
Journal of Convex Analysis 23 (2016), No. 1, 263290 Copyright Heldermann Verlag 2016 About MoreauYosida Regularization of the Minimal Time Crisis Problem Terence Bayen Institut Mathématiques, Université Montpellier, CC 051, 34095 Montpellier Cedex 5, France tbayen@math.univmontp2.fr Alain Rapaport Montpellier SupAgro, 2 Place Pierre Viala, 34060 Montpellier Cedex 2, France rapaport@montpellier.inra.fr We study an optimal control problem where the cost functional to be minimized represents the socalled time of crisis, i.e. the time spent by a trajectory solution of a control system outside a given set K. This functional can be expressed using the characteristic function of K that is discontinuous preventing the use of the standard Maximum Principle. We consider a regularization scheme of the problem based on the MoreauYosida approximation of the indicator function of K. We prove the convergence of an optimal sequence for the approximated problem to an optimal solution of the original problem. We then investigate the convergence of the adjoint vector given by Pontryagin's Principle when the regularization parameter goes to zero. Finally, we provide an example illustrating the convergence property and we compute explicitly an optimal feedback policy and the value function. Keywords: Optimal control, Pontryagin Maximum Principle, Hybrid Maximum Principle, Regularization. [ Fulltextpdf (281 KB)] for subscribers only. 