Analysis of higher order linear differential-algebraic systems

Abstract

We study linear over- or under-determined differential-algebraic systems of order larger than $1$. We analyze the classical procedure of turning the system into a first order system. We show that this approach leads to solutions that may have different smoothness requirements. We derive canonical and condensed forms as well as general existence and uniqueness results for differential-algebraic systems of arbitrary order and index. We also show how to identify exactly those variables for which the order reduction to first order does not lead to extra smoothness requirements. Finally we discuss some consequences for the analysis of matrix polynomials.