Non-convex Quasi-variational Differential Inclusions

Abstract

In this article we discuss the evolution problem known as sweeping process for a class of prox-regular non-convex sets. Assuming that such sets depend continuously on time and state, we prove local and global existence of solutions which are absolutely continuous functions.

Reduction of state dependent sweeping process to unconstrained differential inclusion

“…on [T 0 , T ], u(T 0 ) = u 0 ∈ C(T 0 , u 0 ). (1.2) In the non-convex setting, Chemetov and Monteiro Marques [9] proved the existence of solutions of (1.2) when C(t, u) is a (uniformly) prox-regular set of the Hilbert space H moving in an absolutely continuous way and F(t, x) is a Carathéodory single-valued mapping. Later, Castaing et al showed that the existence of solution for (1.2) with a multimapping F can be established when the sets C(t, u) are convex and F is upper semicontinuous with weakly compact convex values.…”

Delay perturbed state-dependent sweeping process

“…The sequence (u n ) n∈N * being in addition equicontinuous according to (8), this sequence (u n ) n∈N * is relatively compact in C([0, T ], H ), so we can extract a subsequence of (u n ) n∈N * (that we do not relabel) which converges uniformly to u on [0, T ]. By the inequality (8) again there is a subsequence of (u n ) n∈N * (that we do not relabel) which converges 1−L 2 as a Lipschitz constant therein.…”

“…See the monographs [1][2][3][4] and the papers [5][6][7][8][9][10][11][12] for an overview on this area of research. However, there are a few results devoted to differential inclusions governed by nonconvex sweeping processes with delay.…”

Nonconvex Sweeping Process with a Moving Set Depending on the State

“…The first work dealing with a moving set C(t, x) depending on the time and the state has been made in [21] under the convexity assumption for C(t, x). Recently, Chemetov and Monteiro Marques [10] established the first results concerning the situation where the moving set C(t, x), depending on both the time and on the state, is nonconvex. Given a single valued mapping G : [0, T ]×H → H of Carathéodory type (that is, measurable in t and continuous in x), they studied the differential inclusion…”

“…the solution of (II ) is obtained in [10] by applying the Schauder fixed point theorem to an appropriate compact convex subset of the space of continuous mappings from [0, T ] to H . In [7], by means of a generalized version of the Schauder fixed point theorem from [19,29], Castaing et al provided another approach allowing them to prove the existence of a solution when G ≡ {0} and C(t, x) is prox-regular and ball compact; with the same approach, they also obtained an existence result (even in the presence of a delay) when G is the convexvalued multimapping bounded on [0, T ] × C H (−r, 0) (C H (−r, 0) denotes the space of all continuous mappings from [−r, 0] to H ), and C(t, x) is convex and ball compact.…”