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.
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Keywords: Optimal control problems, final and/or intermediate state constraints, maximum principle, Hamilton-Jacobi-Bellman equation, sensitivity analysis
@article{COCV_2021__27_1_A93_0,
author = {Bokanowski, Olivier and D\'esilles, Anya and Zidani, Hasnaa},
title = {Relationship between maximum principle and dynamic programming in presence of intermediate and final state constraints},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021084},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021084/}
}
TY - JOUR AU - Bokanowski, Olivier AU - Désilles, Anya AU - Zidani, Hasnaa TI - Relationship between maximum principle and dynamic programming in presence of intermediate and final state constraints JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021084/ DO - 10.1051/cocv/2021084 LA - en ID - COCV_2021__27_1_A93_0 ER -
%0 Journal Article %A Bokanowski, Olivier %A Désilles, Anya %A Zidani, Hasnaa %T Relationship between maximum principle and dynamic programming in presence of intermediate and final state constraints %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021084/ %R 10.1051/cocv/2021084 %G en %F COCV_2021__27_1_A93_0
Bokanowski, Olivier; Désilles, Anya; Zidani, Hasnaa. Relationship between maximum principle and dynamic programming in presence of intermediate and final state constraints. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 91. doi: 10.1051/cocv/2021084
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This work has been partially supported by a public grant overseen by the French National Research Agency (ANR) through the “iCODE Institute project” funded by the IDEX Paris-Saclay ANR-11-IDEX-0003-02.
