ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - July 2012 - Volume 18, Issue 03  

Research Article

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

Dan Goreaca1 and Oana-Silvia Sereaa2

a1 UniversitéParis-Est Marne-la-Vallée, LAMA, UMR8050, 5, boulevard Descartes, Cité Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée, France

a2 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

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.

(Received February 9 2011)

(Revised May 14 2011)

(Online publication August 17 2011)

Key Words:

  • Dynamic programming principle;
  • essential supremum;
  • HJ equations;
  • occupational measures;
  • $\mathbb{L}^{p}$ approximations

Mathematics Subject Classification:

  • 34A60;
  • 49J45;
  • 49L20;
  • 49L25;
  • 93C15