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Optimal control of piecewise deterministic Markov process. (English) Zbl 0566.93074
A piecewise deterministic Markov process which is characterized by its behaviour between the jumps, $$x_ t=f(x_ t),$$ and by the jump behaviour given by the jump rate $$\lambda$$, the jump height q in the interior and the jump height p on the boundary of the state space, is controlled by strategies depending on the initial state of the deterministic motion and on the time spent since the starting point. The aim is to minimize continuous, jump and terminal costs. An optimality condition is given in the form of a limiting Hamilton-Jacobi-Bellman equation, the existence of optimal controls is derived, and by considering the original problem as a convex mathematical programming problem, the value function is characterized as supremum of smooth subsolutions.
Reviewer: M.Kohlmann

##### MSC:
 93E20 Optimal stochastic control 60J25 Continuous-time Markov processes on general state spaces 49K45 Optimality conditions for problems involving randomness 90C39 Dynamic programming 49L20 Dynamic programming in optimal control and differential games 60J75 Jump processes (MSC2010) 90C25 Convex programming
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