Numerical Analysis For Several Classes Of Nonlinear Delay Differential-Algebraic Equations

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 proceduresIn 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.