Optimal control with state-space constraint. II.

*(English)*Zbl 0619.49013This 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

##### MSC:

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 |