We consider an optimal control on networks in the spirit of the works of Achdou et al. [NoDEA Nonlinear Differ. Equ. Appl. 20 (2013) 413–445] and Imbert et al. [ESAIM: COCV 19 (2013) 129–166]. The main new feature is that there are entry (or exit) costs at the edges of the network leading to a possible discontinuous value function. We characterize the value function as the unique viscosity solution of a new Hamilton-Jacobi system. The uniqueness is a consequence of a comparison principle for which we give two different proofs, one with arguments from the theory of optimal control inspired by Achdou et al. [ESAIM: COCV 21 (2015) 876–899] and one based on partial differential equations techniques inspired by a recent work of Lions and Souganidis [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27 (2016) 535–545].
Accepté le :
DOI : 10.1051/cocv/2018003
Mots-clés : Optimal control, networks, Hamilton-Jacobi equation, viscosity solutions, uniqueness, switching cost
@article{COCV_2019__25__A15_0,
author = {Dao, Manh Khang},
title = {Hamilton-Jacobi equations for optimal control on networks with entry or exit costs},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {25},
year = {2019},
doi = {10.1051/cocv/2018003},
zbl = {1437.49041},
mrnumber = {3963664},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2018003/}
}
TY - JOUR AU - Dao, Manh Khang TI - Hamilton-Jacobi equations for optimal control on networks with entry or exit costs JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2018003/ DO - 10.1051/cocv/2018003 LA - en ID - COCV_2019__25__A15_0 ER -
%0 Journal Article %A Dao, Manh Khang %T Hamilton-Jacobi equations for optimal control on networks with entry or exit costs %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2018003/ %R 10.1051/cocv/2018003 %G en %F COCV_2019__25__A15_0
Dao, Manh Khang. Hamilton-Jacobi equations for optimal control on networks with entry or exit costs. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 15. doi : 10.1051/cocv/2018003. http://www.numdam.org/articles/10.1051/cocv/2018003/
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