Delay-dependent stability of linear multistep methods for DAEs with multiple delays

This paper aims to investigate the asymptotic stability of linear multistep (LM) methods for linear differential-algebraic equations (DAEs) with multiple delays. Based on the argument principle, we first establish the delay-dependent stability criteria of analytic solutions; then, we propose some practically checkable conditions for weak delay-dependent stability of numerical solutions derived by implicit LM methods. Lagrange interpolations are used to compute the delayed terms. Several numerical examples are given to illustrate the theoretical results.     

Asymptotic stability of exact and discrete solutions for neutral multidelay-integro-differential equations

This paper concerns the long-time behavior of the exact and discrete solutions to a class of nonlinear neutral integro-differential equations with multiple delays. Using a generalized Halanay inequality, we give two sufficient conditions for the asymptotic stability of the exact solution to this class of equations. Runge–Kutta methods with compound quadrature rule are considered to discretize this class of equations with commensurate delays. Nonlinear stability conditions for the presented methods are derived. It is found that, under suitable conditions, this class of numerical methods retain the asymptotic stability of the underlying system. Some numerical examples that illustrate the theoretical results are given.     

Stability and stochastic stabilization of numerical solutions of regime-switching jump diffusion systems

This work studies stability and stochastic stabilization of numerical solutions of a class of regime-switching jump diffusion systems. These systems have a wide range of applications in communication systems, flexible manufacturing and production planning, financial engineering and economics because they involve three classes of stochastic factors: white noise, Poisson jump and Markovian switching. This paper focuses on the stability of numerical solutions of the switching jump diffusion systems and examines the conditions under which the Euler–Maruyama (EM) and the backward EM may share the stability of the exact solution. These conditions show that all these three classes of stochastic factors may serve as stabilizing factors and play positive roles for the stability property of both exact and numerical solutions.     

Asymptotic stability analysis of Runge-Kutta methods for nonlinear systems of delay differential equations

Summary. We consider systems of delay differential equations (DDEs) of the form with the initial condition . Recently, Torelli [10] introduced a concept of stability for numerical methods applied to dissipative nonlinear systems of DDEs (in some inner product norm), namely RN-stability, which is the straighforward generalization of the wellknown concept of BN-stability of numerical methods with respect to dissipative systems of ODEs. Dissipativity means that the solutions and corresponding to different initial functions and , respectively, satisfy the inequality , and is guaranteed by suitable conditions on the Lipschitz constants of the right-hand side function . A numerical method is said to be RN-stable if it preserves this contractivity property. After showing that, under slightly more stringent hypotheses on the Lipschitz constants and on the delay function , the solutions and are such that , in this paper we prove that RN-stable continuous Runge-Kutta methods preserve also this asymptotic stability property. Received March 29, 1996 / Revised version received August 12, 1996     

Stability Criteria for Solutions of Systems of Linear Deterministic or Stochastic Delay Difference Equations with Continuous Time

We give spectral and algebraic coefficient criteria (necessary and sufficient conditions) as well as sufficient algebraic coefficient conditions for the Lyapunov asymptotic stability of solutions to systems of linear deterministic or stochastic delay difference equations with continuous time under white noise coefficient perturbations for the case in which all delay ratios are rational. For stochastic systems, mean-square asymptotic stability is studied. The Lyapunov function method is used. Our criteria on algebraic coefficients and our sufficient conditions are stated in terms of matrix Lyapunov equations (for deterministic systems) and matrix Sylvester equations (for stochastic systems).     

STABILITY ANALYSIS OF RUNGE-KUTTA METHODS FOR NONLINEAR SYSTEMS OF PANTOGRAPH EQUATIONS

This paper is concerned with numerical stability of nonlinear systems of pantograph equations. Numerical methods based on （k, l）-algebraically stable Runge-Kutta methods are suggested. Global and asymptotic stability conditions for the presented methods are derived.     

Stability analysis of block boundary value methods for the neutral differential equation with many delays

This paper mainly deals with the asymptotic stability properties of block boundary value methods (B₂VMs) for the neutral differential equation with many delays. For the time lagging arguments, a technique of Lagrange interpolation is considered. Under some certain conditions, it is proved that B₂VMs can preserve the asymptotic stability of exact solutions for the neutral differential equation with many delays if and only if B₂VMs are A-stable for ordinary differential equation. Moreover, some numerical experiments are given to confirm the main conclusion.     

Delay-Dependent Stability of Linear Multistep Methods for Neutral Systems with Distributed Delays

This paper considers the asymptotic stability of linear multistep (LM) methods for neutral systems with distributed delays. In particular, several sufficient conditions for delay-dependent stability of numerical solutions are obtained based on the argument principle. Compound quadrature formulae are used to compute the integrals. An algorithm is proposed to examine the delay-dependent stability of numerical solutions. Several numerical examples are performed to verify the theoretical results.     

Delay-dependent stability of Runge-Kutta methods for neutral systems with distributed delays

The aim of this paper is to analyze the asymptotic stability of Runge-Kutta (RK) methods for neutral systems with distributed delays. With an adaptation of the argument principle, some sufficient criteria for weak delay-dependent stability of numerical solutions are proposed. Several numerical examples are performed to confirm the effectiveness of our theoretical results.     

Almost sure exponential stability of numerical solutions for stochastic delay differential equations

Using techniques based on the continuous and discrete semimartingale convergence theorems, this paper investigates if numerical methods may reproduce the almost sure exponential stability of the exact solutions to stochastic delay differential equations (SDDEs). The important feature of this technique is that it enables us to study the almost sure exponential stability of numerical solutions of SDDEs directly. This is significantly different from most traditional methods by which the almost sure exponential stability is derived from the moment stability by the Chebyshev inequality and the Borel–Cantelli lemma.     

New results for global stability of a class of neutral-type neural systems with time delays

This paper studies the global convergence properties of a class of neutral-type neural networks with discrete time delays. This class of neutral systems includes Cohen–Grossberg neural networks, Hopfield neural networks and cellular neural networks. Based on the Lyapunov stability theorems, some delay independent sufficient conditions for the global asymptotic stability of the equilibrium point for this class of neutral-type systems are derived. It is shown that the results presented in this paper for neutral-type delayed neural networks are the generalization of a recently reported stability result. A numerical example is also given to demonstrate the applicability of our proposed stability criteria.     

Dissipativity and asymptotic stability of nonlinear neutral delay integro-differential equations

This paper is concerned with the dissipativity and asymptotic stability of the theoretical solutions of a class of nonlinear neutral delay integro-differential equations (NDIDEs). We first give a generalization of the Halanay inequality which plays an important role in the study of dissipativity and stability of differential equations. Then, we apply the generalization of the Halanay inequality to NDIDEs and the dissipativity and the asymptotic stability results of the theoretical solution of NDIDEs are obtained. From a numerical point of view, it is important to study the potential of numerical methods in preserving the qualitative behavior of the analytical solutions. Therefore, the results, presented in this paper, provide the theoretical foundation for analyzing the dissipativity and the asymptotic stability of the numerical methods when they are applied to these systems.     

Stability analysis of parabolic partial differential equations with piecewise continuous arguments

This article deals with the analytic and numerical stability of numerical methods for a parabolic partial differential equation with piecewise continuous arguments of alternately retarded and advanced type. First, application of the theory of separation of variables in matrix form and the Fourier method, the necessary and sufficient condition under which the analytic solution is asymptotically stable is derived. Then, the θ‐methods are applied to solve the corresponding initial value problem, the sufficient conditions for the asymptotic stability of numerical methods are obtained. Finally, several numerical examples are presented to support the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 531–545, 2017     

Sufficient criteria are necessary for monotone control volume methods

Control volume methods are prevailing for solving the potential equation arising in porous media flow. The continuous form of this equation is known to satisfy a maximum principle, and it is desirable that the numerical approximation shares this quality. Recently, sufficient criteria were derived guaranteeing a discrete maximum principle for a class of control volume methods. We show that most of these criteria are also necessary. An implication of our work is that no linear nine-point control volume method can be constructed for quadrilateral grids in 2D that is exact for linear solutions while remaining monotone for general problems.     

Analytical and numerical stability of partial differential equations with piecewise constant arguments

This article is concerned with the stability analysis of the analytic and numerical solutions of a partial differential equation with piecewise constant arguments of mixed type. First, by means of the similar technique in Wiener and Debnath [Int J Math Math Sci 15 (1992), 781–788], the sufficient conditions under which the analytic solutions asymptotically stable are obtained. Then, the θ‐methods are used to solve the above‐mentioned equation, the sufficient conditions for the asymptotic stability of numerical methods are derived. Finally, some numerical experiments are given to demonstrate the conclusions.Copyright © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1‐16, 2014     

Global asymptotic stability analysis of discrete-time Cohen–Grossberg neural networks based on interval systems

The global asymptotic stability of discrete-time Cohen–Grossberg neural networks (CGNNs) with or without time delays is studied in this paper. The CGNNs are transformed into discrete-time interval systems, and several sufficient conditions for asymptotic stability for these interval systems are derived by constructing some suitable Lyapunov functionals. The conditions obtained are given in the form of linear matrix inequalities that can be checked numerically and very efficiently by using the MATLAB LMI Control Toolbox. Finally, some illustrative numerical examples are provided to demonstrate the effectiveness of the results obtained.     

Delay equations on time scales: Essentials and asymptotics of the solutions

The paper discusses the notion of a delay dynamic equation on time scales and describes some asymptotic properties of its solutions. The application of the derived results to continuous and discrete time scales presents new qualitative results for delay differential and difference equations. In particular, our approach faciliates the joint investigation of stability properties of the exact equations and their numerical discretizations.     

New stability criteria for a class of neutral systems with discrete and distributed time-delays: an LMI approach

In this paper, a problem of the asymptotic stability for a class of neutral systems with multiple discrete and distributed time-delays is considered. Lyapunov stability theory is applied to guarantee the stability for the systems. New discrete-delay-independent and discrete-delay-dependent stability conditions are derived in terms of the spectral radius and linear matrix inequality. By mathematical analysis, the stability criteria are proved to be less conservative than the ones reported in the current literatures. A numerical example is given to illustrate the availability of the proposed results.     

Stability of the Radau IA and Lobatto IIIC methods for delay differential equations

Recently, we have proved that the Radau IA and Lobatto IIIC methods are P-stable, i.e., they have an analogous stability property to A-stability with respect to scalar delay differential equations (DDEs). In this paper, we study stability of those methods applied to multidimensional DDEs. We show that they have a similar property to P-stability with respect to multidimensional equations which satisfy certain conditions for asymptotic stability of the zero solutions. The conditions are closely related to stability criteria for DDEs considered in systems theory. Received October 8, 1996 / Revised version received February 21, 1997     

Delay-independent stability of Euler method for nonlinear one-dimensional diffusion equation with constant delay

This paper is concerned with delay-independent asymptotic stability of a numerical process that arises after discretization of a nonlinear one-dimensional diffusion equation with a constant delay by the Euler method. Explicit sufficient and necessary conditions for the Euler method to be asymptotically stable for all delays are derived. An additional restriction on spatial stepsize is required to preserve the asymptotic stability due to the presence of the delay. A numerical experiment is implemented to confirm the results.