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Control of the continuity equation with a non local flow

Published online by Cambridge University Press:  24 March 2010

Rinaldo M. Colombo ,
Michael Herty  and
Magali Mercier
Rinaldo M. Colombo
Affiliation:
Department of Mathematics, Brescia University, Via Branze 38, 25133 Brescia, Italy. rinaldo@ing.unibs.it
Michael Herty
Affiliation:
RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany.
Magali Mercier
Affiliation:
Université de Lyon, Université Lyon 1, École Centrale de Lyon, INSA de Lyon, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 Novembre 1918, 69622 Villeurbanne Cedex, France.
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Abstract

This paper focuses on the analytical properties of the solutions to the continuity equation with non local flow. Our driving examples are a supply chain model and an equation for the description of pedestrian flows. To this aim, we prove the well posedness of weak entropy solutions in a class of equations comprising these models. Then, under further regularity conditions, we prove the differentiability of solutions with respect to the initial datum and characterize this derivative. A necessary condition for the optimality of suitable integral functionals then follows.

Keywords

Optimal control of the continuity equationnon-local flows
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 2 , April 2011 , pp. 353 - 379
Copyright
© EDP Sciences, SMAI, 2010

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