The Hybrid Maximum Principle for Optimal Control Problems with Spatially Heterogeneous Dynamics is a Consequence of a Pontryagin Maximum Principle for [math]-Local Solutions

Abstract

SIAM Journal on Control and Optimization, Volume 62, Issue 4, Page 2412-2432, August 2024.
Abstract. The title of the present work is a nod to the paper “The hybrid maximum principle is a consequence of Pontryagin maximum principle” by Dmitruk and Kaganovich [Systems Control Lett., 57 (2008), pp. 964–970]. We investigate a similar framework of hybrid optimal control problems that is also different from Dmitruk and Kaganovich’s. Precisely, we consider a general control system that is described by a differential equation involving a spatially heterogeneous dynamics. In that context, the sequence of dynamics followed by the trajectory and the corresponding switching times are fully constrained by the state position. We prove with an explicit counterexample that the augmentation technique used by Dmitruk and Kaganovich cannot be fully applied to our setting, but we show that it can be adapted by introducing a new notion of local solution to classical optimal control problems and by establishing a corresponding Pontryagin maximum principle. Thanks to this method, we derive a hybrid maximum principle adapted to our setting, with a simple proof that does not require any technical tools (such as implicit function arguments) to handle the dynamical discontinuities.