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Section: New Results

Singular Perturbation for the discounted continuous control of Piecewise Deterministic Markov Processes

Participant : François Dufour.

This work deals with the expected discounted continuous control of piecewise deterministic Markov processes (PDMP's) using a singular perturbation approach for dealing with rapidly oscillating parameters. The state space of the PDMP is written as the product of a finite set and a subset of the Euclidean space Im4 ${\#8477 }^n$ . The discrete part of the state, called the regime, characterizes the mode of operation of the physical system under consideration, and is supposed to have a fast (associated to a small parameter $ \epsilon$>0 ) and a slow behavior. By using a similar approach as developed in the book of Yin (98), the idea in this paper is to reduce the number of regimes by considering an averaged model in which the regimes within the same class are aggregated through the quasi-stationary distribution so that the different states in this class are replaced by a single one. The main goal is to show that the value function of the control problem for the system driven by the perturbed Markov chain converges to the value function of this limit control problem as $ \epsilon$ goes to zero. This convergence is obtained by, roughly speaking, showing that the infimum and supremum limits of the value functions satisfy two optimality inequalities as $ \epsilon$ goes to zero. This enables us to show the result by invoking a uniqueness argument, without needing any kind of Lipschitz continuity condition. These results have been obtained in collaboration with Oswaldo Luis Do Valle Costa from Escola Politécnica da Universidade de São Paulo, Brazil. It has been accepted for publication in Applied Mathematics and Optimization [20] . Part of this work has been presented at the 48th IEEE Conference on Decision and Control, Atlanta, USA, 2010, [29] .


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