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Linearization techniques for $\mathbb{L}^{\infty}$-control problemsand dynamic programming principles in classical and $\mathbb{L}^{\infty}$-control problems

Published online by Cambridge University Press:  17 August 2011

Dan Goreac  and
Oana-Silvia Serea
Dan Goreac
Affiliation:
UniversitéParis-Est Marne-la-Vallée, LAMA, UMR8050, 5, boulevard Descartes, Cité Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée, France
Oana-Silvia Serea
Affiliation:
Université de Perpignan, LAMPS, 52, av. Paul Alduy, 66860 Perpignan and École Polytechnique, CMAP, Route de Saclay, 91128 Palaiseau Cedex, France. oana-silvia.serea@univ-prep.fr

Abstract

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The aim of the paper is to provide a linearization approach to the $\mathbb{L}^{\infty}$-control problems. We begin by proving a semigroup-type behaviour of the set of constraints appearing in the linearized formulation of (standard) control problems. As a byproduct we obtain a linear formulation of the dynamic programming principle. Then, we use the $\mathbb{L}^{p}$ approach and the associated linear formulations. This seems to be the most appropriate tool for treating $\mathbb{L}^{\infty}$ problems in continuous and lower semicontinuous setting.

Keywords

Dynamic programming principleessential supremumHJ equationsoccupational measures$\mathbb{L}^{p}$ approximations

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 3 , July 2012 , pp. 836 - 855
Copyright
© EDP Sciences, SMAI, 2011

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