c 2007 Society for Industrial and Applied Mathematics

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1 SIAM J. CONTROL OPTIM. Vol. 46, No. 2, pp c 2007 Society for Industrial and Applied Mathematics NULL CONTROLLABILITY OF SOME SYSTEMS OF TWO PARABOLIC EUATIONS WITH ONE CONTROL FORCE SERGIO GUERRERO Abstract. In this paper we establish some exact controllability results for systems of two parabolic equations. First, we prove the existence of insensitizing controls for the L 2 norm of the gradient of solutions of linear heat equations. Then, in the worst situation where null controllability for a system of two parabolic equations can hold, we prove this result for some general couplings. Key words. Carleman inequalities, system of parabolic equations, controllability AMS subject classifications. 35K40, 93B05 DOI / Introduction. Let Ω R N (N 1 be a bounded connected open set whose boundary Ω is regular enough. Let T > 0, and let ω Ω be a (small nonempty open subset which will usually be referred to as a control domain. We will use the notation =Ω (0,T and Σ = Ω (0,T. The main objective of this paper is to establish some new controllability results for coupled parabolic equations. The first main result of this paper concerns insensitizing controls. More precisely, we want to insensitize a functional associated with a state system, which is a linear parabolic equation. Let us introduce an open set O Ω, which is called the observatory (or observation open set. In order to state our problem, we introduce the following system: y t Δy + ay + B y = v1 ω + f in, (1 y = 0 on Σ, y t=0 = y 0 + τŷ 0 in Ω. Here, v is the control, y 0 L 2 (Ω, and a R and B R N are constants. Furthermore, we suppose that ŷ 0 is unknown with ŷ 0 L 2 (Ω = 1 and that τ is a small unknown real number. Then, the interpretation of system (1 is that y is the temperature of a body, v is a localized heat source, (where we have access to the body to be chosen, f is another heat source, and the initial state of the body is partially unknown. In general, the functional J τ we would like to insensitize (which is called sentinel has to be differentiable. In this framework, the task is to find a control v such that the influence of the unknown data τŷ 0 is not perceptible for J τ (see (3 below. In the literature, the usual functional is given by the L 2 norm of the state (see [4], [6], or [19], for instance. Here, we are interested in insensitizing the L 2 norm of Received by the editors February 27, 2006; accepted for publication (in revised form October 18, 2006; published electronically April 17, Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Boîte Corrier 187, Cedex 05 Paris, France (guerrero@ann.jussieu.fr. 379

2 380 SERGIO GUERRERO the gradient of the state (solution of (1. Thus, let us introduce the functional (2 J τ (y = y 2 dx dt, O (0,T where y is the solution of (1. Our objective is to find a control v such that the presence of the unknown data is imperceptible for J τ, that is to say, such that J τ (3 τ (y τ=0 =0 ŷ 0 L 2 (Ω such that ŷ 0 L 2 (Ω =1. If this holds, we will say that the control v insensitizes the functional J τ. Usually, insensitizing problems are formulated in an equivalent way as a controllability problems of a cascade system (see, for instance, [17] and [4] for a rigorous deduction of this fact. Indeed, if we consider the adjoint state of (1 (or apply the Lagrange principle, it is very easy to see that condition (3 is equivalent to w t=0 0 in Ω, where w together with z fulfills z t Δz + az + B z = v1 ω + f in, w t Δw + aw B w = ( z1 O in, (4 z =0,w= 0 on Σ, z t=0 = y 0,w t=t = 0 in Ω. Here, we have denoted z y τ=0. Assume for a moment that y 0 L 2 (Ω and f, v L 2 (. Then, it is not difficult to prove that there exists a unique solution (z,w of (4 which belongs to L 2 (0,T; H0 1 (Ω 2 and depends continuously on (y 0,f,v in L 2 (Ω L 2 ( 2. To our best knowledge, the first time this kind of problem was addressed was in [18] for second and fourth order parabolic equations of the heat kind and for the Navier Stokes system. As we said above, all results around this subject concern the functional Jτ (y = y 2 L 2 (O (0,T with y a solution of a parabolic system. In [4], the authors prove the existence of ε-insensitizing controls (i. e., such that J τ (y τ=0 ε for solutions of a semilinear heat system with C 1 and globally Lipschitz nonlinearities. In [6], the author proved the existence of insensitizing controls for the same system. For an extension of this result to more general nonlinearities, see [5] and the references therein. As we shall see in the statement of Theorem 1 below, we will take y 0 0. For a justification of this fact and a possible choice of more general initial conditions, see [6]. Throughout this paper we will suppose that ω O. This is a condition that has always been imposed in the literature where insensitizing controls are concerned. Recently, for the (simpler situation of looking for an ε-insensitizing control and the functional Jτ, it has been demonstrated that this condition is not necessary for solutions of linear heat equations (see [7]. The controllability result for system (4 is given in the following theorem. Theorem 1. Let m>3 be a real number and y 0 0. Then, there exists a constant K 0 > 0 depending on Ω,ω,O,T,a, and B such that for any f L 2 ( satisfying e K0/tm f L 2 ( < +, there exists a control v such that the corresponding solution (w, z of (4 satisfies w t=0 0 in Ω.

3 CONTROLLABILITY OF SYSTEMS OF PARABOLIC EUATIONS 381 Corollary 2. There exists insensitizing controls v of the functional J τ given by (2. Remark 1. The same result holds when a and B are functions which depends only on the time variable t and are in L (0,T. The proof of this fact is direct from that of Theorem 1. Let us briefly explain the difficulties a controllability result for system (4 possesses. For this, we introduce the associated adjoint system: ψ t Δψ + aψ B ψ = (( ϕ1 O in, ϕ t Δϕ + aϕ + B ϕ =0 in, (5 ψ =0,ϕ= 0 on Σ, ψ t=t =0,ϕ t=0 = ϕ 0 in Ω. It is by now a classical fact that the null controllability result we want to prove for system (4 is equivalent to the following observability inequality: (6 e K0/tm ϕ 2 dx dt C ψ 2 dx dt, where m is some positive number and C and K 0 are two positive constants depending on Ω,ω,O,T, a, and B but independent of ϕ 0 (see, for instance, [14] or [11]. The main idea one usually follows in order to prove (6 is a combination of observability inequalities for ψ and ϕ (as solutions of heat equations and trying to eliminate the local term (concentrated in ω (0,T concerning ϕ. The great difficulty one encounters when trying this for system (5 is that no local estimate of the kind ϕ 2 dx dt C Δϕ 2 dx dt, ω ω, can be obtained using local arguments (observe that ω can be taken as small as we want, so we can always suppose that ω Ω =. This means that we have to find another way to locally relate ϕ and ψ. The idea we follow here is to first obtain an observability inequality of the kind (7 e K1/tm ϕ 2 dx dt C Δϕ 2 dx dt. The reason why an estimate like (7 is not easy to prove relies on the fact that no boundary conditions are known for Δϕ. More details about this are given in subsection 2.1, below. Remark 2. Is this result true when the coefficients a and B depend on the space variable? We observe here that, in this situation, not even the following unique continuation property is known: ψ =0 inω (0,T ϕ, ψ 0 in Ω (0,T. As an extension of the result stated in Theorem 1, some insensibilization properties have recently been demonstrated for the more complicated situation of a system

4 382 SERGIO GUERRERO of the Stokes kind. More precisely, we consider the functional J τ, with y the solution of y t Δy + ay + B y + p = v1 ω + f in, y =0 in, y = 0 on Σ, y t=0 = τŷ 0 in Ω, with a and B constants. In [13], the existence of controls insensitizing J τ is established. See also [13] for an extension of this to more general functionals and further controllability results for coupled Stokes-like systems. The second and main objective of this paper is to extend the previous controllability result to more intrinsic coupled parabolic systems. The question is, Which coupling do we need to be able to control the whole system with only one control force? As far as the controllability of strongly coupled parabolic equations is concerned, in the literature the local exact controllability of phase field systems was proved in [3], while the global version was later proved in [1]. For more general coupled parabolic equations with only one control force, some results have been given in [2] and [12]. In this paper, we will concentrate on studying the null controllability of systems of two parabolic equations, where the coupling terms are first order space derivatives in one equation and second order space derivatives in the other. In this situation, we will be again interested in controlling only one of the two equations while driving both states to zero at t = T. We consider, for instance, the case where we control the lower order coupling term equation. We set the following control coupled system: w t Δw + cw + E w = P 2 (t, x; D(zθ 2 in, z t Δz + hz + K z = P 1 (t, x; D(wθ 1 +v1 ω in, (8 w = z = 0 on Σ, w t=0 = w 0, z t=0 = z 0 in Ω, where c, E, h, and K are constants and P i (t, x; D (i =1, 2 is a partial differential operator in the space variables of order i such that (9 P i (t, x; Du = m i,β (t, x x β u, m i,β L (0,T; W 2/i, (Ω β i (that is to say, m 2,β, x (m 2,β,m 1,β, x (m 1,β, x(m 2 1,β L (. In (8, θ i C 2 (Ω (1 i 2. We assume that there exists a nonempty open set ω 2 ω and a constant C>0such that θ 2 C>0inω 2. Observe that, in particular, one can take θ 1 and θ 2 to have a support as small as we want (one can also take θ 1 θ 2 1 in Ω, which is the best possible situation. Also for system (8 we have the existence and uniqueness of solution (w, z. For instance, if v L 2 ( and (w 0,z 0 L 2 (Ω 2, then (w, z L 2 (0,T; H0 1 (Ω 2, which depends continuously on (v, w 0,z 0 L 2 ( L 2 (Ω 2.

5 CONTROLLABILITY OF SYSTEMS OF PARABOLIC EUATIONS 383 Our objective here is to drive both w and z to zero at time T by means of the control v. Accordingly, we consider the corresponding adjoint system: ϕ t Δϕ + cϕ E ϕ =(P1 (t, x; Dψθ 1 in, ψ t Δψ + hψ K ψ =(P2 (t, x; Dϕθ 2 in, (10 ϕ = ψ = 0 on Σ, ϕ t=t = ϕ 0, ψ t=t = ψ 0 in Ω, where P1 and P2 are the adjoint operators of P 1 and P 2, respectively. It is very easy to prove the classical fact that the previous controllability property is equivalent to the following observability inequality: ϕ t=0 2 L 2 (Ω + ψ t=0 2 L 2 (Ω C ψ 2 dx dt, with C = C(Ω,ω,T > 0 independent of (ϕ 0,ψ 0. In order to achieve this, we need the following properties to hold for the differential operator P 2 : (11 and m 2,β are constant (12 u H2 (Ω C P 2 u L2 (Ω u H 2 (Ω H 1 0 (Ω, for some C = C(Ω > 0. Observe that no boundary condition for P 2 ϕ is demanded. Theorem 3. Assume that conditions (11 (12 hold. Then, there exists a control v such that the solution of (8 satisfies w t=t z t=t 0 in Ω. Remark 3. Other boundary conditions can be considered in system (8. For instance, if one imposes Neumann boundary conditions, Theorem 3 also holds when we impose (13 instead of (12. u H 2 (Ω C P2 u L 2 (Ω u H 2 (Ω, u n Σ =0 In general, if one imposes Bw Σ = 0 as a boundary condition for w in (8, Theorem 3 holds if (14 u H 2 (Ω C P 2 u L2 (Ω u H 2 (Ω, Bu Σ =0. Remark 4. One can extend the result stated in Theorem 3 to the case where c, E h, and K are functions which depend on time and belong to L (0,T, no matter which boundary conditions are considered. Remark 5. Instead of the operator P 2, one could have also considered an operator L containing a first order time derivative. Indeed, let Lu = l γ γ t u, l γ constants. γ 1

6 384 SERGIO GUERRERO Then, Theorem 3 holds for L instead of P 2. Observe that the proof in this situation is simpler, since the boundary conditions satisfied by ϕ are also satisfied by L ϕ. Remark 6. A combination of Theorem 3, Remark 3, and Remark 5 yields that when we consider the differential operator u = (l γ γ t + m β x β u, l γ,m β constants γ 1, β 2 instead of P 2, the result stated in Theorem 3 holds as long as u H 2 (Ω C u L 2 (Ω C = C(Ω > 0, for any u H 2 (Ω satisfying the same boundary conditions as w in (8. Finally, we consider the situation where we control the higher order coupling term equation. Thus, let us introduce the following system: p t Δp + cp + E p = P 2 (t, x; D(qθ 4 +v1 ω in, q t Δq + hq + K q = P 1 (t, x; D(p θ 3 in, (15 p = q = 0 on Σ, p t=0 = p 0, q t=0 = q 0 in Ω. Here, p is a vector-valued function and P 2 a vectorial differential operator of order 2 in space such that each component is given by (9 for i = 2. On the other hand, q is a scalar-valued function and P 1 is a divergence-type operator, that is to say, P 1 f R for f a vector-valued function. Finally, θ i C 2 (Ω (3 i 4 and we assume the existence of a nonempty open subset ω 3 ω and a positive constant C such that θ 3 C>0inω 3. For v L 2 ( N and (p 0,q 0 L 2 (Ω N+1, there exists a unique solution (p,q L 2 (0,T; H 1 0 (Ω N+1 which depends continuously on (v, p 0,q 0 L 2 ( N L 2 (Ω N+1. Observe that now we are controlling the first equation. Obviously, the adjoint system associated with (15 is again (10. In order to establish the corresponding null controllability result, this time we need to impose the following conditions on the operator P 1 : (16 and (17 m 1,β are constant u L2 (0,T ;H 1 (Ω C P 1 u L 2 ( N u H1 0 (Ω. The corresponding result in this situation is presented in the following theorem. Theorem 4. Assume conditions (16 (17 are satisfied. Then, there exists a control v such that the solution of (15 satisfies p t=t q t=t 0 in Ω. Once Theorem 3 is demonstrated, one can follow the same ideas in order to prove Theorem 4 by just adapting the corresponding arguments. For the sake of completeness, we present a system for which Theorem 4 applies (for simplicity, we take θ 3 θ 4 1: p t Δp + cp + E p =(P 2,1,...,P 2,N q, +v1 ω in, q t Δq + hq + K q = p in, p = q = 0 on Σ, p t=0 = p 0, q t=0 = q 0 in Ω,

7 CONTROLLABILITY OF SYSTEMS OF PARABOLIC EUATIONS 385 with P 2,j differential operators of order 2 in the x variable satisfying (9. This paper is organized as follows. In section 2, we prove Theorem 1. In subsection 2.1 we prove new Carleman-type estimates, which will be crucial for this proof, and in subsection 2.2 we combine some results and conclude its proof. Finally, in section 3 we prove Theorem Insensitizing controls for the functional J τ. As we saw in the introduction, we can restrict ourselves to proving the null controllability of the coupled system (4, that is to say, Theorem 1. As usual, in order to prove this result we concentrate on the corresponding adjoint system: ϕ t Δψ + aψ B ψ = (( ϕ1 O in, ϕ t Δϕ + aϕ + B ϕ =0 in, (18 ψ =0,ϕ= 0 on Σ, ψ t=t =0,ϕ t=0 = ϕ 0 in Ω, where ϕ 0 L 2 (Ω. As explained in the introduction, in the framework of controllability it is a classical fact that the null controllability property for system (4 is equivalent to the following observability inequality: (19 e K0/tm ψ 2 dx dt C ψ 2 dx dt, for certain positive constants K 0 and C which are independent of ψ 0, and for some positive m. For the proof of (19, we will follow a classical approach consisting of obtaining a suitable weighted-like estimate (the so-called Carleman estimate similar to the observability inequality. For a systematic use of this kind of estimate see, for instance, [14] or [11]. In order to establish this Carleman inequality, we need to define some weight functions: (20 α m (x, t = k(m+1 exp{ m λ η0 } exp{λ(k η 0 + η 0 (x} t m (T t m, αm(t = max α m (x, t =α m Ω (x, t, x Ω ξm(t = min ξ m (x, t =ξ m Ω (x, t, x Ω ξ m (x, t = eλ(k η0 +η 0 (x t m (T t m, where m>3 and k>mare fixed. Here, η 0 C 2 (Ω satisfies (21 η 0 C>0 in Ω\ ω 0, η 0 > 0 in Ω, and η 0 0 on Ω, with ω 0 ω O an open set. The proof of the existence of such a function η 0 is given in [11]. The weights (20 were first considered in [10] in order to obtain Carleman estimates for the three-dimensional micropolar fluid model.

8 386 SERGIO GUERRERO Accordingly, we define I(s, λ; as follows: I(s, λ; g :=s 1 e 2sαm ξ 1 ( m gt 2 + Δg 2 dx dt (22 + sλ 2 e 2sαm ξ m g 2 dx dt + s 3 λ 4 e 2sαm ξm g 3 2 dx dt. Furthermore, we denote by I w (s, λ, the terms in the expression of I(s, λ, concerning the L 2 ( and L 2 (0,T; H0 1 (Ω norms (that is, the integrals appearing in the second line of (22. With this notation, we can prove the following result. Proposition 5. There exists a positive constant C which depends on Ω, ω, and T such that I w (Δϕ+s 2 λ 4 e 3sαm ξm(s 2 2 λ 2 ξm ψ ψ 2 dx dt (23 C(1 + T 2 s 7 λ 8 e 2sαm ξ 7 m ψ 2 dx dt, for any λ C and s C(T 2m + T m. Remark 7. From the Carleman inequality (23, one can readily deduce the observability inequality (19. Indeed, it suffices to combine the fact that ψ t=t 0 with the dissipation of ϕ(t L2 (Ω as t goes to T (see, for instance, [11]. As a consequence, the proof of Theorem 1 is achieved. The proof of Proposition 5 is divided into two steps, which correspond to subsections 2.1 and 2.2. The first, and more important, step deals with the equation satisfied by ϕ (which is independent of ψ. In the second step, we combine both equations in order to conclude the desired inequality (23. Before starting with this, we recall a Carleman estimate which will be essential in our proof. This estimate concerns energy solutions of heat equations with nonhomogeneous Neumann boundary conditions. Lemma 6. Let u 0 L 2 (Ω, f 1 L 2 (, f 2 L 2 ( N, and f 3 L 2 (Σ. Then there exists a constant C(Ω,ω 0 > 0 such that the solution u L 2 (0,T; H 1 (Ω L (0,T; L 2 (Ω of u t Δu = f 1 + f 2 in, u n + f 2.n = f 3 on Σ, u t=t = u 0 in Ω satisfies I w (u C (s 3 λ sλ e 2sαm ξm u 3 2 dx dt e 2sαm f 1 2 dx dt + s 2 λ 2 Σ e 2sα m ξ m f 3 2 dσ dt e 2sαm ξ 2 m f 2 2 dx dt

9 CONTROLLABILITY OF SYSTEMS OF PARABOLIC EUATIONS 387 for any λ C and s C(T 2m + T 2m 1. This lemma was essentially proved in [8]. In fact, the inequality proved in [8] concerned the same weight functions as in (24, but with m = 1. Then one can follow the steps of the proof in [8] (see Theorem 1 in that reference and adapt the arguments by just taking into account that t α m := α m,t CTξ (m+1/m m and tt α m := α m,tt CT 2 ξ (m+2/m m, with C>0independent of s, λ, and T New Carleman estimate for ϕ. Here we deal with the problem ϕ t Δϕ + aϕ + B ϕ =0 in, (24 ϕ = 0 on Σ, ϕ t=0 = ϕ 0 in Ω. Recall that a R and B R N are constants. For this system, we prove the following estimate. Lemma 7. There exists a positive constant C depending on Ω and ω 0 such that (25 I w (Δϕ Cs 3 λ 4 e 2sαm ξm Δϕ 3 2 dx dt, for any λ C and s C(T 2m + T m. Remark 8. Observe that, in particular, we deduce from this inequality the following well-known unique continuation property: (26 Δϕ =0 inω 0 (0,T ϕ 0 in Ω (0,T. As far as we know, it is a new fact that (26 can be quantified in terms of an inequality like (25. On the other hand, we do not know if (26 holds when a and B are not constant with respect to the space variable. Proof of Lemma 7. We first look at the equation satisfied by Δϕ: (Δϕ t Δ(Δϕ+aΔϕ + B Δϕ =0 in. Observe that no boundary conditions are prescribed for Δϕ. At this point, we can apply Lemma 6 (with f 2 0 and deduce the existence of a constant C = C(Ω,ω 0 > 0 such that I w (Δϕ C (s 3 λ 4 e 2sαm ξm Δϕ 3 2 dx dt (27 + sλ Σ e 2sα m ξ m Δϕ n 2 dx dt for any λ C and s C(T 2m + T 2m 1. The next step will be to eliminate the last term in the right-hand side of (27. In order to do this, we introduce the function ϕ := η(tϕ, where (28 η(t =s (1/2 (1/m λe sα m (t (ξ m (1/2 (1/m (t.

10 388 SERGIO GUERRERO In view of (24, it fulfills ϕ t Δϕ + aϕ + B ϕ = η t ϕ in Ω, (29 ϕ =0 on Ω, ϕ t=0 = 0 in Ω. Thanks to (27, we are going to deduce that ϕ is a very regular function. In fact, we have that η t ϕ L 2 (0,T; H 2 (Ω H 1 0 (Ω, since η t Δϕ L 2 (Ω and η t Δϕ L 2 ( CTs (3/2 (1/m λ e sα m (ξ m 3/2 Δϕ L 2 ( Cs 3/2 λ e sα m (ξ m 3/2 Δϕ L 2 (, for s CT m. The square of this last quantity is bounded by the left-hand side of (27 (recall that e sα m is the minimum of e sα m. Consequently, we have that ϕ L 2 (0,T;(H 4 H 1 0 (Ω (see, for instance, [16] and (30 T ϕ 2 L 2 (0,T ;H 4 (Ω = s1 2/m λ 2 e 2sα m (ξ m 1 2/m ϕ 2 H 4 (Ω dt CI w (Δϕ. 0 (31 Taking this into account, by a simple integration by parts we deduce that T s 2 1/m λ 3 e 2sα m (ξ m 2 1/m ϕ 2 H 3 (Ω dt CI w(δϕ. From (30 and (31, we obtain in particular that (32 s (3/2 3/(2m λ 3 T 0 0 e 2sα m (ξ m (3/2 3/(2m Δϕ n 2 L 2 (Σ dt CI w (Δϕ. Since m>3, this justifies that the second term in the right-hand side of (27 is absorbed by the left-hand side. As a conclusion, we obtain the desired inequality ( Carleman estimate for ϕ and conclusion. Finally, we will deal with the particular coupling of ψ and ϕ. First, assuming ϕ is given, we apply a Carleman estimate to the weak solution ψ of (18 (observe that the right-hand side of the equation satisfied by ψ belongs, for instance, to L 2 (0,T; H 1 (Ω, which can be found in [15] (for the explicit dependence with respect to λ and T, see [9]: s 4 λ 6 e 3sαm ξm ψ 4 2 dx dt + s 2 λ 4 e 3sαm ξm ψ 2 2 dx dt (33 C (s 4 λ 6 e 3sαm ξm ψ 4 2 dx dt + s 3 λ 4 e 3sαm ξm ϕ 3 2 dx dt, O (0,T

11 CONTROLLABILITY OF SYSTEMS OF PARABOLIC EUATIONS 389 for any λ C and s C(T 2m + T 2m 1. Observe that we have chosen to apply this result for smaller exponentials, that is to say, for e 3sαm instead of e 2sαm. Then we easily see that the last integral in the right-hand side of (33 is bounded by I w (Δϕ, as long as λ is large enough. In fact, if we denote α m (t = min x Ω α m (x, t and ξ m (t = max x Ω ξ m (x, t, we have T s 3 λ 4 e 3sαm ξm ϕ 3 2 dx dt s 3 λ 4 e 3s αm ( ξ m 3 ϕ 2 L 2 (Ω dt O (0,T 0 T Cs 3 λ 4 e 3s αm ( ξ m 3 Δϕ 2 L 2 (Ω dt Cs3 λ 4 e 2sαm ξm Δϕ 3 2 dx dt, 0 for λ C. Here we have used the fact that ϕ has null trace. Combining this with (25 and (33, we obtain s 2 λ 4 e 3sαm ξm(s 2 2 λ 2 ξm ψ ψ 2 dx dt + I w (Δϕ (34 C (s 3 λ 4 ξ 3 ( m sλ 2 e 3sαm ξ m ψ 2 + e 2sαm Δϕ 2 dx dt, for any λ C and s C(T 2m + T m. Now, since ω 0 O, from the equation satisfied by ψ, wefind Δϕ = ψ t Δψ + aψ B ψ in ω 0 (0,T. Then we plug this into the expression of the last integral in (34 and obtain s 3 λ 4 e 2sαm ξm Δϕ 3 2 dx dt = s 3 λ 4 e 2sαm ξ 3 m(δϕ( ψ t Δψ + aψ B ψ dx dt. We define a positive function θ Cc 2 (ω such that θ 1inω 0. Then the task turns to estimating the following integral: s 3 λ 4 θe 2sαm ξm(δϕ( ψ 3 t Δψ + aψ B ψ dx dt. After several integration by parts (getting all derivatives out of ψ with respect to both space and time, we get s 3 λ 4 θe 2sαm ξm(δϕ( ψ 3 t Δψ + aψ B ψ dx dt = s 3 λ 4 s 3 λ 4 2s 3 λ 4 + s 3 λ 4 θ(e 2sαm ξ 3 m t Δϕ ψ dx dt Δ(θe 2sαm ξ 3 mδϕ ψ dx dt (θe 2sαm ξ 3 m Δϕ ψ dx dt B (θe 2sαm ξ 3 mδϕ ψ dx dt.

12 390 SERGIO GUERRERO Here, we have used the equation satisfied by ϕ and the fact that θ has compact support in ω. Let us do some computations involving the weight functions: for s CT 2m and (e 2sαm ξ 3 m t CTse 2sαm (ξ m 4+1/m, Δ(e 2sαm ξ 3 m Cs 2 λ 2 e 2sαm ξ 5 m, for s CT 2m and λ C. With this, we obtain s 3 λ 4 θe 2sαm ξm(δϕ( ψ 3 t Δψ + aψ B ψ dx dt εi w (Δϕ+C(1 + T 2 s 7 λ 8 e 2sαm ξ 7 m ψ 2 dx dt, which, combined with (34, gives the desired inequality ( Proof of Theorem 3. In this section we will prove Theorem 3. As indicated in the introduction, in order to prove Theorem 4 one can follow the same ideas of the proof of Theorem 3. For simplicity, in this section we will keep the notation η 0 for the function defined in (21. In the present situation, ω 0 will stand for an open set contained in ω 2, which was also contained in ω (see the paragraph between (9 and (10. Throughout this section we will work with the following system (see (10: ϕ t Δϕ + cϕ E ϕ =(P1 (t, x; Dψθ 1 in, ψ t Δψ + hψ K ψ =(P2 ϕθ 2 in, (35 ϕ = ψ = 0 on Σ, ϕ t=t = ϕ 0, ψ t=t = ψ 0 in Ω. Recall that, by (11, P2 is a second order differential operator in space with constant coefficients. In order to prove Theorem 3, it suffices to establish the following observability inequality for the solutions of (35. Proposition 8. There exists C(Ω,ω,T > 0 independent of (ϕ 0,ψ 0 such that (36 ϕ t=0 2 L 2 (Ω + ψ t=0 2 L 2 (Ω C ψ 2 dx dt. As in the previous section, the strategy will consist of proving the corresponding Carleman inequality for system (35. It is presented in the following lemma. Lemma 9. There exists a positive constant C(Ω,ω such that I w (P2 ϕ+s 6 λ 8 e 2sαm ξm ψ 6 2 dx dt (37 C(1 + T 2 s 10 λ 8 for any λ C and s C(T 2m + T m. e 6sαm+4sα m ξ 10 m ψ 2 dx dt

13 CONTROLLABILITY OF SYSTEMS OF PARABOLIC EUATIONS 391 Thanks to (12, the observability inequality (36 readily follows from (37. As in the proof of Lemma 7, we first deal with the heat equation satisfied by ϕ and we try to obtain an independent Carleman inequality, viewing (P 1 (t, x; Dψθ 1 as a right-hand side. More precisely, we consider the heat equation satisfied by P 2 ϕ: (P 2 ϕ t +Δ(P 2 ϕ+cp 2 ϕ + E (P 2 ϕ = Δ((P 1 (t, x; Dψθ 1 in. To P2 ϕ (as solution of the previous heat equation, we apply Lemma 6 and we obtain I w (P2 ϕ (s 3 λ 4 e 2sαm ξm P 3 2 ϕ 2 dx dt ( + sλ e 2sα m ξ P 2 2 ϕ m Σ n + n (P 2 (38 1 (t, x; Dψ dx dt + s 2 λ 2 e 2sαm ξm( Δψ ψ 2 + ψ 2 dx dt, for any λ C and s C(T 2m + T 2m 1. Next, we estimate the boundary term in the right-hand side of (38. To this end, we define ϕ = η(tϕ, with η(t given by (28. This function fulfills the following system: ϕ t +Δϕ + cϕ + E ϕ = η(t(p1 (t, x; Dψθ 1 + η t ϕ in Ω, (39 ϕ =0 on Ω, ϕ t=t = 0 in Ω. Assuming that the right-hand side of (39 belongs to L 2 (0,T; H 2 (Ω, we have ϕ L 2 (0,T;(H 4 H 1 0 (Ω H 1 (0,T; H 2 (Ω (see, for instance, [16] and ϕ 2 L 2 (0,T ;H 4 (Ω + ϕ t 2 L 2 (0,T ;H 2 (Ω (40 C(I w (P2 ϕ+ η(t(p1 (t, x; Dψθ 1 L 2 (0,T ;H 2 (Ω. Here we have used (12. With the same argument as in the previous section, we get T s (3/2 3/(2m λ 3 e 2sα m (ξ m (3/2 3/(2m P2 2 ϕ 0 n dt (41 L 2 (Σ C(I w (P2 ϕ+ η(t(p1 (t, x; Dψθ 1 L 2 (0,T ;H 2 (Ω. Again, since m>3, this justifies that the second term in the right-hand side of (38 is absorbed. As a conclusion, we obtain from (38 (42 I w (P2 ϕ C (s 3 λ 4 e 2sαm ξm P 3 2 ϕ 2 dx dt T + s 1 2/m λ 2 e 2sα m (ξ m 1 2/m (P1 (t, x; Dψθ 1 2 H 2 (Ω dt 0 + s 2 λ 2 e 2sαm ξm( Δψ ψ 2 + ψ 2 dx dt,

14 392 SERGIO GUERRERO for any λ C and s C(T 2m + T m. Now we deal with the equation satisfied by ψ. Thus, we apply the classical Carleman inequality for the heat equation with right-hand side in L 2 (. Let us define I 0 (ψ =s 6 λ 8 e 2sαm ξm ψ 6 2 dx dt + s 4 λ 6 e 2sαm ξm ψ 4 2 dx dt Then we have + s 2 λ 4 e 2sαm ξ 2 m Δψ 2 dx dt. I 0 (ψ C (s 6 λ 8 e 2sαm ξm ψ 6 2 dx dt + s 3 λ 4 e 2sαm ξm P 3 2 ϕ 2 dx dt, for λ C and s C(T 2m + T 2m 1. Combining this with (42, we obtain I 0 (ψ+i w (P2 ϕ C (s 3 λ 4 e 2sαm ξm P 3 2 ϕ 2 dx dt (43 T + s 1 2/m λ 2 e 2sα m (ξ m 1 2/m ψ 2 H 3 (Ω dt 0 + s 6 λ 8 e 2sαm ξm ψ 6 2 dx dt, for λ C and s C(T 2m + T m. Let us now introduce the function ψ = ρ 0 (tψ, with ρ 0 (t =s 1/2 λe sα m (ξ m 1/2. Then ψ satisfies ψ t +Δ ψ + h ψ + K ψ = ρ 0 (t(p2 ϕθ 2 + ρ 0,t ψ in Ω, (44 ψ =0 on Ω, ψ t=t = 0 in Ω. Since the right-hand side belongs to L 2 (0,T; H 1 (Ω, we deduce that ψ L 2 (0,T; H 3 (Ω and ( ψ 2 L 2 (0,T ;H 3 (Ω C sλ 2 e 2sαm ξ m ( (P2 ϕ 2 + P2 ϕ 2 dx dt (45 + T 2 s 3 λ 2 C(I 0 (ψ+i w (P 2 ϕ. e 2sα m (ξ m 3+2/m ψ 2 dx dt Observe that in inequality (45 we did not use all the information we had about ϕ, because in the argument of estimating the normal derivative of P 2 ϕ, we obtained

15 CONTROLLABILITY OF SYSTEMS OF PARABOLIC EUATIONS 393 good estimates for ϕ 2 L 2 (0,T ;H 3 (Ω with the power s2 1/m. We chose not to profit from this for the sake of simplicity. In particular, from (45 we deduce that the second term in the right-hand side of (43 is absorbed. For the moment, we have (46 I 0 (ψ+i w (P2 ϕ+ ψ 2 L 2 (0,T ;H 3 (Ω C (s 3 λ 4 e 2sαm ξm P 3 2 ϕ 2 dx dt + s 6 λ 8 e 2sαm ξm ψ 6 2 dx dt, for λ C and s C(T 2m + T m. We will finally estimate the local term concerning P 2 ϕ. From the equation satisfied by ψ, wehave P 2 ϕ = ψ t +Δψ + hψ + K ψ in ω 0 (0,T (recall that supp θ 2 ω 2 and ω 0 ω 2. This gives s 3 λ 4 e 2sαm ξm P 3 2 ϕ 2 dx dt = s 3 λ 4 e 2sαm ξ 3 m(p 2 ϕ(ψ t +Δψ + hψ + K ψ dx dt. As in the previous section, with the help of a cut-off function θ 0 Cc 2 (ω with θ 0 1 in ω 0, we can integrate by parts with respect to both time and space, and we find s 3 λ 4 e 2sαm ξm P 3 2 ϕ 2 dx dt ε(i w (Δϕ+I 0 (ψ + C(1 + T 2 s 6 λ 6 f(θ 0 e 4sαm+2sα m ξ 6 m ( ψ 2 + ψ 2 dx dt, for some f(θ 0 Cc 1 (ω. Observe that in order to estimate the first term e 2sαm ξmθ 3 0 P2 ϕψ t dx dt, we have integrated in time and used (40 for the integral e 2sαm ξmθ 3 0 P2 ϕ t ψ dx dt. We integrate by parts again and obtain s 3 λ 4 e 2sαm ξm P 3 2 ϕ 2 dx dt ε (I w (P 2 ϕ+i 0 (ψ+s 2 λ 4 e 2sαm ξm Δψ 2 2 dx dt + C(1 + T 2 s 10 λ 8 e 6sαm+4sα m ξ 10 m ψ 2 dx dt.

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