On Linear Passive Complementarity Systems

Abstract

We study the notion of passivity in the context of complementarity systems, which form a class of nonsmooth dynamical systems that is obtained from the coupling of a standard input/output system to complementarity conditions as used in mathematical programming. In terms of electrical circuits, the systems that we study may be viewed as passive networks with ideal diodes. Extending results from earlier work, we consider here complementarity systems with external inputs. It is shown that the assumption of passivity of the underlying input/output dynamical system plays an important role in establishing existence and uniqueness of solutions. We prove that solutions may contain delta functions but no higher-order impulses. Several characterizations are provided for the state jumps that may occur due to inconsistent initialization or to input discontinuities. Many of the results still hold when the assumption of passivity is replaced by the assumption of “passifiability by pole shifting”. The paper ends with some remarks on stability.