We prove new bounds on the control cost for the abstract heat equation, assuming a spectral inequality or uncertainty relation for spectral projectors. In particular, we specify quantitatively how upper bounds on the control cost depend on the constants in the spectral inequality. This is then applied to the heat flow on bounded and unbounded domains modeled by a Schrödinger semigroup. This means that the heat evolution generator is allowed to contain a potential term. The observability/control set is assumed to obey an equidistribution or a thickness condition, depending on the context. Complementary lower bounds and examples show that our control cost estimates are sharp in certain asymptotic regimes. One of these is dubbed homogenization regime and corresponds to the situation where the control set becomes more and more evenly distributed throughout the domain while its density remains constant.
Mots-clés : Observability, null-controllability, spectral inequality, abstract heat equation, control cost, thick sets, homogenization, Schrödinger semigroup
@article{COCV_2020__26_1_A54_0,
author = {Naki\'c, Ivica and T\"aufer, Matthias and Tautenhahn, Martin and Veseli\'c, Ivan},
title = {Sharp estimates and homogenization of the control cost of the heat equation on large domains},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {26},
year = {2020},
doi = {10.1051/cocv/2019058},
mrnumber = {4145245},
zbl = {1451.35241},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2019058/}
}
TY - JOUR AU - Nakić, Ivica AU - Täufer, Matthias AU - Tautenhahn, Martin AU - Veselić, Ivan TI - Sharp estimates and homogenization of the control cost of the heat equation on large domains JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2019058/ DO - 10.1051/cocv/2019058 LA - en ID - COCV_2020__26_1_A54_0 ER -
%0 Journal Article %A Nakić, Ivica %A Täufer, Matthias %A Tautenhahn, Martin %A Veselić, Ivan %T Sharp estimates and homogenization of the control cost of the heat equation on large domains %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2019058/ %R 10.1051/cocv/2019058 %G en %F COCV_2020__26_1_A54_0
Nakić, Ivica; Täufer, Matthias; Tautenhahn, Martin; Veselić, Ivan. Sharp estimates and homogenization of the control cost of the heat equation on large domains. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 54. doi : 10.1051/cocv/2019058. http://www.numdam.org/articles/10.1051/cocv/2019058/
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