Impulsive-Smooth Behaviors *

Abstract

Following the vocabulary due to J.C. Willems, the term ‘impulsive-smooth behavior’ may be used for any subset of the space of vector-valued impulsive-smooth distributions introduced by M.L.J. Hautus. In this paper we axe particularly interested in those behaviors that can be represented by sets of linear differential and algebraic equations. We consider both polynomial representations without auxiliary variables, and first-order representations with auxiliary variables. We show how these representations may be transformed into one another. For both types of representations, we give necessary and sufficient conditions for minimality and describe the extent to which minimal representations are unique. It turns out that, in the present context, ‘controllability at infinity’ is not necessary for minimality.