AVERAGE CONTINUOUS CONTROL OF PIECEWISE DETERMINISTIC MARKOV PROCESSES

This paper deals with the long run average continuous control problem of piecewise deterministic Markov processes (PDMPs) taking values in a general Borel space and with compact action space depending on the state variable. The control variable acts on the jump rate and transition measure of the PDM...

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Bibliographic Details
Published in:SIAM journal on control and optimization 2010-01, Vol.48 (7-8), p.4262-4291
Main Authors: COSTA, O. L. V, DUFOUR, F
Format: Article
Language:English
Subjects:
Boundaries
Costs
Differential equations
Exact sciences and technology
Feedback control systems
Inequalities
Markov analysis
Markov processes
Mathematical analysis
Mathematics
Optimization
Probability and statistics
Probability theory and stochastic processes
Running
Sciences and techniques of general use
Selectors
Studies
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Summary:This paper deals with the long run average continuous control problem of piecewise deterministic Markov processes (PDMPs) taking values in a general Borel space and with compact action space depending on the state variable. The control variable acts on the jump rate and transition measure of the PDMP, and the running and boundary costs are assumed to be positive but not necessarily bounded. Our first main result is to obtain an optimality equation for the long run average cost in terms of a discrete-time optimality equation related to the embedded Markov chain given by the postjump location of the PDMP. Our second main result guarantees the existence of a feedback measurable selector for the discrete-time optimality equation by establishing a connection between this equation and an integro-differential equation. Our final main result is to obtain some sufficient conditions for the existence of a solution for a discrete-time optimality inequality and an ordinary optimal feedback control for the long run average cost using the so-called vanishing discount approach. Two examples are presented illustrating the possible applications of the results developed in the paper. [PUBLICATION ABSTRACT]
ISSN:0363-0129
1095-7138
DOI:10.1137/080718541