Remarks on the control of two-phase Stefan free-boundary problems

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. SIAM J. CONTROL OPTIM.©2022 Society for Industrial and Applied Mathematics Vol. 60, No. 5, pp. 3078--3099 REMARKS ON THE CONTROL OF TWO-PHASE STEFAN FREE-BOUNDARY PROBLEMS\ast RAUL K. C. ARA\' UJO\dagger , ENRIQUE FERN\' ANDEZ-CARA\ddagger , JUAN L\' IMACO\S ,AND DIEGO A. SOUZA\P Abstract. This paper concerns the null controllability of the two-phase 1D Stefan problem with distributed controls. This is a free-boundary problem that models solidification or melting processes. In each phase, a parabolic equation, completed with initial and boundary conditions, must be satisfied; the phases are separated by a phase change interface, where an additional freeboundary condition is imposed (the so-called Stefan condition). We assume that two localized sources of heating/cooling controls act on the system (one in each phase). We prove the following local null controllability result: the temperatures can be steered to zero and, simultaneously, the interface can be steered to a prescribed location provided the initial data and the interface position are sufficiently close to the targets. The ingredients of the proofs are a compactness-uniqueness argument (which gives appropriate observability estimates adapted to constraints) and a fixed-point formulation and resolution of the controllability problem (which gives the result for the nonlinear system). We also prove a negative result corresponding to the case where only one control acts on the system and the interface does not collapse to the boundary. Key words. free-boundary problems, two-phase Stefan problem, null controllability, Carleman inequalities, constrained observability inequalities MSC codes. 35R35, 80A22, 93B05, 93C20 DOI. 10.1137/21M1402261 1. Introduction. The two-phase Stefan problem is a mathematical model, i.e., a coupled system composed of two PDEs and one ODE, used to describe liquid-solid phase transition. This physical phenomenon is frequently found in many processes in science and engineering, for example, continuous casting of steel [2], cryosurgical treatment of cancer [28], analysis of crystal growth [5], and design of lithium-ion batteries [4]. It is also important to highlight that, besides their use in thermodynamics processes, similar systems can be used to model other phenomena, such as analysis and computation of the flux of a fluid on a free surface [32, 21, 29], fluid-solid interaction [7, 26, 30], gas flow through a porous medium [1, 9, 31], and tumor growth [19, 18]. Let us recall the mathematical formulation of the two-phase 1D Stefan problem, and let us formulate the related control problem considered in this paper. \ast Received by the editors March 2, 2021; accepted for publication (in revised form) July 15, 2022; published electronically October 11, 2022. https://doi.org/10.1137/21M1402261 Funding: The first author was partially supported by the project CAPES-MathAmSud: ACIPDE. The second and fourth authors were partially supported by grant PID2020-114976GB- I00 funded by MCIN/AEI/10.13039/501100011033. The fourth author was also partially supported by grant IJC2018-037863-I funded by MCIN/AEI/10.13039/501100011033. \dagger Department of Mathematics, Federal University of Pernambuco, 50740-545 Recife, Brazil, and Institute de Math\'ematiques de Bordeaux, Universit\'e de Bordeaux 351, cours de la Lib\'eration - F 33405 Talence, France ([email protected]e.br). \ddagger Department of Numerical Analysis and Differential Equations, University of Seville, 41080 Seville, Spain, and Department of Mathematics, Federal University of Para\'{\i}ba, UFPB, CEP 58050-085, Jo\~ao Pessoa-PB, Brazil ([email protected]). \S IME, Universidade Federal Fluminense, Niter\'oi 24020140, Brazil ([email protected]). \P Department of Numerical Analysis and Differential Equations, University of Seville, 41080 Seville, Spain ([email protected]). 3078 Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. CONTROL OF TWO-PHASE STEFAN PROBLEMS 3079 Let L > 0, T > 0,and \ell l, \ell 0, \ell r\in (0, L) be given with \ell l< \ell 0< \ell r. We also consider two functions, u0\in H1 0(0, \ell 0) with u0\geq 0 and v0\in H1 0(\ell 0, L) with v0\leq 0, and two open sets, \omega l\subset \subset (0, \ell l) and \omega r\subset \subset (\ell r, L). At each t, the material domain is separated into two parts: x\in [0, \ell (t)) (the liquid phase) and x\in (\ell (t), L] (the solid phase). Here, \ell =\ell (t) is the position of the interface between liquid and solid phases; it satisfies \ell (0) = \ell 0and \ell (t)\in (\ell l, \ell r) for all t. The aim of this paper is to study the controllability properties of the following two-phase Stefan problem: (1) \left\{ ut - dluxx =hl1\omega lin Ql, vt - drvxx =hr1\omega rin Qr, u(0, t) = 0 in (0, T), v(L, t) = 0 in (0, T), u(\cdot ,0) = u0in (0, \ell 0), v(\cdot ,0) = v0in (\ell 0, L), u(\ell (t), t) = v(\ell (t), t) = 0 in (0, T), - \ell \prime (t) = dlux(\ell (t), t) - drvx(\ell (t), t) in (0, T). Here and in what follows, dland drmust be viewed as diffusion coefficients, and we use the notation \left\{ Q:= (0, L)\times (0, T), Ql:= \{ (x, t)\in Q:t\in (0, T), x \in (0, \ell (t))\} , Qr:= \{ (x, t)\in Q:t\in (0, T), x \in (\ell (t), L)\} , \scrO l=\omega l\times (0, T),and \scrO r=\omega r\times (0, T). The main result in this paper is the following. Theorem 1. Let \ell T\in (\ell l, \ell r). Then there exists \delta > 0such that, for any u0\in H1 0(0, \ell 0)with u0\geq 0, any v0\in H1 0(\ell 0, L)with v0\leq 0,and any \ell 0\in (\ell l, \ell r)satisfying \| u0\| H1 0(0,\ell 0)+\| v0\| H1 0(\ell 0,L)+| \ell 0 - \ell T| \leq \delta , there exist controls (hl, hr)\in L2(\scrO l)\times L2(\scrO r)and associated states (u, v, \ell )with \Biggl\{ \ell \in H1(0, T)\cap C1((0, T]), \ell (t)\in (\ell l, \ell r)\forall t\in [0, T], u, ux, ut, uxx \in L2(Ql)and v, vx, vt, vxx \in L2(Qr), such that (2) \ell (T) = \ell T, u(\cdot , T)=0 in (0, \ell T),and v(\cdot , T)=0 in (\ell T, L). Remark 1. We will see in section 5.1 that the maximum principle for parabolic equations implies that null controllability cannot hold if one of the controls (for instance, hr) vanishes and the interface satisfies 0 < \ell (T)< L. However, the possibility of getting a null control result with only one control when one of the phases is allowed to collapse to the boundary, that is, \ell (T) = Lor \ell (T) = 0, is open. For completeness, let us mention some previous works on the control of (1) and similar models. Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 3080 ARA\' UJO, FERN\' ANDEZ-CARA, L\' IMACO, AND SOUZA The analysis of the controllability properties for linear and nonlinear parabolic PDEs defined in cylindrical domains is a classical problem in control theory, and some of the main contributions are in the references [8, 10, 16, 20, 25]. On the other hand, the study of the controllability properties of free-boundary problems for PDEs has not been much explored, although some important results have been obtained in the past few years, especially for one-phase Stefan problems and variants; see [6, 13, 14, 12]. In [22], the authors study the controllability problem for the free-boundary viscous Burgers equation with one moving end point. Regarding the two-phase Stefan problem, the best results to our knowledge concern stabilization. More precisely, it is proved in [23] that, under some assumptions, there exist Neumann boundary controls and associated states (u, v, \ell ) defined for all t > 0 such that lim t\rightarrow \infty \| u(\cdot , t) - \scrT m\| L2(0,\ell (t)) = lim t\rightarrow \infty \| v(\cdot , t) - \scrT m\| L2(0,\ell (t)) = 0 and lim t\rightarrow \infty \ell (t) = \ell T, where \scrT mis a melting/solidification temperature. A natural question is whether or not it is possible to drive both the temperature and the interface to prescribed targets at a finite time. In this paper we give a positive partial answer to this question. Recall that, in [7, 15, 26], a similar problem was considered for a 1D fluid-structure problem, with the following equations on the interface: u(\ell (t), t) = v(\ell (t), t) = \ell \prime (t), vx(\ell (t), t) - ux(\ell (t), t) = m\ell \prime \prime (t) for t\in (0, T). In contrast to previous works on free-boundary controllability, in this paper we deal with situations leading to new difficulties. Let us discuss some of these differences: \bullet Control of two phases. Obviously, the fact that we model a two-phase transition process greatly complicates the structure and properties of the state system and requires an appropriate analysis. \bullet Control of the interface. The aim is to control not only the temperature on both sides but also the interface between liquid and solid regions. This will bring an extra difficulty. The main strategy will rely on linearization, then reformulation as a constrained observability problem, and then resolution of a fixed-point equation. The rest of this paper is organized as follows. In section 2.1, we will reformulate the free-boundary problem as a nonlinear parabolic system in a cylindrical domain. In section 3, we will present an improved observability inequality, which leads to the null controllability for a related linearized system subject to a linear constraint. In section 4, we will give a proof of Theorem 1. To this end, we will apply a fixedpoint argument. Finally, in section 5, we will present some additional comments and questions. 2. Preliminaries. 2.1. Reformulation of the free-boundary problem. First, let us find a suitable diffeomorphism \Phi that transforms the free-boundary problem for the parabolic system (1) into an equivalent problem for a nonlinear parabolic system in a cylindrical domain. To do this, let us fix a function \ell \in H1(0, T)\cap C1((0, T]) such that \ell (t)\in (\ell l, \ell r) for all t\in [0, T],and let us take \sigma > 0 sufficiently small such that \ell l+\sigma < \ell (t) - \sigma and \ell (t) + \sigma < \ell r - \sigma in [0, T]. Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. CONTROL OF TWO-PHASE STEFAN PROBLEMS 3081 Then, for any \ell l+ 2\sigma < y < \ell r - 2\sigma , we build a function m(\cdot , y) : \BbbR \rightarrow \BbbR by linear interpolation of the points (\ell l - \sigma , \ell l - \sigma ), (\ell l+\sigma , \ell l+\sigma ), (y - \sigma , \ell 0 - \sigma ), (y+\sigma , \ell 0+\sigma ), (\ell r - \sigma , \ell r - \sigma ),and (\ell r+\sigma , \ell r+\sigma ) and then extend the extreme segments toward infinity. Specifically, we have the following definition for m(\cdot , y): m(x, y) := \left\{ xif x\leq \ell l+\sigma , \ell l+\sigma +(\ell l - \ell 0+ 2\sigma )(x - \ell l - \sigma ) \ell l+ 2\sigma - yif \ell l+\sigma < x < y - \sigma , x - y+\ell 0if y - \sigma < x < y +\sigma , \ell 0+\sigma +(\ell r - \ell 0 - 2\sigma )(x - y - \sigma ) \ell r - y - 2\sigma if y+\sigma < x < \ell r - \sigma , xif x\geq \ell r - \sigma . Let us now consider a function \eta \in C\infty (\BbbR ) such that supp \eta \subset ( - \sigma , \sigma )\int \sigma - \sigma \eta (x)d x = 1 and \eta (x) = \eta ( - x)\forall x\in \BbbR . Then, we can define a smooth function G:\BbbR \times (\ell l+ 2\sigma , \ell r - 2\sigma )\mapsto \rightarrow \BbbR as follows: G(x, y) := [\eta \ast m(\cdot , y)](x). A simple computation leads to the equalities (3) G(x, \ell 0) = x\forall x\in \BbbR , G(y, y) = \ell 0, \partial xG(y, y) = 1, and \nabla G(x, y) = ([\eta \prime \ast m(\cdot , y)](x),[\eta \ast \partial ym(\cdot , y)](x)) , where \partial ym(x, y) := \left\{ 0 if x\leq \ell l+\sigma , (\ell l - \ell 0+ 2\sigma )(x - \ell l - \sigma ) (\ell l+ 2\sigma - y)2if \ell l+\sigma < x < y - \sigma , - 1 if y - \sigma < x < y +\sigma , (\ell r - \ell 0 - 2\sigma )(x - \ell r+\sigma ) (\ell r - y - 2\sigma )2if y+\sigma < x < \ell r - \sigma , 0 if x\geq \ell r - \sigma . Let us introduce the mapping \Phi : Q\mapsto \rightarrow Q, with \Phi (x, t) := (G(x, \ell (t)), t). It can be seen that \Phi is a diffeomorphism in Q; it coincides with the identity in the regions (0, \ell l+\sigma )\times (0, T) and (\ell r - \sigma , L)\times (0, T ),and, moreover, \Phi (\ell (t), t) = (\ell 0, t) for all t\in [0, T]. Let us introduce the sets Q0,l := (0, \ell 0)\times (0, T ) and Q0,r := (\ell 0, L)\times (0, T),and let us define p:Q0,l \rightarrow \BbbR and q:Q0,r \rightarrow \BbbR , with p(\xi , t) := u(x, t) = u(\Phi - 1(\xi , t)) and q(\xi , t) := v(x, t) = v(\Phi - 1(\xi , t)), where (\xi , t) := \Phi (x, t). Then, we have that the pair (p, q) satisfies (4) \left\{ pt - d\ell lp\xi \xi +b\ell lp\xi =hl1\omega lin Q0,l, qt - d\ell rq\xi \xi +b\ell rq\xi =hr1\omega rin Q0,r, p(0,\cdot ) = q(L, \cdot ) = 0 in (0, T), p(\cdot ,0) = p0in (0, \ell 0), q(\cdot ,0) = q0in (\ell 0, L), p(\ell 0,\cdot ) = q(\ell 0,\cdot ) = 0 in (0, T), dlp\xi (\ell 0, t) - drq\xi (\ell 0, t) = - \ell \prime (t) in (0, T), Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 3082 ARA\' UJO, FERN\' ANDEZ-CARA, L\' IMACO, AND SOUZA where p0:= u0\circ [G(\cdot , \ell 0)] - 1=u0\in H1 0(0, \ell 0), q0:= v0\circ [G(\cdot , \ell 0)] - 1=v0\in H1 0(\ell 0, L), and (5) d\ell l(\cdot , t) := dl\bigl( Gx\bigl( [G(\cdot , \ell (t))] - 1, \ell (t)\bigr) \bigr) 2, d\ell r(\cdot , t) := dr\bigl( Gx\bigl( [G(\cdot , \ell (t))] - 1, \ell (t)\bigr) \bigr) 2, b\ell l(\cdot , t) := Gy\bigl( [G(\cdot , \ell (t))] - 1, \ell (t)\bigr) \ell \prime (t) + dlGxx \bigl( [G(\cdot , \ell (t))] - 1, \ell (t)\bigr) , b\ell r(\cdot , t) := Gy\bigl( [G(\cdot , \ell (t))] - 1, \ell (t)\bigr) \ell \prime (t) + drGxx \bigl( [G(\cdot , \ell (t))] - 1, \ell (t)\bigr) . Remark 2. Since \ell \in H1(0, T), it is not difficult to deduce that (d\ell l, d\ell r)\in L\infty (Q0,l)\times L\infty (Q0,r) and (b\ell l, b\ell r)\in L2(0, T;L\infty (0, \ell 0))\times L2(0, T;L\infty (\ell 0, L)). Moreover, there exist constants K1, K2>0, independent of \ell and T, such that \| (b\ell l, b\ell r)\| L2(L\infty )\times L2(L\infty )\leq K1\| \ell \prime \| L2(0,T )+K2T. This way, we have that Theorem 1 is equivalent to proving a local controllability result for (4). Actually, we will prove the following. Theorem 2. Let \ell T\in (\ell l, \ell r). Then there exists \delta > 0such that, for any p0\in H1 0(0, \ell 0)with p0\geq 0, any q0\in H1 0(\ell 0, L)with q0\leq 0,and any \ell 0\in (\ell l, \ell r)satisfying \| p0\| H1 0(0,\ell 0)+\| q0\| H1 0(\ell 0,L)+| \ell 0 - \ell T| < \delta , there exist controls (hl, hr)\in L2(\scrO l)\times L2(\scrO r)and associated solutions (p, q, \ell )to (4) with \Biggl\{ \ell \in H1(0, T)\cap C1((0, T]), \ell (t)\in (\ell l, \ell r)\forall t\in [0, T], p, p\xi , pt, p\xi \xi \in L2(Q0,l)and q, q\xi , qt, q\xi \xi \in L2(Q0,r), such that \ell (T) = \ell T, p(\cdot , T)=0 in (0, \ell 0),and q(\cdot , T)=0 in (\ell 0, L). 2.2. Well-posedness of the two-phase free-boundary problem. The aim of this section is to prove the local existence and uniqueness for the two-phase freeboundary problem (1). More precisely, we have the following result. Proposition 1. Let L, T > 0and \ell l< \ell 0< \ell rbe given. Then, the system (1) is locally well-posed. In other words, for any (hl, hr)\in L2(\scrO l)\times L2(\scrO r)and (u0, v0)\in H1 0(0, \ell 0)\times H1 0(\ell 0, L), there exist a time 0<\widehat T\leq Tand a unique strong solution to (1) in the time interval (0,\widehat T)such that \Biggl\{ \ell \in H1(0,\widehat T), \ell (0) = \ell 0, \ell (t)\in (\ell l, \ell r)\forall t\in [0,\widehat T], u, ux, ut, uxx \in L2(\widehat Ql)and v, vx, vt, vxx \in L2(\widehat Qr), where \widehat Ql:= \{ (x, t)\in Q:t\in (0,\widehat T), x \in (0, \ell (t))\} and \widehat Qr:= \{ (x, t)\in Q:t\in (0,\widehat T), x \in (\ell (t), L)\} . Thanks to the diffeomorphism \Phi : Q\mapsto \rightarrow Q, introduced in section 2.1, Proposition 1 is equivalent to the local existence and uniqueness of (4). More precisely, Proposition 1 is an immediate consequence of the following result. Proposition 2. Let the conditions of Proposition 1be satisfied. Then, the nonlinear system (4) is locally well-posed; i.e., if (hl, hr)\in L2(\scrO l)\times L2(\scrO r)and (p0, q0)\in Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. CONTROL OF TWO-PHASE STEFAN PROBLEMS 3083 H1 0(0, \ell 0)\times H1 0(\ell 0, L)are fixed, there exist \widehat T\in (0, T)and a unique strong solution in (0,\widehat T)such that \Biggl\{ \ell \in H1(0,\widehat T), \ell (0) = \ell 0, \ell (t)\in (\ell l, \ell r)\forall t\in [0,\widehat T], p, p\xi , pt, p\xi \xi \in L2(\widehat Q0,l)and q, q\xi , qt, q\xi \xi \in L2(\widehat Q0,r), where \widehat Q0,l := (0, \ell 0)\times (0,\widehat T)and \widehat Q0,r := (\ell 0, L)\times (0,\widehat T). Proof. First, let us introduce the spaces XT l:=L2(0,T ;H2(0,\ell 0))\cap H1(0,T;L2(0,\ell 0)) and XT r:=L2(0,T ;H2(\ell 0,L))\cap H1(0,T;L2(\ell 0,L)). For each fixed \ell \in H1(0, T ), we can use the Faedo--Galerkin method to get a unique strong solution (p\ell , q\ell )\in XT l\times XT rto (4)1--(4)6. Moreover, thanks to Remark 2, we find a positive constant C1, independent of p0, q0, hl, hr, \ell , and T, such that (6) \| (p, q)\| 2 XT l\times XT r\leq C1\Bigl( 1 + K+\surd T\Bigr) \Bigl[ 1 + \Bigl( K+\surd T\Bigr) K+T\Bigr] eC1(1+\surd T)(K+\surd T)\Pi 2, where K:= \| \ell \prime \| L2(0,T )and \Pi := \| (p0, q0)\| H1 0\times H1 0+\| (hl, hr)\| L2\times L2. Now, let us assume that \ell l<\widehat \ell l< \ell 0<\widehat \ell r< \ell rand R > 0,and let us introduce the set \scrA R,T := \{ \ell \in H1(0, T) : \widehat \ell l\leq \ell (t)\leq \widehat \ell r\forall t\in [0, T], \ell (0) = \ell 0,\| \ell \prime \| L2(0,T )\leq R\} and the mapping \Lambda : \scrA R,T \mapsto \rightarrow H1(0, T), with \Lambda (\ell ) = \scrL \ell and \scrL \ell (t) := \ell 0 - \int t 0\bigl[ dlp\ell \xi (\ell 0, \tau ) - drq\ell \xi (\ell 0, \tau )\bigr] d\tau , where (p\ell , q\ell )\in XT l\times XT ris the unique strong solution to (4)1--(4)6. It is not difficult to see that \scrA R,T is a nonempty, closed, and convex subset of H1(0, T). Let us check that, for some 0 <\widehat T\leq T, \Lambda satisfies the following assumptions of Banach's fixed-point theorem in \scrA R, \widehat T: \bullet There exists \widetilde T\in (0, T]such that (7) \Lambda (\scrA R,\tau )\subset \scrA R,\tau \forall \tau \in (0,\widetilde T]. Indeed, \scrL \ell (0) = \ell 0. Let us introduce (8) C(T, R) := C1\Bigl( 1 + R+\surd T\Bigr) \Bigl[ 1 + \Bigl( R+\surd T\Bigr) R+T\Bigr] eC1(1+\surd T)(R+\surd T). Then, using the H\"older inequality, (6), and (8), we see that, for some C2>0 independent of T, one has | \scrL \ell (t) - \ell 0| \leq \int t 0| dlp\ell \xi (\ell 0, \tau ) - drq\ell \xi (\ell 0, \tau )| d \tau \leq C2\widetilde T1/2\| (p\ell \xi (\ell 0,\cdot ), q\ell \xi (\ell 0,\cdot ))\| L2(0, \widetilde T)\times L2(0, \widetilde T) \leq C2\widetilde T1/2\| (p\ell , q\ell )\| XT l\times XT r \leq C2\widetilde T1/2C(T, R)1/2\Pi Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 3084 ARA\' UJO, FERN\' ANDEZ-CARA, L\' IMACO, AND SOUZA for all t\in [0,\widetilde T] and any \widetilde T\in (0, T]. On the other hand, since the trace operator \Gamma : XT l\times XT r\mapsto \rightarrow H1/4(0, T)\times H1/4(0, T) defined by \Gamma (p, q) := (p\ell \xi (\ell 0,\cdot ), q\ell \xi (\ell 0,\cdot )) is continuous, thanks to the continuity of the embedding H1/4(0, T)\lhook \rightarrow L4(0, T), we find C3>0 (independent of T) such that \| \scrL \prime \ell \| L2(0, \widetilde T)\leq C3\widetilde T1/4C(T, R)1/2\Pi . It follows easily from the inequalities above that, if \widetilde Tis sufficiently small, (7) holds. \bullet There exists \widehat T\in (0, T]such that \Lambda : \scrA R, \widehat T\mapsto \rightarrow \scrA R, \widehat Tis a contraction. It can be proved that there exists D1=D1(\tau ) such that D1(\tau )\rightarrow 0 as \tau \rightarrow 0 and (9) \| \scrL \ell 1 - \scrL \ell 2\| H1(0,\tau )\leq D1(\tau )\bigl( \| p\ell 1 - p\ell 2\| X\tau l+\| q\ell 1 - q\ell 2\| X\tau r\bigr) \forall \ell 1, \ell 2\in \scrA R,\tau . Furthermore, using standard energy estimates, we get a positive D2=D2(\tau ), similar to (8), such that (10) \| p\ell 1 - p\ell 2\| X\tau l+\| q\ell 1 - q\ell 2\| X\tau r\leq D2(\tau )\| (F\ell l, F\ell r)\| L2( \widetilde Q0,l)\times L2( \widetilde Q0,r), where (11) F\ell l:= (d\ell 1 l - d\ell 2 l)p\ell 2 \xi \xi - (b\ell 1 l - b\ell 2 l)p\ell 2 \xi and F\ell r:= (d\ell 1 r - d\ell 2 r)q\ell 2 \xi \xi - (b\ell 1 r - b\ell 2 r)q\ell 2 \xi and \widetilde Q0,l := (0, \ell 0)\times (0, \tau ) and \widetilde Q0,r := (\ell 0, L)\times (0, \tau ). Then, using the fact that Gand its inverse G - 1(defined in section 2.1) are smooth functions, we see that there exists D3=D3(\tau ) (D3(s) is bounded for 0\leq s\leq T) such that (12) \| (F\ell l, F\ell r)\| L2( \widetilde Q0,l)\times L2( \widetilde Q0,r)\leq D3(\tau )\| (p\ell 2, q\ell 2)\| X\tau l\times X\tau r\| \ell 1 - \ell 2\| H1(0,\tau ). Combining (9)--(12), we deduce that (13) \| \Lambda (\ell 1) - \Lambda (\ell 2)\| H1(0,\tau )\leq E(\tau )\| \ell 1 - \ell 2\| H1(0,\tau ), where E(\tau ) := D1(\tau )D2(\tau )D3(\tau )\| (p\ell 2, q\ell 2)\| X\tau l\times X\tau r. Since (p\ell 2, q\ell 2) is uniformly bounded in X\tau l\times X\tau rfor all 0 < \tau \leq \widetilde Tprovided \ell 2\in \scrA R,\tau , we find that E(s)\rightarrow 0 as s\rightarrow 0. As a consequence, there exists \widehat T\in (0,\widetilde T] such that \Lambda : \scrA R, \widehat T\mapsto \rightarrow \scrA R, \widehat Tis a contraction. Therefore, \Lambda possesses exactly one fixed-point in \scrA R, \widehat T. This ends the proof. 3. Approximate controllability of a linearized system. In this section, we are going to complete a first step in the proof of Theorem 2. More precisely, we are going to prove a controllability result for a suitable (natural) linearization of (4). To do this, let us fix \ell \in C1([0, T]) with \ell (0) = \ell 0and \ell ([0, T]) \subset (\ell l, \ell r),and let us consider the system (14) \left\{ M\ell l(p) = hl1\omega lin Q0,l, M\ell r(q) = hr1\omega rin Q0,r, p(0,\cdot ) = p(\ell 0,\cdot ) = q(\ell 0,\cdot ) = q(L, \cdot ) = 0 in (0, T), p(\cdot ,0) = p0in (0, \ell 0), q(\cdot ,0) = q0in (\ell 0, L), Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. CONTROL OF TWO-PHASE STEFAN PROBLEMS 3085 where the operators M\ell land M\ell rare, respectively, defined by M\ell l(p) := pt - d\ell lp\xi \xi +b\ell lp\xi and M\ell r(q) := qt - d\ell rq\xi \xi +b\ell rq\xi . Also, let us introduce the function \scrL : [0, T]\mapsto \rightarrow \BbbR given by \scrL (t) := \ell 0 - \int t 0 [dlp\xi (\ell 0, \tau ) - drq\xi (\ell 0, \tau )] d\tau . Remark 3. Using the facts that Gand G - 1are smooth and \ell \in C1([0, T]), we can prove that d\ell land b\ell lbelong, respectively, to C1(Q0,l) and C0(Q0,l). Furthermore, the second spatial derivative of d\ell land the first spatial derivative of b\ell lare functions in C0(Q0,l). The same can be obtained for the coefficients d\ell rand b\ell r. The main goal of this section is to obtain a (robust) approximate controllability result for (14) subject to the linear constraint on the states (15) \scrL (T) = \ell T. In other words, we want to find controls (hl, hr)\in L2(\scrO l)\times L2(\scrO r) such that the associated solutions to (14) satisfy (15). Let us first reformulate (15). Thus, consider the auxiliary adjoint system (16) \left\{ (M\ell l)\ast (\psi ) = 0 in Q0,l, (M\ell r)\ast (\zeta ) = 0 in Q0,r, \psi (0,\cdot )=0, \psi (\ell 0,\cdot ) = 1 in (0, T), \zeta (\ell 0,\cdot )=1, \zeta (L, \cdot ) = 0 in (0, T), \psi (\cdot , T) = 0 in (0, \ell 0), \zeta (\cdot , T) = 0 in (\ell 0, L), where the operators (M\ell l)\ast and (M\ell r)\ast are, respectively, defined by (M\ell l)\ast (\psi ) := - \psi t - (d\ell l\psi )\xi \xi - (b\ell l\psi )\xi and (M\ell r)\ast (\zeta ) := - \zeta t - (d\ell r\zeta )\xi \xi - (b\ell r\zeta )\xi . It is not difficult to check that (16) possesses a unique weak solution (\psi \ell , \zeta \ell ), with \psi \ell \in L2(0, T;H1(0, \ell 0)) \cap H1(0, T;H - 1(0, \ell 0)), \zeta \ell \in L2(0, T;H1(\ell 0, L)) \cap H1(0, T;H - 1(\ell 0, L)). A crucial property of (\psi \ell , \zeta \ell ) is the following. Proposition 3. Given R > 0, let us consider the set \scrB R:= \{ \ell \in C1([0, T]); \| \ell \prime \| C0([0,T ]) \leq R\} . Then, there exists a positive constant C0depending only on \ell 0,\ell l, \ell r,\omega l,\omega r,T, and Rsuch that, for any \ell \in \scrB R, one has \| \psi \ell \| L2(\scrO l)+\| \zeta \ell \| L2(\scrO r)\geq C0. Proof. We argue by contradiction. Thus, if the assertion were false, then there would exist \ell 1, \ell 2, . . . and associated pairs (\psi 1, \zeta 1),(\psi 2, \zeta 2), . . . (weak solutions to (16)), such that (17) \| \ell \prime n\| \infty \leq Rand \| \psi n\| L2(\scrO l)+\| \zeta n\| L2(\scrO r)<1 n\forall n\geq 1. Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 3086 ARA\' UJO, FERN\' ANDEZ-CARA, L\' IMACO, AND SOUZA Due to the smoothing effect of parabolic operators and the facts that the (d\ell n l, b\ell n l) are uniformly bounded in C1(Q0,l)\times C0(Q0,l) and the second spatial derivative of d\ell l and the first spatial derivative of b\ell lare uniformly bounded in C0(Q0,l), there exists \sigma > 0 such that \| \psi n\| L2(0,T - \sigma ;H2(0,\ell 0)) +\| \psi n t\| L2(0,T - \sigma ;L2(0,\ell 0)) \leq C\forall n\geq 1, with C > 0 depending only on \ell 0,\ell l,\ell r,T, and R. Consequently, after extraction of a subsequence, we would have \left\{ \ell n\rightarrow \ell strongly in C0([0, T - \sigma ]), \ell n\rightarrow \ell weakly in H1(0, T - \sigma ), \psi n\rightarrow \psi weakly in L2(0, T - \sigma ;H2(0, \ell 0)) \cap H1(0, T - \sigma ;L2(0, \ell 0)), and we would be able to pass to the limit in the equation and in the boundary condition satisfied by \psi nto deduce that (18) \biggl\{ (M\ell l)\ast (\psi ) = 0 in (0, \ell 0)\times (0, T - \sigma ), \psi (0,\cdot )=0, \psi (\ell 0,\cdot ) = 1 in (0, T - \sigma ). But we would also have, by (17), that \psi \equiv 0 in \omega l\times (0, T - \sigma ), which is impossible in view of the unique continuation property and (18)2. This ends the proof. Let us multiply (14)1by \psi \ell , and let us integrate in Q0,l to obtain \int \int \scrO l hl\psi \ell d\xi dt = - \int \ell 0 0 p0(\xi )\psi \ell (\xi , 0) d\xi - \int T 0 dlp\xi (\ell 0, \tau )d\tau .(19) Analogously, multiplying (14)2by \zeta \ell and integrating in Q0,r we get \int \int \scrO r hr\zeta \ell d\xi dt = - \int L \ell 0 q0(\xi )\zeta \ell (\xi , 0) d\xi +\int T 0 drq\xi (\ell 0, \tau )d\tau .(20) It follows from (19)--(20) that a pair of controls (hl, hr)\in L2(\scrO l)\times L2(\scrO r) are such that \scrL (T) = \ell Tif and only if \int \int \scrO l hl\psi \ell d\xi dt +\int \int \scrO r hr\zeta \ell d\xi dt =\ell T - \ell 0 - \int \ell 0 0 p0(\xi )\psi \ell (\xi , 0) d\xi - \int L \ell 0 q0(\xi )\zeta \ell (\xi , 0) d\xi . (21) Accordingly, we see that the role of the auxiliary adjoint problem (16) is to allow a reformulation of the approximate controllability problem for (14) subject to (15) (a constraint on the state) as a control-constrained approximate controllability problem for (14). In section 3.2, we will establish the approximate controllability of (14) subject to the linear constraint (21). Before this, we will need an adequate (improved) observability inequality. 3.1. An improved observability inequality. To do this, let us first consider open sets \omega 0,l \subset \subset \omega l,\omega 0,r \subset \subset \omega l, and let us introduce the weight functions \eta 0,l \in C2([0, \ell 0]) and \eta 0,r \in C2([\ell 0, L]) satisfying \biggl\{ \eta 0,l >0 in (0, \ell 0), \eta 0,l(0) = \eta 0,l(\ell 0) = 0,and | \eta \prime 0,l| >0 in [0, \ell 0]\setminus \omega 0,l, \eta 0,r >0 in (\ell 0, L), \eta 0,r(\ell 0) = \eta 0,r(L)=0,and | \eta \prime 0,r| >0 in [\ell 0, L]\setminus \omega 0,r. Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. CONTROL OF TWO-PHASE STEFAN PROBLEMS 3093 p=y+f, where y\in L2(0, T ;H2(0, \ell 0)) \cap H1(0, T;L2(0, \ell 0)) is the unique strong solution of the problem (39) \left\{ yt - d\ell ly\xi \xi +b\ell ly\xi =Fin Q0,l, y(0,\cdot ) = y(\ell 0,\cdot ) = 0 in (0, T), y(\cdot ,0) = 0 in (0, \ell 0), where F=h\ell l,\varepsilon 1\omega l - ft+d\ell lf\xi \xi - b\ell lf\xi . Now, let \sigma > 0 be such that \omega l\subset \subset (0, \ell l - \sigma ),and, moreover, let G\sigma l:= (\ell l - \sigma , \ell l+ \sigma )\times (0, T)\subset Q0,l. We can easily check that F\in L4(0, T;L4(\ell l - \sigma , \ell l+\sigma )). Therefore, from local parabolic regularity results, we obtain that y\in X4(0, T; (\ell l - \sigma /2, \ell l+\sigma /2)) and \| y\| X4(0,T ;(\ell l - \sigma /2,\ell l+\sigma /2)) \leq C\bigl( \| F\| L4(0,T ;L4(\ell l - \sigma ,\ell l+\sigma )) +\| y\| L2(0,T ;H2(0,\ell 0))\cap H1(0,T ;L2(0,\ell 0))\bigr) , where Cdepends only on \| d\ell l\| \infty ,\| b\ell l\| \infty ,\ell l,\ell 0, and \sigma . Next, using standard parabolic energy estimates and (33), we get \| y\| X4(0,T ;(\ell l - \sigma /2,\ell l+\sigma /2)) \leq C\Bigl( \| (p0, q0)\| W1,4 0\times W1,4 0+| \ell 0 - \ell T| \Bigr) for some C > 0 as above. Here, we have used the fact that \| d\ell l\| \infty and \| b\ell l\| \infty are bounded in terms of R. Finally, using this inequality, the regularity of the trace y(\ell l,\cdot ), the fact that yis a strong solution to (39), and [33, Propositions 9.2.3 and 9.2.5], we conclude that y\in X4(0, T; (\ell l, \ell 0)),and, moreover, (40) \| y\| X4(0,T ;(\ell l,\ell 0)) \leq C\Bigl( \| (p0, q0)\| W1,4 0\times W1,4 0+| \ell 0 - \ell T| \Bigr) for a new C > 0. In a similar way, we can write q=z+g, where g\in X4(0, T; (\ell 0, L)) is a shift function for the initial data q0, and z\in X4(0, T; (\ell 0, \ell r)) satisfies (41) \| z\| X4(0,T ;(\ell 0,\ell r)) \leq C\Bigl( \| (p0, q0)\| W1,4 0\times W1,4 0+| \ell 0 - \ell T| \Bigr) . Then, the estimate in (38) is an immediate consequence of (40)--(41) and the following embedding from [3, Lemma 2.2]: X4(0, T; (\ell l, \ell 0)) \times X4(0, T; (\ell 0, \ell r)) \lhook \rightarrow C1+\kappa ,(1+\kappa )/2(Gl)\times C1+\kappa ,(1+\kappa )/2(Gr), where \kappa = 1/4. Let us introduce the function \theta : [0, T]\mapsto \rightarrow \BbbR , given by (42) \theta (t) = drq\xi (\ell 0, t) - dlp\xi (\ell 0, t). Then, as an immediate consequence of (38), we get that \theta \in C1/8([0, T]),and, moreover, there exists a positive constant C(depending on \ell l, \ell r, \omega l, \omega r, T, and R) such that (43) \| \theta \| C1/8([0,T ]) \leq C\Bigl( \| (p0, q0)\| W1,4 0\times W1,4 0+| \ell 0 - \ell T| \Bigr) . Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 3094 ARA\' UJO, FERN\' ANDEZ-CARA, L\' IMACO, AND SOUZA 4.2. A fixed-point argument. In this section we will achieve the proof of Theorem 2. It will be a consequence of the following uniform approximate controllability result. Theorem 3. Assume that R > 0is given. Then, there exists \delta > 0such that, for any p0\in W1,4 0(0, \ell 0)with p0\geq 0, any q0\in W1,4 0(\ell 0, L)with q0\leq 0, any \ell 0, \ell T\in (\ell l, \ell r)satisfying \| p0\| W1,4 0(0,\ell 0)+\| q0\| W1,4 0(\ell 0,L)+| \ell 0 - \ell T| \leq \delta , and any \varepsilon > 0, there exist controls (h\varepsilon l, h\varepsilon r)\in L2(\scrO l)\times L2(\scrO r)and associated solutions to (4), with (44) \left\{ \ell \varepsilon \in C1([0, T]) and \ell \varepsilon (t)\in (\ell l, \ell r)\forall t\in [0, T],\| \ell \prime \varepsilon \| C0([0,T ]) \leq R, p\varepsilon \in L2(0, T;H2(0, \ell 0)) \cap H1(0, T;L2(0, \ell 0)), q\varepsilon \in L2(0, T;H2(\ell 0, L)) \cap H1(0, T;L2(\ell 0, L)), satisfying the exact-approximate controllability condition (45) \ell \varepsilon (T) = \ell Tand \| (p\varepsilon (\cdot , T), q\varepsilon (\cdot , T))\| L2(0,\ell 0)\times L2(\ell 0,L)\leq \varepsilon . Moreover, the controls can be found satisfying the following uniform estimate with respect to \varepsilon : (46) \| (h\varepsilon l1\omega l, h\varepsilon r1\omega r)\| L2(Q0,l)\times L2(Q0,r )\leq C(\| (p0, q0)\| W1,4\times W1,4+| \ell 0 - \ell T| ) for some positive C(depending on \ell l, \ell r, \omega l, \omega r, T, and R). Proof. Given \ell l<\~ \ell l<\~ \ell r< \ell rand R > 0, we define the set \scrA R:= \{ \ell \in C1([0, T]) : \~ \ell l\leq \ell (t)\leq \~ \ell r\forall t\in [0, T], \ell (0) = \ell 0,\| \ell \prime \| C0([0,T ]) \leq R\} . Obviously, \scrA Ris a nonempty, closed, and convex subset of C1([0, T]). Let us also introduce the mapping \Lambda \varepsilon :\scrA R\mapsto \rightarrow C1([0, T ]), given by \Lambda \varepsilon (\ell ) = \scrL ,with \scrL (t) := \ell 0 - \int t 0 [dlp\xi (\ell 0, \tau ) - drq\xi (\ell 0, \tau )] d\tau , where (p, q) is the state associated to the control pair (h\ell l,\varepsilon , h\ell r,\varepsilon ) constructed as in the proof of Proposition 7 (recall Lemma 1) and, therefore, \scrL (T) = \ell T. Thanks to (42) and (43), we have that \scrL \in C1([0, T]). Let us check that \Lambda \varepsilon satisfies the following conditions of Schauder's fixed-point theorem: \bullet \Lambda \varepsilon is continuous. Indeed, let the \ell n(n\geq 1) and \ell belong to \scrA R, and assume that \ell n\rightarrow \ell in C1([0, T]). We must prove that \Lambda \varepsilon (\ell n)\rightarrow \Lambda \varepsilon (\ell ) in C1([0, T]). To that end, we will first prove that the corresponding solutions to (16) satisfy (47) (\psi \ell n, \zeta \ell n)\rightarrow (\psi \ell , \zeta \ell ) strongly in L2(Q0,l)\times L2(Q0,r). Let f\in L2(0, T;H2(0, \ell 0)) \times H1(0, T;L2(0, \ell 0)) be such that f(0,\cdot ) = 0 and f(\ell 0,\cdot ) = 1 on (0, T),and let us put \psi \ell n= \Psi \ell n+fand \psi \ell = \Psi \ell +f. It is then clear that y\ell n:= \Psi \ell n - \Psi \ell is the unique weak solution to \left\{ (M\ell n l)\ast (y\ell n) = F\ell nin Q0,l, y\ell n(0,\cdot ) = y\ell n(\ell 0,\cdot ) = 0 in (0, T), y\ell n(\cdot ,0) = 0 in (0, \ell 0), Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. CONTROL OF TWO-PHASE STEFAN PROBLEMS 3095 where F\ell n\in L2(0, T;H - 1(0, \ell 0)) is given by F\ell n:=(d\ell l,\xi \xi - d\ell n l,\xi \xi )f+ 2(d\ell l,\xi - d\ell n l,\xi )f\xi + (d\ell l - d\ell n l)f\xi \xi + (b\ell l,\xi - b\ell n l,\xi )f + (b\ell l - b\ell n l)f\xi + ((d\ell l - d\ell n l)\Psi \ell )\xi \xi + ((b\ell l - b\ell n l)\Psi \ell )\xi . Then, using the facts that (d\ell n l, b\ell n l) are uniformly bounded in the space C1(Q0,l)\times C0(Q0,l) and (d\ell n l,\xi \xi , b\ell n l,\xi ) are uniformly bounded in C0(Q0,l)\times C0(Q0,l), the standard parabolic energy estimates, and the regularity of the function G(as well as the regularity of its inverse G - 1), we get that y\ell n\rightarrow 0 strongly in L2(Q0,l) which, in turn, implies \psi \ell n\rightarrow \psi \ell strongly in L2(Q0,l). Analogously, we can prove that \zeta \ell n\rightarrow \zeta \ell strongly in L2(Q0,r). Now, we recall that, for each \varepsilon > 0, there exists a unique (\varphi n T,\varepsilon , \phi n T,\varepsilon ) in L2(0, \ell 0)\times L2(\ell 0, L) that minimizes the functional J\ell n,\varepsilon , defined in (34). Due to the facts that J\ell n,\varepsilon (\varphi n T,\varepsilon , \phi n T,\varepsilon )\leq 0 and the constant appearing in the right side of (27) does not depend on n, we get that the minimizers are uniformly bounded with respect to nin the space L2(0, \ell 0)\times L2(\ell 0, L),and the corresponding (\varphi n \varepsilon , \phi n \varepsilon ), solutions to (26), are uniformly bounded spaces given in (25). Therefore, there exist (\varphi T,\varepsilon , \phi T,\varepsilon ) in L2(0, \ell 0)\times L2(\ell 0, L) and (\varphi \varepsilon , \phi \varepsilon ) in L2(Q0,l)\times L2(Q0,r) such that, at least for a subsequence, one has (48) \left\{ (\varphi n T,\varepsilon , \phi n T,\varepsilon )\rightarrow (\varphi T,\varepsilon , \phi T,\varepsilon ) weakly in L2(0, \ell 0)\times L2(\ell 0, L), (\varphi n \varepsilon (\cdot ,0), \phi n \varepsilon (\cdot ,0))\rightarrow (\varphi \varepsilon (\cdot ,0), \phi \varepsilon (\cdot ,0)) weakly in L2(0, \ell 0)\times L2(\ell 0, L),and (\varphi n \varepsilon , \phi n \varepsilon )\rightarrow (\varphi \varepsilon , \phi \varepsilon ) strongly in L2(Q0,l)\times L2(Q0,r). We will show now that (\varphi T,\varepsilon , \phi T,\varepsilon ) is the unique minimizer of the functional J\ell ,\varepsilon . Indeed, we first note from the convergences in (47) and (48)3that (\BbbP \ell n l(\varphi n \varepsilon ),\BbbP \ell n r(\phi n \varepsilon )) \rightarrow (\BbbP \ell l(\varphi \varepsilon ),\BbbP \ell r(\phi \varepsilon )) strongly in L2(Q0,l)\times L2(Q0,r). Then, using this fact and the weak convergences (48)1,2, we easily get (49) J\ell ,\varepsilon (\varphi T,\varepsilon , \phi T,\varepsilon )\leq lim inf nJ\ell n,\varepsilon (\varphi n T,\varepsilon , \phi n T,\varepsilon ). Now, let (\varphi T, \phi T) be given in L2(0, \ell 0)\times L2(\ell 0, L),and let the (\varphi n, \phi n) be the solutions to the system (26), with \ell replaced by \ell n, for n= 1,2, . . .. Then, using the same ideas that led to (47), we can ensure that the (\varphi n, \phi n) converge strongly in L2(Q0,l)\times L2(Q0,r) to the solution (\varphi , \phi ) to (26) and the (\varphi n(\cdot ,0), \phi n(\cdot ,0)) converge weakly in L2(0, \ell 0)\times L2(\ell 0, L) to (\varphi (\cdot ,0), \phi (\cdot ,0)). Therefore, from (47) and (49) we deduce that (50) J\ell ,\varepsilon (\varphi T,\varepsilon , \phi T,\varepsilon )\leq lim inf nJ\ell n,\varepsilon (\varphi n T,\varepsilon , \phi n T,\varepsilon )\leq lim inf nJ\ell n,\varepsilon (\varphi T, \phi T) = J\ell ,\varepsilon (\varphi T, \phi T). Since (\varphi T, \phi T)\in L2(0, \ell 0)\times L2(\ell 0, L) is arbitrary, we conclude that (\varphi T,\varepsilon , \phi T,\varepsilon ) minimizes J\ell ,\varepsilon . Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 3096 ARA\' UJO, FERN\' ANDEZ-CARA, L\' IMACO, AND SOUZA Now, let us consider, for each n, the pair (h\ell n l,\varepsilon 1\omega l, h\ell n r,\varepsilon 1\omega r) associated by Proposition 7 to \ell n. It follows easily from (47), (48)3, and (50) that (51) (h\ell n l,\varepsilon 1\omega l, h\ell n r,\varepsilon 1\omega r)\rightarrow (h\ell l,\varepsilon 1\omega l, h\ell r,\varepsilon 1\omega r) strongly in L2(\scrO l)\times L2(\scrO r), where (h\ell l,\varepsilon 1\omega l, h\ell r,\varepsilon 1\omega r) is the control corresponding to \ell . Let us denote by (pn \varepsilon , qn \varepsilon ) and (p\varepsilon , q\varepsilon ) the solutions to (14) associated, respectively, to (h\ell n l,\varepsilon 1\omega l, h\ell n r,\varepsilon 1\omega r) and (h\ell l,\varepsilon 1\omega l, h\ell r,\varepsilon 1\omega r). Then, if we set (yn, zn) := (pn \varepsilon - p\varepsilon , qn \varepsilon - q\varepsilon ) and (wn l1\omega l, wn r1\omega r) := (h\ell n l,\varepsilon 1\omega l - h\ell l,\varepsilon 1\omega l, h\ell n r,\varepsilon 1\omega r - h\ell r,\varepsilon 1\omega r), we find that (52) \left\{ yn t - d\ell n lyn \xi \xi +b\ell n lyn \xi =wn l1\omega l+Fn lin Q0,l, zn t - d\ell n rzn \xi \xi +b\ell n rzn \xi =wn r1\omega r+Fn rin Q0,r, yn(0,\cdot ) = yn(\ell 0,\cdot ) = zn(\ell 0,\cdot ) = zn(L, \cdot ) = 0 in (0, T), yn(\cdot ,0) = 0 in (0, \ell 0), zn(\cdot ,0) = 0 in (\ell 0, L), where Fn l:= (d\ell n l - d\ell l)p\varepsilon ,\xi \xi - (b\ell n l - b\ell l)p\varepsilon ,\xi and Fn l:= (d\ell n l - d\ell l)p\varepsilon ,\xi \xi - (b\ell n l - b\ell l)p\varepsilon ,\xi . Recall that (p0, q0)\in W1,4 0(0, \ell 0)\times W1,4 0(\ell 0, L). Therefore, arguing as in section 4.1 and Lemma 1, we first deduce that (Fn l, Fn r)\in L4((\ell l, \ell 0)\times (0, T)) \times L4((\ell 0, \ell r)\times (0, T)) and (yn \xi (\ell 0,\cdot ), zn \xi (\ell 0,\cdot )) \in C1/8([0, T]) \times C1/8([0, T]),and also that \| yn \xi (\ell 0,\cdot )\| C1/8+\| zn \xi (\ell 0,\cdot )\| C1/8 \leq C\bigl( \| (Fn l, Fn r)\| L4(L4)\times L4(L4)+\| (yn, zn)\| L2(H2)\times L2(H2)\bigr) for some C > 0, independent of n. It is not difficult to check that, in this inequality, the first term in the righthand side goes to 0 when n\rightarrow \infty . From standard parabolic estimates applied to (52) and (51), we also have the convergence to zero of the second term in the right-hand side. Therefore, we deduce that (pn \varepsilon ,\xi (\ell 0,\cdot ), qn \varepsilon ,\xi (\ell 0,\cdot )) \rightarrow (p\varepsilon ,\xi (\ell 0,\cdot ), q\varepsilon ,\xi (\ell 0,\cdot )) in C1/8([0, T]), which implies the continuity of \Lambda \varepsilon . \bullet \Lambda \varepsilon is compact. Note that \Lambda \varepsilon (\ell )\prime (t) = \theta (t) for all \ell \in \scrA Rand all t\in [0, T], where \theta is the function defined in (42). Thus, we conclude easily from (43) that \Lambda \varepsilon (\scrA R) is a bounded subset of C1+1/8([0, T]), which is a compact subset of C1([0, T]). \bullet There exists \delta > 0, such that, whenever (p0, q0)\in W1,4 0(0, \ell 0)\times W1,4 0(\ell 0, L) and \| (p0, q0)\| W1,4 0\times W1,4 0+| \ell 0 - \ell T| \leq \delta , \Lambda \varepsilon (\scrA R)\subset \scrA R. Indeed, it follows easily from (43) that there exists C > 0 (depending on \ell l, \ell r, \omega l, \omega r, T, and R) such that | \scrL (t) - \ell 0| \leq CT \Bigl( \| (p0, q0)\| W1,4 0\times W1,4 0+| \ell 0 - \ell T| \Bigr) \forall t\in [0, T] and | \scrL \prime (t)| \leq C\Bigl( \| (p0, q0)\| W1,4 0\times W1,4 0+| \ell 0 - \ell T| \Bigr) \forall t\in [0, T]. Thus, we get the result by taking \delta \leq min \Biggl\{ R C,\ell 0 - \~ \ell l CT ,\~ \ell r - \ell 0 CT \Biggr\} . Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. CONTROL OF TWO-PHASE STEFAN PROBLEMS 3097 Consequently, for initial data p0, q0, and \ell 0satisfying the above conditions, Schauder's fixed-point theorem guarantees that there exists \ell \varepsilon \in \scrA Rsuch that \Lambda \varepsilon (\ell \varepsilon ) = \ell \varepsilon . It is easy to see that this is sufficient to achieve the proof of the result. Now, we are in position to prove Theorem 2. Indeed, since the fixed points \ell \varepsilon and controls (h\varepsilon l, h\varepsilon r) furnished by Theorem 3 are uniformly bounded, respectively, in C1+1/8([0, T]) and L2(\scrO l)\times L2(\scrO r), there exist \ell and (hl, hr) such that, at least for a subsequence, we have (53) \biggl\{ \ell \varepsilon \rightarrow \ell strongly in C1([0, T]) and (h\varepsilon l, h\varepsilon r)\rightarrow (hl, hr) weakly in L2(\scrO l)\times L2(\scrO r). Since the coefficients (d\ell \varepsilon l, b\ell \varepsilon l) and (d\ell \varepsilon r, b\ell \varepsilon r) are uniformly bounded, respectively, in L\infty (Q0,l)\times L\infty (Q0,l) and L\infty (Q0,r)\times L\infty (Q0,r), we can conclude from energy estimates and (53) that there exists (p, q) with (54) \biggl\{ p\varepsilon \rightarrow pweakly in L2(0, T;H2(0, \ell 0)\cap H1 0(0, \ell 0)) \cap H1(0, T;L2(0, \ell 0)), q\varepsilon \rightharpoonup q weakly in L2(0, T;H2(0, \ell 0)\cap H1 0(\ell 0, L)) \cap H1(0, T;L2(\ell 0, L)), where the (p\varepsilon , q\varepsilon ) are associated to the (h\varepsilon l, h\varepsilon r). Then, (p, q) is the solution to (14), associated to (hl, hr). Moreover, from (45), it is clear that \ell (T) = \ell Tand (p(\cdot , T), q(\cdot , T )) = (0,0) on (0, T). Furthermore, as a consequence of (54) and the embeddings H2(0, \ell 0)c \lhook \rightarrow C1([0, \ell 0]) \lhook \rightarrow L2(0, \ell 0) and H2(\ell 0, L)c \lhook \rightarrow C1([\ell 0, L]) \lhook \rightarrow L2(\ell 0, L), we find that, for any given t\in [0, T], the following holds: \ell (t) = lim \varepsilon \rightarrow 0\ell \varepsilon (t) = lim \varepsilon \rightarrow 0\biggl( \ell 0 - \int t 0 [dlp\varepsilon ,\xi (\ell 0, \tau ) - drq\varepsilon ,\xi (\ell 0, \tau )] d\tau \biggr) =\ell 0 - \int t 0 [dlp\xi (\ell 0, \tau ) - drq\xi (\ell 0, \tau )] d\tau . This implies that the Stefan condition (4)7is satisfied by (\ell , p, q) and ends the proof of Theorem 2. 5. Additional comments. 5.1. Lack of controllability with only one control. In the next result it is proved that, if hlor hrvanishes and the interface does not collapse to the boundary, then null controllability cannot hold. Theorem 4. Assume that u0\in H1 0(0, \ell 0)with u0\geq 0,v0\in H1 0(\ell 0, L)with v0\leq 0,and v0\not \equiv 0. Then, if (hl, hr)\in L2(\scrO l)\times L2(\scrO r),hr\equiv 0,and the associated strong solution to (1) satisfies \ell (t)< L for all t\in [0, T], we necessarily have v(\cdot , T)\not \equiv 0in (\ell (T), L). Proof. Let us assume, by contradiction, that (1) is null controllable with hr\equiv 0, i.e., u(\cdot , T)\equiv 0 in (0, \ell (T)) and v(\cdot , T)\equiv 0 in (\ell (T), L). Then, considering the diffeomorphism \Phi and the function q=v\circ \Phi - 1, defined in section 2.1, we get easily that qis the solution to \left\{ qt - d\ell rq\xi \xi +b\ell rq\xi = 0 in Q0,r, q(\ell 0,\cdot ) = q(L, \cdot ) = 0 in (0, T), q(\cdot ,0) = q0in (\ell 0, L), Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 3098 ARA\' UJO, FERN\' ANDEZ-CARA, L\' IMACO, AND SOUZA where q0:= v0\circ [G(\cdot , \ell 0)] - 1\in H1 0(\ell 0, L) and, obviously, q0\leq 0 and q0\not \equiv 0. We also have (55) q(\cdot , T)\equiv 0 in (\ell 0, L). We can apply the weak maximum principle to qin Q0,r. Obviously, this gives q\leq 0 in Q0,r. Thus, if (55) holds, we get, from the strong maximum principle and the fact that q\leq 0 in Q0,r, that q\equiv 0 in Q0,r, which contradicts q0\not \equiv 0. Remark 5. Note that the previous argument also shows that the null controllability for (1) cannot be achieved for solutions which preserve the signs of the initial conditions in each phase region. In other words, in order to drive the solution to zero at time T, the liquid and solid states must penetrate each other before T. 5.2. Boundary controllability and other extensions. We can prove local boundary controllability results similar to Theorem 1. Thus, let us introduce the system (56) \left\{ ut - dluxx = 0 in Ql, vt - drvxx = 0 in Qr, u(0, t) = kl(t), v(L, t) = kr(t) in (0, T), u(\cdot ,0) = u0in (0, \ell 0), v(\cdot ,0) = v0in (\ell 0, L), u(\ell (t), t) = v(\ell (t), t) = 0 in (0, T), - \ell \prime (t) = dlux(\ell (t), t) - drvx(\ell (t), t) in (0, T), where (kl, kr) stands for the boundary control pair. Then, using a domain extension technique and Theorem 1, it is easy to prove that, if u0and v0are sufficiently small, and \ell 0is sufficiently close to \ell T, there exist controls (kl, kr) and associated solutions to (56) that satisfy \ell (T) = \ell T,u(\cdot , T ) = 0 in (0, \ell T) and v(\cdot , T) = 0 in (\ell T, L). Let us finally mention that the arguments and results in this paper can also be used to solve other variants of the two-phase Stefan controllability problem. 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