几类非线性延迟微分代数方程的数值分析
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摘要
延迟微分代数方程常出现在自动控制、化学反应模拟、电力和电路分析、多体动力学、生物、医学、国民经济等许多实际应用问题中.延迟微分代数方程既可看成是包含延迟项的微分代数方程,又可看成带约束条件的延迟微分方程.因此对延迟微分代数方程及其数值方法的研究可继承微分代数方程和延迟微分方程的很多思想与技巧.但延迟微分代数方程因同时具有约束条件和时滞效应,故其许多特性是微分代数方程和延迟微分方程所不具有的,这给其理论研究和数值计算带来了实质困难.目前国内外对微分代数方程数值分析的研究文献较多,对延迟微分代数方程数值分析研究主要集中于线性问题和1-指标问题；对高指标非线性延迟微分代数方程数值分析的研究是困难的,目前国内外仅有少量的研究且大多为常延迟.变分迭代方法因其求近似解析解的灵活高效性而获得了广泛的研究,但国内外还未见到该方法应用于延迟微分代数方程的工作.有关控制系统(主要是线性系统)的输入状态稳定性的研究工作很多,而非线性延迟微分代数控制系统的输入状态稳定性的研究还未见到.
第一章首先介绍了延迟微分代数方程的应用背景；其次介绍了延迟微分代数方程的稳定性及渐近稳定性及其数值方法的稳定性、渐近稳定性及收敛性的研究现状；再次介绍了应用于求解非线性问题的变分迭代方法；最后介绍了控制系统输入状态稳定性的研究情况.
第二章研究了用BDF方法和线性多步方法求解一类2-指标非线性变延迟微分代数方程,获得了稳定的BDF方法、稳定且在无穷远点严格稳定的线性多步方法(时滞部分用拉格朗日插值)的收敛性结果.所获结果将1995年Petzold等人关于常延迟的结果推广到了变延迟.
第三章讨论用单支方法求解一类2-指标非线性变延迟微分代数方程,获得了稳定且在无穷远点严格稳定的单支方法(延迟部分用拉格朗日插值)的整体误差估计.
第四章研究了利用变分迭代方法求非线性变延迟微分代数方程的解析解或近似解析解.变分迭代方法是近年来有大量研究与应用的一类求解近似解析解的方法.该方法利用较少的迭代步可获得较好解析解或近似解析解.本文针对两类不同形式的变延迟微分代数方程,分别构造不同的迭代格式,获得了两类变分迭代公式的收敛性结果.
第五章将Sontag等人提出控制系统的IS-稳定概念和胡广大、刘明珠等人提出控制系统数值方法的IS-稳定概念推广到非线性延迟微分代数控制系统及其数值方法,获得了延迟微分代数控制系统IS-稳定的充分条件,并系统研究了求解非线性延迟微分代数系统Runge-Kutta方法、单支方法的IS-稳定性.
Delay differential-algebraic equations(DDAEs) often arise in automatic con-trol, chemical reaction simulation, power and circuit analysis, multi-body dynam-ics, biology, medical science, economics, etc. DDAEs can be viewed as differential-algebraic equations with delay terms and delay differential equations subject to constraints, so the discussion of DDAEs inherits many ideas and much technique from these of both differential-algebraic equations and delay differential equations. But for DDAEs, the interaction of algebraic constraints with delayed terms gives rise to behaviours which are not seen from differential-algebraic equations and delay differential equations. This brings some substantial difficulties of their the-oretical studies and numerical computation. Thus DDAEs merit a separate inves-tigation in their own right. In present, many literatures on numerical analysis for differential-algebraic equations have appeared, in particular, the researches into numerical analysis for delay-differential-algebraic equations mainly are focused on linear problems and 1-index problems. It is difficult to research numerical anal-ysis for high-index nonlinear delay-differential-algebraic equations, and there are only a few of results at home and abroad, furthermore, most of them are about constant-delay problems. Variational iteration method is extensively studied as it is flexible and efficient to obtain approximate analytic solutions of the solved prob-lems, but no literatures apply this method to delay-differential-algebraic equations at home and abroad. There are many studies on the IS-stability of control systems (mainly linear systems), but there are no literatures on IS-stability of nonlinear delay-differential-algebraic control systems.
In chapter 1, firstly, the author introduces some applications of DDAEs; sec-ondly, introduces the stability and asymptotic stability of DDAEs together with the stability, asymptotic stability and convergence of numerical methods for DDAEs; thirdly, introduces the variational iteration method for nonlinear problems; finally introduces the current research situation of IS-stability for the original control problems and their numerical methods.
In chapter 2, the author studies BDF methods and linear multistep methods (LMMs) for index-2 nonlinear differential-algebraic equations with a variable delay and obtains the convergence results of stable BDF methods, stable and strictly stable at infinity LMMs with Lagrange interpolation procedures. These obtained results extend the corresponding results obtained by Petzold et. al. in 1995 from constant delays to variable delays.
In chapter 3, the author studies one-leg methods for index-2 nonlinear differential-algebraic equations with a variable delay and obtains the convergence results of stable and strictly stable at infinity one-leg methods with Lagrange interpolation procedures
In chapter 4, the author uses the variational iteration method(VIM) to ob-tain the analytical or approximate-analytical solutions of nonlinear differential-algebraic equations with a variable delay. This method is an approximate-analytic method which has been widely discussed until recently. Moreover, we can obtain the analytical or approximate-analytical solutions by fewer iteration step. Accord-ing to the VIM. we ean construet different correction fuectionals for two different forms of DDAEs. and obtain some convergence results of two kinds of VIM.
In chapter 5, the author extends the concepts of IS-stability of control systems proposed by Sontag et. al. and IS-stability of numerical methods for control systems by Hu Guangda. Liu Mingzhu et. al. to nonlinear delay-differential-algebraic control systems. We derive the sufficient conditions, under which delay-differential-algebraic control systems are IS-stable. Based on these conditions, we have studied the IS-stability of Runge-Kutta methods and one-leg methods for nonlinear delay differential-algebraic control systems.
第一章首先介绍了延迟微分代数方程的应用背景；其次介绍了延迟微分代数方程的稳定性及渐近稳定性及其数值方法的稳定性、渐近稳定性及收敛性的研究现状；再次介绍了应用于求解非线性问题的变分迭代方法；最后介绍了控制系统输入状态稳定性的研究情况.
第二章研究了用BDF方法和线性多步方法求解一类2-指标非线性变延迟微分代数方程,获得了稳定的BDF方法、稳定且在无穷远点严格稳定的线性多步方法(时滞部分用拉格朗日插值)的收敛性结果.所获结果将1995年Petzold等人关于常延迟的结果推广到了变延迟.
第三章讨论用单支方法求解一类2-指标非线性变延迟微分代数方程,获得了稳定且在无穷远点严格稳定的单支方法(延迟部分用拉格朗日插值)的整体误差估计.
第四章研究了利用变分迭代方法求非线性变延迟微分代数方程的解析解或近似解析解.变分迭代方法是近年来有大量研究与应用的一类求解近似解析解的方法.该方法利用较少的迭代步可获得较好解析解或近似解析解.本文针对两类不同形式的变延迟微分代数方程,分别构造不同的迭代格式,获得了两类变分迭代公式的收敛性结果.
第五章将Sontag等人提出控制系统的IS-稳定概念和胡广大、刘明珠等人提出控制系统数值方法的IS-稳定概念推广到非线性延迟微分代数控制系统及其数值方法,获得了延迟微分代数控制系统IS-稳定的充分条件,并系统研究了求解非线性延迟微分代数系统Runge-Kutta方法、单支方法的IS-稳定性.
Delay differential-algebraic equations(DDAEs) often arise in automatic con-trol, chemical reaction simulation, power and circuit analysis, multi-body dynam-ics, biology, medical science, economics, etc. DDAEs can be viewed as differential-algebraic equations with delay terms and delay differential equations subject to constraints, so the discussion of DDAEs inherits many ideas and much technique from these of both differential-algebraic equations and delay differential equations. But for DDAEs, the interaction of algebraic constraints with delayed terms gives rise to behaviours which are not seen from differential-algebraic equations and delay differential equations. This brings some substantial difficulties of their the-oretical studies and numerical computation. Thus DDAEs merit a separate inves-tigation in their own right. In present, many literatures on numerical analysis for differential-algebraic equations have appeared, in particular, the researches into numerical analysis for delay-differential-algebraic equations mainly are focused on linear problems and 1-index problems. It is difficult to research numerical anal-ysis for high-index nonlinear delay-differential-algebraic equations, and there are only a few of results at home and abroad, furthermore, most of them are about constant-delay problems. Variational iteration method is extensively studied as it is flexible and efficient to obtain approximate analytic solutions of the solved prob-lems, but no literatures apply this method to delay-differential-algebraic equations at home and abroad. There are many studies on the IS-stability of control systems (mainly linear systems), but there are no literatures on IS-stability of nonlinear delay-differential-algebraic control systems.
In chapter 1, firstly, the author introduces some applications of DDAEs; sec-ondly, introduces the stability and asymptotic stability of DDAEs together with the stability, asymptotic stability and convergence of numerical methods for DDAEs; thirdly, introduces the variational iteration method for nonlinear problems; finally introduces the current research situation of IS-stability for the original control problems and their numerical methods.
In chapter 2, the author studies BDF methods and linear multistep methods (LMMs) for index-2 nonlinear differential-algebraic equations with a variable delay and obtains the convergence results of stable BDF methods, stable and strictly stable at infinity LMMs with Lagrange interpolation procedures. These obtained results extend the corresponding results obtained by Petzold et. al. in 1995 from constant delays to variable delays.
In chapter 3, the author studies one-leg methods for index-2 nonlinear differential-algebraic equations with a variable delay and obtains the convergence results of stable and strictly stable at infinity one-leg methods with Lagrange interpolation procedures
In chapter 4, the author uses the variational iteration method(VIM) to ob-tain the analytical or approximate-analytical solutions of nonlinear differential-algebraic equations with a variable delay. This method is an approximate-analytic method which has been widely discussed until recently. Moreover, we can obtain the analytical or approximate-analytical solutions by fewer iteration step. Accord-ing to the VIM. we ean construet different correction fuectionals for two different forms of DDAEs. and obtain some convergence results of two kinds of VIM.
In chapter 5, the author extends the concepts of IS-stability of control systems proposed by Sontag et. al. and IS-stability of numerical methods for control systems by Hu Guangda. Liu Mingzhu et. al. to nonlinear delay-differential-algebraic control systems. We derive the sufficient conditions, under which delay-differential-algebraic control systems are IS-stable. Based on these conditions, we have studied the IS-stability of Runge-Kutta methods and one-leg methods for nonlinear delay differential-algebraic control systems.
引文
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