(Almost) Everything you always wanted to know about deterministic control problems in stratified domains

Guy  Barles; Emmanuel  Chasseigne; Guy  Barles; Emmanuel  Chasseigne

doi:10.3934/nhm.2015.10.809

Networks and Heterogeneous Media

2015, Volume 10, Issue 4: 809-836. doi: 10.3934/nhm.2015.10.809

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(Almost) Everything you always wanted to know about deterministic control problems in stratified domains

Guy Barles ^{1
,},
Emmanuel Chasseigne ^{1
,}

1.
Laboratoire de Mathématiques et Physique Théorique(UMR CNRS 7350), Fédération Denis Poisson (FR CNRS 2964), Université François Rabelais, Parc de Grandmont, 37200 Tours

Received: 01 February 2015 Revised: 01 August 2015
Primary: 49L20, 49L25; Secondary: 35F21.

We revisit the pioneering work of Bressan & Hong on deterministic control problems in stratified domains, i.e. control problems for which the dynamic and the cost may have discontinuities on submanifolds of $\mathbb{R}^N$. By using slightly different methods, involving more partial differential equations arguments, we $(i)$ slightly improve the assumptions on the dynamic and the cost; $(ii)$ obtain a comparison result for general semi-continuous sub and supersolutions (without any continuity assumptions on the value function nor on the sub/supersolutions); $(iii)$ provide a general framework in which a stability result holds.
- Optimal control,
- discontinuous dynamic,
- stratified domains,
- Bellman equation,
- viscosity solutions
Citation: Guy Barles, Emmanuel Chasseigne. (Almost) Everything you always wanted to know about deterministic control problems in stratified domains[J]. Networks and Heterogeneous Media, 2015, 10(4): 809-836. doi: 10.3934/nhm.2015.10.809

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Abstract

We revisit the pioneering work of Bressan & Hong on deterministic control problems in stratified domains, i.e. control problems for which the dynamic and the cost may have discontinuities on submanifolds of $\mathbb{R}^N$. By using slightly different methods, involving more partial differential equations arguments, we $(i)$ slightly improve the assumptions on the dynamic and the cost; $(ii)$ obtain a comparison result for general semi-continuous sub and supersolutions (without any continuity assumptions on the value function nor on the sub/supersolutions); $(iii)$ provide a general framework in which a stability result holds.

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