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

TL;DR

This paper establishes new bounds on the control cost for the heat equation on large domains, linking spectral inequalities to control efficiency, with applications to homogenization and unbounded domains.

Contribution

It provides quantitative bounds on control costs based on spectral inequalities, extending to unbounded domains and including homogenization effects.

Findings

Control cost bounds depend explicitly on spectral inequality constants.

Results are sharp in certain asymptotic regimes, including homogenization.

Application to heat flow with potential on large and unbounded domains.

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 Schr\"odinger 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 that the control set becomes more and more evenly distributed throughout the domain while its density remains constant.