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 equation, non-local flows
@article{COCV_2011__17_2_353_0,
author = {Colombo, Rinaldo M. and Herty, Michael and Mercier, Magali},
title = {Control of the continuity equation with a non local flow},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {353--379},
year = {2011},
publisher = {EDP Sciences},
volume = {17},
number = {2},
doi = {10.1051/cocv/2010007},
mrnumber = {2801323},
zbl = {1232.35176},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2010007/}
}
TY - JOUR AU - Colombo, Rinaldo M. AU - Herty, Michael AU - Mercier, Magali TI - Control of the continuity equation with a non local flow JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2011 SP - 353 EP - 379 VL - 17 IS - 2 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2010007/ DO - 10.1051/cocv/2010007 LA - en ID - COCV_2011__17_2_353_0 ER -
%0 Journal Article %A Colombo, Rinaldo M. %A Herty, Michael %A Mercier, Magali %T Control of the continuity equation with a non local flow %J ESAIM: Control, Optimisation and Calculus of Variations %D 2011 %P 353-379 %V 17 %N 2 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2010007/ %R 10.1051/cocv/2010007 %G en %F COCV_2011__17_2_353_0
Colombo, Rinaldo M.; Herty, Michael; Mercier, Magali. Control of the continuity equation with a non local flow. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 2, pp. 353-379. doi: 10.1051/cocv/2010007
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