Value function for regional control problems via dynamic programming and Pontryagin maximum principle

TL;DR

This paper develops a method to analyze regional deterministic optimal control problems with discontinuities, using duplication techniques to relate their value functions to classical control problems, and extends sensitivity relations to these regional problems.

Contribution

It introduces a duplication technique to represent regional control problems as classical problems in higher dimensions, enabling analysis of their value functions and regularity.

Findings

Value function is the minimum over structured classical control problems.

Duplication technique acts as a desingularization of the value function.

Regularity of the regional value function matches that of the lifted classical problem.

Abstract

In this paper we focus on regional deterministic optimal control problems, i.e., problems where the dynamics and the cost functional may be different in several regions of the state space and present discontinuities at their interface. Under the assumption that optimal trajectories have a locally finite number of switchings (no Zeno phenomenon), we use the duplication technique to show that the value function of the regional optimal control problem is the minimum over all possible structures of trajectories of value functions associated with classical optimal control problems settled over fixed structures, each of them being the restriction to some submanifold of the value function of a classical optimal control problem in higher dimension.The lifting duplication technique is thus seen as a kind of desingularization of the value function of the regional optimal control problem. In turn, we extend to regional optimal control problems the classical sensitivity relations and we prove that the regularity of this value function is the same (i.e., is not more degenerate) than the one of the higher-dimensional classical optimal control problem that lifts the problem.