Optimal control of a nonconvex perturbed sweeping process

Abstract

The paper concerns optimal control of discontinuous differential inclusions of the normal cone type governed by a generalized version of the Moreau sweeping process with control functions acting in both nonconvex moving sets and additive perturbations. This is a new class of optimal control problems in comparison with previously considered counterparts where the controlled sweeping sets are described by convex polyhedra. Besides a theoretical interest, a major motivation for our study of such challenging optimal control problems with intrinsic state constraints comes from the application to the crowd motion model in a practically adequate planar setting with nonconvex but prox-regular sweeping sets. Based on a constructive discrete approximation approach and advanced tools of first-order and second-order variational analysis and generalized differentiation, we establish the strong convergence of discrete optimal solutions and derive a complete set of necessary optimality conditions for discrete-time and continuous-time sweeping control systems that are expressed entirely via the problem data. MSC:49J52; 49J53; 49K24; 49M25; 90C30The sweeping process was introduced by Jean-Jacques Moreau in the 1970s (see [29]) in the forṁ( 1.1) where C(t) is a (Lipschitz or absolutely) continuous moving convex set, and where the normal cone N to it is understood in the sense of convex analysis for which Moreau was one of the creators and major players. The original Moreau's motivation came mainly from applications to elastoplasticity, but it has been well recognized over the years that the sweeping process is important for many other applications to various problems in mechanics, hysteresis systems, traffic equilibria, social and economic modelings, etc.; see, e.g., [14,20,21,22,23,33,36] and the references therein. Due to the maximal monotonicity of the normal cone operator in convex analysis, the sweeping system (1.1) is described by a dissipative discontinuous differential inclusion and can formally be related to control theory for dynamical systems governed by differential inclusions of the typeẋ ∈ F (x), which has been broadly developed in variational analysis and optimal control; see, e.g., the books [11,27,37] with their extensive bibliographies. However, the results of the latter theory, obtained under certain Lipschitzian assumptions on F , are not applicable to the discontinuous sweeping process (1.1). Moreover, it is well known that the Cauchy problem for (1.1) admits a unique solution, which excludes any optimization and control of the sweeping process in form (1.1) with a given moving set C(t).The authors of [12] introduced a control version of the sweeping process by inserting control actions into the moving set C(t) with considering its polyhedral evolutionx ≤ b i (t), i = 1, . . . , m , u i (t) = 1 for all t ∈ [0, T ], (1.2)where optimal control functions u i (t) and b i (t) ought to be selected in order to minimize some cost functional. Formulated in this way optimization models for the controlled sweeping proces...

Optimal Control of Sweeping Processes in Robotics and Traffic Flow Models

“…Of course, the main mathematical difference between the mathematical descriptions of these two models is the space dimension. On the other hand, the available results for planar crowd motion models developed in [12,13] are not applicable to either of the models considered in Sections 3 and 4 due to the unconstrained nature of the previously obtained developments.…”

Applications of optimal control of a nonconvex sweeping process to optimization of the planar crowd motion model

“…This program has been basically realized in our recent preprint [6]. For completeness and reader's convenience we present in Section 2 the main results of [6], which are applied here to the planar crowd motion model; it was our major motivation for [6].…”

“…This program has been basically realized in our recent preprint [6]. For completeness and reader's convenience we present in Section 2 the main results of [6], which are applied here to the planar crowd motion model; it was our major motivation for [6]. Then Section 3 contains applications of the obtained results for the sweeping optimal control to an optimization problem for the planar crowd motion model while Section 4 is devoted to solving the latter problem with two participants.…”

“…Optimal control of nonpolyhedral sweeping processes. The purpose of this section is to overview major results of our preprint [6] and adapt them for the subsequent applications to the planar crowd motion model. We first recall some notions of variational analysis and generalized differentiation that are systematically used in below.…”

“…It is proved in [22, Proposition 2.9] that assumptions (H2) and (H3) with the constants therein guarantee the η-uniform prox-regularity of C(t) from 1.4 for each t ∈ [0, T ] with η := α M 3 β . We also mention the following well-posedness result established in [13,Theorem 1], which says that for any u(•) ∈ W Note that the latter additional assumptions (including the convexity of ) are not needed for deriving the main results of [6] on necessary optimality conditions for problem (P ). What actually is needed is a certain local relaxation stability when a given local minimizer for (P ) maintains its local optimality under the convexification of (P ) with respect to velocity variables.…”