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Issue RAIRO-Oper. Res.
Volume 43, Number 2, April-June 2009
Page(s) 201 - 214
DOI 10.1051/ro/2009012
Published online 28 April 2009

RAIRO-Oper. Res. 43 (2009) 201-214
DOI: 10.1051/ro/2009012

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

Mustapha Serhani1 and Nadia Raïssi2

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

Received January 30, 2006. Accepted February 2, 2009. Published online 28 April 2009

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, ($\alpha$), 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 $(\alpha)$. 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 ($\alpha$), 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.


Mathematics Subject Classification. 49J24, 49K24, 49N15, 49L99, 49J52.

Key words: Optimal control, duality, HJB equation, proximal supersolution, proximal subdifferential.


© EDP Sciences, ROADEF, SMAI 2009


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