Nonconvex Duality and Semicontinuous Proximal Solutions of HJB Equation in Optimal Control

Mustapha Serhani; Nadia Raïssi

RAIRO - Operations Research (2009)

  • Volume: 43, Issue: 2, page 201-214
  • ISSN: 0399-0559

Abstract

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In this work, we study an optimal control problem dealing with differential inclusion. Without requiring Lipschitz condition of the set valued map, it is very hard to look for a solution of the control problem. Our aim is to find estimations of the minimal value, (α), of the cost function of the control problem. For this, we construct an intermediary dual problem leading to a weak duality result, and then, thanks to additional assumptions of monotonicity of proximal subdifferential, we give a more precise estimation of (α). On the other hand, when the set valued map fulfills the Lipshitz condition, we prove that the lower semicontinuous (l.s.c.) proximal supersolutions of the Hamilton-Jacobi-Bellman (HJB) equation combined with the estimation of (α), lead to a sufficient condition of optimality for a suspected trajectory. Furthermore, we establish a strong duality between this optimal control problem and a dual problem involving upper hull of l.s.c. proximal supersolutions of the HJB equation (respectively with contingent supersolutions). Finally this strong duality gives rise to necessary and sufficient conditions of optimality.

How to cite

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Serhani, Mustapha, and Raïssi, Nadia. "Nonconvex Duality and Semicontinuous Proximal Solutions of HJB Equation in Optimal Control." RAIRO - Operations Research 43.2 (2009): 201-214. <http://eudml.org/doc/250653>.

@article{Serhani2009,
abstract = { In this work, we study an optimal control problem dealing with differential inclusion. Without requiring Lipschitz condition of the set valued map, it is very hard to look for a solution of the control problem. Our aim is to find estimations of the minimal value, (α), of the cost function of the control problem. For this, we construct an intermediary dual problem leading to a weak duality result, and then, thanks to additional assumptions of monotonicity of proximal subdifferential, we give a more precise estimation of (α). On the other hand, when the set valued map fulfills the Lipshitz condition, we prove that the lower semicontinuous (l.s.c.) proximal supersolutions of the Hamilton-Jacobi-Bellman (HJB) equation combined with the estimation of (α), lead to a sufficient condition of optimality for a suspected trajectory. Furthermore, we establish a strong duality between this optimal control problem and a dual problem involving upper hull of l.s.c. proximal supersolutions of the HJB equation (respectively with contingent supersolutions). Finally this strong duality gives rise to necessary and sufficient conditions of optimality. },
author = {Serhani, Mustapha, Raïssi, Nadia},
journal = {RAIRO - Operations Research},
keywords = {Optimal control; duality; HJB equation; proximal supersolution; proximal subdifferential.; optimal control; proximal subdifferential},
language = {eng},
month = {4},
number = {2},
pages = {201-214},
publisher = {EDP Sciences},
title = {Nonconvex Duality and Semicontinuous Proximal Solutions of HJB Equation in Optimal Control},
url = {http://eudml.org/doc/250653},
volume = {43},
year = {2009},
}

TY - JOUR
AU - Serhani, Mustapha
AU - Raïssi, Nadia
TI - Nonconvex Duality and Semicontinuous Proximal Solutions of HJB Equation in Optimal Control
JO - RAIRO - Operations Research
DA - 2009/4//
PB - EDP Sciences
VL - 43
IS - 2
SP - 201
EP - 214
AB - In this work, we study an optimal control problem dealing with differential inclusion. Without requiring Lipschitz condition of the set valued map, it is very hard to look for a solution of the control problem. Our aim is to find estimations of the minimal value, (α), of the cost function of the control problem. For this, we construct an intermediary dual problem leading to a weak duality result, and then, thanks to additional assumptions of monotonicity of proximal subdifferential, we give a more precise estimation of (α). On the other hand, when the set valued map fulfills the Lipshitz condition, we prove that the lower semicontinuous (l.s.c.) proximal supersolutions of the Hamilton-Jacobi-Bellman (HJB) equation combined with the estimation of (α), lead to a sufficient condition of optimality for a suspected trajectory. Furthermore, we establish a strong duality between this optimal control problem and a dual problem involving upper hull of l.s.c. proximal supersolutions of the HJB equation (respectively with contingent supersolutions). Finally this strong duality gives rise to necessary and sufficient conditions of optimality.
LA - eng
KW - Optimal control; duality; HJB equation; proximal supersolution; proximal subdifferential.; optimal control; proximal subdifferential
UR - http://eudml.org/doc/250653
ER -

References

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