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Optimal control with state-space constraint. II. (English) Zbl 0619.49013
This paper deals with the Hamilton-Jacobi-Bellman (HJB) equation associated to the optimal control of a certain class of jump processes, namely piecewise deterministic processes. In this case, the HJB equation involves a non-local operator. For processes constrained to a closed domain, and as in the case of deterministic processes, previously analyzed by the author in the first part of this work [ibid. 24, 552-561 (1986; Zbl 0597.49023)], solutions of viscosity type are considered and the following results are proved: a) There is at most one constrained viscosity solution of the HJB equation. b) If the optimal cost function v is bounded uniformly continuous (BUC) and a relation of dynamic programming type holds, then v is a constrained viscosity solution. c) Under suitable conditions, v is BUC and a relation of dynamic programming type is verified by v.
Reviewer: R.Gonzalez

49L20 Dynamic programming in optimal control and differential games
60J60 Diffusion processes
93E20 Optimal stochastic control
35J65 Nonlinear boundary value problems for linear elliptic equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
49K45 Optimality conditions for problems involving randomness
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