same initial condition. Let us change the initial condition of the response system to be $\left[{\hat{x}}_{1}(s),{\hat{x}}_{2}(s)\right]\,=[-1.2,\ -3.0],\ s\in [-1,0]$. Then, under the designed resilient DOF controller, the plot of $ \begin{eqnarray*}\gamma (t)=\sqrt{{\int }_{0}^{t}{e}^{{\rm{T}}}(\sigma ){Q}_{1}e(\sigma ){\rm{d}}\sigma /{\int }_{0}^{t}{\omega }^{{\rm{T}}}(\sigma )\omega (\sigma ){\rm{d}}\sigma }\end{eqnarray*}$ and the state trajectories of the drive-response systems are shown in figures 4 and 5, respectively. It can be seen that γ (t)<γ=0.1084 and the chaos synchronization of the drive-response systems is achieved very quickly.
In this paper, a resilient DOF controller for ${{ \mathcal H }}_{\infty }$ synchronization of a class of chaotic Hopfield networks with the time-varying delay has been designed. By employing the Lyapunov functional method together with some recent inequalities, a delay-dependent criterion for the ${{ \mathcal H }}_{\infty }$ synchronization has been derived. Then, with the help of some decoupling techniques, sufficient conditions for the existence of the resilient DOF controller have been developed for both the time-varying and constant time-delay cases.
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,1,4, Muyun Fang(方木云)11School of Computer Science & Technology, 
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