连续的龙格库塔方法对多延迟量微分代数方程的渐进稳定性(英文)

Asymptotic stability of continuous Runge-Kutta methods for differential-algebraic equations with several delays

延迟微分代数系统(DDAEs)是具有时滞影响和代数约束的微分系统,为计算机辅助设计、化学反应模拟、线路分析、最优控制、实时仿真以及管理系统等科学与工程应用问题提供了有效的数学模型.中立型多延迟微分代数系统是一种结构较复杂的DDAEs,因为它不仅含有多个延迟项,而且还包含有未知函数的导数.然而,由于延迟微分代数方程的复杂性,只有极少数延迟微分方程能获得其理论解的精确解析表达式.因此,研究延时微分代数方程的数值解法显得十分重要.而在数值解的研究中,有效可靠的算法及算法的数值稳定性研究,又是必须首先面对的问题.研究了连续的龙格库塔方法对多延迟量微分代数方程的渐进稳定性,并证明了这种方法在系数矩阵都是上三角形的假设下是渐进稳定的,这种假设对有广泛应用的Hessenberg DDAEs是正确的.

Delay differential-algebraic equations（DDAEs）,which has both delay and algebraic constraints,provides effective mathematical model for scientific and engineering applications problems,such as computer-aided design,chemical process simulation circuit analysis,optimal control,real-time simulation and management system.Neutral delay differential-algebraic equations with several delay terms is a kind of DDAEs with more complicated structure since it not only has several delay terms but also the derivatives of unknow function.However,because of the complexity of DDAEs,it becomes quite difficult to obtain the numerical methods for DDAEs and they have become one of the most chief and principal methods to solve DDAEs.Moreover,effective and reliable computational methods and the stability of numerical methods are the key problems we have to face in the research of the numerical solution.This paper develops the continuous Runge-Kutta methods for the differential-algebraic equations with several delays and proves that the methods is asymptotically stable under the assumption that the coefficient matrices are all upper triangular.This assumption is true for Hessenberg DDAEs which have a wide range of applications.