Optimal control of piecewise deterministic markov process

The trajectories of piecewise deterministic Markov processes are solutions of an ordinary (vector)differential equation with possible random jumps between the different integral curves. Both continuous deterministic motion and the random jumps of the processes are controlled in order to minimize the expected value of a performance functional consisting of continuous, jump and terminal costs. A limiting form of the Hamilton-Jacobi-Bellman partial differential equation is shown to be a necessary and sufficient optimality condition. The existence of an optimal strategy is proved and acharacterization of the value function as supremum of smooth subsolutions is also given. The approach consists of imbedding the original control problem tightly in a convex mathematical programming problem on the space of measures and then solving the latter by dualit