Archiv der Mathematik

, Volume 111, Issue 1, pp 85–99 | Cite as

Sharp geometric condition for null-controllability of the heat equation on \(\mathbb {R}^d\) and consistent estimates on the control cost

  • Michela Egidi
  • Ivan VeselićEmail author


In this note we study the control problem for the heat equation on \(\mathbb {R}^d\), \(d\ge 1\), with control set \(\omega \subset \mathbb {R}^d\). We provide a necessary and sufficient condition (called \((\gamma , a)\)-thickness) on \(\omega \) such that the heat equation is null-controllable in any positive time. We give an estimate of the control cost with explicit dependency on the characteristic geometric parameters of the control set. Finally, we derive a control cost estimate for the heat equation on cubes with periodic, Dirichlet, or Neumann boundary conditions, where the control sets are again assumed to be thick. We show that the control cost estimate is consistent with the \(\mathbb {R}^d\) case.


Control theory Heat equation Observability Spectral inequality Control cost Thick sets 

Mathematics Subject Classification

35Q93 93Bxx 


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Technische Universität DortmundDortmundGermany

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