On the lack of null-controllability of the heat equation on the half-line

Abstract

Abstract. We consider the linear heat equation on the half-line with a Dirichlet boundary control. We analyze the null-controllability problem. More precisely, we study the class of initial data that may be driven to zero in finite time by means of an appropriate choice of the L 2 boundary control. We rewrite the system on the similarity variables that are a common tool when analyzing asymptotic problems. Next, the control problem is reduced to a moment problem which turns out to be critical since it concerns the family of real exponentials {e jt } j≥1 in which the usual summability condition on the inverses of the eigenvalues does not hold. Roughly speaking, we prove that controllable data have Fourier coefficients that grow exponentially for large frequencies. This result is in contrast with the existing ones for bounded domains that guarantee that every initial datum belonging to a Sobolev space of negative order may be driven to zero in an arbitrarily small time.

Carleman Estimates for a Class of Degenerate Parabolic Operators

“…The theory has also been extended to semilinear problems (see, for example, [2,3,12,15,21,24,25]) and to equations in unbounded domains (see, for example, [13,33,34]; see also [31,41]). For the Stokes and Navier-Stokes equations we also refer the reader to [4,10,11,18,22,23,26,27,28].…”

Backward uniqueness results for some parabolic equations in an infinite rod

“…Regarding controllability aspects, Theorem 1.1 is strongly linked to the work [30], which proves that the solution of (1) cannot be controlled to zero when the initial datum y 0 belongs to the weighted Sobolev space L 2 (R * + , exp(x 2 /4)dx) (and y 0 ≡ 0). Such problem has also been studied from the observability point of view, which is a dual property of the controllability one.…”

“…Concerning Theorem 1.2. The controllability properties of (2) were analyzed in [30] for initial data y 0 ∈ L 2 (R * + ), and it was shown that there is no non-trivial initial condition in L 2 (R * + ) which can be driven to 0 with control functions u ∈ L 2 (0, T ) (in fact, [30] focuses of the equation…”

Exact null internal controllability for the heat equation on unbounded convex domains

“…As regards the case where O is unbounded, which is the main objective of this work, this remained largely open. In some special cases, however, (O half-space), the boundary controllability was discussed in [14], [15].…”