Consider a star-shaped network of strings. Each string is governed by the wave equation. At each boundary node of the network there is a player that performs Dirichlet boundary control action and in this way influences the system state. At the central node, the states are coupled by algebraic conditions in such a way that the energy is conserved. We consider the corresponding antagonistic game where each player minimizes a certain quadratic objective function that is given by the sum of a control cost and a tracking term for the final state. We prove that under suitable assumptions a unique Nash equilibrium exists and give an explicit representation of the equilibrium strategies.
Keywords: Vibrating string, boundary control, network, Nash equilibrium, game, pipeline network, gas transport
Gugat, Martin 1 ; Steffensen, Sonja 1
@article{COCV_2018__24_4_1789_0,
author = {Gugat, Martin and Steffensen, Sonja},
title = {Dynamic boundary control games with networks of strings},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1789--1813},
year = {2018},
publisher = {EDP Sciences},
volume = {24},
number = {4},
doi = {10.1051/cocv/2017082},
zbl = {1415.49026},
mrnumber = {3922441},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2017082/}
}
TY - JOUR AU - Gugat, Martin AU - Steffensen, Sonja TI - Dynamic boundary control games with networks of strings JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 1789 EP - 1813 VL - 24 IS - 4 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2017082/ DO - 10.1051/cocv/2017082 LA - en ID - COCV_2018__24_4_1789_0 ER -
%0 Journal Article %A Gugat, Martin %A Steffensen, Sonja %T Dynamic boundary control games with networks of strings %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 1789-1813 %V 24 %N 4 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2017082/ %R 10.1051/cocv/2017082 %G en %F COCV_2018__24_4_1789_0
Gugat, Martin; Steffensen, Sonja. Dynamic boundary control games with networks of strings. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1789-1813. doi: 10.1051/cocv/2017082
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