Quasi Differential Quotients

TL;DR

This paper introduces Quasi Differential Quotients ($QDQ$s), a new form of generalized differentiation for set-valued maps, which enhances open mapping results and aids in optimal control theory applications.

Contribution

It defines $QDQ$s as a novel class of generalized derivatives, extending previous concepts and enabling stronger theorems and optimal control results involving nonsmooth vector fields.

Findings

$QDQ$s allow non-punctured open mapping results.

They facilitate stronger set-separation theorems.

They are useful in deriving maximum principles in optimal control.

Abstract

We explore basic properties and some applications of Quasi Differential Quotients ($QDQ$s) and the related $QDQ$-approximating multi-cones. A $QDQ$, which is a special kind of H.Sussmann's Approximate Generalized Differential Quotient ($AGDQ$), consists in a notion of generalized differentiation for set-valued maps. $QDQ$s have the advantage over $AGDQ$s of allowing a genuine, non-punctured, Open Mapping result, so implying stronger set-separation theorems. They have already proved quite useful in the investigation of some connections occurring between infimum gap phenomena and the normality of minima. Moreover, $QDQ$-approximating multi-cones are fit in optimal control to deduce Maximum Principles that involve (set-valued) Lie brackets of nonsmooth vector fields.