Abstract. The paper addresses a new class of optimal control problems governed by the dissipative and discontinuous differential inclusion of the sweeping/Moreau process while using controls to determine the best shape of moving convex polyhedra in order to optimize the given Bolza-type functional, which depends on control and state variables as well as their velocities. Besides the highly non-Lipschitzian nature of the unbounded differential inclusion of the controlled sweeping process, the optimal control problems under consideration contain intrinsic state constraints of the inequality and equality types. All of this creates serious challenges for deriving necessary optimality conditions. We develop here the method of discrete approximations and combine it with advanced tools of first-order and second-order variational analysis and generalized differentiation. This approach allows us to establish constructive necessary optimality conditions for local minimizers of the controlled sweeping process expressed entirely in terms of the problem data under fairly unrestrictive assumptions. As a by-product of the developed approach, we prove the strong W 1,2 -convergence of optimal solutions of discrete approximations to a given local minimizer of the continuous-time system and derive necessary optimality conditions for the discrete counterparts. The established necessary optimality conditions for the sweeping process are illustrated by several examples.

Abstract. We consider the free endpoint Mayer problem for a controlled Moreau process, the control acting as a perturbation of the dynamics driven by the normal cone, and derive necessary optimality conditions of Pontryagin's Maximum Principle type. The results are also discussed through an example. We combine techniques from [19] and from [6], which in particular deals with a different but related control problem. Our assumptions include the smoothness of the boundary of the moving set C(t), but, differently from [6], do not require strict convexity. Rather, a kind of inward/outward pointing condition is assumed on the reference optimal trajectory at the times where the boundary of C(t) is touched. The state space is finite dimensional.

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Closed sets $K\subset \mathbb R^{n}$ satisfying an external sphere condition with uniform radius (called $\varphi$-convexity, proximal smoothness, or positive reach) are considered. It is shown that for $\mathcal H^{n-1}$-a.e. $x\in \partial K$ the proximal normal cone to $K$ at $x$ has dimension one. Moreover if $K$ is the closure of an open set satisfying a (sharp) nondegeneracy condition, then the De Giorgi reduced boundary is equivalent to $\partial K$ and the unit proximal normal equals $\mathcal H^{n-1}$-a.e. the (De Giorgi) external normal. Then lower semicontinuous functions $f:\mathbb R^{n}\rightarrow \mathbb R\cup\{ +\infty\}$ with $\varphi$-convex epigraph are shown, among other results, to be locally $BV$ and twice $\mathcal L^{n}$-a.e. differentiable; furthermore, the lower dimensional rectifiability of the singular set where $f$ is not differentiable is studied. Finally we show that for $\mathcal L^{n}$-a.e. $x$ there exists $\delta (x )>0$ such that $f$ is semiconvex on $B(x,\delta(x))$. We remark that such functions are neither convex nor locally Lipschitz, in general. Methods of nonsmooth analysis and of geometric measure theory are used

A minimal time problem with linear dynamics and convex target is considered. It is shown, essentially, that the epigraph of the minimal time function T (•) is ϕ-convex (i.e., it satisfies a kind of exterior sphere condition with locally uniform radius), provided T (•) is continuous. Several regularity properties are derived from results in [G. Colombo and A. Marigonda, Calc. Var. Partial Differential Equations, 25 (2005), pp. 1-31], including twice a.e. differentiability of T (•) and local estimates on the total variation of DT .