Local Null Controllability of a 1D Stefan Problem

Abstract

The purpose of this article is to give a new proof of a null controllability result for a 1D free-boundary problem of the Stefan kind for a heat PDE. We introduce a method based on local inversion that, in contrast with other previous arguments, does not rely on any compactness property and can be generalized to higher dimensions.

Controllability of One-Dimensional Viscous Free Boundary Flows

“…The controllability aspects of one-dimensional, parabolic freeboundary problems similar to (1.1) have been addressed in several recent works (see e.g. [13,17,16,19]). In [13,17], Fernández-Cara et al consider the one-phase Stefan problem…”

“…The conclusion for (1.4) follows by means of a Schauder fixed-point argument applied to the map −→ 0 − 0 v x (τ, (τ )) dτ in an appropriate subspace of C 1 ([0, T ]). In [16], the authors obtain the same local controllability result by means of a different technique, which relies on a transformation to a fixed domain, a linear controllability test and an inverse function argument. Our strategy of proof for the controllability of both states of (1.1) has some resemblance to that in [16], but with several important technical differences.…”

“…In [16], the authors obtain the same local controllability result by means of a different technique, which relies on a transformation to a fixed domain, a linear controllability test and an inverse function argument. Our strategy of proof for the controllability of both states of (1.1) has some resemblance to that in [16], but with several important technical differences. Moreover, with small adjustments, the control strategy we present herein also yields a local null-controllability result for both the solution and the free boundary of the Stefan problem (1.4), namely (T ) = * and v(T, •) = 0 in (0, * ) whenever v 0 and 0 − * are small enough.…”

“…Proof. The proof for estimating the first three norms follows standard energy estimate arguments, and we refer to [16,Lemma 3.4] for details. To obtain the weighted L ∞ (H 1 )-estimate, we note that by interpolation…”

Recovering the initial condition in the one-phase Stefan problem

“…The uniqueness of the inversion has been established by several authors under different smoothness assumptions on the boundary influx and the initial condition [24,4,6,14]. The problem can also be related to the controllability of parabolic or hyperbolic free-boundary problems [12,11,1,16,2].…”