Hamilton-Jacobi-Bellman equations with fast gradient-dependence.

We investigate existence, uniqueness, and regularity properties for a class of H-J-B equations arising in non-linear control problems with unbounded controls. These equations involve Hamiltonians which are superlinear in the adjoint variable, and they have been already studied in the case when the growth in the adjoint variable is, in a sense, uniform with respect to the state variable. For instance, this is the case of the linear-quadratic problem. On the contrary, our results concern Hamiltonians that are superlinear in the adjoint variable, possibly not uniformly with respect to the state variable. Actually, this is the general situation one has to deal with when considering optimal control problems with a nonlinear dynamics (e.g. by slightly perturbing the linear quadratic problem). We also investigate situations where the fast growth of the Hamiltonian in the adjoint variable degenerates into a very discontinuity. Such Hamiltonians arise quite naturally in those optimal control problems where, roughly speaking, the dynamics and the cost display the same growth in the control variable.