Dynamic Programming for Some Optimal Control Problems with Hysteresis

We study an innnite horizon optimal control problem for a system with two state variables. One of them has the evolution governed by a controlled ordinary diierential equation and the other one is related to the latter by a hysteresis relation, represented here by either a play operator or a Prandtl-Ishlinskii operator. By dynamic programming, we derive the corresponding (discontinuous) rst order Hamilton-Jacobi equation, which in the rst case is of nite dimension and in the second case is of innnite dimension. In both cases we prove that the value function is the only bounded uniformly continuous viscosity solution of the equation.