Multifunctions of Bounded Variation, Preliminary Version I

R. B. Vinter

Published 2014-11-01Version 1

Consider control systems described by a differential equation with a control term or, more generally, by a differential inclusion with velocity set F(t,x). Certain properties of state trajectories can be derived when, in addition to other hypotheses, it is assumed that $F(t,x)$ is merely measureable w.r.t. the time variable. But sometimes a refined analysis requires the imposition of stronger hypotheses regarding the t dependence of F(t,x). Stronger forms of necessary conditions for state trajectories that minimize a cost can derived, for example, if it is hypothesized that $F(t,x)$ is Lipschitz continuous w.r.t. time. It has recently become apparent that interesting addition properties of state trajectories can still be derived, when the Lipschitz continuity hypothesis is replaced by the weaker requirement that $F(t,x)$ has bounded variation w.r.t. t. This paper introduces a new concept of multifunctions $F(t,x)$ that have bounded variation w.r.t. time near a given state trajectory, of special relevance to control system analysis. Properties of such multifunctions are derived Their significance of illustrated by an application to sensitivity analysis.