A general Hamilton-Jacobi framework for non-linear state-constrained control problems

Albert Altarovici; Olivier Bokanowski; Hasnaa Zidani

ESAIM: Control, Optimisation and Calculus of Variations (2013)

  • Volume: 19, Issue: 2, page 337-357
  • ISSN: 1292-8119

Abstract

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The paper deals with deterministic optimal control problems with state constraints and non-linear dynamics. It is known for such problems that the value function is in general discontinuous and its characterization by means of a Hamilton-Jacobi equation requires some controllability assumptions involving the dynamics and the set of state constraints. Here, we first adopt the viability point of view and look at the value function as its epigraph. Then, we prove that this epigraph can always be described by an auxiliary optimal control problem free of state constraints, and for which the value function is Lipschitz continuous and can be characterized, without any additional assumptions, as the unique viscosity solution of a Hamilton-Jacobi equation. The idea introduced in this paper bypasses the regularity issues on the value function of the constrained control problem and leads to a constructive way to compute its epigraph by a large panel of numerical schemes. Our approach can be extended to more general control problems. We study in this paper the extension to the infinite horizon problem as well as for the two-player game setting. Finally, an illustrative numerical example is given to show the relevance of the approach.

How to cite

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Altarovici, Albert, Bokanowski, Olivier, and Zidani, Hasnaa. "A general Hamilton-Jacobi framework for non-linear state-constrained control problems." ESAIM: Control, Optimisation and Calculus of Variations 19.2 (2013): 337-357. <http://eudml.org/doc/272900>.

@article{Altarovici2013,
abstract = {The paper deals with deterministic optimal control problems with state constraints and non-linear dynamics. It is known for such problems that the value function is in general discontinuous and its characterization by means of a Hamilton-Jacobi equation requires some controllability assumptions involving the dynamics and the set of state constraints. Here, we first adopt the viability point of view and look at the value function as its epigraph. Then, we prove that this epigraph can always be described by an auxiliary optimal control problem free of state constraints, and for which the value function is Lipschitz continuous and can be characterized, without any additional assumptions, as the unique viscosity solution of a Hamilton-Jacobi equation. The idea introduced in this paper bypasses the regularity issues on the value function of the constrained control problem and leads to a constructive way to compute its epigraph by a large panel of numerical schemes. Our approach can be extended to more general control problems. We study in this paper the extension to the infinite horizon problem as well as for the two-player game setting. Finally, an illustrative numerical example is given to show the relevance of the approach.},
author = {Altarovici, Albert, Bokanowski, Olivier, Zidani, Hasnaa},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {state constraints; optimal control problems; nonlinear controlled systems; Hamilton-Jacobi equations; viscosity solutions; exact penalization; finite horizon problem; infinite horizon control problem; two-player game problem},
language = {eng},
number = {2},
pages = {337-357},
publisher = {EDP-Sciences},
title = {A general Hamilton-Jacobi framework for non-linear state-constrained control problems},
url = {http://eudml.org/doc/272900},
volume = {19},
year = {2013},
}

TY - JOUR
AU - Altarovici, Albert
AU - Bokanowski, Olivier
AU - Zidani, Hasnaa
TI - A general Hamilton-Jacobi framework for non-linear state-constrained control problems
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2013
PB - EDP-Sciences
VL - 19
IS - 2
SP - 337
EP - 357
AB - The paper deals with deterministic optimal control problems with state constraints and non-linear dynamics. It is known for such problems that the value function is in general discontinuous and its characterization by means of a Hamilton-Jacobi equation requires some controllability assumptions involving the dynamics and the set of state constraints. Here, we first adopt the viability point of view and look at the value function as its epigraph. Then, we prove that this epigraph can always be described by an auxiliary optimal control problem free of state constraints, and for which the value function is Lipschitz continuous and can be characterized, without any additional assumptions, as the unique viscosity solution of a Hamilton-Jacobi equation. The idea introduced in this paper bypasses the regularity issues on the value function of the constrained control problem and leads to a constructive way to compute its epigraph by a large panel of numerical schemes. Our approach can be extended to more general control problems. We study in this paper the extension to the infinite horizon problem as well as for the two-player game setting. Finally, an illustrative numerical example is given to show the relevance of the approach.
LA - eng
KW - state constraints; optimal control problems; nonlinear controlled systems; Hamilton-Jacobi equations; viscosity solutions; exact penalization; finite horizon problem; infinite horizon control problem; two-player game problem
UR - http://eudml.org/doc/272900
ER -

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