Robust stability analysis of stochastic neural networks with Markovian jumping parameters and probabilistic time‐varying delays

This article discusses the issue of robust stability analysis for a class of Markovian jumping stochastic neural networks (NNs) with probabilistic time‐varying delays. The jumping parameters are represented as a continuous‐time discrete‐state Markov chain. Using the stochastic stability theory, properties of Brownian motion, the information of probabilistic time‐varying delay, the generalized Ito's formula, and linear matrix inequality (LMI) technique, some novel sufficient conditions are obtained to guarantee the stochastical stability of the given NNs. In particular, the activation functions considered in this article are reasonably general in view of the fact that they may depend on Markovian jump parameters and they are more general than those usual Lipschitz conditions. The main features of this article are described in the following: first one is that, based on generalized Finsler lemma, some improved delay‐dependent stability criteria are established and the second one is that the nonlinear stochastic perturbation acting on the system satisfies a class of Lipschitz linear growth conditions. By resorting to the Lyapunov–Krasovskii stability theory and the stochastic analysis tools, sufficient stability conditions are established using an efficient LMI approach. Finally, two numerical examples and its simulations are given to demonstrate the usefulness and effectiveness of the proposed results. © 2014 Wiley Periodicals, Inc. Complexity 21: 59–72, 2016     

Robust Stability Criterion for Discrete‐Time Nonlinear Switched Systems with Randomly Occurring Delays via T–S Fuzzy Approach

This article investigates exponential stability of uncertain discrete‐time nonlinear switched systems with parameter uncertainties and randomly occurring delays via Takagi–Sugeno fuzzy approach. The randomness of time‐varying delay is characterized by introducing a Bernoulli stochastic variable that follows certain probability distribution. By adopting the average dwell‐time approach with Lyapunov–Krasovskii functional and using convex reciprocal lemma, delay‐dependent sufficient conditions for exponential stability of the switched fuzzy system are derived in terms of linear matrix inequalities (LMIs), which can be solved readily using any LMI solvers. Finally, illustrative examples are provided to demonstrate the effectiveness of the proposed approach. © 2014 Wiley Periodicals, Inc. Complexity 20: 49–61, 2015     

Robust asymptotic state estimation of Takagi–Sugeno fuzzy Markovian jumping Hopfield neural networks with mixed interval time‐varying delays

In this paper, the Takagi–Sugeno (T–S) fuzzy model representation is extended to the state estimation of uncertain Markovian jumping Hopfield neural networks with mixed interval time‐varying delays. The main purpose is to estimate the neuron states, through available output measurements such that for all admissible time delays, the dynamics of the estimation error are globally asmptotically stable in the mean square. Based on the Lyapunov–Krasovskii functional and stochastic analysis approach, several delay‐dependent robust state estimators for such T–S fuzzy Markovian jumping Hopfield neural networks can be achieved by solving a linear matrix inequality (LMI), which can be easily facilitated by using some standard numerical packages. Finally a numerical example is provided to demonstrate the effectiveness of the proposed method. Copyright © 2011 John Wiley & Sons, Ltd.     

Large‐time behaviour of solutions to non‐linear wave equations: higher‐order asymptotics

The Cauchy problems for the Korteweg–de Vries–Burgers equation and the Benjamin–Bona– Mahony–Burgers equation are studied. Using subtle estimates of solutions to the linearized equations, the higher‐order terms of the asymptotic expansion as of solutions are derived. Copyright © 1999 John Wiley & Sons, Ltd.     

Robust guaranteed cost control for discrete‐time systems via partially delay‐dependent controller with linear fractional uncertainties

In this article, a partially delay‐dependent controller is designed to analyze the guaranteed performance analysis of a class of uncertain discrete‐time systems with time‐varying delays. By constructing suitable Lyapunov–Krasovskii Functional (LKF), sufficient conditions are derived to ensure the system to be robustly stochastically stable in mean square sense by using Wirtinger‐based inequality and convex reciprocal lemma. The proper cost function is chosen to guarantee an adequate level of performance. The derived conditions are expressed in terms of linear matrix inequalities (LMIs) which can be easily solved by LMI Toolbox in MATLAB. Further, the advantage of employing the obtained results is illustrated via numerical examples. © 2016 Wiley Periodicals, Inc. Complexity 21: 113–122, 2016     

Traveling wave solutions of n‐dimensional delayed reaction–diffusion systems and application to four‐dimensional predator–prey systems

This paper deals with the existence of traveling wave solutions for n‐dimensional delayed reaction–diffusion systems. By using Schauder's fixed point theorem, we establish the existence result of a traveling wave solution connecting two steady states by constructing a pair of upper–lower solutions that are easy to construct. As an application, we apply our main results to a four‐dimensional delayed predator–prey system and obtain the existence of traveling wave solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Stability analysis of impulsive delayed switched systems and applications

In this paper, a general class of impulsive delayed switched systems is considered. By employing the Lyapunov–Razumikhin method and some analysis techniques, we established several global asymptotic stability and global exponential stability criteria for the considered impulsive delayed switched systems, which improve and extend some recent works. As an application, the result of global exponential stability are used to study a class of uncertain linear switched systems with time‐varying delays. Several LMI‐based conditions are proposed to guarantee the global robust stability and global exponential stabilization. The designed memoryless state feedback controller can be easily checked by the LMI toolbox in Matlab. Moreover, the dwell time constraint is imposed for the switching law. Finally, two numerical examples and their simulations are given to show the effectiveness of our proposed results. Copyright © 2012 John Wiley & Sons, Ltd.     

Stability of the Crank–Nicolson–Adams–Bashforth scheme for the 2D Leray‐alpha model

We consider the stability of an efficient Crank–Nicolson–Adams–Bashforth method in time, finite element in space, discretization of the Leray‐α model. We prove finite‐time stability of the scheme in L², H¹, and H², as well as the long‐time L‐stability of the scheme under a Courant‐Freidrichs‐Lewy (CFL)‐type condition. Numerical experiments are given that are in agreement with the theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1155–1183, 2016     

Zero‐viscosity‐capillarity limit to rarefaction waves for the 1D compressible Navier–Stokes–Korteweg equations

In this paper, we study the zero viscosity and capillarity limit problem for the one‐dimensional compressible isentropic Navier–Stokes–Korteweg equations when the corresponding Euler equations have rarefaction wave solutions. In the case that either the effects of initial layer are ignored or the rarefaction waves are smooth, we prove that the solutions of the Navier–Stokes–Korteweg equation with centered rarefaction wave data exist for all time and converge to the centered rarefaction waves as the viscosity and capillarity number vanish, and we also obtain a rate of convergence, which is valid uniformly for all time. These results are showed by a scaling argument and elementary energy analysis. Copyright © 2016 John Wiley & Sons, Ltd.     

Dynamic analysis of Markovian jumping impulsive stochastic Cohen–Grossberg neural networks with discrete interval and distributed time-varying delays

In this paper, the dynamic analysis problem is considered for a new class of Markovian jumping impulsive stochastic Cohen–Grossberg neural networks (CGNNs) with discrete interval and distributed delays. The parameter uncertainties are assumed to be norm bounded and the discrete delay is assumed to be time-varying and belonging to a given interval, which means that the lower and upper bounds of interval time-varying delays are available. Based on the Lyapunov–Krasovskii functional and stochastic stability theory, delay-interval dependent stability criteria are obtained in terms of linear matrix inequalities. Some asymptotic stability criteria are formulated by means of the feasibility of a linear matrix inequality (LMI), which can be easily calculated by LMI Toolbox in Matlab. A numerical example is provided to show that the proposed results significantly improve the allowable upper bounds of delays over some existing results in the literature.     

A moving Kriging‐based MLPG method for nonlinear Klein–Gordon equation

In this paper, the meshless local Petrov–Galerkin approximation is proposed to solve the 2‐D nonlinear Klein–Gordon equation. We used the moving Kriging interpolation instead of the MLS approximation to construct the meshless local Petrov–Galerkin shape functions. These shape functions possess the Kronecker delta function property. The Heaviside step function is used as a test function over the local sub‐domains. Here, no mesh is needed neither for integration of the local weak form nor for construction of the shape functions. So the present method is a truly meshless method. We employ a time‐stepping method to deal with the time derivative and a predictor–corrector scheme to eliminate the nonlinearity. Several examples are performed and compared with analytical solutions and with the results reported in the extant literature to illustrate the accuracy and efficiency of the presented method. Copyright © 2016 John Wiley & Sons, Ltd.     

One‐level Newton–Krylov–Schwarz algorithm for unsteady non‐linear radiation diffusion problem

In this paper, we present a parallel Newton–Krylov–Schwarz (NKS)‐based non‐linearly implicit algorithm for the numerical solution of the unsteady non‐linear multimaterial radiation diffusion problem in two‐dimensional space. A robust solver technology is required for handling the high non‐linearity and large jumps in material coefficients typically associated with simulations of radiation diffusion phenomena. We show numerically that NKS converges well even with rather large inflow flux boundary conditions. We observe that the approach is non‐linearly scalable, but not linearly scalable in terms of iteration numbers. However, CPU time is more important than the iteration numbers, and our numerical experiments show that the algorithm is CPU‐time‐scalable even without a coarse space given that the mesh is fine enough. This makes the algorithm potentially more attractive than multilevel methods, especially on unstructured grids, where course grids are often not easy to construct. Copyright © 2004 John Wiley & Sons, Ltd.     

Operator time‐splitting techniques combined with quintic B‐spline collocation method for the generalized Rosenau–KdV equation

In this article, the generalized Rosenau–KdV equation is split into two subequations such that one is linear and the other is nonlinear. The resulting subequations with the prescribed initial and boundary conditions are numerically solved by the first order Lie–Trotter and the second‐order Strang time‐splitting techniques combined with the quintic B‐spline collocation by the help of the fourth order Runge–Kutta (RK‐4) method. To show the accuracy and reliability of the proposed techniques, two test problems having exact solutions are considered. The computed error norms L₂ and L_∞ with the conservative properties of the discrete mass Q(t) and energy E(t) are compared with those available in the literature. The convergence orders of both techniques have also been calculated. Moreover, the stability analyses of the numerical schemes are investigated.     

A predictor‐‐corrector scheme for the sine‐Gordon equation

A predictor–corrector scheme is developed for the numerical solution of the sine‐Gordon equation using the method of lines approach. The solution of the approximating differential system satisfies a recurrence relation, which involves the cosine function. Pade' approximants are used to replace the cosine function in the recurrence relation. The resulting schemes are analyzed for order, stability, and convergence. Numerical results demonstrate the efficiency and accuracy of the predictor–corrector scheme over some well‐known existing methods. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 133–146, 2000     

Inverse Sturm–Liouville spectral problem on symmetric star‐tree

A spectral problem for the Sturm–Liouville equation on the edges of an equilateral regular star‐tree with the Dirichlet boundary conditions at the pendant vertices and Kirchhoff and continuity conditions at the interior vertices is considered. The potential in the Sturm–Liouville equation is a real–valued square summable function, symmetrically distributed with respect to the middle point of any edge. If {λ_j}is a sequence of real numbers, necessary and sufficient conditions for {λ_j}to be the spectrum of the problem under consideration are established. Copyright © 2013 John Wiley & Sons, Ltd.     

A Petrov–Galerkin spectral method for fourth‐order problems

In this paper, we consider the Petrov–Galerkin spectral method for fourth‐order elliptic problems on rectangular domains subject to non‐homogeneous Dirichlet boundary conditions. We derive some sharp results on the orthogonal approximations in one and two dimensions, which play important roles in numerical solutions of higher‐order problems. By applying these results to a fourth‐order problem, we establish the H²‐error and L²‐error bounds of the Petrov–Galerkin spectral method. Numerical experiments are provided to illustrate the high accuracy of the proposed method and coincide well with the theoretical analysis. Copyright © 2013 John Wiley & Sons, Ltd.     

Full‐Order observer design for nonlinear complex large‐scale systems with unknown time‐varying delayed interactions

This article is concerned with the problem of state observer for complex large‐scale systems with unknown time‐varying delayed interactions. The class of large‐scale interconnected systems under consideration is subjected to interval time‐varying delays and nonlinear perturbations. By introducing a set of argumented Lyapunov–Krasovskii functionals and using a new bounding estimation technique, novel delay‐dependent conditions for existence of state observers with guaranteed exponential stability are derived in terms of linear matrix inequalities (LMIs). In our design approach, the set of full‐order Luenberger‐type state observers are systematically derived via the use of an efficient LMI‐based algorithm. Numerical examples are given to illustrate the effectiveness of the result. © 2014 Wiley Periodicals, Inc. Complexity 21: 123–133, 2015     

Stabilized semi‐implicit spectral deferred correction methods for Allen–Cahn and Cahn–Hilliard equations

Stabilized semi‐implicit spectral deferred correction methods are constructed for the time discretization of Allen–Cahn and Cahn–Hilliard equations. These methods are unconditionally stable, lead to simple linear system to solve at each iteration, and can achieve high‐order time accuracy with a few iterations in each time step. Ample numerical results are presented to demonstrate the effectiveness of the stabilized semi‐implicit spectral deferred correction methods for solving the Allen–Cahn and Cahn–Hilliard equations. Copyright © 2013 John Wiley & Sons, Ltd.     

Finite‐time flocking problem of a Cucker–smale‐type self‐propelled particle model

In this article, we propose a Cucker–Smale‐type self‐propelled particle model with continuous non‐Lipschitz protocol. We show that the flocking can occur in finite time if the communication rate function satisfies a lower bound condition. Both our theoretical and numerical results uncover a power‐law relationship between the convergence time and the number of individuals. Our result implies that the individuals in groups with high density can transit rapidly to ordered collective motion. We also investigate the influence of control parameter on the convergence speed. © 2015 Wiley Periodicals, Inc. Complexity 21: 354–361, 2016