Viscosity solutions of two classes of coupled Hamilton-Jacobi-Bellman equations

This paper studies viscosity solutions of two sets of linearly coupled Hamilton-Jacobi-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 are allowed to depend on both the continuous state and the current state of the Markov chain. The paper establishes the existence and uniqueness of viscosity solutions for these two sets of HJB equations, whose Hamiltonian structures are different from the standard ones.