Sharp estimates and homogenization of the control cost of the heat equation on large domains

¹ University of Zagreb, Faculty of Science, Department of Mathematics, Zagreb, Croatia.
² Queen Mary University of London, School of Mathematical Sciences, London, UK.
³ Technische Universität Chemnitz, Fakultät für Mathematik, Chemnitz, Germany.
⁴ Technische Universität Dortmund, Fakultät für Mathematik, Dortmund, Germany.

^* Corresponding author: martin.tautenhahn@mathematik.tu-chemnitz.de

Received: 17 December 2018
Accepted: 19 September 2019

Abstract

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.