Semicontinuous viscosity solutions to mixed boundary value problems with degenerate convex Hamiltonians.

We introduce a Boundary Value Problem, (BVP), for a class of Hamilton– Jacobi–Bellman equations with degenerate, possibly discontinuous Hamiltonians and lower semicontinuous boundary conditions. This (BVP) includes exit–time control problems for control systems with controls bounded in Lβ norm (β ≥ 1). We prove uniqueness of the semicontinuous viscosity solution and give a representation formula in terms of the value function of the associated control problem. With small changes in the Hamiltonian, (BVP) allows also to represent a different class of control problems, including e.g. the problem of minimizing the Lβ norm of the control steering an initial condition to a given target in a fixed interval of time.