Controllability of linear and semilinear non-diagonalizable parabolic systems

ESAIM: COCV 21 (2015) 1178–1204 ESAIM: Control, Optimisation and Calculus of Variations DOI: 10.1051/cocv/2014063 www.esaim-cocv.org CONTROLLABILITY OF LINEAR AND SEMILINEAR NON-DIAGONALIZABLE PARABOLIC SYSTEMS ∗ Enrique Fern´ andez-Cara1, Manuel Gonz´ alez-Burgos1and Luz de Teresa2 Abstract. This paper is concerned with the controllability of some (linear and semilinear) nondiagonalizable parabolic systems of PDEs. We will show that the well known null controllability properties of the classical heat equation are also satisfied by these systems at least when there are as many scalar controls as equations and some (maybe technical) conditions are satisfied. We will also show that, in some particular situations, the number of controls can be reduced. The minimal amount is then determined by a Kalman rank condition. Mathematics Subject Classification. 93B05, 35K20. Received March 15, 2012. Revised April 29, 2014. Published online July 6, 2015. 1. Introduction This paper deals with the controllability properties of some non-diagonalizable parabolic systems. Let Ω⊂RNbe a non-empty regular and bounded domain, let us fix T>0andletussetQ:= Ω×(0,T) and Σ:= ∂Ω ×(0,T). We will first consider the linear system ⎧ ⎨ ⎩ yt−AΔy =M(x, t)y+Bv1ωin Q, y=0 on Σ, y(x, 0) = y0(x)inΩ, (1.1) where ω⊂Ωis a (small) open subdomain, A∈L(Rn),M∈L∞(Q;L(Rn)),B∈L(Rn;Rn)andy0∈L2(Ω;Rn). Here, v=(v1,...,v n)∗is the control, to be determined for example in L2(ω×(0,T); Rn), while y= (y1,...,y n)∗is the state. Of course, the most interesting situation appears when we are able to control the system for large nand small n, since this means that we govern the behavior of many equations with few Keywords and phrases. Null controllability, parabolic, non-diagonalizable. ∗Supported by Grant MTM2010-15592 of D.G.E.S. (Spain) and project IN101013 of D.G.A.P.A. (Mexico). 1Dpto, E.D.A.N., Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain. [email protected] & [email protected] 2Instituto de Matem´aticas, Universidad Nacional Aut´onoma de M´exico, Circuito Exterior, C.U. 04510 D.F. M´exico, Mexico. [email protected] Article published by EDP Sciences c EDP Sciences, SMAI 2015 CONTROLLABILITY OF LINEAR AND SEMILINEAR NON-DIAGONALIZABLE PARABOLIC SYSTEMS 1179 controls. However, we will see that the coupling through the non-diagonalizable matrix Ain the higher order terms of the operator introduces serious difficulties to control with n<n. The following assumption will be assumed throughout this paper: ∃a0>0 such that Aξ ·ξ≥a0|ξ|2∀ξ∈Rn. (1.2) Notice that, if (1.2) is satisfied, for every v∈L2(ω×(0,T); Rn) and every y0∈L2(Ω;Rn), (1.1) possesses a unique weak solution y,with y∈L2(0,T;H1 0(Ω;Rn)) ∩C0([0,T]; L2(Ω;Rn)); see Section 2. For maybe technical reasons, we will also assume that The dimensions of the Jordan blocks of the canonical form of Aare ≤4. (1.3) It will be said that (1.1)isnull-controllable at time Tif, for any y0∈L2(Ω;Rn), there exists v∈L2(ω× (0,T); Rn) such that the associated solution satisfies y(x, T )=0 in Ω. (1.4) Since (1.1) is linear, this is equivalent to the exact controllability to the trajectories at time T,thatistosay, to the following property: for any trajectory y(i.e. any weak solution to (1.1) corresponding to an initial state y0∈L2(Ω;Rn) and the control v≡0) and any y0∈L2(Ω;Rn), there exists v∈L2(ω×(0,T); Rn) such that the associated solution satisfies y(x, T )=y(x, T )inΩ. (1.5) Consequently, the null controllability of (1.1) also implies approximate controllability, i.e. the fact that, for any y0,y d∈L2(Ω;Rn)andanyε>0, there exists v∈L2(ω×(0,T); Rn) such that the associated solution satisfies y(·,T)−ydL2≤ε. The controllability properties of similar scalar problems are nowadays well known; see for instance [18,19,22,26,31,32]. To be precise, let us consider the following control system ⎧ ⎨ ⎩ zt−Δz =u1ωin Q, z=0 on Σ, z(x, 0) = z0(x)in Ω. (1.6) Then, for every Ω,ωand T,(1.6) is null-controllable at time T;see[26,31]. To our knowledge, almost all the papers in the literature devoted to the controllability of parabolic non-scalar systems of PDEs deal with distributed controls; see for instance [3–5,14,27–29,35]. In these papers, most results have been established for 2 ×2 systems, with the control exerted only on one equation. The best achievements in this context seem to be those in [4,5,28]. In [28], the authors study a cascade parabolic system of nequations (n≥2) controlled with one single distributed control. In [4,5], the authors provide necessary and sufficient conditions for the controllability of n×nparabolic linear systems with constant or time-dependent coefficients. The analysis of similar boundary controllability problems has been the objective of [7,13,20]. A review of all these results can be found in [8]. It is an interesting fact that, in the framework of the controllability of coupled parabolic systems, new (and possibly counter-intuitive) phenomena arise: minimal time of controllability and dependence of the controllability result on the position of the control domain ω;see[9–11,15]. 1180 E. FERN´ ANDEZ-CARA ET AL. Let us recall one of the main results proved in [4]. Consider the problem ⎧ ⎨ ⎩ yt−Δy =My+Bv1ωin Q y=0 on Σ y(·,0) = y0in Ω (1.7) where M∈L(Rn;Rn), B∈L(Rn;Rn)(withn, n≥1) and y0∈L2(Ω;Rn). Let [M|B] be the following matrix in L(Rn×n;Rn): [M|B]=[B|MB|M2B|... |Mn−1B].(1.8) Then the following holds: The linear system (1.7)is null controllable if and only the so called Kalman’s rank condition rank [M|B]=n is satisfied. In that case, null controllability holds at time T>0. In this paper, our first main result is the following. Theorem 1.1. Let A,M=M(x, t)and Bbe as above. Assume that (1.2)and (1.3)are satisfied and, also, n≥n, rank B=n. (1.9) Then (1.1)is null controllable. In practice, (1.9) means that there are many scalar controls in the system (at least as many as scalar states) and, moreover, their “action” through Bcan have any direction in the n-dimensional space Rn. The proof relies on a (new) global Carleman inequality that can be obtained for the solutions to the related adjoint systems ⎧ ⎨ ⎩ −ϕt−A∗Δϕ =M(x, t)∗ϕin Q ϕ=0 on Σ ϕ(x, T )=ϕT(x)inΩ (1.10) where ϕT∈L2(Ω;Rn); see Lemma 2.3. We have tried to explain that assumption (1.3) is necessary for this argument in Remark 2.4. At present, we have not been able to exclude it from the hypotheses. Remark 1.2. Observe that assumptions (1.3)and(1.9) are sufficient to ensure the null controllability property of (1.1) for any diffusion and coupling matrices Aand M=M(x, t). Assumption (1.9) is not necessary for proving the null controllability of system (1.1). Indeed, in [6,27] the authors prove the (local) null controllability result of phase-field models by one control force (n=2,n=1). Remark 1.3. If n<nand Mis a L∞matrix-valued function, even when Ais a multiple of the identity, new phenomena can arise. More precisely, in [15] the authors prove that the approximate controllability property of a 2 ×2 linear system with A=Idepends on the position of the control set ω. On the other hand, it is established in [10] that, in the same framework, the null controllability result holds when the control time T>0 is greater than a minimal time T0which depends on the coefficients of M. The null controllability result fails when T<T 0. Now, let us introduce a locally Lipschitz-continuous function f:Rn→ Rnand let us consider the semilinear system ⎧ ⎨ ⎩ yt−AΔy =f(y)+Bv1ωin Q, y=0 on Σ, y(x, 0) = y0(x)inΩ. (1.11) CONTROLLABILITY OF LINEAR AND SEMILINEAR NON-DIAGONALIZABLE PARABOLIC SYSTEMS 1181 Again, we can speak of the null, exact to the trajectories or approximate controllability properties of (1.11). For instance, it will be said that (1.11) is exactly controllable to the trajectories at time Tif, for any y0∈ L2(Ω;Rn) and any weak solution ycorresponding to v≡0, there exists a control v∈L2(ω×(0,T); Rn)and an associated solution yto (1.11) such that (1.5) holds. Now, this property is not equivalent but stronger than null controllability. On the other hand, it is not difficult to see that it also implies approximate controllability. Our second main result in this paper is the following: Theorem 1.4. Let Aand Bbe as in Theorem 1.1. Asume that (1.2),(1.3)and (1.9)hold and f:Rn→ Rnis globally Lipschitz-continuous. (1.12) Then (1.11)is exactly controllable to the trajectories at any time T. This result can be deduced from Theorem 1.1 (or, more precisely, from the Carleman inequality in Lem. 2.3 below) using arguments that are nowadays well known; see [23]. For completeness, we will provide the proof in Section 3. We will also see that the asumption (1.12) can be weakened so that, in particular, some slightly superlinear systems are controllable and, in fact, the action of the control can serve to avoid blow-up before t=T. Let us come back to (1.1) and let us consider the particular case in which Mis constant. In this situation, it is possible to obtain controllability results also for n≤n,providedBsatisfies appropriate conditions. More precisely, let us denote by λ1,λ 2,... the eigenvalues of the Dirichlet Laplacian in Ωand let us recall the notation (1.8). We then have the following theorem, which is the third main result in this paper: Theorem 1.5. Let us assume that A, M ∈L(Rn),B∈L(Rn;Rn)and (1.2)and (1.3)hold. Then (1.1)is null controllable at time Tif and only if the following condition is satisfied: rank [λiA−M|B]=n∀i≥1.(1.13) The same result was established in [5] in the case in which Ais diagonalizable. There, an observability inequality for a linear (adjoint) system is shown to be implied by a property satisfied by the solutions to a high order scalar PDE. The proof of Theorem 1.5 uses similar arguments; the details are given in Section 4. Notice that, in order to see whether or not (1.13) holds, one only has to check a finite amount of inequalities. This is because the λigo to +∞as i→+∞and, consequently, for ilarge enough, they are outside the solution set of any algebraic equation of the form det Z(λ)=0, where Z(λ)isaminorof[λiA−M|B]. An example of system that fulfills the assumptions of Theorem 1.5 is the following linearized two-phase solidification model, see [33,34]: ⎧ ⎨ ⎩ θt−Δθ =1ut+2wt+m11θ+v1ω, ut−Δu =βθ +m22u+m23w, wt−Δw =βθ +m32u+m33w. (1.14) Here, we assume that β,theiand the mij are positive constants. The unknowns θ,uand wcan be interpreted as the temperature and two phase-field functions associated with two different kinds of solidification processes. It is not difficult to see that (1.14)canbewrittenintheform(1.1), by replacing utand vtin the first PDE. The result is: ⎛ ⎝θt ut wt⎞ ⎠−A⎛ ⎝Δθ Δu Δw ⎞ ⎠=M⎛ ⎝θ u w⎞ ⎠+⎛ ⎝v1ω 0 0⎞ ⎠, 1182 E. FERN´ ANDEZ-CARA ET AL. where A=⎛ ⎝112 01 0 00 1 ⎞ ⎠,M=⎛ ⎝m11 +β(1+2)1m22 +2m32 1m23 +2m33 βm 22 m23 βm 32 m33 ⎞ ⎠. A simple computation shows that the condition (1.13)inTheorem1.5 is in this case independent of λiand is satisfied if and only if m22 +m23 =m32 +m33. In the sequel, C,C0,C1, ... and Rare used to denote generic positive constants. Frequently, it will be convenient to specify the particular data on which they depend. The rest of the paper is organized as follows. In the next section, we present the proof of Theorem 1.1. As mentioned above, the main tool for this proof is a Carleman inequality for the solutions to (1.10). This is established by combining carefully appropriate Carleman estimates for similar scalar problems. In Section 3, we give the proof of Theorem 1.4. As for similar scalar problems, this relies on a fixed-point argument. More precisely, we rewrite the controllability problem for (1.11) as a fixed-point equation for an adequate mapping. It will be seen that (1.12) (or some other assumption of this kind) is needed to bound uniformly the solutions, which justifies its inclusion in the result. In Section 4, we give the proof of Theorem 1.5. As mentioned above, the main ideas for the proof have been adapted from [4,5]; the main estimates (again leading to appropriate observability inequalities) are established noting that the components of the solutions to the adjoint system solve a scalar PDE that is of the n-th order in tand −Δ. Finally, Section 5deals with some final comments and open questions. 2. Proof of Theorem 1.1 In the reminder of this paper, we will denote by dmax the maximal dimension of a Jordan block of the canonical form of A. By hypothesis, dmax ≤4. The starting point is a basic global Carleman inequality for the solutions to scalar (real-valued and complexvalued) parabolic equations. Thus, let α0=α0(x) be a function satisfying α0∈C2(Ω),α 0>0inΩ, α0=0 on ∂Ω, |∇α0|>0inΩ\ω. (2.1) Such a function exists, see [26]. Let us set ξ(x, t)= eλα0(x) t(T−t),α(x, t)=e(λ+μ)α0L∞−eλα0(x) t(T−t),ρ(x, t)=e α(x,t),(2.2) where λ>0andμ>0. The following notation will be used in order to abridge the estimates: Im(s, λ;ψ):=Q ρ−2s(sξ)m−4(|ψt|2+|Δψ|2) +(sξ)m−2λ2|∇ψ|2+(sξ)mλ4|ψ|2 and Im,ω(s, λ;ψ):=ω×(0,T ) ρ−2s(sξ)mλ4|ψ|2 CONTROLLABILITY OF LINEAR AND SEMILINEAR NON-DIAGONALIZABLE PARABOLIC SYSTEMS 1183 for all s, λ > 0, for any integer mand for any sufficiently regular function ψ=ψ(x, t)withvaluesinR,Cor Rn. Let us consider the linear backwards in time scalar system ⎧ ⎨ ⎩ −ψt−Δψ =gin Q, ψ=0 on Σ, ψ(x, T )=ψT(x)inΩ, (2.3) where g∈L2(Q)andψT∈L2(Ω). In the following result, due to Fursikov and Imanuvilov [26], we recall the basic global Carleman estimates for the solutions to (2.3). Lemma 2.1. For any integer m, there exist constants sm,λmand Cmsuch that, for any s≥smand any λ≥λm, the solutions to (2.3)satisfy Im(s, λ;ψ)≤CmIm,ω(s, λ;ψ)+Q ρ−2s(sξ)m−3|g|2.(2.4) Furthermore, λmand Cmonly depend on m,Ωand ωand smcan be taken of the form sm=σm(T+T2), where σmonly depends on m,Ωand ω. For a detailed justification of the existence and properties of λm,smand Cm, see the proof of Lemma 1.3 in [24]. Secondly, let us consider the similar complex-valued system ⎧ ⎨ ⎩ −ψt−(a+ib)Δψ =gin Q, ψ=0 on Σ, ψ(x, T )=ψT(x)inΩ, (2.5) where now a, b ∈R,a>0, g∈L2(Q;C)andψT∈L2(Ω;C). It is also possible to deduce global Carleman estimates for the solutions to (2.5). They are given in the following lemma, whose proof is essentially given in Fu [25]. Lemma 2.2. For any integer m, there exist constants sm,λmand Cmsuch that, for any s≥smand any λ≥λm, the solutions to (2.5)satisfy (2.4).Furthermore,λmand Cmonly depend on m,Ω,ω,aand band smcan be taken of the form sm=σm(T+T2),whereσmonly depends on m,Ω,ω,aand b. Again, the existence and properties of λm,smand Cmare justified by the arguments in [24]. Let us set g=g1+ig2and ψT=ηT+iζTin (2.5). By writting the solutions in the form ψ=η+iζ,wesee that they can also be regarded as solutions to the 2 ×2system −ηt ζt−a−b ba Δη Δζ =g1 g2, together with Dirichlet boundary and initial conditions for ηand ζ. Let us denote by M∞the norm of Min L∞(Q;L(Rn)). Now, we present a Carleman estimate for the solutions to the non-scalar (adjoint) problem (1.10): Lemma 2.3. Let the assumptions in Theorem 1.1 be satisfied. For any integer m, there exist constants s m,λ m and C msuch that, for any s≥s mand any λ≥λ m, the solutions to (1.10)satisfy Im−3(s, λ;ϕ)≤C mIm,ω(s, λ;ϕ).(2.6) Furthermore, C monly depends on m,Ω,ωand A,λ monly depends on m,Ω,ω,Aand M∞and s mcan be taken of the form s m=σ m(T+T2),whereσ monly depends on m,Ω,ω,Aand M∞. 1184 E. FERN´ ANDEZ-CARA ET AL. Proof. In this proof, we will denote by C0a generic positive constant only depending on Ω,ωand A. First, notice that it can be assumed that Ais written in the canonical form. Indeed, there exists a non-singular matrix P∈L(Cn) such that A=PJP−1for some J∈L(Cn)oftheform J=diag(J1,...,J s), where the Jiare the Jordan blocks associated to the eigenvalues μiof A. By hypothesis, we have (1.2)andthis implies that, for all i, Ji=⎡ ⎢ ⎣ μi1 μi1 μi1 μi ⎤ ⎥ ⎦(2.7) or a matrix with the same shape and smaller dimension, with Re μi>0. The solutions to (1.10) can be put in correspondance with the solutions to ⎧ ⎪ ⎨ ⎪ ⎩ −ψt−J∗Δψ =P∗M(x, t)∗(P∗)−1ψin Q, ψ=0 on Σ, ψ(x, T )=P∗ϕT(x)inΩ, (2.8) through the change of variable ϕ=(P∗)−1ψand, obviously, it suffices to prove (2.6)forψ. For instance, let us assume that, in (2.8), the first four PDEs correspond to the same block and let us write them in the form −ψ1,t −μΔψ1= n  j=1 ˜ M1j(x, t)ψj, −ψ2,t −μΔψ2= n  j=1 ˜ M2j(x, t)ψj+Δψ1, −ψ3,t −μΔψ3= n  j=1 ˜ M3j(x, t)ψj+Δψ2, −ψ4,t −μΔψ4= n  j=1 ˜ M4j(x, t)ψj+Δψ3,(2.9) where the ˜ Mij (x, t) stand for the components of the matrix P∗M(x, t)∗(P∗)−1. Let us write (2.4)forψ1,ψ2,ψ3and ψ4respectively with m= 3, 2, 1 and 0. The following is found for all large sand λ: I3(s, λ;ψ1)≤C0I3,ω(s, λ;ψ1)+M2 ∞Q ρ−2s|ψ|2 , I2(s, λ;ψ2)≤C0I2,ω(s, λ;ψ2)+M2 ∞Q ρ−2s(sξ)−1|ψ|2+Q ρ−2s(sξ)−1|Δψ1|2 , I1(s, λ;ψ3)≤C0I1,ω(s, λ;ψ3)+M2 ∞Q ρ−2s(sξ)−2|ψ|2+Q ρ−2s(sξ)−2|Δψ2|2 , I0(s, λ;ψ4)≤C0I0,ω(s, λ;ψ4)+M2 ∞Q ρ−2s(sξ)−3|ψ|2+Q ρ−2s(sξ)−3|Δψ3|2 .(2.10) CONTROLLABILITY OF LINEAR AND SEMILINEAR NON-DIAGONALIZABLE PARABOLIC SYSTEMS 1185 Therefore, an appropriate linear combination of the left hand sides can be used to control and absorb all the second-order terms in the right. Indeed, it is clear that (sξ)n1≤C(sξ)n2whenever n1≤n2. Accordingly, if s≥σ0(T+T2)forsomeσ0only depending on Ωand ω,wehave: I3(s, λ;ψ1)+I2(s, λ;ψ2)+I1(s, λ;ψ3)+I0(s, λ;ψ4) ≤C0I3,ω(s, λ;ψ1)+I2,ω(s, λ;ψ2)+ I1,ω(s, λ;ψ3)+I0,ω(s, λ;ψ4)+C0M2 ∞Q ρ−2s|ψ|2. In order to absorb as many terms as possible in the last integral in the right hand side, we do as follows: (1) We take s≥s3+C0M2/3 ∞T2and λ≥λ3; this makes it possible to skip |ψ1|2. (2) Then, we take sand λas before and also satisfying s≥s2+C0M∞T2and λ≥λ2;inthisway,wecan suppress |ψ2|2. (3) Then, with sand λas in the previous step and also satisfying s≥s1+C0M2 ∞T2and λ≥λ2, we can also skip |ψ3|2and, finally, (4) We choose sand λas in the previous step and also satisfying s≥s0and λ≥λ0+C0M1/2 ∞,inordertoskip |ψ4|2. Hence, there exist s 3and λ 3(as in the statement) such that, for all s≥s 3and λ≥λ 3, one has: I3(s, λ;ψ1)+I2(s, λ;ψ2)+I1(s, λ;ψ3)+I0(s, λ;ψ4) ≤C0I3,ω(s, λ;ψ1)+I2,ω(s, λ;ψ2)+ I1,ω(s, λ;ψ3)+I0,ω(s, λ;ψ4)+C0M2 ∞Q ρ−2s j≥5 |ψj|2.(2.11) Obviously, similar estimates can also be obtained for the ψicorresponding to any other Jordan block of equal or lower dimension. It is also clear that, if we choose sand eventually λas indicated, after addition, we get in the left hand side terms that can absorb all the zero-order terms in the right. Therefore, I0(s, λ;ψ)≤C0I3,ω(s, λ;ψ) for all s≥s 3,λ≥λ 3. This proves the lemma for m=3. With similar computations, it is possible to prove (2.6) for any other integer m. We skip the details, that can be easily deduced from the previous argument.  Remark 2.4. From the proof of this lemma, we see that the best possible choices of s mand λ mare as follows (recall that dmax denotes the maximal dimension of a Jordan block for A): •If dmax =1(i.e. Ais diagonalizable), then we can choose s m=sm+C0M2/3 ∞T2and λ m=λm. •If dmax =2,wecantakes m=sm+C0M∞T2and λ m=λm. •If dmax = 3, then we need s m=sm+C0M2 ∞T2and λ m=λm. •Finally, if dmax =4,wehavetotakes m=smand λ m=λm+C0M2 ∞. Remark 2.5. It is also clear that a proof of (2.6) of the same kind cannot work when dmax ≥5. Indeed, if (for instance) the first block is of dimension 5, the associated components ψ1, ..., ψ5are coupled through second-order terms and we must add to (2.9) a fifth PDE: −ψ5,t −μΔψ5= n  j=1 ˜ M5j(x, t)ψj+Δψ4. 1186 E. FERN´ ANDEZ-CARA ET AL. If we look for an estimate of ψ5and we try to incorporate a new Carleman inequality to (2.10), in view of the term Δψ4in the right hand side, we are forced to take m=−1. But then the zero-order term for ψ5that we obtain in the left is Q ρ−2s(sξ)−1λ4|ψ5|2 and this is not sufficient to control the similar zero-order term in the right coming from the first Carleman inequality in (2.10). We can now achieve the proof of Theorem 1.1. In the remainder of this section, R(resp. C) denotes various positive constants only depending on Ω,ω,A and M∞(resp. Ω,ω,A,M∞and T). First, recall that, in view of classical arguments, the null controllabilty of (1.1) is equivalent to the observability of (1.10), that is, to the estimate ϕ(·,0)2 L2≤Cω×(0,T ) |B∗ϕ|2(2.12) for any solution to (1.10); for a detailed explanation, see for instance [24]. In view of the assumption (1.9), this is also equivalent to the simpler estimate ϕ(·,0)2 L2≤Cω×(0,T ) |ϕ|2.(2.13) Therefore, let us check that the Carleman inequality (2.6), together with the usual parabolic energy estimates, imply (2.13)forsomeC. Indeed, let us take (for example) m=3,s=s 3and λ=λ 3in (2.6). In view of the energy estimates −d dtϕ2 L2+2a0∇ϕ2 L2≤Rϕ2 L2, we find that ϕ(·,0)2 L2≤2 TeRT 3T/4 T/4 ϕ(·,t)2 L2dt. (2.14) The left hand side of (2.6) is bounded from below as follows: I0(s 3,λ  3;ϕ)≥Q ρ−2s 3λ4 3|ϕ|2 ≥R3T/4 T/4 e−2s 3α∗(t)ϕ(·,t)2 L2dt, (2.15) where α∗(t):=max Ω α(x, t)≤R T2· Since s 3has the form s 3=R(T+T2), the following is obtained: I0(s 3,λ  3;ϕ)≥e−R(1+ 1 T)3T/4 T/4 ϕ(·,t)2 L2dt and this inequality, together with (2.14), yields: I0(s 3,λ  3;ϕ)≥e−R(1+T+1 T)ϕ(·,0)2 L2. CONTROLLABILITY OF LINEAR AND SEMILINEAR NON-DIAGONALIZABLE PARABOLIC SYSTEMS 1193 where the pij are the components of Id ∂t+A∗Δ+M∗.Thus, P(∂t,Δ)ϕ1= σ∈Pnpσ(2),2...p σ(n),n(pσ(1),1ϕ1) =− σ∈Pnpσ(2),2...p σ(n),n( n  j=2 pσ(1),jϕj) =− n  j=2  σ∈Pnpσ(1),jpσ(2),2...p σ(n),nϕj:= − σ∈Pn ˜pjϕj. But all the operators in this last sum vanish, since each of them can be written as the determinant of a square matrix with two columns that are identical. Consequently, we certainly have P(∂t,Δ)ϕ1=0. Obviously, this argument also holds for ϕ2,...,ϕ n.Thus,wecanwrite(4.10) for any component of B∗ϕ. This gives the following inequality for all j, k ≥0andall=1,...,n : Q ρ−2s|(B∗((−Δ)k∂j tϕ))|2=Q ρ−2s|(−Δ)k∂j t(B∗ϕ)|2 ≤C(k,j)ω×(0,T ) |(B∗φ)|2. Let us fix k≥0. By introducing ˆρ(t):=max Ω ρ(x, t), and recalling (4.4), we get Q ˆρ(t)−2s|(−Δ)kK∗ϕ|2≤C n  =1 Q ρ−2s|(B∗((−Δ)k∂j tϕ)|2 ≤C n  =1 C(k,j)ω×(0,T ) |(B∗ϕ)|2 ≤Cω×(0,T ) |B∗ϕ|2. This yields (4.3). 4.3. Conclusion and end of the proof of Theorem 1.5 In view of (4.2)and(4.3)fork=2(n−1)2, the following holds for any solution to (1.10) associated to a final data ϕTsatisfying (4.13): Q ˆρ(t)−2s|ϕ|2=T 0 ˆρ(t)−2sΩ |ϕ(x, t)|2dxdt ≤R(k)Q ˆρ(t)−2s|(−Δ)k(K∗ϕ)|2 ≤CR(k)ω×(0,T ) |B∗ϕ|2. 1194 E. FERN´ ANDEZ-CARA ET AL. In particular, Q ˆρ(t)−2s|ϕ|2≤Cω×(0,T ) |B∗ϕ|2. By density, it is obvious that this inequality remains true for the solutions to (1.10) corresponding to arbitrary final data in L2(Ω;Rn). Hence, arguing as in the final part of the proof of Theorem 1.1 in Section 2,weget (4.1), which shows that, under the assumptions of Theorem 1.5,(1.1) is null-controllable. 4.4. Proof of Lemma 4.1 We will adapt the arguments in [5]. The fact that Ais not diagonalizable introduces some nontrivial complications and the computations and estimates are more involved; but the idea is similar. We will prove (4.10) by induction on kand j. 4.4.1. Step 1: Proof of (4.10)for k=j=0 Let us see that, if sand λare large enough, one has I12(s, λ;φ)≤C(0,0) ω(0,0)×(0,T) (t(T−t))−m(0,0)ρ−2s|φ|2(4.14) for some m(0,0), C(0,0) and ω(0,0). Again,itcanbeassumedthatAis in the Jordan canonical form. We then have for some p≥1 Id ∂t+A∗Δ+M∗=⎡ ⎢ ⎢ ⎢ ⎣ H1(∂t,Δ)M∗ 21 ... M∗ p1 M∗ 12 H2(∂t,Δ)... M∗ p2 . . .. . ..... . . M∗ 1pM∗ 2p...H p(∂t,Δ) ⎤ ⎥ ⎥ ⎥ ⎦, wherewehaveintroducedthenon-scalaroperatorsHi(∂t,Δ):=Id ∂t+J∗ iΔ+M∗ ii,theJ∗ iare Jordan blocks, i.e. each of them is of the form (2.7)forsomeμi∈Cand the Mij provide the corresponding block decomposition of M. The PDE (4.12) can be written in the form p  i=1 det Hi(∂t,Δ)φ=F(φ).(4.15) In the terms in F(φ) we find the composition of at most p−2 operators of the kind det Hj(∂t,Δ) applied to φ. Let us introduce the functions ψi,with ψ1=φ, ψ2=detH1(∂t,Δ)ψ1, ..., ψ p=detHp−1(∂t,Δ)ψp−1. Then we can rewrite (4.15) as the following system for the ψi: ⎧ ⎪ ⎨ ⎪ ⎩ det Hp(∂t,Δ)ψp=F(φ), det Hp−1(∂t,Δ)ψp−1=ψp, ... ... ... det H1(∂t,Δ)φ=ψ2. (4.16) Recall that, by hypothesis, we also have φ=ψ2=...=ψp=0 on Σ. (4.17) We will now provide global estimates of ψpand its derivatives in terms of local estimates of ψpand (lower order) estimates of F(φ); then, global estimates of ψp−1and its derivatives in terms of local estimates of ψp−1 CONTROLLABILITY OF LINEAR AND SEMILINEAR NON-DIAGONALIZABLE PARABOLIC SYSTEMS 1195 and lower order estimates of ψp;etc. And, finally, global estimates of φand its derivatives in terms of local estimates of φand (lower order) estimates of ψ2. An appropriate combination of these estimates will lead to an inequality where we find, in the left hand side, global weighted integrals of φ,ψ2,... and,in the right, only local integrals and a global weighted integral of |F(φ)|2. Thus, let us consider the first PDE in (4.16). For instance, assume that Jpis a Jordan block of dimension r associated to the complex eigenvalue αwith α>0andletusdenotebyη1,...,η rthe diagonal components of Mpp; by assumption, r≤4. Then, this PDE can be rewritten in the form r  i=1 (∂t+αΔ +ηi)ψp=F(φ)−G(ψp),(4.18) where G(ψp) is a linear combination of partial derivatives of ψp. Let us introduce the new variables ζ1=ψp,ζ 2=(∂t+αΔ +ηr)ζ1, ..., ζ r=(∂t+αΔ +η2)ζr−1. Now, we can rewrite (4.18) as a first-order system for the ζi: ⎧ ⎪ ⎨ ⎪ ⎩ (∂t+αΔ +η1)ζr=F(φ)−G(ψp), (∂t+αΔ +η2)ζr−1=ζr, ... ... ... (∂t+αΔ +ηr)ζ1=ζ2. (4.19) Again, we have information on the ζion the lateral boundary: ζ1=...=ζr=0 on Σ. (4.20) In the “worst” possible case, we have r=4and(4.19)and(4.20) respectively read ⎧ ⎪ ⎨ ⎪ ⎩ (∂t+αΔ +η1)ζ4=F(φ)−G(ψp), (∂t+αΔ +η2)ζ3=ζ4, (∂t+αΔ +η3)ζ2=ζ3, (∂t+αΔ +η4)ζ1=ζ2, (4.21) and ζ1=ζ2=ζ3=ζ4=0 on Σ. (4.22) Notice that |G(ψp)|2is bounded by a sum of squares of derivatives of ψp. More precisely, we have |G(ψp)|2≤ CIG(ψp), with IG(ψp):= 3  a=0 |(−Δ)aψp|2+ 4  j=1 2  b=0 |(−Δ)b(∂t+αΔ +ηj)ψp|2 + 4  j,k=1 1  c=0 |(−Δ)c(∂t+αΔ +ηj)(∂t+αΔ +ηk)ψp|2. In view of the Carleman estimates (2.4) (established in Lem. 2.2) applied to the functions ζi,wehave: I3(s, λ;ζ4)≤CQ ρ−2s|F(φ)|2+IG(ψp)+CI3,ω0(s, λ;ζ4), λ4I6(s, λ;ζ3)≤CQ ρ−2s(sξ)3λ4|ζ4|2+Cλ4I6,ω0(s, λ;ζ3), λ8I9(s, λ;ζ2)≤CQ ρ−2s(sξ)6λ8|ζ3|2+Cλ8I9,ω0(s, λ;ζ2), λ12I12(s, λ;ζ1)≤CQ ρ−2s(sξ)9λ12|ζ2|2+Cλ12I12,ω0(s, λ;ζ1), for any λ≥λ0and s≥s0=σ0(T+T2). 1196 E. FERN´ ANDEZ-CARA ET AL. Consequently, an appropriate linear combination of the terms in the left hand sides absorbes the global weighted integrals of |ζ4|2,|ζ3|2and |ζ2|2.Moreprecisely,fors≥s0and λ≥λ0,weget: I3(s, λ;ζ4)+λ4I6(s, λ;ζ3)+λ8I9(s, λ;ζ2)+λ12I12(s, λ;ζ1) ≤C(I3,ω0(s, λ;ζ4)+λ4I6,ω0(s, λ;ζ3)+λ8I9,ω0(s, λ;ζ2)+λ12I12,ω0(s, λ;ζ1)) +CQ ρ−2s|F(φ)|2+CQ ρ−2sIG(ψp).(4.23) The next task will be to add some extra terms on the left hand side of the previous inequality. To this end, we reason as follow. We apply −Δto the second PDE in (4.21) and we use Lemma 2.1 for the resulting equation with m= 2. Notice that this is possible, since −Δζ3=0onΣ. We find: I2(s, λ;Δζ3)≤Cω0×(0,T) ρ−2s(sξ)2λ4|Δζ3|2)+CQ ρ−2s(sξ)−1|Δζ4|2 ≤Cλ4I6(s, λ;ζ3)+I3(s, λ;ζ4). Observe that the previous argument can be applied, this time, to the third and fourth PDE in (4.21). Applying −Δand using Lemma 2.1 for the corresponding equations with m=5andm= 8, we deduce: λ4I5(s, λ;Δζ2)≤Cω0×(0,T ) ρ−2s(sξ)5λ8|Δζ2|2)+CQ ρ−2s(sξ)2λ4|Δζ3|2 ≤Cλ8I9(s, λ;ζ2)+λ4I6(s, λ;ζ3) and λ8I8(s, λ;Δζ1)≤Cω0×(0,T) ρ−2s(sξ)8λ12|Δζ1|2)+CQ ρ−2s(sξ)5λ8|Δζ2|2 ≤Cλ12I12(s, λ;ζ1)+λ8I8(s, λ;ζ2). Then, we can add all these new terms to the left hand side of (4.23) and, for a new positive constant C, obtain I3(s, λ;ζ4)+λ4I6(s, λ;ζ3)+λ8I9(s, λ;ζ2)+λ12I12(s, λ;ζ1) +I2(s, λ;Δζ3)+λ4I5(s, λ;Δζ2)+λ8I8(s, λ;Δζ1) ≤C(I3,ω0(s, λ;ζ4)+λ4I6,ω0(s, λ;ζ3)+λ8I9,ω0(s, λ;ζ2)+λ12I12,ω0(s, λ;ζ1)) +CQ ρ−2s|F(φ)|2+CQ ρ−2sIG(ψp).(4.24) We can continue the previous process and add better global terms on the left hand side of (4.24). Thus, if we apply (−Δ)2to the third PDE in (4.21) and we use again Lemma 2.1 for the corresponding equations with m=1,weget I1(s,λ;(−Δ)2ζ2)≤Cω0×(0,T) ρ−2ssξλ4|(−Δ)2ζ2|2)+CQ ρ−2s(sξ)−2|Δζ3|2 ≤Cλ4I5(s, λ;Δζ2)+I2(s, λ;Δζ3). The previous argument, this time applied to the last PDE in (4.21), also gives λ4I4(s, λ;(−Δ)2ζ1)≤Cλ8I8(s, λ;Δζ1)+λ8I5(s, λ;Δζ2). CONTROLLABILITY OF LINEAR AND SEMILINEAR NON-DIAGONALIZABLE PARABOLIC SYSTEMS 1197 Putting the previous inequalities in (4.24), we obtain I3(s, λ;ζ4)+λ4I6(s, λ;ζ3)+λ8I9(s, λ;ζ2)+λ12I12(s, λ;ζ1) +I2(s, λ;Δζ3)+λ4I5(s, λ;Δζ2)+λ8I8(s, λ;Δζ1) +I1(s, λ;(−Δ)2ζ2)+λ4I4(s, λ;(−Δ)2ζ1) ≤C(I3,ω0(s, λ;ζ4)+λ4I6,ω0(s, λ;ζ3)+λ8I9,ω0(s, λ;ζ2)+λ12I12,ω0(s, λ;ζ1)) +CQ ρ−2s|F(φ)|2+CQ ρ−2sIG(ψp).(4.25) Finally, let us take (−Δ)3to the last PDE of (4.21). Again, from the regularity assumptions on φwe have (−Δ)3ζ1=0onΣ. So, we can apply Lemma 2.1 with m= 0 to the resulting PDE and deduce I0(s, λ;(−Δ)3ζ1)≤Cλ4I4(s, λ;(−Δ)2ζ1)+I1(s, λ;(−Δ)2ζ2). This inequality together with (4.25)provides I3(s, λ;ζ4)+λ4I6(s, λ;ζ3)+λ8I9(s, λ;ζ2)+λ12I12(s, λ;ζ1) +I2(s, λ;Δζ3)+λ4I5(s, λ;Δζ2)+λ8I8(s, λ;Δζ1) +I1(s, λ;(−Δ)2ζ2)+λ4I4(s, λ;(−Δ)2ζ1) +I0(s, λ;(−Δ)3ζ1) ≤C(I3,ω0(s, λ;ζ4)+λ4I6,ω0(s, λ;ζ3)+λ8I9,ω0(s, λ;ζ2)+λ12I12,ω0(s, λ;ζ1)) +CQ ρ−2s|F(φ)|2+CQ ρ−2sIG(ψp), for a new positive constant C. Let us denote by Jtot(s, λ;ζ) the sum of the terms on the left hand side of the previous inequality (with ζ=(ζ1,ζ 2,ζ 3,ζ 4)). Then, the previous inequality can be written as Jtot(s, λ;ζ)≤C(I3,ω0(s, λ;ζ4)+λ4I6,ω0(s, λ;ζ3)+λ8I9,ω0(s, λ;ζ2) +λ12I12,ω0(s, λ;ζ1)) + CQ ρ−2s|F(φ)|2+Q ρ−2sIG(ψp),(4.26) with s≥s1=σ1(T+T2)andλ≥λ1. Let ω(0,0) be an open set satisfying ω0⊂⊂ ω(0,0) ⊂⊂ ω1.Fromnowon,wefixs≥s1and λ≥λ1and we try to replace the local terms in (4.26) corresponding to ζ2,ζ3and ζ4by a term of the form (ψp=ζ1) Cλ1ω(0,0)×(0,T) ρ−2s(sξ)2|ψp|2, where 1and 2are nonnegative integers. We need some lengthy computations, but using the cascade structure of the system (4.21), the process is clear. For instance, let us see what can be done with the local term corresponding to ζ4. First, we introduce an open subset ω0,withω0⊂⊂ ω0⊂⊂ ω(0,0), and a cut-off function χ=χ(x)with χ∈C∞ 0(ω0), χ≥0andχ≡1inω0and we write ζ4=(∂t+αΔ +η2)ζ3(see (4.21)) I3,ω0(s, λ;ζ4)≤λ4ω0×(0,T) ρ−2s(sξ)3χ|ζ4|2 =λ4ω0×(0,T) ρ−2s(sξ)3χ(∂t+αΔ +η2)ζ3ζ4=I1+I2+I3. 1198 E. FERN´ ANDEZ-CARA ET AL. Let us take μin (2.2)insuchawaywehave e(λ+μ)α0L∞−eλα0(x)≥eλα0(x)≥1,∀x∈Ω. Then, we also have |∇ ρ−2s(sξ)|≤Cλρ−2s(sξ)+1,|Δρ−2s(sξ)|≤Cλ2ρ−2s(sξ)+2, ∂t(ρ−2s(sξ))≤Cρ−2s(sξ)+2 (4.27) for any (x, t)∈Qand s≥s1and λ≥λ1. Using the previous inequalities, we deduce I1=−λ4ω0×(0,T) ∂tρ−2s(sξ)3χζ3ζ4−λ4ω0×(0,T ) ρ−2s(sξ)3χζ3∂tζ4 ≤Cλ4ω0×(0,T) ρ−2s(sξ)5χζ3ζ4+λ4ω0×(0,T ) ρ−2s(sξ)3χζ3∂tζ4 ≤εI3(s, λ;ζ4)+C ελ8ω0×(0,T ) ρ−2s(sξ)7|ζ3|2, with ε>0. We can also bound I2=αλ4ω0×(0,T) Δρ−2s(sξ)3χζ4ζ3 =αλ4ω0×(0,T ) [Δρ−2s(sξ)3ζ4+2∇ρ−2s(sξ)3·∇ ζ4]χζ3 +αλ4ω0×(0,T ) ρ−2s(sξ)3Δζ4χζ3+... where the dots contain terms of lower order. Now, we apply the Cauchy-Schwarz inequality and inequality (4.27). So, I2≤εI3(s, λ;ζ4)+C ελ8ω0×(0,T ) ρ−2s(sξ)7|ζ3|2. Finally, I3≤εI3(s, λ;ζ4)+C ελ4ω0×(0,T ) ρ−2s(sξ)3|ζ3|2. Putting the previous inequalities together, we get I3,ω0(s, λ;ζ4)≤εI3(s, λ;ζ4)+C ελ4ω0×(0,T ) ρ−2s(sξ)7|ζ3|2. The previous inequality is valid for any ε>0, s≥s1and λ≥λ1. Coming back to (4.26), if we take εsmall enough, we obtain Jtot(s, λ;ζ)≤C(λ8I7,ω0(s, λ;ζ3)+λ8I9,ω0(s, λ;ζ2)+λ12I12,ω0(s, λ;ζ1)) +CQ ρ−2s|F(φ)|2+Q ρ−2sIG(ψp) for a new positive constant C. CONTROLLABILITY OF LINEAR AND SEMILINEAR NON-DIAGONALIZABLE PARABOLIC SYSTEMS 1199 Observe that the previous reasoning can be applied twice in order to eliminate the local terms corresponding to ζ3and ζ2. The resulting inequality is Jtot(s, λ;ζ)≤Cλ1ω1×(0,T) ρ−2s(sξ)2|ψp|2+CQ ρ−2s|F(φ)|2+Q ρ−2sIG(ψp),(4.28) with s≥s1=σ1(T+T2)andλ≥λ1. In inequality (4.28)ω1is a new open subset satisfying ω0⊂⊂ ω1⊂⊂ ω(0,0) and 1and 2are nonnegative intergers. Now, taking into account that the operators ∂t+αΔ +ηicommute, we see that (4.18)(forr=4)canbe rewritten equivalently in the form 4  i=1 (∂t+αΔ +ησ(i))ψp=F(φ)−G(ψp), where σis any permutation in P4. Hence, we can introduce the new variables ζσ 1=ψp,ζ σ 2=(∂t+αΔ +ησ(4))ζσ 1, ..., ζσ 4=(∂t+αΔ +ησ(2))ζσ 3 and we can also write (4.18) as a similar first-order system for the ζσ i: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ (∂t+αΔ +ησ(1))ζσ 4=F(φ)−G(ψp). (∂t+αΔ +ησ(2))ζσ 3=ζσ 4, (∂t+αΔ +ησ(3))ζσ 2=ζσ 3, (∂t+αΔ +ησ(4))ζσ 1=ζσ 2. (4.29) Again, we have: ζσ 1=ζσ 2=ζσ 3=ζσ 4=0 on Σ. (4.30) Arguing as before, we obtain an estimate like (4.28) where, now, we have in the left global weighted integrals of ζσ 1=ψp,ζσ 2,ζσ 3and ζσ 4and, in the right, terms concerning F(φ)andIG(ψp): Jtot(s, λ;ζσ)≤Cλ1ω1×(0,T) ρ−2s(sξ)2|ψp|2 +CQ ρ−2s|F(φ)|2+Q ρ−2sIG(ψp). In this inequality we have used the notation ζσ=(ζσ 1,ζσ 2,ζσ 3,ζσ 1). Let us denote by Itot(s, λ;ψp) the sum of all these left hand sides, obtained for all σ∈P 4.Then Itot(s, λ;ψp)≤Cλ1ω1×(0,T) ρ−2s(sξ)2|ψp|2 +CQ ρ−2s|F(φ)|2+Q ρ−2sIG(ψp). Observe that all the terms in IG(ψp) except |(−Δ)3ψp|2are also in the left multiplied by weights of the form (sξ)aρ−2swith a>0. Consequently, for sufficiently large s, these terms are absorbed and we find: Itot(s, λ;ψp)≤Cλ1ω1×(0,T ) ρ−2s(sξ)2|ψp|2 +CQ ρ−2s|F(φ)|2+CQ ρ−2s|(−Δ)3ψp|2. 1200 E. FERN´ ANDEZ-CARA ET AL. Finally the global term corresponding to ρ−2s|(−Δ)3ψp|2can be absorbed by the term I0(s, λ;(−Δ)3ψp) (appearing in the expression of Itot(s, λ;ψp)) taking λlarge enough. Hence, Itot(s, λ;ψp)≤Cλ1ω1×(0,T) ρ−2s(sξ)2|ψp|2 +CQ ρ−2s|F(φ)|2,∀s≥s2=σ2(T+T2),λ≥λ2, for some n(0,0),m(0,0) ≥1. From now on, we fix λ=λ2. Let us now consider the second PDE in (4.16). Arguing in the same way (and assuming again that we are in the worst possible situation, associated to a block of dimension 4), we deduce the following estimate for ψp−1: I12(s, λ;ψp−1)≤CQ ρ−2s|ψp|2+ω1×(0,T ) (sξ)2ρ−2s|ψp−1|2. The corresponding similar estimate also holds for ψp−2,etc. Thus, after addition and taking into account that ψ1=φand the global integrals of ψp,..., ψ2in the right hand side are smaller than the terms in the left, we get an estimate for all the ψi: p  i=1 I12(s, λ;ψi)≤CQ ρ−2s|F(φ)|2+ω1×(0,T ) (sξ)2ρ−2s|φ|2+ p  i=2 ω1×(0,T ) (sξ)2ρ−2s|ψi|2. Again, using the cascade structure of system (4.16), all the local integrals in the right can be absorbed by the left hand side, with the exception of the local weighted integral of |φ|2. All we have to do is to enlarge the open set ω1and argue like in the passage from (4.26)to(4.28). Therefore, the following is obtained: p  i=1 I12(s, λ;ψi)≤CQ ρ−2s|F(φ)|2+ω(0,0)×(0,T) (sξ)m(0,0)ρ−2sχ|φ|2. However, we see that, taking into account that the operators det Hi(∂t,Δ)commute,(4.16) can also be writtenintheform p  i=1 det Hσ(i)(∂t,Δ)φ=F(φ), where σis any permutation in Pn. This means that another equivalent formulation of (4.12)is ⎧ ⎪ ⎨ ⎪ ⎩ det Hσ(p)(∂t,Δ)ψσ p=F(φ), det Hσ(p−1)(∂t,Δ)ψσ p−1=ψσ p, ... ... ... det Hσ(1)(∂t,Δ)φ=ψσ 2, and we can also get an estimate of the same form where, now, we have in the left global weighted integrals of φ,ψσ 2, ..., ψσ p. Recall that F(φ) is a sum of terms where, at most, p−2 operators of the kind det Hj(∂t,Δ) are applied to φ.Sinceσis arbitrary in Pn, using all these possible estimates together and arguing as above, it becomes also clear that the terms containing |F(φ)|2can be controlled by the terms in the left. This proves (4.14). CONTROLLABILITY OF LINEAR AND SEMILINEAR NON-DIAGONALIZABLE PARABOLIC SYSTEMS 1201 4.4.2. Step 2: Induction on kand j Now, let us assume that (4.10) is true for any k=0,1,...,k,anyj=0,1,...,j and any solution to (4.12) satisfying the assumptions of the lemma and let us prove (4.10) (for instance) with kreplaced by k+1;the proof with the same kand jreplaced by j+ 1 is essentially the same. Since ˆ φ:= (−Δ)φalso satisfies (4.11)and(4.12), we have by hypothesis I12(s, λ;(−Δ)k∂j tˆ φ)≤C(k,j)ω(k,j)×(0,T ) (t(T−t))−m(k,j)ρ−2s|ˆ φ|2, that is, I12(s, λ;(−Δ)k+1∂j tφ)≤C(k,j)ω(k,j)×(0,T ) (t(T−t))−m(k,j)ρ−2s|Δφ|2.(4.31) Let us set ˜m=m(k,j)and˜ω=ω(k,j), let ω∗be an open set satisfying ˜ω⊂⊂ ω∗⊂⊂ ωand let χ∗=χ∗(x) be a new cut-off function, with χ∗∈C∞ 0(ω∗), χ∗≥0andχ∗≡1in˜ω. Then, for some integer m∗≥m(k,j), one has ω(k,j)×(0,T ) (t(T−t))−m(k,j) ρ−2s|Δφ|2 ≤ω∗ ×(0,T) (t(T−t))−m(k,j) ρ−2sχ∗|Δφ|2 =ω∗ ×(0,T) (t(T−t))−m(k,j) ρ−2sχ∗φ(−Δ)2φ+... ≤ω∗ ×(0,T) (t(T−t))−m∗ ρ−2sχ∗ |φ|2+CI12(s, λ;Δφ)+... ≤ω∗ ×(0,T) (t(T−t))−m∗ ρ−2sχ∗ |φ|2 +Cω∗ ×(0,T) (t(T−t))−m(1,0) ρ−2s|φ|2+..., where the dots denote again lower order terms. It is thus clear that there exist m(k+1,j), C(k+1,j)andω(k+ 1,j) such that I12(s, λ;(−Δ)k+1∂j tφ)≤C(k+1,j) ω(k+1,j)×(0,T ) (t(T−t))−m(k+1,j) ρ−2s|φ|2. This ends the proof of the lemma.  5. Further comments and open questions This section is devoted to make some comments on extensions of the previous results and, also, to report some related open problems. The first question is whether the hypothesis (1.3) can be eliminated or at least weakened in Theorems 1.1 and 1.4. This is not clear; of course, as noticed in Remark 2.4, without new tools for the proof of (2.13), different from global (scalar) Carleman estimates, it seems very difficult to obtain the same results for dmax ≥5. In the one-dimensional case, when the matrix Mdoes not depend on t, a possible alternative is the reformulation of the null controllability problem for (1.1) as a moment problem. This has been done in some recent papers (see for instance [7,11,13,20]) given boundary controllability characterizations for some coupled parabolic systems. Recall that the analysis and methods in Section 2can be performed for the more general coupled systems ⎧ ⎪ ⎨ ⎪ ⎩ yt−ASy =M(x, t)y+ N k=1 Wk(x, t)∂ky+v1ωin Q, y=0 on Σ, y(x, 0) = y0(x)inΩ 1202 E. FERN´ ANDEZ-CARA ET AL. and ⎧ ⎪ ⎨ ⎪ ⎩ yt−ASy =M(x, t)y+ N k=1 ∂k(Wk(x, t)y)+v1ωin Q, y=0 on Σ, y(x, 0) = y0(x)inΩ, where Asatisfies (1.2)and dmax ≤2,(5.1) Sis a second-order partial differential operator of the form Sz := N  i,j=1 aij(x, t)∂i∂jz with aij =aji ∈W1,∞(Q) for all i, j and N  i,j=1 aij (x, t)ηiηj≥a0|ξ|2∀η∈RN,∀(x, t)∈Q, a0>0 and Wk∈L∞(Q;L(Rn;Rn)) for all k=1,...,N. This also leads to controllability results for semilinear systems of a more general class than (1.11). More precisely, let us consider the system ⎧ ⎨ ⎩ yt−AΔy =F(y,∇y)+v1ωin Q, y=0 on Σ, y(x, 0) = y0(x)inΩ, (5.2) where F:Rn×Rn×N→ Rnis given. We then have the following result: Theorem 5.1. Let Aand Bbe as in Theorem 1.1 and asume that (1.2)and (5.1)hold and F:Rn×Rn×N→ Rnis globally Lipschitz-continuous. Then (5.2)is exactly controllable to the trajectories. The proof is similar to the proof of Theorem 1.4. The details are left to the reader; they rely on the ideas in [30], see the proof of Theorem 1.1 in [17]. Of course, it is again unknown whether the assumption dmax ≤2 can be suppressed. For general linear systems of the kind (1.1), it is an open question to characterize those n≤nand B∈ L(Rn;Rn) such that null controllability holds. Up to now, this is known only for constant matices M,as indicated in Theorem 1.5;see Remarks 1.2 and 1.3;see also [1,2] for some results in this direction and the recent paper [16], where the authors have introduced other techniques that could shed some light to this question. It is also meaningful to consider boundary controllability problems for systems similar to (1.1)and(1.11). For instance, it makes sense to analyze the null controllability of ⎧ ⎨ ⎩ yt−AΔy =M(x, t)yin Q, y=Bv 1γon Σ, y(x, 0) = y0(x)inΩ, (5.3) where γ⊂∂Ω is a non-empty set, with controls vfor instance in L∞(γ×(0,T); Rn). It also makes sense to analyze the exact controllability to the trajectories for the semilinear system ⎧ ⎨ ⎩ yt−AΔy =f(y)inQ, y=Bv 1γon Σ, y(x, 0) = y0(x)inΩ. (5.4)