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Optimal control of a class of piecewise deterministic processes

Published online by Cambridge University Press:  30 July 2013

M. ANNUNZIATO  and
A. BORZÌ
M. ANNUNZIATO
Affiliation:
Dipartimento di Matematica, Università degli Studi di Salerno, Via Giovanni Paolo II, 132 - 84084 Fisciano (SA), Italy email: mannunzi@unisa.it
A. BORZÌ
Affiliation:
Institut für Mathematik, Universität Würzburg, Emil-Fischer-Strasse 30, 97074 Würzburg, Germany email: alfio.borzi@mathematik.uni-wuerzburg.de
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Abstract

A new control strategy for a class of piecewise deterministic processes (PDP) is presented. In this class, PDP stochastic processes consist of ordinary differential equations that are subject to random switches corresponding to a discrete Markov process. The proposed strategy aims at controlling the probability density function (PDF) of the PDP. The optimal control formulation is based on the hyperbolic Fokker–Planck system that governs the time evolution of the PDF of the PDP and on tracking objectives of terminal configuration with a target PDF. The corresponding optimization problems are formulated as a sequence of open-loop hyperbolic optimality systems following a model predictive control framework. These systems are discretized by first-order schemes that guarantee positivity and conservativeness of the numerical PDF solution. The effectiveness of the proposed computational control framework is validated considering PDP with dichotomic noise.

Keywords

Optimal control problems involving partial differential equationsHybrid systemsFokker–Planck equationsStability and convergence of numerical methodsMarkov renewal processes

Information

Type
Papers
Information
European Journal of Applied Mathematics , Volume 25 , Issue 1 , February 2014 , pp. 1 - 25
Copyright
Copyright © Cambridge University Press 2013 

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