Manuel Torres-Valdebenito

arXiv2024

A fixed-point approach to history-dependent sweeping processes

In this paper, we study the well-posedness of state-dependent and state-independent sweeping processes driven by prox-regular sets and perturbed by a history-dependent operator. Our approach, based on an enhanced version of Gronwall's lemma and fixed-point arguments, provides an efficient framework for analyzing sweeping processes. In particular, our findings recover all existing results for the class of Volterra sweeping processes and provide new insights into history-dependent sweeping processes. Finally, we apply our theoretical results to establish the well-posedness of a viscoelastic model with long memory.

math.OC↗

arXiv2024

Galerkin-like Method for Integro-Differential Inclusions with applications to Volterra Sweeping processes

In this paper, we develop the Galerkin-like method to address first-order integro-differential inclusions. Under compactness or monotonicity conditions, we obtain new results for the existence of solutions for this class of problems, which generalize existing results in the literature and provide new insights for differential inclusions with an unbounded right-hand side. The effectiveness of the proposed approach is illustrated by presenting new existence results for nonconvex state-dependent Volterra sweeping processes, where the right-hand side is unbounded, and the classical theory of differential inclusions is not applicable. This is the first result of its kind. The paper concludes with an application to the existence of an optimal control problem governed by nonconvex state-dependent Volterra sweeping processes in finite dimensions.

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