Abstract
In this paper we formulate new general optimality conditions for impulsive control of piecewise-deterministic processes. We prove continuity of the value functions for optimal stopping, discounted impulsive control, and impulsive control with long run average cost. We study conditions for optimal and nearly optimal policies for the corresponding impulsive control problems. We give variational formulations of the optimality conditions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. Barles, Deterministic impulse control problems,SIAM J. Control Optim.,23 (1985), 419–432.
A. Bensoussan and J.-L. Lions,Applications des Inequations Variationelles en Controle Stochastique, Dunod, Paris, 1978.
A. Bensoussan and J.-L. Lions,Controle Impulsionnel et Inequations Quasi-Variationnelles, Dunod, Paris, 1982.
O. L. V. Costa and M. H. A. Davis, Approximations for optimal stopping of a piecewise-deterministic process,Math. Control Signals Systems,1 (1988), 123–146.
O. L. V. Costa and M. H. A. Davis, Impulsive control of piecewise-deterministic processes,Math. Control Signals Systems,2 (1989), 187–206.
M. H. A. Davis, Piecewise-deterministic Markov processes: A general class of nondiffusion stochastic models,J. Roy. Statist. Soc. B,46 (1984), 353–388.
E. Dynkin,Markov Processes, Springer-Verlag, Berin, 1965.
D. Gatarek, On value functions for impulsive control of piecewise-deterministic processes,Stochastics,32 (1990), 27–52.
D. Gatarek, On first-order quasi-variational inequalities with integral terms,Appl. Math. Optim.,24 (1991), 85–98.
D. Gatarek, A note on impulsive control of Feller processes with costly information,Apl. Mat.,35 (1990), 51–59.
D. Gatarek, Impulsive control of piecewise-deterministic processes with long run average cost, submitted toStochastics.
D. Gatarek and Ł. Stettner, On the compactness method in general ergodic impulsive control of Markov processes,Stochastics,31 (1990), 15–25.
U. S. Gugerli, Optimal stopping of piecewise-deterministic Markov processes,Stochastics,19 (1986), 221–236.
S. Lenhart, Viscosity solutions associated with switching control problems for piecewise-deterministic processes,Houston J. Math.,13 (1987) 405–426.
S. Lenhart, Viscosity solutions associated with impulsive control problems for piecewise-deterministic processes, preprint.
S. Lenhart and Y.-C. Liao, Integro-differential equations associated with optimal stopping time of piecewise-deterministic processes,Stochastics,15 (1985), 183–207.
M. Robin, Controle Impulsionnel des Processus de Markov, Thesis, Universite Paris IX, 1978.
A. A. Yushkevich, Continuous-time Markov decision processes with interventions,Stochastics,9 (1983), 235–274.
J. Zabczyk, Stopping problems in stochastic control,Proceedings of the International Congress of Mathematicians, Warsaw, August 16–24, 1983.
Author information
Authors and Affiliations
Additional information
This work was done during the author's stay at University of Bonn under grant DAAD 314/104/007/0.
Rights and permissions
About this article
Cite this article
Gatarek, D. Optimality conditions for impulsive control of piecewise-deterministic processes. Math. Control Signal Systems 5, 217–232 (1992). https://doi.org/10.1007/BF01215846
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01215846