Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficientes

ESAIM: Control, Optimisation and Calculus of Variations June 2002, Vol. 8, 621–661 URL: http://www.emath.fr/cocv/ DOI: 10.1051/cocv:2002047 EXACT CONTROLLABILITY TO TRAJECTORIES FOR SEMILINEAR HEAT EQUATIONS WITH DISCONTINUOUS DIFFUSION COEFFICIENTS Anna Doubova1, A. Osses2and J.-P. Puel3 Abstract. The results of this paper concern exact controllability to the trajectories for a coupled system of semilinear heat equations. We have transmission conditions on the interface and Dirichlet boundary conditions at the external part of the boundary so that the system can be viewed as a single equation with discontinuous coefficients in the principal part. Exact controllability to the trajectories is proved when we consider distributed controls supported in the part of the domain where the diffusion coefficient is the smaller and if the nonlinear term f(y) grows slower than |y|log3/2(1 + |y|) at infinity. In the proof we use null controllability results for the associate linear system and global Carleman estimates with explicit bounds or combinations of several of these estimates. In order to treat the terms appearing on the interface, we have to construct specific weight functions depending on geometry. Mathematics Subject Classification. 35B37. Received October 23, 2001. Revised February 7, 2002. 1. Introduction and hypothesis Let Ω ⊂RN,N≥1 be a bounded connected open set with boundary Γ of class C2. Let ω⊂Ωbea nonempty open subset and T>0. We will use the following notation: Q=Ω×(0,T), Σ = Γ ×(0,T). For any p∈[1,+∞], we will denote by ||·|| pthe usual norm in Lp(Q). There are two different situations that will be analyzed in this paper. More precisely, let Ω0and Ω1be a partition of Ω in two non empty open sets such that Case 1: Ω0⊂⊂ Ω,Ω1=Ω\Ω0(see Fig. 1, left); (1) Case 2: Ω1⊂⊂ Ω,Ω0=Ω\Ω1(see Fig. 1, right).(2) Keywords and phrases: Carleman inequalities, controllability, transmission problems. 1Departamento E.D.A.N., Universidad de Sevilla, Tarfia s/n, 41012 Sevilla, Spain and ´ Ecole Polytechnique, 91128 Palaiseau Cedex, France; e-mail: [email protected], [email protected]ique.fr This work has been partially supported by D.G.E.S., Spain, Grants PB98–1134. 2Departamento de Ingener´ıa Matem´atica, Facultad de Ciencias de F´ısicas y Matem´aticas, Universidad de Chile, Casilla 170/3 - Correo 3, Santiago, Chile and Centro de Modelamiento Matem´atico, UMR 2071 CNRS-Uchile; e-mail: [email protected] This work has been partially supported by FONDECYT grants No. 1000955 and 7000955. 3LaboratoiredeMath´ematiques Appliqu´ees, Universit´e de Versailles Saint-Quentin, 45 avenue des ´ Etats Unis, 78035 Versailles Cedex, France and ´ Ecole Polytechnique, 91128 Palaiseau Cedex, France; e-mail: [email protected].fr c EDP Sciences, SMAI 2002 622 A. DOUBOVA, A. OSSES AND J.-P. PUEL Ω0 Ω1 S− S+ Γ n nSS+ Ω0 Ω1 Γ n n Figure 1. Two geometrical cases covered in this paper depending on Ω0⊂⊂ ΩorΩ 1⊂⊂ Ω. We denote by S=Ω0∩Ω1the interface, which will be supposed of class C2and by nthe outward unit normal to Ω1at the points of Sand also the outward unit normal to Ω at the points of Γ. Let S+(resp. S−)bethe part of Scorresponding to the positive (resp. negative) direction of the normal n. Remark 1.1. The two cases mentioned above are not exhaustive, we do not treat other possible geometrical situations in this paper. In both cases mentioned above, we will consider the following transmission problem for semilinear heat equation                    ∂ty−div(a0(x)∇y)+f(y)=v1ω+g0in Ω0×(0,T), ∂ty−div(a1(x)∇y)+f(y)=v1ω+g1in Ω1×(0,T), y|S+×(0,T)=y|S−×(0,T ), a0(x)∂ny|S+×(0,T)=a1(x)∂ny|S−×(0,T), y=0,on Σ y(x, 0) = y0in Ω. (3) Here f:R→Ris a locally Lipschitz-continuous function, ∂nydenotes the outward normal derivative to Ω1, y0∈L2(Ω) and v∈Lr(0,T;Lr(ω)), gi∈Lr(0,T;Lr(Ωi)), i=0,1withrsuch that    1 r+N 2r<1ifN≥2, r=2 ifN=1. (4) Remark 1.2. We could in fact consider v∈Lp(0,T;Lq(ω)), gi∈Lp(0,T;Lq(Ωi)), i=0,1with1/p +N/(2q) <1 in order to have L∞solutions, but in the sake of simplicity we take p=q=r. Remark 1.3. Without loss of generality we can assume y0∈L∞(Ω). Otherwise, taking v=0fort∈(0,δ), δ>0 and thanks to the regularizing effect of parabolic equations, y(δ)∈L∞(Ω) for some δ>0 [21,22]. In (3), y=y(x, t) is the state and v=v(x, t) is the control which acts on the system through ωsince 1ωis the characteristic function of the set ω. We will assume that the diffusion coefficient in (3) satisfies the following: ai∈C2(Ωi)fori=0,1, a0|S+6=a1|S−.(5) System (3) represents the coupling between two parabolic semilinear equations whose diffusion coefficient has a jump. At the interface S, we impose the continuity of the solution yand also of the fluxes. CONTROLLABILITY FOR HEAT EQUATION WITH DISCONTINUOUS COEFFICIENTS 623 Let us set a(x)=a0(x)ifx∈Ω0, a1(x)ifx∈Ω1.(6) We also set g(x)=g0(x)ifx∈Ω0, g1(x)ifx∈Ω1.(7) Taking into account notations (6) and (7), problem (3) can be written in the divergence form (with discontinuous diffusion coefficients) as follows:      ∂ty−div(a(x)∇y)+f(y)=v1ω+gin Q, y=0 on Σ, y(x, 0) = y0in Ω. (8) We will require ato satisfy a(x)≥α>0a.e.inΩ (9) and the following additional hypothesis: a0|S+≤a1|S−.(10) We assume that for each η>0, there exists Cη>0 such that  f(s)−f(s0) s−s0 2/3 ≤Cη+ηlog(1 + |s−s0|)∀s, s0∈R.(11) Let us also consider an “ideal” trajectory y∗, solution of the problem (without control)      ∂ty∗−div(a(x)∇y∗)+f(y∗)=gin Q, y∗=0 on Σ, y(x, 0)∗=y∗ 0in Ω (12) where y∗ 0∈L2(Ω) and g∈Lr(0,T;Lr(Ω)), with ras in (4). We know that under conditions (9) and (11), problem (12) possesses exactly one local solution in time (cf. [21] and [22]). Moreover, we can say that there exists atimeT∗>0, such that for T<T ∗, the solution y∗of (12) satisfies y∗∈C0([0,T]; L2(Ω)) ∩L∞(δ, T ;L∞(Ω)), for every δ>0. The main goal of this paper is to analyze the controllability properties of (8). Definition 1.1. We say that (8) is exactly controllable to the trajectories if, for any trajectory y∗solution of (12) and for any initial condition y0∈L2(Ω), for every T<T ∗, there exists a control v∈Lr(0,T;Lr(ω)) such that (8) has a solution yon (0,T) satisfying y(x, T )=y∗(x, T )inΩ.(13) 624 A. DOUBOVA, A. OSSES AND J.-P. PUEL Definition 1.2. System (8) is said null controllable at time Tif, for each y0∈L2(Ω), there exists v∈ Lr(0,T;Lr(ω)) such that the corresponding initial boundary problem (8) admits a solution y∈C0([0,T]; L2(Ω)) satisfying y(x, T )=0 inΩ.(14) For linear problems, it is easy to see that the notions of null controllability and exact controllability to the trajectories are equivalent, but this is not true for nonlinear systems. Definition 1.3. It will be said that (8) is approximately controllable in L2(Ω) at time Tif, for any y0∈L2(Ω), any yd∈L2(Ω) and any ε>0, there exists a control v∈Lr(0,T;Lr(ω)) such that the corresponding initial boundary problem (8) possesses a solution y∈C0([0,T]; L2(Ω)), with ky(·,T)−ydkL2(Ω) ≤ε. (15) In the case in which the diffusion coefficients are sufficiently regular, the controllability of linear and semilinear parabolic systems has been analyzed in several recent papers. Among them, let us mention [1,5,11,13,15–17], and [8] concerning null controllability [9,12,13,25] and [8] for approximate controllability [16] and [13] for exact controllability to the trajectories. 2. Main result 2.1. Geometric hypothesis and main result In order to state the main result of this work, we need the following geometrical conditions. Condition 2.1 (corresponding to case (1)).We assume that there exists a vector field ζ:Ω17→ RN,ζ∈ C1(Ω1), such that ζ(x)·n(x)<0∀x∈Γ,(16) ζ(x)·n(x)>0∀x∈S, (17) ζ(x)6=0 ∀x∈Ω1(18) and if we consider the characteristics associated to ζ    dx(t) dt=ζ(x(t)),t>0, x(0) = x0, (19) with x0∈Γ, we also assume that for some time T1>0and for every x0∈Γ,thereexistst1(x0)<T 1such that the solution x(t)of (19) verifies x(t)∈Ω1for 0<t<t 1(x0) (20) and x(t1(x0)) ∈Sfor x0∈Γ.(21) Remark 2.1. Condition 2.1 implies that Γ and Sare isotopic, but it is not clear whether isotopy is sufficient to ensure this condition. Remark 2.2. Notice that Condition 2.1 is fulfilled for usual domains, see for example the cases of Figure 2. CONTROLLABILITY FOR HEAT EQUATION WITH DISCONTINUOUS COEFFICIENTS 625 00 00 00 11 11 11 0000 0000 0000 0000 1111 1111 1111 1111 0000 0000 0000 1111 1111 1111 0000 0000 0000 1111 1111 1111 000 000 000 111 111 111 ( a )( b ) ( c ) ( d ) ( e ) Figure 2. Condition 2.1 is fulfilled in situations (a, c, e) but not in (b) and (d). The boundary Γ is represented by a solid line and the interface Sby a dashed line, the dashed region represents Ω0and the black dot the location of the control zone. 00000 00000 00000 00000 11111 11111 11111 11111 (a) 00000 00000 00000 00000 11111 11111 11111 11111 (b) 00000 00000 00000 00000 11111 11111 11111 11111 (c) 00000 00000 00000 00000 11111 11111 11111 11111 (d) 00000 00000 00000 00000 11111 11111 11111 11111 (e) 000 000 000 111 111 111 000 000 000 111 111 111 000 000 000 111 111 111 000 000 111 111 Figure 3. Condition 2.2 is fulfilled in situations (a–d) but not in (e) with the same notations as in the previous figure. Condition 2.2 (corresponding to case (2)).We assume that there exist two disjoint open sets O1,O2⊂⊂ Ω1 (with always a unit outward normal n) and vectors fields ξi: Ω17→ RN,ξi∈C1(Ω1),i=1,2, such that ξi(x)·n(x)>0∀x∈S, ξi(x)·n(x)>0∀x∈∂Oi,i=1,2, ξi(x)6=0 ∀x∈Ω1\Oi (22) and for the characteristics associated to ξi    dxi(t) dt=−ξi(xi(t)),t>0, xi(0) = xi 0, (23) with xi 0∈S, we assume also that for some time Ti 2>0,andforallxi 0∈S,thereexiststi 2(xi 0)<T i 2such that the solution xi(t)of (23) verifies xi(t)∈Ω1\Oifor 0<t<t i 2(xi 0) and xi(ti 2(xi 0)) ∈∂Oifor xi 0∈S,i=1,2. Remark 2.3. Notice that this hypothesis is essentially Condition 2.1 written for the case (2). It is also fulfilled in usual geometrical cases, see for example the cases in Figure 3. 626 A. DOUBOVA, A. OSSES AND J.-P. PUEL The aim of this paper is to prove the following theorem: Theorem 2.1. Assume that in problem (8) the coefficient asatisfies (5, 6, 9, 10), fis a locally Lipschitzcontinuous function satisfying (11) or Condition 2.1in case (1) or Condition 2.2 in case (2) are fulfilled. If ω∩Ωi 06=∅, for each connected component Ωi 0of Ω0, then for each case (1) or (2, 8) is exactly controllable to the trajectories. The idea of the proof of Theorem 2.1 is the following. With a simple change of variables we reduce the problem of exact controllability to the trajectories for (8) to null controllability for a still nonlinear similar transmission problem. For this null controllability result we use approximate controllability to the zero state for an associated linear transmission problem with controls in Lr(0,T;Lr(ω)) for ras in (4) and then we apply a fixed point method. For this we need explicit estimates on the cost of approximate controllability which is obtained from observability inequalities (see Props. 4.1 and 4.2). These estimates are deduced from global Carleman inequalities. In case (1), we use one single global Carleman inequality (see Th. 3.3) with a suitable weight function, whose construction is presented in Lemma 3.1. Case (2) is more complicated and we have to combine two different global Carleman inequalities (see Th. 3.4) with two appropriate weight functions whose construction are given in Lemma 3.2. The growth condition of the non linear term fis analyzed using the arguments of [13]. The idea of combining the controllability of a linearized system and a fixed point argument in the proof is rather general. It was introduced in [23] in the context of the boundary controllability of the semilinear wave equation. For other controllability results proved in a similar way, see for instance [9,13,15] and [8]. In the proofs we will suppose that Ω0and Ω1are connected sets and we assume the simpler hypothesis ω∩Ω06=∅. Otherwise the weight functions for Carleman inequalities are constructed analogously on each connected component of Ω0and Ω1. The paper is organized as follows. In Section 3 we deduce global Carleman inequalities, that we use for proving the main result. Section 4 is devoted to obtain some observability estimates. In Section 5, we prove Theorem 2.1. Finally, in Section 6, we give an explicit construction of suitable weight functions, needed for the global Carleman inequalities. 2.2. Some consequences and extensions 1. Observe that, the controllability result holds if the control acts in the part of the domain where the diffusion coefficient is smaller. To our knowledge, this result is the first one in the literature related to exact controllability to the trajectories when the diffusion coefficients are discontinuous. 2. In the case s0=0andf(0) = 0, notice that assumption (11) can be simply read as follows: lim |s|→+∞ f(s) |s|log3/2(1 + |s|)=0.(24) The proof of Theorem 2.1 also gives the result of null controllability for (8) with a suitable hypothesis on g under the hypothesis and the same geometrical cases considered in Theorem 2.1 by taking condition (24) instead of (11). 3. Notice that approximate controllability for a linear transmission problem is always true and it is independent of the choice of the part of the domain where the control acts as a consequence of unique continuation property. Nonlinear problem (8) with fgrowing as in (11) is still approximately controllable under the conditions of Theorem 2.1. This is due to the fact that approximate controllability in this case can be proved as a consequence of exact controllability to the trajectories. This idea is taken from [12], where approximate controllability for semilinear heat equations is obtained in such a way. CONTROLLABILITY FOR HEAT EQUATION WITH DISCONTINUOUS COEFFICIENTS 627 4. We can also consider in (8) the more general case in which the diffusion coefficients are represented by a real symmetric uniformly elliptic matrix A,i.e. there exists a constant α>0 such that A(x, ξ, ξ)= N X i,j=1 Aij ξiξj≥α|ξ|2∀ξ∈RN,for a.e. x∈Ω (25) and Ais regular in each Ωi,i=0,1. In this case, condition (10) has to be replaced by det A An ·nS ≥0,(26) where [ ]Sdenotes the jump across S. Until now, null controllability for semilinear parabolic systems (in the divergence form) has been analyzed when the diffusion coefficients are sufficiently regular. More precisely, when A=(Aij ), i, j =1,... ,N with Aij ∈C1,2(Q) (see [15]). 2.3. Open problems related to Theorem 2.1 1. If ω⊂Ω1we do not know whether or not system (8) is exactly controllable to the trajectories. Is in this case null controllability also an open problem. 2. In [13], it is proved that even in the case of regular diffusion coefficients, for each β>2, there exist functions f=f(s)withf(0) = 0 and lim |s|→∞ |f(s)| |s|logβ(1 + |s|)=αwith α>0, (27) such that the corresponding system (for the semilinear heat equation) is not null-controllable for any T>0. In view of point 2 in Section 2.2, we see that, when fsatisfies (27) with 3/2≤β≤2, null-controllability of (8) is an open question. 3. On the other hand, it is proved in [13], that also in the case of regular diffusion coefficients, for each β>2, there exist functions fsatisfying (27) such that the corresponding system (for the semilinear heat equation) is not approximately controllable for all T>0. Then, approximate controllability for the transmission problem (8) with 3/2≤β≤2, is also an open question. 4. An abstract result due to Russell [20] shows that boundary exact controllability for the wave equation implies boundary exact null controllability for the heat equation with the same type of control and geometry. This result is proved in the case of smooth coefficients. If we consider this principle still true in the case of non smooth coefficients, the geometrical hypothesis that we consider here seems to be too restrictive in the case N=1(cf. [7] for the controllability of the corresponding wave equation) but not for N≥2. 5. In [14], it is proved null controllability result for the one-dimensional linear heat equation like ρ(x)− (a(x)zx)x+m(x)z= 0 with only BV coefficients without any assumption on the control zone. However, the proof is definitely strictly one dimensional relying on the corresponding one for the wave equation and null controllability result is true if the potential mdepends only on space variable, but not on time. Then, even in the one-dimensional case it is not clear how to treat with this method a similar nonlinear problem. 628 A. DOUBOVA, A. OSSES AND J.-P. PUEL 3. Global Carleman inequalities In this section we will deduce two global Carleman inequalities, that we need for the proof of Theorem 2.1. For this purpose, we will introduce suitable weight functions. Let us first consider the situation of case (1) (see Fig. 1, left). The first weight function is given by the following result: Lemma 3.1. Assume that we have the geometrical situation of case (1) (see Fig. 1). Assume that the function adefined in (5, 6) satisfies (9, 10) and that Condition 2.1 holds. If ω∩Ω06=∅then for every open set ω0⊂⊂ ω∩Ω0there exists a function e β∈C0(Ω),e βi=e β|Ωi∈C2(Ωi),i=0,1,e β>0in Ω,suchthat e β=0 on Γ,(28) ∂ne β<0on Γ,(29) e β=1 on S, (30) ∂ne β0>0,∂ ne β1>0on S, (31) a0∂ne β0=a1∂ne β1on S(32) and |∇e β|>0in Ω\ω0.(33) The proof of Lemma 3.1 will be given in Section 6. Now, we consider the geometrical case (2) (see Fig. 1, right). For the second Carleman inequality, which we will use to treat the situation 2, we need two suitable weight functions. We have the following result: Lemma 3.2. Assume that we have the geometrical situation of case (2) (see Fig. 1). Assume that the function adefined in (5, 6) satisfies (9, 10) and that there exist two open disjoint sets O1,O2⊂⊂ Ω1verifying Condition 2.2. Let Biand e Bi,i=1,2be balls such that B1⊂⊂ e B1⊂⊂ O 1and B2⊂⊂ e B2⊂⊂ O 2.Ifω∩Ω06=∅then for every open set ω0⊂⊂ ω∩Ω0there exist two functions e β1and e β2such that e β1(x)=(e β0(x)if x∈Ω0, e β1 1(x)if x∈Ω1,e β2(x)=(e β0(x)if x∈Ω0, e β2 1(x)if x∈Ω1,(34) with the following properties: e β0∈C2(Ω0),e β0>0in Ω0, e β0=0on Γ,∂ ne β0<0on Γ,(35) ∂ne β0>0on S, e β0=2on S, (36) |∇e β0|>0in Ω0\ω0.(37) And for i=1,2,e βi 1∈C2(Ω1),e βi 1>0in Ω1, e βi 1=e β0=2 on S, (38) a0∂ne β0=a1∂ne βi 1on S, i =1,2,(39) e β1 1≥2e β2 1in e B2,(40) e β2 1≥2e β1 1in e B1,(41) CONTROLLABILITY FOR HEAT EQUATION WITH DISCONTINUOUS COEFFICIENTS 629 and |∇e βi 1|>0in Ω1\Bi,i=1,2.(42) The proof of Lemma 3.2 will also be given in Section 6. Remark 3.1. Notice that in geometrical case (2) (Ω1⊂⊂ Ω) it is impossible to have a function e βwhich is constant on Sand such that ∇e β6=0inΩ 1. Let us consider the functions β=e β+K, β=5 4max Ω β, (43) with K>0 such that K≥5max Ωe β, and e βis given by Lemma 3.1. Let λbe a sufficiently large positive constant that only depends on Ω and ω. It will be fixed later on. For t∈(0,T) and following [16] and [12], we introduce the following functions: ϕ(x, t)= eλβ(x) t(T−t),η(x, t)=eλβ −eλβ t(T−t)·(44) Notice that ∇η=−λϕ∇β, ∇ϕ=λϕ∇β. (45) Let us set Z0={q:q∈C2(Ωi×[0,T]),i=0,1,q|S+×(0,T)=q|S−×(0,T ), a0∂nq|S+×(0,T)=a1∂nq|S−×(0,T ),q=0 on Σ}· We have the following Carleman estimate: Theorem 3.3. Assume that ω∩Ω06=∅,asatisfies (5, 6, 9) and (10) and Condition 2.1 in case (1) is fulfilled. There exists λ1(Ω,ω,a)>0such that for each λ>λ 1there exists a positive constant Cthat only depends on Ω, ωand a,ands1(λ)>0so that the following estimate holds s3ZZQ e−2sηt−3(T−t)−3|q|2dxdt+sZZQ e−2sηt−1(T−t)−1|∇q|2dxdt ≤C s3ZZω×(0,T) e−2sηt−3(T−t)−3|q|2dxdt +ZZQ e−2sη|∂tq+ div(a(x)∇q)|2dxdt (46) for all q∈Z0and s≥s1. Moreover, s1is of the form s1=σ1(Ω,ω,a,λ)(T2+T),whereσ1is a positive constant that only depends on Ω,ω,aand λ. Proof of the Theorem 3.3. In the sequel, Cwill stand for a generic positive constant only depending on Ω, ω and a, whose value can change from line to line. We will also use the usual convention of repeated indices. Let us assume q∈Z0and s>0. We set f=∂tq+ div(a(x)∇q) 636 A. DOUBOVA, A. OSSES AND J.-P. PUEL and we will chose ε>0 sufficiently small. Moreover, we have |∂tϕ|=|T−2t| t2(T−t)2eλβ ≤C(Ω,ω)Tϕ 2,(89) |∂tη|=|T−2t|(eλβ −eλβ) t2(T−t)2≤C(Ω,ω)Teλβ t2(T−t)2≤C(Ω,ω)Te2λβ t2(T−t)2≤C(Ω,ω)Tϕ 2,(90) |∂2 ttη|=2|T2−3Tt+3t2|(eλβ −eλβ) t3(T−t)3≤14T2eλβ t3(T−t)3≤C(Ω,ω)T2e2λβ t3(T−t)3≤C(Ω,ω)T2ϕ3.(91) In (90) and (91), we have used that eλβ ≤e2λβ. This is implied by the fact that β=5 4max Ω β≤2min Ω β, (92) which is a consequence of the choice of Kin (43). Taking into account, equations (82, 85, 89, 90) and (91) we deduce |I210|≤C(Ω,ω,a)Ts2λ2ZZQ ϕ3|ψ|2dxdt, (93) |I310|≤C(Ω,ω,a)T2sZZQ ϕ3|ψ|2dxdt, (94) |I320|≤C(Ω,ω,a)Ts2λ2ZZQ ϕ3|ψ|2dxdt, (95) |I330|≤C(Ω,ω,a)Ts2λ2ZZQ ϕ3|ψ|2dxdt. (96) On the other hand, from (53) we can write that kfsk2 2≤ke−sηfk2 2+C(Ω,ω,a)s2λ4ZZQ ϕ2|ψ|2≤ke−sηfk2 2+C(Ω,ω,a)s2λ4T2ZZQ ϕ3|ψ|2.(97) Using (80, 86–88, 93–96) and (97) in (79), we obtain: kM1ψk2 2+kM2ψk2 2+Cs3λ4ZT 0ZΩ\ω0 ϕ3|ψ|2dxdt+Csλ2ZT 0ZΩ\ω0 ϕ|∇ψ|2dxdt +2s3λ3ZT 0ZS ϕ3|a∂nβ|2[∂nβ]S|ψ|2dσdt+2sλ ZT 0ZS ϕ|a∂nψ|2[∂nβ]Sdσdt ≤ke−sηfk2 2+Csλ3T4ZT 0ZS ϕ3|a∂nβ|2[∂nβ]S|ψ|2dσdt+εsλ ZT 0ZS ϕ|a∂nψ|2[∂nβ]Sdσdt +CsλZZQ ϕ|∇ψ|2dxdt+εsλ2ZZQ ϕ|∇ψ|2dxdt+Cs3λ3ZZQ ϕ3|ψ|2dxdt +Cs2(λ4T2+λ2T) ZZQ ϕ3|ψ|2dxdt+Cs(T2+λ4T4)ZZQ ϕ3|ψ|2dxdt. CONTROLLABILITY FOR HEAT EQUATION WITH DISCONTINUOUS COEFFICIENTS 637 From this, for λ≥λ1(Ω,ω,a)≥λ0(Ω,ω,a), with λ1not depending on T,andforεsmall enough, we can write kM1ψk2 2+kM2ψk2 2+s3λ4ZZQ ϕ3|ψ|2dxdt+sλ2ZZQ ϕ|∇ψ|2dxdt +2s3λ3ZT 0ZS ϕ3|a∂nβ|2[∂nβ]S|ψ|2dσdt≤Cke−sηfk2 2 +s3λ4ZT 0Zω0 ϕ3|ψ|2dxdt+sλ2ZT 0Zω0 ϕ|∇ψ|2dxdt+s(T2+λ4T4)ZZQ ϕ3|ψ|2dxdt +s2(λ4T2+λ2T) ZZQ ϕ3|ψ|2dxdt+sλ3T4ZT 0ZS ϕ3|a∂nβ|2[∂nβ]S|ψ|2dσdt#. We take now s≥σ0(Ω,ω,a,λ)(T2+T), then we also have kM1ψk2 2+kM2ψk2 2+s3λ4ZZQ ϕ3|ψ|2dxdt+sλ2ZZQ ϕ|∇ψ|2dxdt ≤C"ke−sηfk2 2+s3λ4ZT 0Zω0 ϕ3|ψ|2dxdt+sλ2ZT 0Zω0 ϕ|∇ψ|2dxdt#.(98) Let us deduce from (98) that (46) holds for all s≥s1where s1=σ1(Ω,ω,a,λ)(T2+T). Recall that ψ=e −sηq. Then, ∂xiψ=e −sη(∂xiq−s∂xiηq)=e −sη(∂xiq+sλϕ∂xiβq). So we can write that e−sη∂xiq=∂xiψ−sλe−sηϕ∂xiβq. Consequently, we find the following: sλ2ZZQ e−2sηϕ|∇q|2dxdt=sλ2ZZQ ϕ|∇ψ−e−sηsλϕ∇βq|2dxdt ≤Csλ2ZZQ ϕ|∇ψ|2dxdt+C(Ω,ω)s3λ4ZZQ e−2sηϕ3|q|2dxdt. Then, from (98) we have s3λ4ZZQ e−2sηϕ3|q|2dxdt+sλ2ZZQ e−2sηϕ|∇q|2dxdt ≤C(Ω,ω,a)"ke−sηfk2 2+s3λ4ZT 0Zω e−2sηϕ3|q|2dxdt+sλ2ZT 0Zω0 e−2sηϕ|∇q|2dxdt#. (99) In order to conclude the proof of the Carleman inequality (46) it is sufficient to prove that sλ2ZT 0Zω0 e−2sηϕ|∇q|2dxdt≤C"ke−sηfk2 2+s3λ4ZT 0Zω e−2sηϕ3|q|2dxdt +s2(λ4T2+λ2T)ZT 0Zω e−2sηϕ3|q|2dxdt+s(λ3T4+λ3T2+λ2T3)ZT 0Zω e−2sηϕ3|q|2dxdt#. (100) 638 A. DOUBOVA, A. OSSES AND J.-P. PUEL In fact, combining (99) and (100), we deduce that the global Carleman estimate (46) is true for s≥σ1(Ω,ω,a,λ) (T2+T3/2+T). Notice that it is possible to drop the term in T3/2since T3/2≤1/2(T2+T). In order to obtain (100), we consider a function ρ∈C∞ 0(ω) such that ρ≡1inω0and ρ≥0. We consider ω⊂Ω0and the estimates obtained below remain true for larger ω. Multiplying by sλe−2sηρϕq the equation ∂tq+ div(a∇q)=f and integrating in ω×(0,T), we obtain sλ2 2ZT 0Zω e−2sηρϕ∂t|q|2+sλ2ZT 0Zω e−2sηρϕ div(a∇q)qdxdt=sλ2ZT 0Zω e−2sηρϕfq dxdt. (101) In (101), the second term, can be written after integration by parts as follows: sλ2ZT 0Zω e−2sηρϕ ∂xi(a∂xiq)qdxdt=−sλ2ZT 0Zω e−2sηρϕa|∇q|2dxdt−sλ2 2ZT 0Zω ∂xi(e−2sηρϕ)a∂ xi|q|2dxdt. Then, from (101) we deduce sλ2ZT 0Zω0 e−2sηϕ|∇q|2dxdt≤C(a)sλ2 2ZT 0Zω e−2sηρϕ∂t|q|2dxdt+C(a)sλ2ZT 0Zω e−2sηρϕfq dxdt +C(a)sλ2ZT 0Zω e−2sη∂2 xixi(e−2sηρϕ)|q|2dxdt.(102) Let us consider the first term of the right hand side of (102). We have X4=sλ2 2ZT 0Zω e−2sηρϕ∂t|q|2dxdt=−sλ2 2ZT 0Zω ∂t(e−2sηρϕ)|q|2dxdt =s2λ2ZT 0Zω e−2sηρϕ∂tη|q|2dxdt−sλ2 2ZT 0Zω e−2sηρ∂tϕ|q|2dxdt. (103) Using now (89) and (90) in (103) we obtain |X4|≤Cs2λ2TZT 0Zω e−2sηρϕ3|q|2dxdt+Csλ2T3ZT 0Zω e−2sηρϕ3|q|2dxdt. (104) On the other hand, for the third term of the right hand side of (102), we can write sλ2ZT 0Zω ∂2 xixi(e−2sηρϕ)|q|2dxdt≤Cs3λ4ZT 0Zω e−2sηϕ3ρ|q|2dxdt +C(s2λ4T2+sλ3T4+sλ3T2)ZT 0Zω e−2sηϕ3ρ|q|2dxdt. (105) CONTROLLABILITY FOR HEAT EQUATION WITH DISCONTINUOUS COEFFICIENTS 639 This is a consequence of the particular form of ηand ϕ. Indeed, after the following calculation ∂xi(e−2sηρϕ)=e −2sη [∂xiρϕ +ρ∂xiϕ−2sρϕ∂xiη], ∂2 xixj(e−2sηρϕ)=e −2sη −2sϕ∂xjη∂xiρ−2sρ∂xjη∂xiϕ+4s2ϕρ∂xjη∂xiη +ϕ∂2 xixjρ+∂xiρ∂xjϕ+∂xjρ∂xiϕ+ρ∂2 xixjϕ−2sϕ∂xjρ∂xiη−2sρ∂xjϕ∂xiη−2sρϕ∂2 xixjηi and using the following straightforward estimates |ϕ∂xjη|≤λϕ2|∇β|≤Cλϕ2≤CT2λϕ3, |∂xjη∂xiϕ|≤λ2ϕ2|∇β|2≤Cλ2ϕ2≤CT2λ2ϕ3, |ϕ∂xjη∂xiη|≤λ2ϕ3|∇β|2≤Cλ2ϕ3, |∂xjϕ|≤λϕ|∇β|≤Cλϕ ≤CT4λϕ3 |∂2 xixjϕ|≤λϕ|∆β|+λ2ϕ2|∇β|2≤C(λϕ +λ2ϕ2)≤C(T4λ+T2λ2)ϕ3 |ϕ∂2 xixjη|≤λϕ2|∆β|+λ2ϕ3|∇β|2≤CT2λϕ3+Cλ2ϕ3 it is not difficult to see that (105) holds. On the other hand we have sλ2ZT 0Zω e−2sηρϕfq dxdt≤Cke−sηfk2 2+Cs2λ4ZT 0Zω e−2sηϕ2ρ2|q|2dxdt ≤Cke−sηfk2 2+Cs2λ4T2ZT 0Zω e−2sηϕ3|q|2dxdt. (106) Using (104, 105) and (106) in (102) we get (100). As we mentioned above, this ends the proof of Carleman inequality (46) of Theorem 3.3. The situation of Case 2 is quite different. Let us consider the functions βi=e βi+Ki, βi=5 4max Ω βi,for i=1,2,(107) with Ki>0 such that Ki≥5max Ωe βi,and e βiis given by the Lemma 3.2. We also introduce the following weight functions: ϕi(x, t)= eλβi(x) t(T−t),η i(x, t)=eλβi−eλβi t(T−t),i=1,2.(108) Our second Carleman estimate is the following: Theorem 3.4. Assume that ω∩Ω06=∅,asatisfies (5, 6, 9) and (10) and Condition 2.2 in case (2) is fulfilled. There exists λ2(Ω,ω,a)>0so that for each λ>λ 2there exists a positive constant Cthat only depends on Ω, 640 A. DOUBOVA, A. OSSES AND J.-P. PUEL ω,O1,O2and a,ands6(λ)>0so that the following estimate holds s3ZZQ (e−2sη1+e −2sη2)t−3(T−t)−3|q|2dxdt+sZZQ (e−2sη1+e −2sη2)t−1(T−t)−1|∇q|2dxdt ≤Cs3ZZω×(0,T ) (e−2sη1+e −2sη2)t−3(T−t)−3|q|2dxdt +CZZQ (e−2sη1+e −2sη2)|∂tq+ div(a(x)∇q)|2dxdt (109) for all q∈Z0and s≥s6. Moreover, s6is of the form s6=σ6(Ω,ω,O1,O2,a,λ)(T2+T),whereσ6is a positive constant that only depends on Ω,ω,O1,O2,aand λ. Proof of Theorem 3.4. In order to obtain (109), we will apply the global Carleman inequality (46) from Theorem 3.3 and the properties of weight functions (108). We observe, that from Lemma 3.2, we know that ∇βican vanish only in ω0and Bifor i=1,2, where the open subsets B1and B2are fixed balls defined in Lemma 3.2. Taking into account these statements, we can use two weight functions given by (108) and write two Carleman estimates like (46). More precisely, there exist a positive constant Cand s1that only depends on Ω ωand a, such that s3ZZQ e−2sηit−3(T−t)−3|q|2dxdt+sZZQ e−2sηit−1(T−t)−1|∇q|2dxdt ≤C s3ZZω×(0,T ) e−2sηit−3(T−t)−3|q|2dxdt+s3ZZ e Bi×(0,T) e−2sηit−3(T−t)−3|q|2dxdt +ZZQ e−2sηi|∂tq+ div(a(x)∇q)|2dxdt (110) for i=1,2, for all q∈Z0and s≥s1. Moreover, s1is of the form s1=σ1(Ω,ω,a,λ)(T2+T). Let us show that from (110), using the properties of the functions β1and β2, we can deduce the Carleman estimate (109). For this, it will be sufficient to see that for each C>0, there exists s4such that e−2sη2≥2Ce−2sη1in e B1,(111) e−2sη1≥2Ce−2sη2in e B2(112) for s≥s4=σ4(Ω,ω,O1,O2,λ)T2. Indeed, adding the two Carleman inequality (110), we deduce s3ZZQ (e−2sη1+e −2sη2)t−3(T−t)−3|q|2dxdt+sZZQ (e−2sη1+e −2sη2)t−1(T−t)−1|∇q|2dxdt ≤Cs3ZZω×(0,T ) (e−2sη1+e −2sη2)t−3(T−t)−3|q|2dxdt +C s3ZZ e B1×(0,T) e−2sη1t−3(T−t)−3|q|2dxdt+s3ZZ e B2×(0,T) e−2sη2t−3(T−t)−3|q|2dxdt! +CZZQ (e−2sη1+e −2sη2)|∂tq+ div(a(x)∇q)|2dxdt (113) for s≥s5=max(s1,s 4). CONTROLLABILITY FOR HEAT EQUATION WITH DISCONTINUOUS COEFFICIENTS 641 On the other hand, according to (111) and (112), we obtain that Cs3ZZ e B1×(0,T) e−2sη1t−3(T−t)−3|q|2dxdt+Cs3ZZ e B2×(0,T) e−2sη2t−3(T−t)−3|q|2dxdt ≤s3 2ZZ e B1×(0,T) e−2sη2t−3(T−t)−3|q|2dxdt+s3 2ZZ e B2×(0,T) e−2sη1t−3(T−t)−3|q|2dxdt ≤s3 2ZZQ (e−2sη1+e −2sη2)t−3(T−t)−3|q|2dxdt (114) for s≥s6=σ6(Ω,ω,O1,O2,λ)T2. Combining (113) and (114), we find (109). To conclude the proof, let us justify (111) and (112). By construction, we have that β2 1≥2β1 1in e B1.(115) Using (115), we can deduce that for all λ≥1 there exists a positive constant α, which only depends on Ω, ω, O1,O2such that η1−η2≥αη1in e B1.(116) Indeed, equation (116) is a consequence of the following: η1−η2=eλβ2 1−eλβ1 1 t(T−t)≥e2λβ1 1−eλβ1 1 t(T−t)≥αeλβ1−eλβ1 1 t(T−t)=αη1in e B1. Then, from (116) we obtain that, for each C>0, there exists s2such that e−2sη2 e−2sη1≥e2sαη1≥e2sα min η1≥2Cin e B1 for s≥s2=σ2(Ω,ω,O1,O2)T2. This is exactly the inequality (111) for s≥s2. By similar arguments, using the fact that β1 1≥2β2 1in O2,(117) it is easy to see that, for each C>0, there exists s3such that e−2sη1 e−2sη2≥2Cin e B2 for s≥s3=σ3(Ω,ω,O1,O2)T2. Then, equation (112) holds for s≥s3. Taking now s4=max(s2,s 3)we get (111) and (112) for s≥s4. This ends the proof of Theorem 3.4. 4. Observability inequalities and technical results In this section we will deduce some observability estimates as a consequence of appropriate global Carleman inequalities and regularizing effect of the heat equation. This will be needed to prove the null controllability result for a linear transmission problem with controls in Lr(0,T;Lr(ω)) with rsufficiently large, such that (4) holds. 642 A. DOUBOVA, A. OSSES AND J.-P. PUEL Let us consider the following linear (adjoint) transmission problem:      −∂tq−div(a(x)∇q)+bq =0in Q, q=0 onΣ, q(x, T )=qTin Ω, (118) where asatisfies (5, 6, 9) and (10), b∈L∞(Q), and qT∈L2(Ω). First of all, let us prove the observability estimate with L2(0,T;L2(ω))-norm in the right hand side. This can be used to deduce null controllability result (and estimates) for linear transmition problem with bounded potential, with controls in L2(0,T;L2(ω)). We have the following: Proposition 4.1. Assume that ω∩Ω06=∅and that Condition 2.1 (resp. Condition 2.2) in case (1) (resp. in case (2)) are fulfilled. Then for any asatisfying (5, 6, 9, 10), b∈L∞(Q)and qT∈L2(Ω), there exists a positive constant Cthat only depends on Ω,ωand a(resp. Ω,ωO1,O2and a), such that ||q(·,0)||2 L2(Ω) ≤exp C1+ 1 T+T||b||∞+||b||2/3 ∞ZZω×(0,T ) |q|2dxdt, (119) where qis the solution to the corresponding system (118). For simplicity, we only present the proof of Proposition 4.1 for the situation (1). We just note, that the proof corresponding to the situation (2) is similar, it suffices to take into account the different estimates (in space) for the weight functions that appear in the global Carleman inequality (109). Thus we also obtain the constants depending on O1and O2. Proof of Proposition 4.1. We will use global Carleman inequality (46) from Theorem 3.3 and some estimates for the weight functions. Let band qTbe given and let qbe the solution to (118). Step 1: We will first see that ZZΩ×(T/4,3T/4) |q|2dxdt≤exp C1+ 1 T+||b||2/3 ∞ZZω×(0,T) |q|2dxdt. (120) By density, we can write (46) for qbeing the solution of (118). This gives s3ZZQ e−2sηt−3(T−t)−3|q|2dxdt≤C s3ZZω×(0,T) e−2sηt−3(T−t)−3|q|2dxdt +ZZQ e−2sη|bq|2dxdt(121) for all s≥s1. We can estimate the second term in the right as follows: ZZQ e−2sη|bq|2dxdt≤2−6T6||b||2 ∞ZZQ e−2sηt−3(T−t)−3|q|2dxdt. Thus, we deduce from (121) that ZZQ e−2sηt−3(T−t)−3|q|2dxdt≤CZZω×(0,T ) e−2sηt−3(T−t)−3|q|2dxdt(122) CONTROLLABILITY FOR HEAT EQUATION WITH DISCONTINUOUS COEFFICIENTS 643 provided s≥s7=maxs1,CT2||b||2/3 ∞.(123) On the other hand, it can be easily verified that e−2sηt−3(t−T)−3≤26T−6exp −CsT−2∀(x, t)∈Q(124) and e−2sηt−3(t−T)−3≥16 33 T−6exp −CsT−2∀(x, t)∈Ω×[T/4,3T/4],(125) whenever s≥s8=max(s7,CT2) (constants Cin (124) and (125) may be different). If we analyze the structure of the constants s7and s8,we see that s8≤s9,wheres9is of the form s9=σ9(Ω,ω,a)T+T2+T2||b||2/3 ∞.(126) Let us fix the constant s=s9. We write (122) for s=s9taking into account (124) and (125) and we deduce that (120) is satisfied for any solution qof (118). Step 2: Let us now prove that ||q(x, 0)||2 2≤exp C1 T+T||b||∞ZZΩ×(T/4,3T/4) |q|2dxdt. (127) The estimate (127) together with (120) leads to the desired observability inequality (119). Let θ0∈C1[0,1] be a function such that, 0 ≤θ0≤1, θ0= 1 in [0,1/4], θ0= 0 in [3/4,1]. Now, we consider a function θ(t)=θ0(t/T ) and we write (118) for θ(t)q.Weobtain      −∂t(θq)−div(a(x)∇(θq)) + b(θq)=−q∂tθin Ω ×(0,3T/4), θq =0 on ∂Ω×(0,3T/4), θq(x, 3T/4) = 0 in Ω. (128) Multiplying (128) by θq and integrating in Ω, we have −1 2 d dtZΩ |θq|2dx+ZΩ a|∇(θq)|2dx=−ZΩ b|θq|2dx−ZΩ θ(∂tθ)|q|2dx∀t≥0. Thus, −d dtZΩ |θq|2dx+2ZΩ a|∇(θq)|2dx≤2||b||∞ZΩ |θq|2dx+2ZΩ θ|∂tθ||q|2dx and −d dtexp (2||b||∞t)ZΩ |θq|2dx≤2exp(2||b||∞t)ZΩ θ|∂tθ||q|2dx(129) 644 A. DOUBOVA, A. OSSES AND J.-P. PUEL for all t≥0. Integrating this inequality with respect to time in [0,t]witht∈[3T/4,T], we obtain ZΩ |q(x, 0)|2dx≤Zt 0 exp (2||b||∞t)2ZΩ |q|2θ∂tθdx ≤2C Texp 3T 2||b||∞ZZΩ×(T/4,3T/4) |q|2dxdt. (130) In (130), we have used the fact that θ≤1and|∂tθ|=|∂tθ0(t/T )|/T ≤C/T. This justifies the estimate (127) and ends the proof of Proposition 4.1. As we mentioned above, to analyze the controllability for nonlinear problem (8) we need the controls in Lr(0,T;Lr(ω)) for rsufficiently large (1/r +N/(2r)<1). For this we are going to prove a refined version of the observability inequality (119), i.e. with Lr0(0,T;Lr0(ω))-norm in the right hand side, where r0is the dual exponent to r.Wehave: Proposition 4.2. Assume that ω∩Ω06=∅and that Condition 2.1 (resp. Condition 2.2) in case (1) (resp. in case (2)) are fulfilled. Then for any asatisfying (5, 6, 9) and (10), b∈L∞(Q),qT∈L2(Ω) and any r0 sufficiently small, there exist a positive constant Cthat only depends on Ω,ω,a,r0and N(resp. Ω,ωO1,O2, a,r0and N) and a positive constant e Cdepending on Ω,ω,a(resp. Ω,ωO1,O2and a) such that ||q(·,0)||2 L2(Ω) +ZZQ e−2s e CT−1/(T−t)(T−t)−3|q|2dxdt≤exp [CH(T,||b||∞)] ZZω×(0,T) |q|r0dxdt!2/r0 (131) for all s≥σ(Ω,ω,a)(T2+T+T2kbk2/3 ∞),whereσis a positive constant depending on Ω,ωand a(resp. Ω,ω, a,O1,O2), H(T,||b||∞)is given by H(T,||b||∞)=1+ 1 T+T+(T+T1/2)||b||∞+||b||2/3 ∞(132) and qis the solution to the corresponding system (118). As before, for simplicity, we only present the proof corresponding to case (1). We take into account the estimates for the weight functions in (109) for treatment of case (2). In the sequel, σ(Ω,ω,a) will stand for a generic positive constant only depending on Ω, ωand a, whose value can change from line to line. Let us first prove the following technical lemma: Lemma 4.3. Let eωbe a nonempty open set such that eω⊂⊂ ω. Then, for any asatisfying (5, 6) and (9), b∈L∞(Q),qT∈L2(Ω) and any r0sufficiently small, there exists C=C(Ω,ω,a)>0such that ZZ e ω×(0,T) e−2sηt−3(T−t)−3|q|2dxdt≤CT−3TαK(T,kbk∞)γe−CsT−2 ZZω×(0,T ) |q|r0dxdt!2/r0 (133) for all s≥σ(Ω,ω,a)T2,whereσis a positive constant depending on Ω,ωand a,α,γare positive numbers only depending of N,K(T,kbk∞)is given by K(T,kbk∞)=1+T1/2(1 + kbk∞)+T−5/2(s+T2) (134) and qis the solution to the corresponding system (118). CONTROLLABILITY FOR HEAT EQUATION WITH DISCONTINUOUS COEFFICIENTS 645 ProofofLemma4.3. Let eωbe a nonempty open set such that eω⊂⊂ ω. Notice that without loss of generality we can consider ω⊂⊂ Ω0or ω⊂⊂ Ω1with a smooth boundary and the estimates obtained below remain true for a larger ω. Let us consider a function θ∈D(ω), such that θ=1ineω.Weset w(x, t)=θ(x)ϕ(x, t)q(x, t), where qis the solution of (118) and ϕis given by ϕ(x, t)= e−sη t3/2(T−t)3/2·(135) Notice that w(T)=w(0) = 0. Taking into account (118), we deduce that wsatisfy the following problem:      −∂tw−div(a(x)∇w)=−bθqϕ −θq∂tϕ−2a∇(θϕ)·∇q−div(a∇(θϕ))qin Q, w=0 on Σ, w(x, T )=0 in Ω. (136) For simplicity of the computations, we put ew(x, t)= ew(x, T −t)for(x, t)∈Q. In a similar way, we introduce the functions ea,eb,eϕand eq.Thenwehave      ∂tew−div(ea(x)∇ew)=−ebθeqeϕ+θeq∂teϕ−2ea∇(θeϕ)·∇eq−div(ea∇(θeϕ))eqin Q, ew=0 onΣ, ew(x, 0) = 0 in Ω. (137) On the other hand, let zbe the solution of the problem      −∂τz−div(a(x)∇z)=0 inω×(0,t), z=0 on∂ω ×(0,t), z(x, t)=ψin ω, (138) where ψ∈L2(Ω) is given. Multiplying (137) by zand integrating in ωand in τ∈(0,t), we obtain the following for t∈(0,T): (ew(t),z(t)) = Zt 0Zω (−ebθ eϕ+θ∂teϕ−div(ea∇(θeϕ)))eqz dxdτ−2Zt 0Zωea∇(θeϕ)·∇eqzdxdτ ≤C(1 + kbk∞)Zt 0 |eϕ|keqkLr0(ω)kzkLr(ω)dτ+CZt 0 |eϕ|keqkLr0(ω)k∇zkLr(ω)dτ +Zt 0 |∂teϕ|keqkLr0(ω)kzkLr(ω)dτ, (139) where Cis a positive constant depending on ω,eω(i.e. on ω)anda. In (139), we have used that |∇eϕ|≤C|eϕ| and that |∆eϕ|≤C|eϕ|. Notice that, since the diffusion coefficients are sufficiently regular in ωand thanks to the regularizing effect of the heat equation (cf. [19] and [6]), we know that for all t>0and1≤p, q ≤+∞the following holds: kS(t)ukLp(ω)≤Ct−N 2(1 q−1 p)kukLq(ω)∀u∈Lq(ω),(140) kS(t)ukW1,p(ω)≤Ct−N 2(1 q−1 p)−1 2kukLq(ω)∀u∈Lq(ω),(141) where {S(t):t≥0}denotes the semigroup generated by the heat equation with Dirichlet boundary conditions. 652 A. DOUBOVA, A. OSSES AND J.-P. PUEL Step 1: Let us consider a trajectory y∗, solution of the problem (12) without control. We introduce the change of variable p=y−y∗,whereyis a solution of (8). Then, we obtain that      ∂tp−div(a(x)∇p)+f(y∗+p)−f(y∗)=v1ωin Q, p=0 on Σ, p(x, 0) = p0in Ω, (176) where p0=y0−y∗ 0. Theorem 2.1 will be proved if we show that, for each p0∈L2(Ω), there exists v∈ Lr(ω×(0,T)) such that p(x, T )=0 inΩ.(177) We will first consider the case in which p0∈L∞and f∈C1in R.Wedenotebyhthe following function: h(a, s)=   f(a+s)−f(a) sif s6=0, f0(a)ifs=0. Then his continuous. Thanks to hypothesis (11) we know that for each η>0, there exists Cη>0 (depending only of ηand the function f) such that |h(y∗(x, t),s)|2/3≤Cη+ηlog(1 + |s|)∀s∈R,∀(x, t)∈Q. (178) Step 2: Let η>0andR>0 be given positive constants whose values will be fixed later on. Let us fix a time TR=minnT,||h||−2/3 L∞(−R,R),||h||−1/3 L∞(−R,R)o·(179) For simplicity, in the sequel, we will refer only to the case (1), but we also take into account the dependence of the constants corresponding to the case (2). Step 3:(a) We consider the truncation function TR:R7→ R,which is given as follows: TR(s)=sif |s|≤R, Rsgn (s) otherwise. For each z∈L2(Q), we consider the linear system      ∂tp−div(a(x)∇p)+h(y∗(x, t),TR(z))p=v1ωin Ω ×(0,TR), p=0 on Γ×(0,TR), p(x, 0) = p0in Ω. (180) Notice that (180) is of the form (164), with b=h(y∗,TR(z)) ∈L∞(Q). Then we can apply the arguments of the proof of Theorem 5.1 to (180). In fact, we will apply this result in a time interval (0,TR), where TRis given by (179). This is a key point in this proof that will drive to appropriate estimates (the idea is taken from [13] and it has been applied later in [8]). (b) More precisely, for every ε>0, let us consider the functional Jεof the form (169). Arguing as in the proof of Theorem 5.1, we obtain the existence of a control vε z∈Lr(ω×(0,TR)), minimizing the Lr(ω×(0,TR))- norm of the form vε z=sgn(qε z)|qε z|r0−1kqε zkLr0(ω×(0,T))1ω,(r0>1) with qε zthe solution of the corresponding CONTROLLABILITY FOR HEAT EQUATION WITH DISCONTINUOUS COEFFICIENTS 653 problem (118), such that the solution pε zof (180) with v=vε zsatisfies kpε z(·,TR)kL2(Ω) ≤ε. (181) Then, from (172) we have ||vε z||Lr(ω×(0,TR)) ≤C1(Ω,ω,a,TR,||h||L∞(−R,R))||p0||L2(Ω) (182) where C1Ω,ω,a,TR,||h||L∞(−R,R)=e C(Ω,ω,a)  1+ 1 TR+TR+(TR+TR1/2)||h||L∞(−R,R)+||h||2/3 L∞(−R,R)  .(183) Using now the definition of TR, we deduce ||vε z||Lr(ω×(0,TR)) ≤eC2(Ω,ω,a,T )  1+||h||2/3 L∞(−R,R)  ||p0||L2(Ω).(184) Moreover, thanks to (178), from (184) for each z∈L2(Q) we obtain the following: ||vε z||Lr(ω×(0,TR)) ≤eC2(Ω,ω,a,T )(1+Cη+ηlog(1+R))||p0||L2(Ω) =C3(Ω,ω,a,T,η,p 0)(1 + R)ηC2(Ω,ω,a,T ).(185) Here we have used the fact that from (178) we can easily write that khk2/3 L∞(−R,R)≤Cη+ηlog(1 + |R|).(186) (c)Foreachε>0andz∈L2(Q) we introduce the mapping Λ : L2(Q)7→ L2(Q) defined as follows: for each z∈L2(Q), Λ(z)=pε z,wherepε zis the the solution of (180) satisfying (181) with v=vε zconstructed in the point (b) of this step. In fact, Λ is of the following form z∈L2(Q)7→ TR(z)∈L∞(Q)7→ h(y∗,TR(z)) ∈L∞(Q)7→ vε z∈Lr(ω×(0,TR)) 7→ pε z∈L2(Q). Arguing as in [9], we apply Schauder’s theorem and we deduce for each ε>0 the existence of a fixed point pε (associated to vε) of Λ which verifies kpz(·,TR)kL2(Ω) ≤ε. (187) Notice that we have used that the solution pε zof (180) is bounded (uniformly in z)inL2(0,TR;H1 0(Ω)) and its time derivative ∂tpε zis bounded in L2(0,TR;H−1(Ω)). (d) Let pεbe a fixed point of Λ associated to the control vεconstructed as above. Since (185) holds for vε, then vεis bounded in Lr(ω×(0,TR)) uniformly in ε,pεis bounded in L2(0,TR;H1 0(Ω)) and ∂tpε zis bounded in L2(0,TR;H−1(Ω)). For an appropriate subsequence, we deduce that as ε→0 vε→¯vRweakly in Lr(ω×(0,TR)),(188) where ¯vR∈Lr(ω×(0,TR)) also satisfies (185), and pε→¯pRweakly in L2(0,TR;H1 0(Ω)), ∂tpε→∂t¯pRweakly in L2(0,TR;H−1(Ω)), 654 A. DOUBOVA, A. OSSES AND J.-P. PUEL where ¯pRis the solution of the following problem:      ∂t¯pR−div(a(x)∇¯pR)+h(y∗(x, t),TR(¯pR))¯pR=¯vR1ωin Ω ×(0,TR), ¯pR=0 on Γ×(0,TR), ¯pR(x, 0) = p0in Ω. (189) Then pε(TR)→¯pR(TR)inL2(Ω) and since we have (187) for all ε>0, we also have ¯pR(TR)=0.(190) On the other hand, since ¯vR∈Lr(0,TR;Lr(ω)), with ras in (4), we can write (cf. for example [2] and [3]) that ||¯pR||∞≤eTR||h(y∗,TR(¯pR))||∞||p0||∞+TReTR||h(y∗,TR(¯pR))||∞k¯vRkLr(0,T R;Lr(ω)) .(191) Using again the definition of TRand also taking into account (185) and (186), we deduce from (191) that ¯pR verifies ||¯pR||∞≤eC4(Ω,ω,a,T )  1+||h||2/3 L∞(−R,R)  ||p0||∞+k¯vRkLr(0,T R;Lr(ω)) ≤C5(Ω,ω,a,T,η,p 0)(1 + R)ηC6(Ω,ω,a,T ).(192) Notice that in (192) the constant C5is independent of Rand the constant C6is independent of ηand R. Let us extend by zero ¯pRand ¯vRto the whole cylinder Q=Ω×(0,T) and for simplicity, we still call them ¯pRand ¯vR. It is clear that (192) holds and that ¯vRis such that ¯pR(T)=0. (e) In order to conclude the proof of this theorem, it is sufficient to check that for ηand Rsuitably chosen, ¯pR (defined on Ω ×(0,T)) satisfies k¯pRk∞≤R. (193) Then we can say that TR(¯pR)=¯pR. Of course, this implies the existence of a control v∈Lr(0,T;Lr(ω)) such that the solution of (176) satisfies (177). Indeed, from (192) we can choose η=1/(2C6)andR>0 such that C5(Ω,ω,a,T,p 0)(1 + R)ηC6(Ω,ω,a,T )<R. Then we obtain (193). This proves Theorem 2.1 when p0∈L∞(Ω) and f∈C1(R). We just mention that we treat the case in which fis only locally Lipschitz continuous as for example in [9] and [13] using approximations of fby C1functions. Then, in this case we deduce the existence of a control v∈Lr(0,T;Lr(ω)) such that the corresponding solution to (176) verifies (177). Finally, if p0∈L2(Ω), for δ>0 sufficiently small we set v≡0fort∈(0,δ). Using the regularizing effect of the heat equation (see, for example [21] and [22]), we deduce that the corresponding (local) solution p of (176) satisfies p(·,δ)∈L∞(Ω). Then, we argue as above for pin the interval [δ, T ] and we obtain a control v∈Lr(0,T;Lr(ω)) such that (177) holds. This ends the proof of Theorem 2.1. CONTROLLABILITY FOR HEAT EQUATION WITH DISCONTINUOUS COEFFICIENTS 655 6. Proofs of Lemma 3.1 and Lemma 3.2 In this section we will present the construction of the weight functions we used for our global Carleman inequalities. Proof of the Lemma 3.1. We will proceed in several steps. Step 1: Let ζand xbe vector fields verifying Condition 2.1. First, we will construct a function e β1∈C1(Ω1), such that e β1>0inΩ 1,(194) e β1=0 onΓ,(195) ∂ne β1<0onΓ,(196) e β1=1 onS, (197) ∂ne β1>0onS, (198) ∇e β16=0 inΩ1.(199) For t∈[0,t 1(x0)], let us introduce the following change of variables: τ=t t1(x0),τ∈[0,1],ex(τ)=x(t). Observe that, if we take x∈Ω1, then there exists x0∈Γ such that x=x(t, x0)fort∈[0,t 1(x0)] or, in other words, there exists τ∈[0,1] such that ex(0) = x0∈Γ, ex(τ)=x(t)andex(1) = x(t1(x0)) ∈S. Moreover dex dτ=dx dtt1(x0)=ζ(x(t))t1(x0).(200) Let us set e β1(x(t)) = e β1(ex(τ)) = τ, τ ∈[0,1].(201) This function verifies the following properties: de β1 dτ(x) = 1 for all x∈Ω1,(202) 0<e β1(x)<1 for all x∈Ω1,(203) e β1(x) = 0 for all x∈Γ,(204) and e β1(x)=e β1(ex(1)) = 1 for all x∈S. (205) On the other hand, from (200) and (202), for all x∈Ω1,wehave de β1 dτ(x(τ)) = ∇e β1(ex(τ))dex dτ(τ)=∇e β1(x)ζ(x(t))t1(x0)=1,(206) therefore ∇e β1(x)6=0 forallx∈Ω1. 656 A. DOUBOVA, A. OSSES AND J.-P. PUEL Moreover, using (200, 205) and (206), we can write that ∇e β1(x)dex dτ=∇e β1(x)·ndex dτ·n=∇e β1(x)·n(ζ(x)·n)t1(x0) = 1 for all x∈S. Then, taking into account (17), we deduce that ∇e β1(x)·n>0 for all x∈S. This means (198). By the similar way, it is easy to check (196). Thus we also have (199). In order to obtain e β1∈C2(Ω1), we just notice that we can approximate the function of class C1, which we have constructed above by an other function of class C2(that we keep calling e β1), such that it still satisfies the properties (194–199). Let us now consider the diffusion coefficients ai,i=0,1 such that (5) holds and let ω0⊂⊂ ω∩Ω0. Step 2:Forε>0 small enough, we set Uε(S)={x:x∈Ω0,dist(x, S)<ε}· We can construct a function α0in Uε(S), such that α0∈C2(Uε(S)) and α0=1 onS, ∂nα0>0onS, α0>0inUε(S),∇α06=0 inUε(S)(207) and a0∂nα0=a1∂ne β1on S. (208) Now, we extend this function in Ω0to a function that we call again α0,withα0∈C2(Ω0)andα0>0inΩ 0. Step 3: Thanks to the Morse theorem, we deduce that there exists a sequence of Morse functions θk,k≥1 (functions with isolated critical points i.e. their gradient vanishes only in a finite number of points), such that θk→α0in C2(Ω0)ifk→+∞.(209) If θkis close enough to α0, the points where ∇θkvanishes can not be in Uε(S). Moreover, we can assume that for some δ>0wehave |∇α0|≥δ>0inUε(S).(210) Let us construct a Morse function µ∈C2(Ω0), such that µ=1 onS, ∂nµ>0onS(211) a0∂nµ=a1∂ne β1on S(212) and ∇µ6=0 inUε(S).(213) For this, we consider ϕ∈D(Uε(S)) and ϕ=1inUε0(S), with 0 <ε 0<ε.Weset µk(x)=θk(x)+ϕ(x)(α0(x)−θk(x)). CONTROLLABILITY FOR HEAT EQUATION WITH DISCONTINUOUS COEFFICIENTS 657 It is clear that µk=α0in Uε0(S).(214) Then, the function µksatisfies (211) and (212). Moreover, we have ∇µk=∇θkin Ω0\Uε(S) and ∇µk=∇θk+ϕ(∇α0−∇θk)+∇ϕ(α0−θk)inUε(S).(215) Then, using (209) and (210) in (215), we deduce that there exist a positive number k0=k0(δ) such that, if k≥k0we have |∇µk|≥|∇θk|−2kϕkC1kα0−θkkC1≥δ 2in Ω0∩Uε(S). We choose k≥k0and we set µ(x)=µk(x). Then, µis a Morse function whose gradient vanishes only in the set of points where the gradient of θkvanishes. This, together with (214), implies that (211) and (213) hold. Step 4: On the other hand, arguing as in [15], we can deduce that there exists a mapping g:Ω 07→ Ω0which is a diffeomorphism on Ω0, which leaves invariant Uε(S) and transports the points where the gradient of µ vanishes in ω0.Weset e β0(x)=µ(g(x)). Then, equation (33) holds. Thanks to the properties (211) of the function µ, we also have (30, 31) and (32). This ends the proof of Lemma 3.1. Proof of the Lemma 3.2. Assume that we are in the situation of Case (2). Let ω0⊂⊂ ω∩Ω0be an arbitrary fixed open subset of Ω0. Step 1: We assume that there exist O1,O2⊂⊂ Ω1two open disjoint subsets, such that Condition 2.2 holds between Ω1and each one of two sets O1and O2. Then, as in the first part of the proof of Lemma 3.1, we construct two functions βi 1∈C1(Ω1\Oi), βi 1>0inΩ1\Oi,i=1,2, such that βi 1=2 onS,i=1,2, ∂nβi 1>0onS,i=1,2, ∂nβi 1>0on∂Oi,i=1,2, βi 1=1 on∂Oi,i=1,2, ∇βi 16=0 inΩ1\Oi,i=1,2, (216) where nstands for the unit exterior normal to Ω1and Oi,i=1,2. Step 2: Let e Biand Bi,i=1,2 be balls such that B1⊂⊂ e B1⊂⊂ O 1and B2⊂⊂ e B2⊂⊂ O 2. We will present only the construction of e β1 1. The second function will be obtained by the same arguments. Let us set Wε={x:x∈O 1,dist(x, ∂O1)<ε}· First, we observe that since ∂nβ1 1>0on∂O1and β1 1=1on∂O1, we construct a function β1 1∈C1(Wε), such that there exists δ>0 such that β1 1≤1inWε,0<β 1 1≤1−4δon ∂Wε\∂O1and ∇β1 16=0inWε. 658 A. DOUBOVA, A. OSSES AND J.-P. PUEL Then, we can extend this function by a function still called β1 1∈C1(O1), such that 0<β 1 1≤1−3δin O1\Wεand ∇β1 16=0 inWε. Now, we approximate β1 1by Morse functions in such a way that 0<β 1 1≤1−2δin O1\Wε, (217) wherewekeepthenameβ1 1for this approximation. The gradient of this function vanishes only in a finite number of points. As we already mentioned in the Step 3 of this proof, we can deduce the existence of a diffeomorphism on O1, which keeps invariant Wεand transports the points where the gradient of β1 1vanishes in B1. We obtain then a new function (that we keep on calling β1 1) such that β1 1∈C1(Ω1), β1 1>0inΩ 1and ∇β1 16=0 inΩ1\B1. (218) Moreover, from (217) we obtain that for δ>0wehave β1 1≤1−δin B1. (219) Analogously, we construct a function β2 1∈C1(Ω1), β2 1>0inΩ 1, which verifies ∇β2 16=0 inΩ1\B2(220) and β2 1≤1−δin B2. (221) Step 3: Let us finally prove that the properties (40) and (41) are satisfied. For this, we will see that it is possible to modify β1 1(resp. β2 1)inB1(resp. B2) in order to obtain the conditions (40) and (40). We will be able to do this without changing the values of these functions in O1\B1and O2\B2. For simplicity, we will present the details of the construction of only one of such a function, because the same arguments will be valid for the other one. Let us define a new function e β1 1as follows: e β1 1(x)=(β1 1(x)ifx∈Ω1\O1, β1 1(x)n(x)if x∈ O1,(222) with n(x)=1 β1 1(x)p ,(223) where p∈Nwill be fixed later on. We can write that e β1 1=β1 1n=e nlog β1 1in O1. (224) Since β1 1=1on∂O1,wehavethatn=1on∂O1and then, from (224) we deduce that e β1 1=β1 1=1 on∂O1. (225) CONTROLLABILITY FOR HEAT EQUATION WITH DISCONTINUOUS COEFFICIENTS 659 Moreover, we have that ∇e β1 1=∇β1 1on ∂O1(226) and that the gradient of e β1 1vanishes only in B1, where the gradient of β1 1is zero, i.e. ∇e β1 16=0 inΩ1\B1. (227) Indeed, using (224) and (223), we have ∇e β1 1=β1 1n∇nlog β1 1+n∇β1 1 β1 1=β1 1n∇β1 1"−plog β1 1 (β1 1)p+1 +1 (β1 1)p+1 #·(228) Taking into account (217), it is easy to deduce from (228), that (226) and (227) hold. In order to modify the values of e β1 1in B1, we first use (219) and we obtain that n≥1 1−δp · Next, from (224) we deduce that e β1 1≤(1 −δ)(1/(1−δ))p.(229) On the other hand, we know that the second function that we constructed in the Step 5 of this proof satisfies 0<β 2 1≤¯ β2 1=max B1 β2 1>0inB1. Choosing now plarge enough, we can deduce from the estimate (229) the following: e β1 1≤1 2¯ β2 1in B1. This gives (41). The same arguments applied to the function β2 1lead to the existence of a new function e β2 1∈C1(Ω1), e β2 1>0inΩ 1, such that satisfies (38, 40) and (42) for i=2. To conclude this step, we observe that we can approximate the functions of class C1already constructed by functions of class C2, preserving the properties of the functions e β1 1and e β2 1. Step 4:Forε, ε0>0 small enough, we set Vε(Γ) = {x:x∈Ω0,dist(x, Γ) <ε} and Vε0(S)={x:x∈Ω0,dist(x, S)<ε 0}· We can locally construct a function α0in Vε(Γ), such that α0∈C2(Vε(Γ)) and α0=0 onΓ,∂ nα0>0onΓ, α0>0inVε(Γ),∇α6=0 inVε(Γ).(230) 660 A. DOUBOVA, A. OSSES AND J.-P. PUEL On the other hand, in Vε0(S) we construct another function, which for simplicity, we also will denote by α0, such that α0∈C2(Vε0(S)) and α0=2 onS, ∂nα0>0onS, α0>0inVε0(S),∇α06=0 inVε0(S)(231) and a0∂nα0=a1∂nβi 1on S, i =1,2.(232) Now, we extend both functions in Ω0, to a function which we keep on calling α0, with the following properties: α0∈C2(Ω0),α 0>0inΩ 0, α0=0 onΓ,∂ nα0>0onΓ, α0=2 onS, ∂nα0>0onS. (233) Step 5: In the sequel, we will use same arguments as for the proof of Lemma 3.1. Thanks to the Morse theorem, we deduce that there exists a sequence of Morse functions θk,k≥1 (functions with isolated critical points i.e. their gradient vanishes only in a finite number of points), such that θk→α0in C2(Ω0)ifk→+∞.(234) If θkis close enough to α0, the points where ∇θkvanishes can not be in Vε(Γ) ∪Vε0(S). Moreover, we can assume that for some δ>0wehave |∇α0|≥δ>0inVε(Γ) ∪Vε0(S).(235) We can construct a Morse function µ∈C2(Ω0), such that µ=0 onΓ,∂ nµ<0onΓ,(236) µ=2 onS, ∂nµ>0onS(237) and ∇µ6=0 inVε(Γ) ∪Vε0(S).(238) Indeed, it suffices to consider ϕ∈D(Vε(Γ) ∪Vε0(S)) and ϕ= 1 in a neighborhood of Γ ∪Sand to define µk(x)=θk(x)+ϕ(x)(α0(x)−θk(x)). Arguing as in the proof of Lemma 3.1, we can choose k≥k0and µ(x)=µk(x)insuchawaythatµis a Morse function with gradient vanishing only in the points contained in the set of points where the gradient of θk vanishes and satisfying the previous properties. Step 6: Finally, we can deduce that there exists a mapping g:Ω7→ Ω which is a diffeomorphism on Ω, which leaves invariant Vε(Γ) ∪Vε0(S) and transports the points where the gradient of µvanishes in ω0.Weset e β0(x)=µ(g(x)). This ends the proof of Lemma 3.2. 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