Optimal sampled-data controls with running inequality state constraints: Pontryagin maximum principle and bouncing trajectory phenomenon

Abstract

In the present paper we derive a Pontryagin maximum principle for general nonlinear optimal sampled-data control problems in the presence of running inequality state constraints. We obtain, in particular, a nonpositive averaged Hamiltonian gradient condition associated with an adjoint vector being a function of bounded variation. As a well known challenge, theoretical and numerical difficulties may arise due to the possible pathological behavior of the adjoint vector (jumps and singular part lying on parts of the optimal trajectory in contact with the boundary of the restricted state space). However, in our case with sampled-data controls, we prove that, under certain general hypotheses, the optimal trajectory activates the running inequality state constraints at most at the sampling times. Due to this so-called bouncing trajectory phenomenon, the adjoint vector experiences jumps at most at the sampling times (and thus in a finite number and at precise instants) and its singular part vanishes. Taking advantage of these informations, we are able to implement an indirect numerical method which we use to solve three simple examples.