A relaxation result for state constrained inclusions in infinite dimension

Helene  Frankowska; Elsa M. Marchini; Marco  Mazzola

doi:10.3934/mcrf.2016.6.113

Article Contents

2016, Volume 6, Issue 1: 113-141. Doi: 10.3934/mcrf.2016.6.113

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A relaxation result for state constrained inclusions in infinite dimension

1.
CNRS, Institut de Mathématiques de Jussieu - Paris Rive Gauche, Sorbonne Universités, UPMC Univ Paris 06, Univ Paris Diderot, Sorbonne Paris Cité, Case 247, 4 Place Jussieu, 75252 Paris
2.
Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano
3.
UPMC Univ Paris 06, Institut de Mathématiques de Jussieu - Paris Rive Gauche, Sorbonne Universités, CNRS, Univ Paris Diderot, Sorbonne Paris Cité, Case 247, 4 Place Jussieu, 75252 Paris

Received: November 30, 2014

Revised: June 30, 2015

Published: February 2016

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Abstract

In this paper we consider a state constrained differential inclusion $\dot x\in \mathbb A x+ F(t,x)$, with $\mathbb A$ generator of a strongly continuous semigroup in an infinite dimensional separable Banach space. Under an ``inward pointing condition'' we prove a relaxation result stating that the set of trajectories lying in the interior of the constraint is dense in the set of constrained trajectories of the convexified inclusion $\dot x\in \mathbb A x+ \overline{\textrm{co}}F(t,x)$. Some applications to control problems involving PDEs are given.

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Mathematics Subject Classification: 34G25, 34K09, 49J45.

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