Desingularization of the Sweeping Process Mapping

TL;DR

This paper extends the desingularization of the coderivative mapping for sweeping processes beyond o-minimal structures, linking it to orbit behavior and integrability conditions, thus broadening the theoretical framework.

Contribution

It characterizes the existence of desingularization for the coderivative of sweeping processes without relying on o-minimal geometry, using orbit behavior and talweg function integrability.

Findings

Desingularization of the coderivative is characterized by orbit behavior.

The results do not depend on o-minimal geometry.

Connections are made with the integrability of the talweg function.

Abstract

In [9], the celebrated K{\L}-inequality has been extended from definable functions $f:\mathbb{R}^{n}\rightarrow\mathbb{R} $ to definable multivalued maps $S:\mathbb{R}\rightrightarrows\mathbb{R}^{n}$, by establishing that the co-derivative mapping $D^{\ast}S$ admits a desingularization around every critical value. As was the case in the gradient dynamics, this desingularization yields a uniform control of the lengths of all bounded orbits of the corresponding sweeping process $-\dot{\gamma}(t)\in N_{S(t)}(\gamma(t))$. In this paper, working outside the framework of o-minimal geometry, we characterize the existence of a desingularization for the coderivative in terms of the behaviour of the sweeping process orbits and the integrability of the talweg function. These results are close in spirit with the ones in [3], where characterizations for the desingularization of the (sub)gradient of functions had been obtained.