NONCONVEX DUALITY AND SEMICONTINUOUS PROXIMAL SOLUTIONS OF HJB EQUATION IN OPTIMAL CONTROL

In this work, we study an optimal control problem deal-ing with differential inclusion. Without requiring Lipschitz conditionof the set valued map, it is very hard to look for a solution of thecontrol problem. Our aim is to find estimations of the minimal value,( α), of the cost function of the control problem. For this, we constructan intermediary 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). On the otherhand, when the set valued map fulfills the Lipshitz condition, we provethat the lower semicontinuous (l.s.c.) proximal supersolutions of theHamilton-Jacobi-Bellman (HJB) equation combined with the estima-tion of ( α), lead to a sufficient condition of optimality for a suspectedtrajectory. Furthermore, we establish a strong duality between this op-timal control problem and a dual problem involving upper hull of l.s.c.proximal supersolutions of the HJB equation (respectively with contin-gent supersolutions). Finally this strong duality gives rise to necessaryand sufficient conditions of optimality