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...

The paper addresses an optimal control problem for a perturbed sweeping process of the rateindependent hysteresis type described by a controlled "play-and stop" operator with separately controlled perturbations. This problem can be reduced to dynamic optimization of a state-constrained unbounded differential inclusion with highly irregular data that cannot be treated by means of known results in optimal control theory for differential inclusions. We develop the method of discrete approximations, which allows us to adequately replace the original optimal control problem by a sequence of well-posed finite-dimensional optimization problems whose optimal solutions strongly converge to that of the controlled perturbed sweeping process. To solve the discretized control systems, we derive effective necessary optimality conditions by using second-order generalized differential tools of variational analysis that explicitly calculated in terms of the given problem data.

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The paper concerns the study and applications of a new class of optimal control problems governed by a perturbed sweeping process of the hysteresis type with control functions acting in both play-and-stop operator and additive perturbations. Such control problems can be reduced to optimization of discontinuous and unbounded differential inclusions with pointwise state constraints, which are immensely challenging in control theory and prevent employing conventional variation techniques to derive necessary optimality conditions. We develop the method of discrete approximations married with appropriate generalized differential tools of modern variational analysis to overcome principal difficulties in passing to the limit from optimality conditions for finite-difference systems. This approach leads us to nondegenerate necessary conditions for local minimizers of the controlled sweeping process expressed entirely via the problem data. Besides illustrative examples, we apply the obtained results to an optimal control problem associated with of the crowd motion model of traffic flow in a corridor, which is formulated in this paper. The derived optimality conditions allow us to develop an effective procedure to solve this problem in a general setting and completely calculate optimal solutions in particular situations.