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

We consider an infinite strip ΩL=(0,2πL)d−1×R, d≥2, L>0, and study the control problem of the heat equation on ΩL with Dirichlet or Neumann boundary conditions, and control set ω⊂ΩL. 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 ΩL” and implies that the set ω cannot be concentrated in a particular region of ΩL. 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.