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

We consider an infinite strip $\Omega_L=(0,2\pi L)^{d-1}\times\mathbb{R}$, $d\geq 2$, $L>0$, and study the control problem of the heat equation on $\Omega_L$ with Dirichlet or Neumann boundary conditions, and control set $\omega\subset\Omega_L$. 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 $\omega$. This is referred to as "thickness with respect to $\Omega_L$" and implies that the set $\omega$ cannot be concentrated in a particular region of $\Omega_L$. 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 $\omega$ and the time $T$.