Sharp estimates and homogenization of the control cost of the heat equation on large 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 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 that the control set becomes more and more evenly distributed throughout the domain while its density remains constant.

arXiv comment28 pages, 3 figures

References in corpus (5)

• Sharp geometric condition for null-controllability of the heat equation on ${\mathbb{R}}^d$ and consistent estimates on the control cost
• How violent are fast controls for Schroedinger and plate vibrations ?
• Sufficient criteria and sharp geometric conditions for observability in Banach spaces
• Scale-free unique continuation estimates and Logvinenko-Sereda Theorems on the torus
• Observability of the heat equation, geometric constants in control theory, and a conjecture of Luc Miller

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• The reflection principle in the control problem of the heat equation
• Final state observability estimates and cost-uniform approximate null-controllability for bi-continuous semigroups
• Control problem for quadratic parabolic differential equations with sparse sensor sets of finite volume or anisotropically decaying density
• Quantitative unique continuation for spectral subspaces of Schrödinger operators with singular potentials
• Unique continuation estimates for Baouendi--Grushin equations on cylinders
• Sturm-Liouville Problems And Global Bounds By Small Control Sets And applications to quantum graphs
• A unified observability result for non-autonomous observation problems