Oriented Calmness and Sweeping Process Dynamics

Abstract

Daniilidis and Drusviatskiy, in 2017, extended the celebrated Kurdyka–Łojasiewicz inequality from definable functions to definable multivalued maps by establishing that the coderivative mapping 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. 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 behavior of the sweeping process orbits and the integrability of the talweg function. These results are close in spirit with the ones in Bolte et al., 2010, in which characterizations for the desingularization of the (sub)gradient of functions is obtained. Funding: A. Daniilidis was supported by the Austrian Science Fund [Grant FWF P-36344N], Fondo Nacional de Desarrollo Científico y Tecnológico [Grant 1211217], and Centro de Modelamiento Matemático [Grant FB210005]. S. Tapia was supported by Agencia Nacional de Investigación y Desarrollo (Chile)-Programa de Fortalecimiento de Capital Humano Académico Doctorado Nacional [Grant 2018-21181905].