A Class Of Delayed Stability Of Differential-algebraic Equations And Numerical Methods

Differential-algebraic equations(DAE) are shown to play an important role in mathemati-cal models regarding a wide variety of scientific and engineering applications including multi-body mechanics, optimal control, electrical design, chemically reacting systems, biology andbiomedicine. Historically, the understanding of DAE has progressed from simpler cases to moregeneral ones. In particular, the definitions of key concepts such as index and solvability haveevolved over several years and denote the differences between the DAE and the ordinary differen-tial equations. For example, being the solution of a DAE may actually imply that some portions ofy are more differentiable than other portions which may be only continuous.The minimum num-ber of differentiation steps required in the procedure a DAE being transformed to an ODE is theindex. It should be note that we are not recommending the series of differentiations and coordinatechanges as a general solution procedure, but the number of such differentiation steps turns out tobe an important quantity in understanding the behavior of the numerical methods.We treat two classes of DAE system. DAE with index less than2and DAE with index greaterthan or equal to2. The first class of DAE encountered in practical applications either are index-1or, if higher index, can be expressed as a simple combination of Hessenberg system. They containsimplicit ODE with no restrictions; ODE with one restriction, which can be transformed to ODEwith no restrictions after a step of differentiation. They are linear or nonlinear systems whichare usually encountered. The second class of DAE, linear or nonlinear, implies more than onerestrictions, more differentiation steps are needed to transform them to ODE with no restrictions.In this paper, we study the stability and the numerical solution of the first class of DAE withtime-delay, here, time delays appear in the variables of an unknown function so that a DAE systemis converted to a DDAE or NDDAE.In chapter2and3, we study the stability of differential system and neutral system with manydelays. Results including stable region, boundary criterions. The key is to exclude the unstableregion by seeking the boundaries geometrically. We obtain two criteria through the evaluationsof a harmonic function on the boundary of a certain region. Our results are more general than[17]' stability analysis of single delay differential equations, In [8,9], the equation that has beendiscussed is multi-delay differential equation, but equations we discuss have different delays inthe entries of unknown vectors, thus our stability analysis will be more complex.In chapter4and5, we discuss DDAE with one restriction. Applying Drazin inverse to numer-ical solutions for DDAE, We get a solution expression to the difference format from a differentialequation in accordance using classical four-stage Runge-Kutta methods. Comparing with [43], our code is more convenient. We also denote that it is difficult to deal with DDAE. The main rea-son is that, this kind of DDAE with delays and accompanied by an algebraic restriction, are to besolved on a linear or nonlinear manifold of a product space according to the concept of solvability.We see an example. For the DDAE, suppose its restriction isg(y1, y2)=y21(t)+y22(t) R2=0, y=(y1, y2)T,(R>0))We have to find solutions of DDAE on a circle ofy2221+y2=RTwo algorithms are obtained and numerical tests are illustrated to show the evaluations of the errorand difficulties on the manifold as well.In chapter5, we study the stability of two step Runge-Kutta methods, give a sufficient condi-tion and apply the results to a practical problem in [64].Although there has been much research work on stability of linear DDAE and their numericalmethods, it is yet more difficult to study the conditions of stability results for nonlinear DDAE. Inchapter6, we give sufficient conditions of the stability and asymptotic stability of the nonlinearDDAE, as well as the implicit Euler methods.In the following chapter1, we give a simple introduction of DAE, and the outline of the work.