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

Guy Barles; Ariela Briani; Emmanuel Trélat

doi:10.3934/mcrf.2018021

Article Contents

2018, Volume 8, Issue 3&4: 509-533. Doi: 10.3934/mcrf.2018021

This issue Previous Article Existence for nonlinear finite dimensional stochastic differential equations of subgradient type Next Article Necessary conditions for infinite horizon optimal control problems with state constraints

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

1.
Institut Denis Poisson, Université de Tours, Faculté des Sciences et Techniques, Parc de Grandmont, 37200 Tours, France
2.
Sorbonne Université, Université Paris-Diderot SPC, CNRS, Inria, Laboratoire Jacques-Louis Lions, équipe CAGE, F-75005 Paris, France

^* Corresponding author: Emmanuel Trélat
^* Corresponding author: Emmanuel Trélat

Received: October 2017

Revised: June 2018

Published: September 2018

This work was partially supported by the ANR HJnet ANR-12-BS01-0008-01

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Abstract

In this paper we consider regional deterministic finite-dimensional optimal control problems, where the dynamics and the cost functional depend on the region of the state space where one is and have discontinuities at their interface.

Under the assumption that optimal trajectories have a locally finite number of switchings (i.e., 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 establish sensitivity relations for regional optimal control problems and we prove that the regularity of the value function of such problems is the same (i.e., is not more degenerate) than the one of the higher-dimensional classical optimal control problem that lifts the problem.

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Mathematics Subject Classification: Primary: 49K20, 49K15; Secondary: 35F21.

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Figure 1. Structure 1-2

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Figure 2. Structure 1-$\mathcal{H}$-2

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Figure 3. Going "to the left" is not optimal

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Figure 4. The trajectory 1-$\mathcal{H}$-2

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Figure 5. The trajectory 1-2

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