Convex Analysis of Minimal Time and Signed Minimal Time Functions

TL;DR

This paper explores minimal time and signed minimal time functions in locally convex topological vector spaces, providing new theoretical insights and subdifferential formulas under convexity assumptions.

Contribution

It introduces a novel notion of closedness for target sets and develops subdifferential formulas for signed minimal time functions in a general convex setting.

Findings

Established subdifferential formulas for signed minimal time functions.

Introduced a new concept of closedness of target sets in LCTV spaces.

Extended the theory of minimal time functions to a broader convex framework.

Abstract

In this paper we first consider the class of minimal time functions in the general setting of locally convex topological vector (LCTV) spaces. The results obtained in this framework are based on a novel notion of closedness of target sets with respect to constant dynamics. Then we introduce and investigate a new class of signed minimal time functions, which are generalizations of the signed distance functions. Subdifferential formulas for the signed minimal time and distance functions are obtained under the convexity assumptions on the given data.