Continuity/constancy of the Hamiltonian function in a Pontryagin maximum principle for optimal sampled-data control problems with free sampling times

Abstract

In a recent paper by Bourdin and Trélat, a version of the Pontryagin maximum principle (in short, PMP) has been stated for general nonlinear finite-dimensional optimal sampled-data control problems. Unfortunately, their result is only concerned with fixed sampling times, and thus, it does not take into account the possibility of free sampling times. The present paper aims to fill this gap in the literature. Precisely, we establish a new version of the PMP that can handle free sampling times. As in the aforementioned work by Bourdin and Trélat, we obtain a first-order necessary optimality condition written as a nonpositive averaged Hamiltonian gradient condition. Furthermore, from the freedom of choosing sampling times, we get a new and additional necessary optimality condition which happens to coincide with the continuity of the Hamiltonian function. In an autonomous context, even the constancy of the Hamiltonian function can be derived. Our proof is based on the Ekeland variational principle. Finally, a linear–quadratic example is numerically solved using shooting methods, illustrating the possible discontinuity of the Hamiltonian function in the case of fixed sampling times and highlighting its continuity in the instance of optimal sampling times.