Discrete Approximations of a Controlled Sweeping Process

Abstract

The paper is devoted to the study of a new class of optimal control problems\ud governed by the classical Moreau sweeping process with the new feature that the polyhe-\ud dral moving set is not fixed while controlled by time-dependent functions. The dynamics of\ud such problems is described by dissipative non-Lipschitzian differential inclusions with state\ud constraints of equality and inequality types. It makes challenging and difficult their anal-\ud ysis and optimization. In this paper we establish some existence results for the sweeping\ud process under consideration and develop the method of discrete approximations that allows\ud us to strongly approximate, in the W^{1,2} topology, optimal solutions of the continuous-type\ud sweeping process by their discrete counterparts

Optimal control of the sweeping process over polyhedral controlled sets

“…To employ and justify the method of discrete approximations in deriving necessary optimality conditions for the control sweeping process, we need to make sure that for all τ ∈ [0, T ] and all k ∈ N sufficiently large each problem (P τ k ) admits an optimal solution. Despite the finite-dimensionality, this issue is nontrivial for (P τ k ) due to the possible nonclosedness of the feasible solution set to this problem because of the dynamic constraints (3.20) generated by the normal cone to the moving set in (3.1); see [12,Example 4.5]. To overcome such a possibility, we employed in [12] the Positive Linear Independence Constraint Qualification (PLICQ) for the given i.l.m.z(·) in the original problem (P ) formulated as follows: (3.26) where the collection of the active constraint indices i ∈ I(x(t),ū(t),b(t)) is defined by…”

A Maximum Principle for the Controlled Sweeping Process

“…Not mentioning some works scattered in the mechanical engineering literature, some early theoretical results appeared in [20] (a Hamilton-Jacobi characterization of the value function, with C constant, later generalized in [14]) and in [17,18] (existence and discrete approximation of optimal controls, in the related framework of rate independent processes). More recently, the papers [11,12,13] are devoted to the case where the control acts on the moving set, which in turn is required to have a polyhedral structure. In particular, [13] contains a set of necessary conditions for local minima which are derived by passing to the limit along suitable discrete approximations.…”

Optimal control of a rate-independent evolution equation via viscous regularization

“…Optimal control of finite-dimensional rate-independent processes has been considered in Brokate [1987Brokate [ , 1988; Brokate and Krejčí [2013] and we witness an increasing interest for the optimal control of sweeping processes, see Castaing et al [2014]; Colombo et al [2012Colombo et al [ , 2015Colombo et al [ , 2016. In the infinite-dimensional setting, the available results are scant.…”