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Nonconvex Duality and Semicontinuous Proximal Solutions of HJB Equationin 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; [email protected]
Nadia Raïssi
Affiliation:
EIMA, Département de Mathématiques et d'Informatique, Faculté des Sciences, Université Ibn Tofail, B.P. 133, Kénitra, Maroc; [email protected]
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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 isvery hard to look for a solution of the control problem. Our aim isto find estimations of the minimal value, (α), of the costfunction of the control problem. For this, we construct anintermediary dual problem leading to a weak duality result, andthen, thanks to additional assumptions of monotonicity of proximalsubdifferential, we give a more precise estimation of (α). Onthe other hand, when the set valued map fulfills the Lipshitzcondition, we prove that the lower semicontinuous (l.s.c.) proximalsupersolutions of the Hamilton-Jacobi-Bellman (HJB) equationcombined with the estimation of (α), lead to a sufficientcondition of optimality for a suspected trajectory. Furthermore, weestablish a strong duality between this optimal control problem anda dual problem involving upper hull of l.s.c. proximalsupersolutions of the HJB equation (respectively with contingentsupersolutions). Finally this strong duality gives rise to necessaryand 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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