Multifunctions of bounded variation

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 time 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 significant 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. time. 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, and their significance of illustrated by an application to sensitivity analysis.

$$L^{\infty }$$ L ∞ estimates on trajectories confined to a closed subset, for control systems with bounded time variation

“…Linear L ∞ estimates have been previously established by Frankowska and Rampazzo [14], under similar, but non-equivalent, hypotheses concerning the timedependence of the differential inclusion, which also cover some of these cases. Properties of control systems with bounded variation time dependence are also explored in [16] and [23].…”