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

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.