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

Published online by Cambridge University Press:  28 April 2009

Mustapha Serhani  and
Nadia Raïssi
Mustapha Serhani
Affiliation:
Département d'économie, FSJES, Université My Ismail, B.P. 3102, Toulal, Meknès, Maroc; mserhani@hotmail.com
Nadia Raïssi
Affiliation:
EIMA, Département de Mathématiques et d'Informatique, Faculté des Sciences, Université Ibn Tofail, B.P. 133, Kénitra, Maroc; n.raissi@lycos.com
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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.

Keywords

Optimal controldualityHJB equationproximal supersolutionproximal subdifferential.
Type
Research Article
Information
RAIRO - Operations Research , Volume 43 , Issue 2 , April 2009 , pp. 201 - 214
Copyright
© EDP Sciences, ROADEF, SMAI, 2009

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