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.

Classification: 35L65, 49K20, 93C20

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}, publisher = {EDP-Sciences}, volume = {17}, number = {2}, year = {2011}, pages = {353-379}, doi = {10.1051/cocv/2010007}, zbl = {1232.35176}, mrnumber = {2801323}, language = {en}, url = {http://www.numdam.org/item/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, Volume 17 (2011) no. 2, pp. 353-379. doi : 10.1051/cocv/2010007. http://www.numdam.org/item/COCV_2011__17_2_353_0/

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