On the null-controllability of the heat equation in unbounded domains

By Luc Miller. 2004

68 citations recorded by the source corpus.

We make two remarks about the null-controllability of the heat equation with Dirichlet condition in unbounded domains. Firstly, we give a geometric necessary condition (for interior null-controllability in the Euclidean setting) which implies that one cannot go infinitely far away from the control region without tending to the boundary (if any), but also applies when the distance to the control region is bounded. The proof builds on heat kernel estimates. Secondly, we describe a class of null-controllable heat equations on unbounded product domains. Elementary examples include an infinite strip in the plane controlled from one boundary and an infinite rod controlled from an internal infinite rod. The proof combines earlier results on compact manifolds with a new lemma saying that the null-controllability of an abstract control system and its null-controllability cost are not changed by taking its tensor product with a system generated by a non-positive self-adjoint operator.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/1f5af8f53d0630a9a7039d15f10a874e1eba123d

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In Computational Theory and Mathematics

In Control and Systems Engineering

Citations in the corpus

DEGENERATE PARABOLIC OPERATORS OF KOLMOGOROV TYPE WITH A GEOMETRIC CONTROL CONDITION (2015, background): “1 Null controllability of the heat equation The null and approximate controllabilities of the heat equation are essentially well understood subjects for both linear and semilinear equations, for bounded or unbounded domains [3, 27, 30, 32, 33, 34, 37, 44, 46, 48, 51, 52, 62, 63] and also with discon”
Null controllability of Grushin-type operators in dimension two (2011, background): “The null and approximate controllability of the heat equation are essentially well understood subjects for both linear and semilinear equations, and for bounded or unbounded domains (see, for instance, [15], [19], [21], [22], [23], [27], [31], [32], [36], [39], [40], [45], [46]).”
Some results on controllability for linear and nonlinear heat equations in unbounded domains (2007, methodology): “Secondly, by means of examples 4 and 5, we generalize the positive results on null controllability of the heat equation obtained in [21].”
Carleman estimates for one-dimensional degenerate heat equations (2006, background): “Controllability properties of nondegenerate parabolic equations have been widely studied in recent years; concerning the heat equation, see in particular [2, 5, 10, 13, 14, 15, 16, 17, 19, 24, 25, 27, 28, 29, 32, 33]; concerning the Stokes and Navier-Stokes equations, see in particular [3, 18, 19, 2”
Observable set, observability, interpolation inequality and spectral inequality for the heat equation in Rn (2017, background): “For the heat equation in an unbounded domain, [37] imposed a condition, in terms of the Gaussian kernel, on the set ω so that the observability inequality does not hold.”
Controllability and Observability of Partial Differential Equations: Some Results and Open Problems (2007, result): “Similar results hold also, roughly, when the uncontrolled domain is unbounded but the distance to the controlled region is uniformly bounded ([22], [151]).”
On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations (2012, background): “In fact, global Carleman estimates yield a precise knowledge of the “cost” of the control function in the linear 1The problem of null-controllability of the heat equation in the case where Ω is unbounded is entirely different [43,44,46].”