Null-controllability and control cost estimates for the heat equation on unbounded and large bounded domains

Abstract

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.

A uniform bound on costs of controlling semilinear heat equations on a sequence of increasing domains and its application

“…The question whether or not a given control system is null controllable and obtaining upper bounds for the associated null control costs, with respect to the time interval or control regions, are important topics in control theory, both for linear partial differential control equations and abstract linear control systems. We refer the reader to [1,3,9,10,17,18,25,26,28] and references therein for a wider discussion on this activated research field.…”

“…We say a set as an equidistributed set in R N if it contains a union of suitably distributed balls of fixed radius. Recently, there are many beautiful existing results on the quantitative unique continuation for general elliptic operators on equidistributed sets (c.f., [10,17,23] and references therein). Meanwhile, we will consider the varying domains Ω n as an approximation of R N , for the simplicity.…”

“…Recently, the authors in [23] (see also [10]) showed in a linear and abstract framework that if the sequence of null controls associated to an exhaustion of an unbounded domain converges, then the solutions converge in the same way to the limiting problem on the unbounded domain. This result allows to infer the null-controllability on unbounded domain by studying the control problem on a sequence of bounded domains.…”

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

“…Remark 1.7. For background in control theory suitable for our context we refer the reader for instance to [6,14]. In particular it is well known that by duality between observability and null-controllability, Theorem 1.6 implies null-controllability of the spherical heat equation…”

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

“…For Euclidean domains which can be exhausted by rectangles, i.e. half‐spaces and orthants, and for cones with angle , , null‐controllability of the heat equation with control region given by the intersection of a thick set and the domain, and corresponding control cost estimates have been established in [7, Section 2]. More generally, the authors show that if the heat equation on a domain in symmetric with respect to a hyperplane is null‐controllable, then so is the heat equation on the symmetric parts.…”