Second order conditions for a control problem with discontinuous cost

Laurent Pfeiffer (1) , Térence Bayen (2)

1 COMMANDS - Controle, Optimisation, modèles, Méthodes et Applications pour les Systèmes Dynamiques non linéaires
2 LMA - EA2151 Laboratoire de Mathématiques d'Avignon

In this paper, we consider the problem of minimizing the total time spent by a controlled dynamics outside a constraint set K. Also known as time of crisis, one essential feature of this problem is the discontinuity of the involved integral cost with respect to the state. We first relate this optimal control problem to a mixed initial-final problem with smooth data. Applying the classical theory of optimality conditions to the auxiliary (smooth) problem, we obtain as a main result second order necessary optimality conditions for the time of crisis. Considering the partition of the state space made out of K and its complementary, we notice that the problem can be seen as a particular case of a hybrid problem. Our analysis is thus a first step toward a second order analysis for the more general class of hybrid problems.

Domains: Optimization and Control [math.OC]

Laurent Pfeiffer, Térence Bayen. Second order conditions for a control problem with discontinuous cost. IEEE Conference on Decision and Control CDC2019, Dec 2019, Nice, France. ⟨hal-02413725⟩