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

Abstract

We consider an infinite strip normalΩL=(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⊂normalΩL. 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.

Spherical Logvinenko-Sereda-Kovrijkine type inequality and null-controllability of the heat equation on the sphere

“…• if M is the torus, a cube, or an infinite strip and H = −∆ is an appropriate self-adjoint realization of the Laplacian [9,5]; 16,4] (and [3] for a treatment of the partial harmonic oscillator); • if the domain M admits an appropriate covering and H is an operator such that every function in the range of the spectral projections satisfies a Bernstein-type inequality [7]. In the present paper we adapt Kovrijkine's approach to prove a spectral inequality for the Laplace-Beltrami operator on the sphere of radius R > 0, that is…”

The Reflection Principle in the Control Problem of the Heat Equation

“…As far as bounds on the control cost are concerned, one classically asks for the type of time dependency, which has been largely explored in [5,10,11,16,18,19,21,27], see also the references therein. However, the dependency of the control cost on geometric parameters of and has only recently been addressed in [6,7,9,19], see also [17] for previous results. Especially the work [19] focused on these dependencies, which have been exploited in several asymptotic regimes there in order to discuss homogenization.…”

“…However, the considerations from [26] currently allow to apply this approach only in the case of Dirichlet boundary conditions. Remark 2.9 In the recent work [6], it has been shown that the system (2.1) on the strip 1 is null-controllable in any time 0 if and only if is a thick set (which can be arbitrarily changed outside the strip), and an explicit control cost bound has been provided.…”