Boundary-value problems for systems of Hamilton-Jacobi-Bellman inclusions with constraints

We study in this paper boundary-value problems for systems of Hamilton-Jacobi-Bellman first-order partial differential equations and variational inequalities, the solutions of which are constrained to obey viability constraints. They are motivated by some control problems (such as impulse control) and financial mathematics. We shall prove the existence and uniqueness of such solutions in the class of closed set-valued maps by giving a precise meaning to what a solution means in this case. We shall also provide explicit formulas for this problem. When we deal with Hamilton-Jacobi-Bellman equations, we obtain the existence and uniqueness of Frankowska contingent episolutions. We shall deduce these results from the fact that the graph of the solution is the viable-capture basin of the graph of the boundary conditions under an auxiliary system and then from their properties and their characterizations proved in [J.-P. Aubin, SIAM J. Control Optim., 40 ( 2001), pp. 853-881].