Improved delay-dependent stability criterion for neural networks with time-varying delay

In this paper, the problem of delay-dependent asymptotic stability criterion for neural networks with time-varying delay has been considered. A new class of Lyapunov functional which contains a triple-integral term is constructed to derive some new delay-dependent stability criteria. The obtained criteria are less conservative because free-weighting matrices method and a convex optimization approach are considered. Finally, numerical examples are given to illustrate the effectiveness of the proposed method.     

A blow‐up result for Kirchhoff‐type equations with high energy

In this paper, we consider the following Kirchhoff type equation: with initial condition and zero Dirichlet boundary condition. We established sufficient conditions on the initial data with arbitrarily high energy such that the solution blows up in finite time. This result generalizes and improves the earlier results in the literature. Copyright © 2010 John Wiley & Sons, Ltd.     

The Properties of Solutions for a Generalized b-Family Equation with Peakons

This paper deals with the Cauchy problem for a shallow water equation with high-order nonlinearities, y _t +u ^m+1 y _x +bu ^m u _x y=0, where b is a constant, $m\in \mathbb{N}$ , and we have the notation $y:= (1-\partial_{x}^{2}) u$ , which includes the famous Camassa–Holm equation, the Degasperis–Procesi equation, and the Novikov equation as special cases. The local well-posedness of strong solutions for the equation in each of the Sobolev spaces $H^{s}(\mathbb{R})$ with $s>\frac{3}{2}$ is obtained, and persistence properties of the strong solutions are studied. Furthermore, although the $H^{1}(\mathbb{R})$ -norm of the solution to the nonlinear model does not remain constant, the existence of its weak solutions in each of the low order Sobolev spaces $H^{s}(\mathbb{R})$ with $1<s<\frac{3}{2}$ is established, under the assumption $u_{0}(x)\in H^{s}(\mathbb{R})\cap W^{1,\infty}(\mathbb{R})$ . Finally, the global weak solution and peakon solution for the equation are also given.     

随机时滞控制系统的均方BIBO稳定性

本文研究了具有时滞和非线性扰动的随机控制系统的均方有界输入-有界输出(BIBO)稳定.首先,探讨了具有离散时滞和非线性扰动的随机系统的均方BIBO稳定性问题,在此基础上,进一步研究带有离散时滞和分布时滞以及非线性扰动的随机系统的均方BIBO稳定性.通过设计合理的控制器,建立合适的Lyapunov泛函,结合Riccati矩阵方程,得到时滞依赖的均方BIBO稳定性条件.     

Novel delay-dependent asymptotical stability of neutral systems with nonlinear perturbations

This paper address the asymptotical stability of neutral systems with nonlinear perturbations. Some novel delay-dependent asymptotical stability criteria are formulated in terms of linear matrix inequalities (LMIs). The resulting delay-dependent stability criteria are less conservative than the previous ones, owing to the introduction of free-weighting matrices, based on a class of novel augment Lyapunov functionals. Numerical examples are given to demonstrate that the derived conditions are much less conservative than those given in the literature.     

Design of PD controller for master-slave synchronization of Lur’e systems with time-delay

In this paper, design of PD controller for master-slave synchronization of Lur’e systems with time-delay has been considered. A new delay-dependent synchronization criterion is derived based on the integral inequality method and free weighting matrices. The criteria are given in terms of LMIs, which can be easily solved by LMI Toolbox in Matlab. A numerical example is given to illustrate the effectiveness of the method.     

Cluster synchronization of linearly coupled complex networks via linear and adaptive feedback pinning controls

Cluster synchronization is investigated for complex networks via linear and adaptive feedback control strategies. It is shown that two different controllers can be designed to achieve the cluster synchronization. Unlike most existing papers, we need not nondelayed and delayed coupling matrices to be symmetric or irreducible. Finally, two examples are given to illustrate the effectiveness of the proposed methods.     

Pinning lag synchronization of complex dynamical networks with known state time-delay and unknown channel time-delay

In this paper, a linear and adaptive feedback pinning strategy is used to study the lag synchronization of complex dynamical networks with known state time-delay and unknown channel time-delay. Firstly, based on the Lyapunov stability theory, a novel Lyapunov functional, which involves the estimated error \(\hat{e}_{i}(t)\) rather than the general synchronization error \(e_{i}(t)\), is constructed. Secondly, in view of the unknown information of the channel time-delay, two available pinning controllers are designed such that the considered networks achieve lag synchronization. Furthermore, by a proper adaptation mechanism, we estimate the unknown channel time-delay successfully under the case that the initial value of the estimated channel time-delay is larger than true channel time-delay, i.e., \(\hat{\tau }(0)>\tau \) and the another case \(\hat{\tau }(0)<\tau \). Finally, the effectiveness and correctness of the lag synchronization criteria are verified through two simulation experiments.