Asymptotic stability of general linear methods for systems of linear neutral delay differential-algebraic equations

This paper is concerned with numerical stability of general linear methods (GLMs) for a system of linear neutral delay differential-algebraic equations. A sufficient and necessary condition for asymptotic stability of GLMs solving such system is derived. Based on this main result, we further investigate the asymptotic stability of linear multistep methods, Runge–Kutta methods, and block θ-methods, respectively. Numerical experiments confirm our theoretical result.     

Stability analysis of extended block boundary value methods for linear neutral delay integro-differential equations

This paper is concerned with the stability of extended block boundary value methods (B₂VMs) for the linear neutral delay integro-differential-algebraic equations (NDIDAEs) and the linear neutral delay integro-differential equations (NDIDEs). It is proved that every A-stable B₂VM can preserve the asymptotic stability of the exact solution of NDIDAEs under some certain conditions. A necessary and sufficient condition of the B₂VMs to be asymptotically stable for NDIDEs is also obtained. A few numerical experiments confirm the expected results.     

GPG-stability of Runge-Kutta methods for generalized delay differential systems

The GP_G-stability of Runge-Kutta methods for the numerical solutions of the systems of delay differential equations is considered. The stability behaviour of implicit Runge-Kutta methods (IRK) is analyzed for the solution of the system of linear test equations with multiple delay terms. After an establishment of a sufficient condition for asymptotic stability of the solutions of the system, a criterion of numerical stability of IRK with the Lagrange interpolation process is given for any stepsize of the method.     

A modified lobatto collocation for linear boundary value problems of differential-algebraic equations

The Lobatto collocation method is modified for efficiently solving linear boundary value problems of differential-algebraic equations with index 1. The stability and superconvergence of this method are established. Numerical implementations are discussed and a numerical example is given.     

Stability of two-step Runge–Kutta methods for neutral delay differential-algebraic equations

This paper develops the two-step Runge–Kutta methods (TSRKs) for the neutral delay differential-algebraic equations (NDDAEs) and proves that the TSRKs are asymptotically stable for linear NDDAEs under the assumption that the coefficient matrices are all upper triangular, which is necessary to formulate the stability result. The discussions are supported by the numerical experiments.     

非线性延迟积分微分方程线性多步法的渐近稳定性

本文研究了非线性延迟积分微分方程线性多步法的渐近稳定性.证明了在约束网格下,带有复合求积公式A-稳定的线性多步法能够保持解析解的渐近稳定性.文章最后,数值试验验证了本文的结论.     

Efficient classes of Runge-Kutta methods for two-point boundary value problems

The standard approach to applying IRK methods in the solution of two-point boundary value problems involves the solution of a non-linear system ofn×s equations in order to calculate the stages of the method, wheren is the number of differential equations ands is the number of stages of the implicit Runge-Kutta method. For two-point boundary value problems, we can select a subset of the implicit Runge-Kutta methods that do not require us to solve a non-linear system; the calculation of the stages can be done explicitly, as is the case for explicit Runge-Kutta methods. However, these methods have better stability properties than the explicit Runge-Kutta methods. We have called these new formulas two-point explicit Runge-Kutta (TPERK) methods. Their most important property is that, because their stages can be computed explicity, the solution of a two-point boundary value problem can be computed more efficiently than is possible using an implicit Runge-Kutta method. We have also developed a symmetric subclass of the TPERK methods, called ATPERK methods, which exhibit a number of useful properties.     

Implicit-system approach to the robust stability for a class of singularly perturbed linear systems

Asymptotic stability and the complex stability radius of a class of singularly perturbed systems of linear differential-algebraic equations (DAEs) are studied. The asymptotic behavior of the stability radius for a singularly perturbed implicit system is characterized as the parameter in the leading term tends to zero. The main results are obtained in direct and short ways which involve some basic results in linear algebra and classical analysis, only. Our results can be extended to other singular perturbation problems for DAEs of more general form.     

Implementing Radau IIA Methods for Stiff Delay Differential Equations

This article discusses the numerical solution of a general class of delay differential equations, including stiff problems, differential-algebraic delay equations, and neutral problems. The delays can be state dependent, and they are allowed to become small and vanish during the integration. Difficulties encountered in the implementation of implicit Runge–Kutta methods are explained, and it is shown how they can be overcome. The performance of the resulting code – RADAR5 – is illustrated on several examples, and it is compared to existing programs. Received October 12, 2000     

Stability of BDF methods for nonlinear volterra integral equations with delay

This paper deals with the stability analysis of the backward differential formulas (or BDF methods) for nonlinear Volterra integral equations with delay (VIDEs). The presented approach is based on a nonclassical Lipschitz condition. In particular, the criteria on the global and the asymptotic stability of the methods are given.     

Further remarks on asymptotic stability and set invariance for linear delay-difference equations

We studied the existence of positively invariant sets for linear delay-difference equations. In particular, we regarded two strong stability notions: robust (with respect to delay parameter) asymptotic stability for the discrete-time case and delay-independent stability for the continuous-time case. The correlation between these stability concepts is also considered. Furthermore, for the delay-difference equations with two delay parameters, we provided a computationally efficient numerical routine which is necessary to guarantee the existence of contractive sets of Lyapunov–Razumikhin type. This condition also appears to be necessary and sufficient for the delay-independent stability and sufficient for the robust asymptotic stability.     

Asymptotic stability of distributed parameter systems with feedback controls

In this paper, the problem of stability in distributed parameter systems with feedback controls is formulated directly in the framework of partial differential equations without resorting to further approximations. Sufficient conditions for Lyapunov asymptotic stability are derived for particular classes of systems with distributed, mixed distributed, and boundary control laws, and also for systems with time delay. The applications of the main results are illustrated by examples.     

灰色随机线性时滞系统的渐近稳定性

首先提出了灰色随机线性时滞系统及其渐近稳定性的概念;然后,利用矩阵理论和随机微分时滞方程解的渐近收敛定理及李雅普诺夫函数,研究了灰色随机线性时滞系统的渐近稳定性,得到了随机淅近稳定的几个充分性条件;最后,通过数值例子说明了所得结果在实际应用中的方便性和有效性.     

On initial value problems in differential-algebraic equations and their numerical treatment

For more than ten years numerical integration methods for initial value problems in differential-algebraic equations (DAE's) have been used — with different success. However, the necessary analysis of these methods as well as of the DAE's themselves is only in its beginning. This work does not aim at providing new methods for DAE's, but at analyzing methods known from the literature and software, respectively. A series of results related to the stability, consistency and Liapunov stability of various methods is presented. In particular, a trapezoidal rule as well as certain multistep methods may become unstable, so that codes using these methods will fail in general. The positive results may stimulate the practical use of higher order methods instead of the BDF's for suitable classes of DAE's.     

Stability of solutions of delay functional integro-differential equations and their discretizations

In this paper we study asymptotic stability and contractivity properties of solutions of a class of delay functional integro-differential equations. These results form the basis for obtaining insight into the analogous properties of numerical solutions generated by continuous Runge-Kutta or collocation methods, where these methods are applied to a suitable reformulation of the given initial-value problem.     

Convergence Results of One-Leg and Linear Multistep Methods for Multiply Stiff Singular Perturbation Problems

One-leg methods and linear multistep methods are two class of important numerical methods applied to stiff initial value problems of ordinary differential equations. The purpose of this paper is to present some convergence results of A-stable one-leg and linear multistep methods for one-parameter multiply stiff singular perturbation problems and their corresponding reduced problems which are a class of stiff differential-algebraic equations. Received April 14, 2000; revised June 30, 2000     

On a class of positive linear differential equations with infinite delay

We give an explicit criterion for positivity of the solution semigroup of linear differential equations with infinite delay and a Perron-Frobenius type theorem for positive equations. Furthermore, a novel criterion for the exponential asymptotic stability of positive equations is presented. Finally, we provide a sufficient condition for the exponential asymptotic stability of positive equations subjected to structured perturbations. A simple example is given to illustrate the obtained results.     

Robust stability and controllability of stochastic differential delay equations with Markovian switching

In this paper, we investigate the almost surely asymptotic stability for the nonlinear stochastic differential delay equations with Markovian switching. Some sufficient criteria on the controllability and robust stability are also established for linear stochastic differential delay equations with Markovian switching.