On null‐controllability of the heat equation on infinite strips and control cost estimate

We consider an infinite stripΩ L = ( 0 , 2 π L ) d − 1 × R , d ≥ 2 , L > 0 , and study the control problem of the heat equation on Ω L with Dirichlet or Neumann boundary conditions, and control set ω ⊂ Ω L . We provide a sufficient and necessary condition for null‐controllability in any positive time T > 0 , which is a geometric condition on the control set ω. This is referred to as “thickness with respect to Ω L ” and implies that the set ω cannot be concentrated in a particular region of Ω L . We compare the thickness condition with a previously known necessity condition for null‐controllability and give a control cost estimate which only shows dependence on the geometric parameters of ω and the time T .