Multifunctions of Bounded Variation, Preliminary Version I

TL;DR

This paper explores the properties of control systems with multifunctions of bounded variation in time, introducing new concepts and analyzing their significance for control system analysis and sensitivity studies.

Contribution

It introduces a new concept of multifunctions with bounded variation in time, extending the analysis of control systems beyond Lipschitz continuity assumptions.

Findings

Derived properties of multifunctions with bounded variation in time.

Demonstrated the importance of these properties in sensitivity analysis.

Showed potential for refined control system analysis.

Abstract

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 measurable w.r.t. the time variable $t$. 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. $t$. 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. $t$ near a given state trajectory, of special relevance to control system analysis. Properties of such multifunctions are derived and their significance is illustrated by an application to sensitivity analysis.