Optimal control of finite horizon type for a multidimensional delayed switching system

We consider a ¯nite horizon optimal control problem for an ODE system, with trajectories presenting a delayed two-values switching along a ¯xed direction. In particular the system exhibits hysteresis. Due to the pres- ence of the switching component of the trajectories, several de¯nitions of value functions are possible. None of these value functions is in general continuous. We prove that, under general hypotheses, the \least value function", i.e. the value function of the more relaxed problem, is the unique lower semicontinuous viscosity solution of two suitably coupled Hamilton-Jacobi-Bellman equations. Such a coupling involves boundary conditions in the viscosity sense.