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 refered 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 .
We provide an abstract framework for a Logvinenko-Sereda type theorem, where the classical compactness assumption on the support of the Fourier transform is replaced by the assumption that the functions under consideration belong to a spectral subspace associated with a finite energy interval for some lower semibounded self-adjoint operator on a Euclidean L 2-space. Our result then provides a bound for the L 2-norm of such functions in terms of their L 2-norm on a thick subset with a constant explicit in the geometric and spectral parameters. This recovers previous results for functions on the whole space, hyperrectangles, and infinite strips with compact Fourier support and for finite linear combinations of Hermite functions and allows to extend them to other domains. The proof follows the approach by Kovrijkine and is based on Bernsteintype inequalities for the respective functions, complemented with a suitable covering of the underlying domain.
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