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 $\left(\alpha \right)$. 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.

Keywords: optimal control, duality, HJB equation, proximal supersolution, proximal subdifferential

@article{RO_2009__43_2_201_0, author = {Serhani, Mustapha and Ra{\"\i}ssi, Nadia}, title = {Nonconvex duality and semicontinuous proximal solutions of {HJB} equation in optimal control}, journal = {RAIRO - Operations Research - Recherche Op\'erationnelle}, pages = {201--214}, publisher = {EDP-Sciences}, volume = {43}, number = {2}, year = {2009}, doi = {10.1051/ro/2009012}, mrnumber = {2527863}, zbl = {1162.49004}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/ro/2009012/} }

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 - Recherche Opérationnelle PY - 2009 SP - 201 EP - 214 VL - 43 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ro/2009012/ DO - 10.1051/ro/2009012 LA - en ID - RO_2009__43_2_201_0 ER -

%0 Journal Article %A Serhani, Mustapha %A Raïssi, Nadia %T Nonconvex duality and semicontinuous proximal solutions of HJB equation in optimal control %J RAIRO - Operations Research - Recherche Opérationnelle %D 2009 %P 201-214 %V 43 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ro/2009012/ %R 10.1051/ro/2009012 %G en %F RO_2009__43_2_201_0

Serhani, Mustapha; Raïssi, Nadia. Nonconvex duality and semicontinuous proximal solutions of HJB equation in optimal control. RAIRO - Operations Research - Recherche Opérationnelle, Volume 43 (2009) no. 2, pp. 201-214. doi : 10.1051/ro/2009012. http://archive.numdam.org/articles/10.1051/ro/2009012/

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