Well-posedness of fractional Moreau’s sweeping processes of Caputo type

Abstract

In this paper, we study the well-posedness (in the sense of existence and uniqueness of a solution) of a sweeping process involving fractional order derivative of Caputo type in Hilbert spaces. The convex constraint set C(t) is assumed to satisfy a Hölder continuous variation. The notion of a solution of this fractional sweeping process is proposed, the existence and uniqueness of such solution are proved and a stability result is provided, as well. More precisely, we present new arguments for the proof of existence of the fractional sweeping process. It is still based on the ideas of the catching-up algorithm of J.J. Moreau (see Kunze and Monteiro Marques (2000) and Moreau (1977)). We establish a catching-up algorithm for fractional sweeping process. This new algorithm gives us a sequence of approximate solutions. The technical problems are to prove that we can extract a weakly convergent subsequence, and to check that this limit function is a solution of the fractional sweeping process. Here we propose a new approach to overcome these problems. The existence of solutions follows, to some extent, from an important differential inequality on the Caputo fractional derivative of the squared norm CD0α‖u(t)‖2≤2u(t),CD0αu(t), which was studied in Aguila-Camacho et al. (2014), Gomoyunov (2018) and Kamenskii et al. (2021), that applied to two approximate solutions, by the monotonicity of normal cone NC(t)(⋅) and the minus sign on the right hand side, yields that their distance is nonincreasing. This reasoning allows us to prove that the sequence of approximate solutions is of the fractional sweeping process, converging to a solution. Our approach is constructive and the algorithm is implemented numerically. The results of the paper are applied in the study of an application to non-regular electrical circuit containing nonsmooth electronic device like diode and fractional inductor, which represent an additional novelty of our paper. Finally, two numerical simulations to valid the theoretical results are given.