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

Abstract

We consider an infinite strip $Ω_L=(0,2πL)^{d-1}\times\mathbb{R}$, $d\geq 2$, $L>0$, and study the control problem of the heat equation on $Ω_L$ with Dirichlet or Neumann boundary conditions, and control set $ω\subsetΩ_L$. 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 $T$.