Optimal control of piecewise deterministic nonlinear systems with controlled transitions: viscosity solutions, their existence and uniqueness

The paper studies viscosity solutions of two sets of Hamilton-Jacob-Bellman (HJB) equations (one for finite horizon and the other one for infinite horizon) which arise in the optimal control of nonlinear piecewise deterministic systems where the controls could be unbounded. The controls enter through the system dynamics as well as the transitions for the underlying Markov chain process, and have access to both the continuous state and the current state of the Markov chain. The two HJB equations associated with this problem are coupled partial differential equations, as a result of which their Hamiltonian structures are different from the standard ones. The paper establishes the existence and uniqueness of their viscosity solutions, and derives explicit structures for the optimum controllers by using such viscosity solutions.