The Stability Of Nonlinear Delay Differential Algebraic Equations And The Application Of Implicit Euler Method

Differential-algebraic equations(DAEs),which has algebraic constraints, arises in a wide variety of scientific and engineering applications, including real-time simulation,circuit analysis,optimal control,computer-aided design, chemical process simulation and management system.In many case, we should know some past states of the system, so it converts to the study of delay differential-algebraic equations(DDAEs). This kind of DDAEs with more complicated structure for it has not only delay terms but also the derivatives of delay terms. it becomes quite difficult to obtain the analytic solutions. Therefore, using numerical methods to solve this kind of system has become an important means, and the premise for application of numerical methods is to study various properties of theoretical solution,which has a very important theoretical and practical significance.In this paper, we investigate a class of nonlinear delay differential-algebraic equations.x(t) = f(x(t), x(t- Ï„), y(t), y(t- Ï„)), t > 0,(Ï„ > 0)(0.0.3)0 = ?(x(t), x(t- Ï„), y(t)), t > 0,(0.0.4)First,Sufficient conditions for the theory of stability and asymptotic stability of the equations are given. Then,Proof that these conditions can be applied conveniently to nonlinear equations. We also show that the implicit Euler methods are stable and asymptotically stable. Last, some numerical experiments are carried out by implicit Euler methods to demonstrate the conclusions.