# 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

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topSerhani, 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 -

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