Deterministic state-constrained optimal control problems without controllability assumptions

Olivier Bokanowski; Nicolas Forcadel; Hasnaa Zidani

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 4, page 995-1015
  • ISSN: 1292-8119

Abstract

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In the present paper, we consider nonlinear optimal control problems with constraints on the state of the system. We are interested in the characterization of the value function without any controllability assumption. In the unconstrained case, it is possible to derive a characterization of the value function by means of a Hamilton-Jacobi-Bellman (HJB) equation. This equation expresses the behavior of the value function along the trajectories arriving or starting from any position x. In the constrained case, when no controllability assumption is made, the HJB equation may have several solutions. Our first result aims to give the precise information that should be added to the HJB equation in order to obtain a characterization of the value function. This result is very general and holds even when the dynamics is not continuous and the state constraints set is not smooth. On the other hand we study also some stability results for relaxed or penalized control problems.

How to cite

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Bokanowski, Olivier, Forcadel, Nicolas, and Zidani, Hasnaa. "Deterministic state-constrained optimal control problems without controllability assumptions." ESAIM: Control, Optimisation and Calculus of Variations 17.4 (2011): 995-1015. <http://eudml.org/doc/272777>.

@article{Bokanowski2011,
abstract = {In the present paper, we consider nonlinear optimal control problems with constraints on the state of the system. We are interested in the characterization of the value function without any controllability assumption. In the unconstrained case, it is possible to derive a characterization of the value function by means of a Hamilton-Jacobi-Bellman (HJB) equation. This equation expresses the behavior of the value function along the trajectories arriving or starting from any position x. In the constrained case, when no controllability assumption is made, the HJB equation may have several solutions. Our first result aims to give the precise information that should be added to the HJB equation in order to obtain a characterization of the value function. This result is very general and holds even when the dynamics is not continuous and the state constraints set is not smooth. On the other hand we study also some stability results for relaxed or penalized control problems.},
author = {Bokanowski, Olivier, Forcadel, Nicolas, Zidani, Hasnaa},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {optimal control problem; state constraints; Hamilton-Jacobi equation; Hamilton-Jacobi-Bellman equation},
language = {eng},
number = {4},
pages = {995-1015},
publisher = {EDP-Sciences},
title = {Deterministic state-constrained optimal control problems without controllability assumptions},
url = {http://eudml.org/doc/272777},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Bokanowski, Olivier
AU - Forcadel, Nicolas
AU - Zidani, Hasnaa
TI - Deterministic state-constrained optimal control problems without controllability assumptions
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2011
PB - EDP-Sciences
VL - 17
IS - 4
SP - 995
EP - 1015
AB - In the present paper, we consider nonlinear optimal control problems with constraints on the state of the system. We are interested in the characterization of the value function without any controllability assumption. In the unconstrained case, it is possible to derive a characterization of the value function by means of a Hamilton-Jacobi-Bellman (HJB) equation. This equation expresses the behavior of the value function along the trajectories arriving or starting from any position x. In the constrained case, when no controllability assumption is made, the HJB equation may have several solutions. Our first result aims to give the precise information that should be added to the HJB equation in order to obtain a characterization of the value function. This result is very general and holds even when the dynamics is not continuous and the state constraints set is not smooth. On the other hand we study also some stability results for relaxed or penalized control problems.
LA - eng
KW - optimal control problem; state constraints; Hamilton-Jacobi equation; Hamilton-Jacobi-Bellman equation
UR - http://eudml.org/doc/272777
ER -

References

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