Existence of Solutions to Differential Inclusions and to Time Optimal Control Problems in the Autonomous Case

Existence of vector minimizers for nonconvex 1-dim integrals with almost convex Lagrangian

“…Indeed, one needs to impose more than just 0-convexity in order to obtain, for general dimension, the same operational possibilities; namely almost convexity. This concept was born, for multifunctions, in collaboration with Arrigo Cellina in the paper [7], to prove existence of solutions to nonconvex differential inclusions and to time-optimal control problems, using reparametrizations. This technique of reparametrizations has been used by Arrigo Cellina and collaborators, during the last decade, to prove also Lipschitz properties and existence results for minimizers of convex noncoercive Lagrangians (see e.g.…”

Almost mixed semi-continuous perturbation of Moreau's sweeping process

“…On another side, Cellina and Ornelas [6] have introduced a new class of righthand side for which the Cauchy problem admits a solution, namely the almost convex sets (see the definition below). In our previous work [1], we have considered an evolution inclusion governed by the Moreau's sweeping process subject to almost convex perturbations, that is external forces applied on the system.…”

“…Let [a, b] ⊂ D(t) be any interval, assume that on this interval there exists two integrable functions λ 1 (·) and λ 2 (·), such that 0 ≤ λ 1 (t) ≤ 1 ≤ λ 2 (t). In addition, assume that λ 1 (·) > 0 a.e., using the same technique as in [2] and [6], there exist two measurable subsets of [a, b], having characteristics functions χ 1 and χ 2 such that χ 1 + χ 2 = χ [a,b] and an absolutely continuous function γ :…”

An Overview on Existence of Vector Minimizers for Almost Convex 1—dim Integrals

“…But the results (obtained in [2]) which we present in this overview deal with n > 1 ; and since in this vector case the hypothesis of 0-convexity does not suffice to guarantee existence of minimizers, we have used instead almost convexity. This concept was born, for multifunctions, in the paper [4], to prove existence of solutions to nonconvex differential inclusions and to time-optimal control problems, using reparametrizations.…”

“…Two techniques have been combined in [2] to prove these results. The first is the above cited reparametrizations, used by A. Cellina and collaborators during the last decade : starting in [5], to prove existence of minimizers for integrals with Existence of Vector Minimizers for Almost Convex 1−dim Integrals 119 convex noncoercive lagrangians ; and continuing in the above stated [4] and other references (which may be seen in [2]).…”