The simulation problem for high order linear differential systems

摘要

In this paper we consider linear constant coefficient ordinary differential equations of arbitrary order of the form R(ddt)w=M(ddt)f with R and M given, but otherwise arbitrary, polynomial matrices. To these equations we associate a behavior B defined as the set of all vector-valued distributions w, f that solve the equations themselves. We think of f as a given distribution, while w is to be found in such a way that the equations defining the behavior are satisfied. Alongside R(ddt)w=M(ddt)f we also consider initial conditions obtained by imposing the value at time t=0 of a linear combination of the variables w and their derivatives, yielding S(ddt)w(0)=Ta with S a polynomial matrix, T a fixed real matrix, and a a real vector. The three main issues we address are:(i)Solvability of the behavior, meaning the problem of finding conditions that assure that corresponding to a (or any) given vector distribution f, a trajectory w exists such that (w,f) belong to B. This question is formalized and answered in Section 4.(ii)The index of the behavior, which means investigating how the smoothness of the given distribution f is related to smoothness of the solution w. Section 5 is dedicated to this issue.(iii)Compatibility of initial conditions, in which case the initial conditions S(ddt)w(0)=Ta are considered alongside the behavior B. We first check that S(ddt)w(0) is well defined. If it is, we provide conditions under which there exists a (unique) distribution w such that w, f belong to B and such that S(ddt)w(0)=Ta are also satisfied for a (or any) given a. This problem is addressed in Section 6.We show how our results generalize to behaviors defined by equations of arbitrary order classical properties of behaviors defined by first order systems such as Eddtw+Aw=f.