Relationship between maximum principle and dynamic programming in presence of intermediate and final state constraints^*

¹ Université de Paris, Laboratoire Jacques-Louis Lions (LJLL), 75005 Paris, France.
² Sorbonne-Université, CNRS, LJLL, 75005 Paris, France.
³ Unité de Mathématiques Appliquées (UMA), Ensta ParisTech, 828 Bd des Maréchaux, 91762 Palaiseau Cedex, France.
⁴ Normandie Univ, INSA Rouen Normandie, LMI (EA 3226 - FR CNRS 3335), 76000 Rouen, France.

^** Corresponding author: olivier.bokanowski@math.univ-paris-diderot.fr

Received: 16 September 2020
Accepted: 4 August 2021

Abstract

In this paper, we consider a class of optimal control problems governed by a differential system. We analyze the sensitivity relations satisfied by the co-state arc of the Pontryagin maximum principle and the value function that associates the optimal value of the control problem to the initial time and state. Such a relationship has been already investigated for state-constrained problems under some controllability assumptions to guarantee Lipschitz regularity property of the value function. Here, we consider the case with intermediate and final state constraints, without any controllability assumption on the system, and without Lipschitz regularity of the value function. Because of this lack of regularity, the sensitivity relations cannot be expressed with the sub-differentials of the value function. This work shows that the constrained problem can be reformulated with an auxiliary value function which is more regular and suitable to express the sensitivity of the adjoint arc of the original state-constrained control problem along an optimal trajectory. Furthermore, our analysis covers the case of normal optimal solutions, and abnormal solutions as well.