Résumé
We study an optimal control problem where the cost functional to be minimized represents the so-called 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 preventingthe use of the standard Maximum Principle. We consider a regularization scheme of the problem basedon the Moreau-Yosida approximation of the indicator function of K. We prove the convergence of anoptimal sequence for the approximated problem to an optimal solution of the original problem. We theninvestigate the convergence of the adjoint vector given by Pontryagin's Principle when the regularizationparameter goes to zero. Finally, we provide an example illustrating the convergence property and wecompute explicitly an optimal feedback policy and the value function.