On the controllability of a free-boundary problem for the 1D heat equation

On the controllability of a free-boundary problem for the 1D heat equation E. Fern´andez-Cara∗, J. Limaco†, S. B. de Menezes‡ Abstract This paper deals with the local null control of a free-boundary problem for the classical 1D heat equation with distributed controls, locally supported in space. In the main result we prove that, if the final time Tis fixed and the initial state is sufficiently small, there exist controls that drive the state exactly to rest at time t=T. Keywords: null controllability, free-boundary problems, 1D heat equation, Carleman estimates. Mathematics Subject Classification: 93B05, 93C20, 35K20, 35R35. 1 Introduction. The main result Assume that L0>0, T > 0, 0 <a<b<L∗< L0and y0∈L∞(0, L0) are given. In this paper, we consider free-boundary problems of the following kind: Find L∈C1([0, T]) with L(t)>0for all tand a function y=y(x, t)such that L(0) = L0,(1)        yt−yxx =v1ω,(x, t)∈QL, y(0, t) = 0, y(L(t), t) = 0, t ∈(0, T), y(x, 0) = y0(x), x ∈(0, L0) (2) and, moreover, yx(L(t), t) = −L0(t), t ∈(0, T).(3) Here, we have used the notation QL={(x, t) : x∈(0, L(t)), t ∈(0, T)}, ω = (a, b); ∗Dpto. E.D.A.N., Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain, [email protected]. Partially supported by grant MTM2013-41286-P (DGI-MINECO, Spain) and CAPES (Brazil) †Inst. Matem´atica - Universidade Federal Fluminense, RJ, Brasil, [email protected]. ‡Dpto. Matem´atica - Universidade Federal do Cear´a, CE, Brasil, [email protected]. 1 as usual, 1ωdenotes the characteristic function of ω. In (1)–(3), vis a control and (L, y) is an associated state. Recall that, for any y0∈H1 0(0, L0) and v∈L2(ω×(0, T)), there exists at least one local in time solution to (2)–(3). In other words, there exist T∗∈(0, T] and a couple (L, y) with        L∈C1([0, T∗)), L(t)≥L∗in [0, T∗), y, yx, yxx, yt∈L2(Q∗ L),with Q∗ L:= {(x, t) : x∈(0, L(t)), t ∈(0, T∗)}, L(0) = L0, y(x, 0) = y0(x) in (0, L0), such that the PDE in (2) is satisfied in Q∗ Lin the distributional sense and the boundary conditions and (3) are fulfilled in (0, T∗). Furthermore, either T∗=T, or at least one of the following identities hold: lim t→T− ∗ L(t) = bor lim t→T− ∗ |L0(t)|= +∞. This is established in [7]; see also [16]. Free-boundary problems similar to (1)–(3) are motivated by many different applications: •Solidification processes and, in particular, the so called Stefan problem, see [11, 12]. •The analysis and computation of free surface flows, see [18, 22, 25]. •Fluid-solid interaction, see [3, 21, 24]. •Gas flow through porous media, see [1, 6, 23]. •Tumor growth and other problems from mathematical biology, see [13, 14], etc. Note that, for a fixed control v, (1) and (2) are not enough to identify the state (L, y). We need an additional information and this is furnished by (3). In many of these areas, this condition is completely natural. For instance, in tumor growth modelling, ycan be viewed as a pressure (cells are pushed towards the low density regions) and (3) says that, on the moving tumor boundary, the growth speed is proportional to the pressure gradient. Of course, this is a version of the well known Darcy’s law for porous media. The main goal in this paper is to analyze the null controllability of (1)–(3). By definition, it will be said that (1)–(3) is null-controllable if, for any y0∈H1 0(0, L0), there exist a control v∈L2(ω×(0, T)), a function L∈C1([0, T]) and an associated solution y=y(x, t) satisfying (1)– (3) and y(x, T)=0, x ∈(0, L(T)).(4) Notice that, if Land (v, y) solve (1)–(4) and (for instance) y(·, T)∈H2(0, L(T)), then L0(T) = 0. Consequently, the null controllability of (1)–(3) is a useful property from the viewpoint of control theory: roughly speaking, if it is fulfilled, we only have to “work” during a finite time interval in order to get the desired behavior of the system. The controllability of linear and nonlinear PDEs has been the objective of a lot of papers the last years. In the context of the linear and semilinear heat equation, the main contributions have been 2 obtained in [8, 20, 5, 15, 10, 4]. On the other hand, to our knowledge, for parabolic free-boundary problems, controllability questions have not been considered in depth; see however [2, 17]. In the sequel, Cdenotes a generic positive constant; C0,C1, etc. are other positive (specific) constants; when it makes sense, the extension by zero in space of any function fis denoted by f∗. The main result in this paper is the following: Theorem 1.1 Let us assume that L0>0,T > 0and 0< a < b < L∗< L0< B are given. Then (1)–(3) is locally null-controllable. More precisely, there exists ε > 0such that, if y0∈H1 0(0, L0) and ky0kH1 0(0,L0)≤ε, there exist Land (v, y), with L∈C1([0, T]), L∗≤L(t)≤B, v ∈L2(ω×(0, T)), y∗∈C0([0, T]; H1 0(0, B)), that satisfy (1)–(3) and (4). The plan of the proof is the following: •First, we prove that there exists ε > 0 such that, whenever ky0kH1 0(0,L0)≤ε, for each β > 0 there exist uniformly bounded Lβand (vβ, yβ) satisfying (1)–(3) and kyβ(·, T)kL2(0,L(T)) ≤β. (5) This is achieved by solving an appropriate fixed point equation in a closed convex set M ⊂ C1([0, T]): L= Λβ(L), L ∈ M. •Then, we take limits as β→0. Thus, from the estimates deduced for Lβand (vβ, yβ), we see that, at least for a subsequence, we have Lβ→Lstrongly in C1([0, T]), vβ→vweakly in L2(ω×(0, T)), y∗ β→y∗strongly in C0([0, T]; H1 0(0, B)), where (v, y) is a control-state pair satisfying (1)–(3) and (4). The rest of this paper is organized as follows. In Section 2, we prove that, for any L∈C1([0, T]) satisfying L∗≤L≤B, L(0) = L0,(6) the linear system (2) is approximately and null-controllable. We also prove that the solutions to (2) satisfy an additional regularity property. Section 3 is devoted to prove Theorem 1.1. In Section 4, we present some additional comments and we give some indications on future work. Finally, Section 5 is an Appendix where we sketch the proof of a Carleman estimate. 3 2 Some controllability results for the classical heat equation in a non-cylindrical domain In this section, we assume that L0>0, T > 0 and 0 <a<b<L∗< L0< B are given. We fix y0∈H1 0(0, L0) and we assume that L∈C1([0, T]) is a prescribed function satisfying (6); in particular, note that L(t)> b for all t∈[0, T ]. Throughout this paper, we will use the notation NL:= kL0k∞, N0:= ky0kH1 0(0,L0).(7) 2.1 The problems and the results Let us consider the linear system        yt−yxx =v1ω,(x, t)∈QL, y(0, t)=0, y(L(t), t) = 0, t ∈(0, T ), y(x, 0) = y0(x), x ∈(0, L0). (8) For every v∈L2(ω×(0, T)) and every y0∈H1 0(0, L0), there exists exactly one solution to (8), with y, yx, yxx, yt∈L2(QL) and, consequently, y∗∈C0([0, T]; H1 0(0, B)). Indeed, an appropriate change of variable allows to rewrite (8) as a similar problem for a parabolic PDE of the form zτ−zξξ +h(ξ, τ)zξ=w in a cylindrical domain, with a bounded coefficient hand a square-integrable right hand side w; see the precise definitions of ξand τbelow, after (21). The following controllability result is satisfied by (8): Theorem 2.1 For any y0∈H1 0(0, L0)and any β > 0there exist pairs (vβ, yβ), with vβ∈L2(ω×(0, T)), y∗ β∈C0([0, T]; H1 0(0, B)), satisfying (8) and kyβ(·, T)kL2(0,L(T)) ≤β. (9) Furthermore, vβcan be found such that kvβkL2(ω×(0,T )) ≤C1ky0kH1 0(0,L0),(10) where C1depends on NL,L∗,B,ωand T, but is independent of β. An immediate consequence is the null controllability of (8): 4 Corollary 2.1 For any y0∈H1 0(0, L0), there exist pairs (v, y), with v∈L2(ω×(0, T)), y∗∈C0([0, T]; H1 0(0, B)), satisfying (8) and (4). Furthermore, vcan be found such that kvkL2(ω×(0,T )) ≤C1ky0kH1 0(0,L0),(11) where C1only depends on NL,L∗,B,ωand T. The proof of Theorem 2.1 follows rather standard arguments. The main tool is a global Carleman estimate for the solution to the adjoint system of (8), that is given by        −ϕt−ϕxx =g(x, t) (x, t)∈QL, ϕ(0, t)=0, ϕ(L(t), t)=0, t ∈(0, T), ϕ(x, T) = ϕT(x), x ∈(0, L(T)), (12) where g∈L2(QL) and ϕT∈L2(0, L(T)). This will be established in the next section. 2.2 A Carleman estimate In the sequel, we will denote by ΣLthe lateral boundary of QL: ΣL={(x, t) : x= 0 or x=L(t),0< t < T }. We begin with a simple result: Lemma 2.1 Let ω0be a non-empty open set with ω0⊂ω. There exists a function α0∈C1(QL) with α0,xx ∈C0(QL)such that            α0(x, t) = 0 ∀(x, t)∈ΣL, |α0,x|>0in QL\(ω0×(0, T)) and α0(x, t) = 1 −x−b L(t)−b∀x∈(b, L(t)),∀t∈[0, T]. For the proof, it suffices to take (for instance) α0(x, t) :=                        x a,if 0 ≤x < a, 1 + p2(x−a) b−a,b−a 2a,if a≤x < a+b 2, 1 + p2(b−x) b−a,b−a 2(L(t)−b),if a+b 2≤x < b, L(t)−x L(t)−b,if b≤x≤L(t), where p(w, z) := zw + (10 −6z)w3+ (8z−15)w4+ (6 −3z)w5. 5 In particular, notice that |α0,x| ≥ ηin QL\(ω0×(0, T)), where 1 η= max a, max 0≤t≤T(L(t)−b). Let us introduce the weights ξand αand the functions α1and γ, with ξ(x, t):= eλα1(x,t) γ(t), α(x, t):= e2λkα1k∞−eλα1(x,t) γ(t), α1(x, t):=α0(x, t)+1, γ(t):=tk(T−t)k, where λ > 0 and k≥2 are real numbers. We will present now a Carleman estimate for the solution ϕof the adjoint system (12). It is inspired by the ideas in Fursikov-Imanovilov [15] and is contained in the following theorem: Theorem 2.2 Let α0,α,γand ξbe the functions defined above. There exist positive constants λ0,s0and C0, only depending on NL,L∗,B,ωand T, such that, for any s≥s0, any λ≥λ0, any g∈L2(QL)and any ϕT∈L2(0, L(T)), one has ZZQL e−2sα(sξ)−1|ϕt|2+|ϕxx|2+λ2(sξ)|ϕx|2+λ4(sξ)3|ϕ|2dx dt +ZT 0 e−2sα(L(t),t)λs ξ(L(t), t)|ϕx(L(t), t)|2dt +ZT 0 e−2sα(0,t)λs ξ(0, t)|ϕx(0, t)|2dt ≤C0 ZZQL e−2sα |g|2dx dt +ZZω×(0,T ) e−2sαλ4(sξ)3|ϕ|2dx dt!, (13) where ϕis the corresponding solution to (12). The proof is given in the Appendix (see Section 5). 2.3 An observability inequality and the controllability properties of (8) The first consequence of Theorem 2.2 is an observability inequality. We will consider the homogeneous adjoint system        −ϕt−ϕxx = 0,(x, t)∈QL, ϕ(0, t)=0, ϕ(L(t), t)=0, t ∈(0, T), ϕ(x, T) = ϕT(x), x ∈(0, L(T)), (14) where ϕT∈L2(0, L(T)). Proposition 2.1 There exists C2>0, only depending on NL,L∗,B,ωand T, such that, for any ϕT∈L2(0, L(T)), the associated solution to (14) satisfies ZL0 0 |ϕ(x, 0)|2dx ≤C2ZZω×(0,T ) |ϕ|2dx dt. (15) Proof: Let us take λ=λ0and s=s0in (13). Then ZZQL e−2s0αξ3|ϕ|2dx dt ≤CZZω×(0,T ) e−2s0αξ3|ϕ|2dx dt 6 and, consequently, Z3T/4 T/4ZL(t) 0 |ϕ|2dx dt ≤CZ3T/4 T/4ZL(t) 0 e−2s0αξ3|ϕ|2dx dt ≤CZZω×(0,T ) |ϕ|2dx dt. (16) On the other hand, by multiplying the PDE in (14) by ϕand integrating in (0, L(t)), we get the identities −1 2 d dt ZL(t) 0 |ϕ|2dx!+1 2L0(t)|ϕ(L(t), t)|2+ZL(t) 0 |ϕx|2dx = 0 ∀t∈(0, T). Since ϕ(L(t), t)≡0, we deduce that d dt ZL(t) 0 |ϕ(x, t)|2dx!≥0 and, integrating in time, we deduce that ZL(0) 0 |ϕ(x, 0)|2dx ≤ZL(t) 0 |ϕ(x, t)|2dx ∀t∈(0, T) and T 2ZL(0) 0 |ϕ(x, 0)|2dx ≤Z3T/4 T/4ZL(t) 0 |ϕ(x, t)|2dx dt. (17) From (16) and (17), we find (15) and the proof is done. 2 By duality, using classical arguments, it is not difficult to deduce from Proposition 2.1 the controllability result in Theorem 2.1; see [5]. Thus, let y0∈H1 0(0, L0) and β > 0 be given and let us introduce the function JL,β, with      JL,β(ϕT) := 1 2ZZω×(0,T ) |ϕ|2dx dt +βkϕTkL2(0,L(T)) + (ϕ(·,0), y0)L2(0,L0) ∀ϕT∈L2(0, L(T)), (18) where, for each ϕT∈L2(0, L(T)), we have denoted by ϕthe unique solution to (14). Then, JL,β possesses a unique minimizer ˆϕT βand the “best” control is given by vβ= ˆϕβω×(0,T ), where ˆϕβis the solution to (14) associated to ˆϕT β. Furthermore, from the inequality JL,β( ˆϕT β)≤JL,β(0) = 0 and (15), we deduce that 1 2ZZω×(0,T ) |ˆϕβ|2dx dt +βkˆϕT βkL2(0,L(T)) ≤ −( ˆϕβ(·,0), y0)L2(0,L0) ≤1 4C2 kˆϕβ(·,0)k2 L2(0,L0)+C2ky0k2 L2(0,L0) ≤1 4ZZω×(0,T ) |ˆϕβ|2dx dt +C2ky0k2 L2(0,L0) 7 and, consequently, kvβk2 L2(ω×(0,T )) ≤ZZω×(0,T ) |ˆϕβ|2dx dt + 2βkˆϕT βkL2(0,L(T)) ≤4C2ky0k2 L2(0,L0).(19) This way, we find the estimates (10) and (11), where the constant C1only depends on NL,L∗, B,ωand T; see for instance [15, 9] for more details. 2.4 A regularity property Let (v, y) be a control-state pair furnished by Theorem 2.1 or Corollary 2.1 and let us introduce the set RL:= QL∩ { (x, t) : x>b0}, where b<b0< L∗. We will see in this section that, for some κ∈(0,1/2] only depending on NL, L∗,B,ω,Tand N0, one has y∈C1+κ,κ/2 x,t (RL),(20) where C1+κ,κ/2 x,t (RL) is the space of functions z∈C0(RL) that possess continuous partial derivatives with respect to xin RLand satisfy sup (x,t),(x0,t0)∈RL|z(x, t)−z(x0, t0)| |x−x0|+|t−t0|κ/2+|zx(x, t)−zx(x0, t0)| |x−x0|κ+|t−t0|κ/2<+∞. Indeed, let δ > 0 be sufficiently small to have b<b0−δ < b0+δ < L∗. In view of Theorems 10.1 and 11.1 in [19], pp. 204–211, there exists κ∈(0,1/2] depending on NL,L∗,B,ω,Tand N0such that y∈C1+κ,κ/2 x,t ([b0−δ, b0+δ]×[0, T]).(21) Let us now introduce the following change of variables in RL: ξ=1 L(t)−b0(b0(L(t)−x) + L0(x−b0)) , τ = (L0−b0)2Zt 0 1 (L(t)−b0)2ds and let us set z(ξ, τ) := y(x, t). Then, it is not difficult to see that zis well defined in the rectangle R∗, where R∗:= (b0, L0)×(0, SL), SL:= (L0−b0)2ZT 0 1 (L(t)−b0)2ds and, moreover,        zτ−zξξ +h(ξ, τ)zξ= 0,(ξ, τ)∈R∗, z(b0, τ) = y(b0, t), z(L0, τ)=0, τ ∈(0, SL), z(ξ, 0) = y0(ξ), ξ ∈(0, L0), where h∈C0(R∗) and khk∞is bounded by a constant only depending on NL,b0,L0and T. Consequently, taking again into account Theorems 10.1 and 11.1 in [19] and the facts that y satisfies (21) and z(b0, τ)≡y(b0, t), we deduce that z∈C1+κ,κ/2 ξ,τ (R∗), 8 whence we get (20). Now, let the function VLbe given by VL(t) := yx(L(t), t)∀t∈[0, T].(22) Obviously, VL∈Cκ([0, T]). Furthermore, from the estimates in the proofs of Theorems 10.1 and 11.1 in [19], we also have kVLkCκ([0,T ]) ≤C3ky0k∞,(23) where C3only depends on NL,L∗,B,ω,Tand N0and is non-decreasing with respect to NL and N0. 3 Proof of the main result We will first see that there exists ε > 0 such that, if N0=ky0kH1 0(0,L0)≤ε, for any small β > 0, there exist uniformly bounded Lβ∈C1([0, T]) and control-pairs (vβ, yβ) satisfying (1)–(3) and (9). To this purpose, we will use a fixed point argument. Thus, let β > 0 and R > 0 be given, let us set M:= {`∈C1([0, T]) : L∗≤`≤B, `(0) = L0, N`:= k`0k∞≤R} and let us introduce the mapping Λβ:M 7→ C1([0, T]), defined as follows: for each `∈ M, L= Λβ(`) if and only if L(t) = L0−Zt 0 yx(`(s), s)ds ∀t∈[0, T],(24) where yis the solution to the linear system        yt−yxx =v1ω,(x, t)∈Q`, y(0, t) = 0, y(`(t), t)=0, t ∈(0, T ), y(x, 0) = y0(x), x ∈(0, L0), (25) and vis the β-control of minimal norm in L2(ω×(0, T)) associated to `, i.e. the unique solution to the following extremal problem:      Minimize ZZω×(0,T ) |v|2dx dt Subject to v∈L2(ω×(0, T )),(25),(9). (26) Let us see that, if N0is small enough (depending on Rbut not on β), we can apply Schauder’s Theorem to Λβin Mand deduce the existence of a fixed point. Obviously, Mis a non-empty bounded, closed, convex subset of C1([0, T]). In view of Theorem 2.1 and the strict convexity of the norm in L2(ω×(0, T)), (26) possesses exactly one solution for each `∈ M and the mapping Λβis well-defined. •Let us see that Λβis continuous. 9