Boundary controllability of parabolic coupled equations

Boundary controllability of parabolic coupled equations Enrique Fern´ andez-Cara∗ , Manuel Gonz´ alez-Burgos ∗ and Luz de Teresa† Abstract This paper is concerned with the boundary controllability of non-scalar linear parabolic systems. More precisely, two coupled one-dimensional parabolic equations are considered. We show that, in this framework, boundary controllability is not equivalent and is more complex than distributed controllability. In our main result, we provide necessary and sufficient conditions for the null controllability. 1 Introduction This paper deals with the controllability properties of some systems of two coupled one-dimensional parabolic equations where the control is exerted at one boundary point for all times. Thus, let us fix T > 0 and let us consider the linear system      yt−yxx =Ay in Q= (0,1) ×(0, T), y(0,·) = Bv, y(1,·) = 0 in (0, T), y(·,0) = y0in (0,1), (1) where A∈L(R2) and B∈R2are given and y0∈H−1(0,1)2. Here, v∈L2(0, T) is a control function (to be determined) and y= (y1, y2)∗is the state variable. Observe that, for every v∈L2(0, T) and y0∈H−1(0,1)2, (1) admits a unique weak solution (defined by transposition) that satisfies y∈L2(Q)2∩C0([0, T]; H−1(0,1)2); see Section 2. It will be said that (1) is approximately controllable in H−1(0,1)2at time Tif, for any y0, yd∈ H−1(0,1)2and any ε > 0, there exists a control function v∈L2(0, T) such that the associated solution satisfies ky(·, T)−ydkH−1(0,1) ≤ε. On the other hand, it will be said that (1) is null controllable at time Tif, for each y0∈ H−1(0,1)2, there exists a control v∈L2(0, T) such that the associated solution satisfies y(·, T) = 0 in H−1(0,1)2.(2) Since (1) is linear, this second property is equivalent to the exact controllability to the trajectories at time T, that is to say, to the following property: for any trajectory by(i.e. any solution to (1) corresponding to v≡0 and by0∈H−1(0,1)2) and any y0∈H−1(0,1)2, there exists a control v∈L2(0, T) such that the associated solution to (1) satisfies y(·, T) = by(·, T) in H−1(0,1)2. ∗Dpto, E.D.A.N., Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain. Supported by grant MTM200607932 of the D.G.E.S. (Spain). E-mails: [email protected], [email protected]. †Instituto de Matem´aticas, Universidad Nacional Aut´onoma de M´exico, Circuito Exterior, C.U. 04510 D.F., M´exico. Supported by project IN102799 of D.G.A.P.A. (Mexico). E-mail: [email protected]. 1 The controllability properties of similar scalar problems are nowadays well known; see for instance [8], [21], [7], [18], [12] and [11]. To be precise, let Ω ⊂RNbe a nonempty regular bounded open set with N≥1, let ω⊂Ω be a nonempty open subset, and let γ⊂∂Ω be a nonempty relative open set. Let us consider the following scalar problems:      yt−∆y=v1ωin Ω ×(0, T), y= 0 on ∂Ω×(0, T), y(·,0) = y0in Ω (3) and      yt−∆y= 0 in Ω ×(0, T), y=v1γon ∂Ω×(0, T), y(·,0) = y0in Ω. (4) Here, 1ωand 1γare, respectively, the characteristic functions of ωand γ,y0∈L2(Ω) is given and vis the control. Under the previous assumptions, for every Ω, ω,γand T, both systems (3) and (4) are approximately controllable in L2(Ω) and also null controllable at any time T(see for instance [18] and [12]). In fact, the boundary controllability results for system (4) can be easily obtained from the corresponding distributed controllability results for system (3) and viceversa. We will see that the situation is quite different for similar non-scalar systems. There are not many works devoted to the controllability of parabolic systems of PDEs. To our knowledge, all them deal with distributed controls, exerted on a small open set ω; see for instance [23], [6], [2], [5], [13], [14], [15], [3] and [4]. In these papers, almost all the results have been established for 2 ×2 systems where the control is exerted on the first equation. The most general results in this context seem to be those in [14], [3] and [4]. In [14], the authors study a cascade parabolic system of nequations (n≥2) controlled with one single distributed control. In [3] and [4], the authors provide necessary and sufficient conditions for the controllability of n×n parabolic linear systems with constant or time-dependent coefficients. It is worth mentioning that, in [17], an approximate boundary controllability result is obtained for a particular system of two parabolic coupled equations as a consequence of a unique continuation principle. The result is valid in several dimensions but only for a very particular kind of coupling. It is also interesting to recall the boundary controllability results for a system of two wave equations obtained by Alabau-Boussouira in [1]. For completeness, let us recall the main result proved in [3] and [4] for the problem      yt−∆y=Ay +Bv1ωin Ω ×(0, T ), y= 0 on ∂Ω×(0, T), y(·,0) = y0in Ω, (5) where A∈L(Rn;Rn), B∈L(Rm,Rn) (with n, m ≥1) and y0∈L2(Ω)n. It is the following: Let [A|B]be the following matrix in L(Rn×m;Rn): [A|B] = [B|AB |A2B| ··· |An−1B]. Then, (5) is null controllable if and only the so called Kalman’s rank condition rank [A|B] = n is satisfied. In that case, null controllability holds at any time T > 0. In this paper, our main aim is to characterize the boundary controllability properties of (1) (a system of 2 equations) when we apply just one control on a part of the boundary. Our main result is the following: 2 Theorem 1.1. Let A∈L(R2;R2)and B∈R2be given and let us denote by µ1and µ2the eigenvalues of A. Then (1) is exactly controllable to the trajectories at any time T > 0if and only if rank [B|AB] = 2 (6) and π−2(µ1−µ2)6=j2−k2∀k, j ∈Nwith k6=j. (7) In view of Theorem 1.1, we find two different situations: when the matrix Ain (1) has one double real eigenvalue or a couple of conjugate complex eigenvalues, (6) is a necessary and sufficient condition for the null controllability at any time (as in the distributed case); otherwise, if Ahas two different real eigenvalues, an additional condition is needed for null controllability, independently of the vector Bwe are considering. As a consequence of this result, we observe that the Kalman’s rank condition is necessary, but not sufficient, for the boundary controllability of (1). This is a crucial discrepancy between boundary and distributed controllability for coupled parabolic systems and shows that, for a given system, these two properties can be independent. The proof of Theorem 1.1 is based on the proof of Fattorini and Russell [8] of the boundary controllability of the one-dimensional heat equation. They reduce the task to construct a biorthogonal family in L2(0,∞) to a given family of exponential functions and, then, to deduce appropriate estimates of the corresponding norms. Recall that two families {pn:n≥1}and {qn:n≥1}in L2(0,∞) are said to be biorthogonal in this space if (pn, qk)L2(0,∞)=δnk ∀n, k ≥1. In our case, we have to construct and estimate appropriately in L2(0,∞) a family that must be biorthogonal to a larger set of functions. We use techniques similar to those in [8], but adapted to this new situation. The constructed family is then used, together with (6) and (7), to prove an observability inequality for the solutions to the adjoint system. As a consequence, we get the null controllability of (1). On the other hand, we prove that (6) and (7) are necessary by analyzing some particular systems that serve as counter-examples to unique continuation. The rest of the paper is organized as follows. In the next Section, we give some basic and preliminary results concerning the existence of a solution and the controllability properties of (1); the proofs of some of them are postponed to Appendix A and Appendix B. In Section 3, we present some results related to the Fattorini-Russell method. In particular, we give details on the construction and estimates of certain biorthogonal families and we show how they can be used to prove some inequalities. In Section 4 we prove Theorem 1.1. Finally, Section 5 deals with some further results and open problems. The main results in this paper have been announced in [10]. 2 Preliminary results This Section is devoted to establish some results for (1) that will be needed in the proof of Theorem 1.1. In the sequel, Cdenotes a generic positive constant; sometimes, we will make emphasis on the dependence of Con T, by writing C(T). We will also use the following notation: k·kXstands for the norm of the normed space Xor Xm, with m≥2; also, k·kLp(X)stands for the norm in Lp(0, T;X) (p≥1). We begin by clarifying what is a solution by transposition to (1). To this end, let us consider the linear backwards in time problem      −ϕt−ϕxx =A∗ϕ+gin Q, ϕ(0,·) = 0, ϕ(1,·) = 0 in (0, T), ϕ(·, T) = 0 in (0,1), (8) 3 where g∈L2(Q)2. It is well known that, for every g∈L2(Q)2, (8) possesses exactly one (strong) solution ϕ∈L2(0, T;H2(0,1)2)∩C0([0, T]; H1 0(0,1)2). Hence, the following definition makes sense: Definition 2.1. Let y0∈H−1(0,1)2and v∈L2(0, T)be given. It will be said that y∈L2(Q)2is a solution by transposition to (1) if, for each g∈L2(Q)2, one has ZZQ y·g dx dt =hy0, ϕ(·,0)i+ZT 0 B·ϕx(0, t)v(t)dt, (9) where ϕis the solution to (8) associated to gand h·,·i stands for the usual duality pairing between H−1(0,1)2and H1 0(0,1)2. Thus, one has: Proposition 2.2. Assume that y0∈H−1(0,1)2and v∈L2(0, T)are given. Then (1) admits a unique solution by transposition ythat satisfies:          y∈L2(Q)2∩C0([0, T]; H−1(0,1)2), yt∈L2(0, T; (D(−∆)0)2), yt−yxx =Ay in L2(0, T; (D(−∆)0)2), y(·,0) = y0in H−1(0,1)2and kykL2(Q)+kytkL2(D(−∆)0)≤Cky0kH−1(0,1) +kvkL2(0,T ). In the sequel, it will be said that yis the state associated to y0and v. Results of this kind are well known. For completeness, we recall the proof of Proposition 2.2 in Appendix A. Now, let us consider the adjoint of system (1):      −ϕt−ϕxx =A∗ϕin Q, ϕ(0,·) = 0, ϕ(1,·) = 0 in (0, T), ϕ(·, T) = ϕ0in (0,1), (10) where ϕ0∈H1 0(0,1)2. In the sequel, the solution to (10) will be called the adjoint state associated to ϕ0. The controllability of (1) can be characterized in terms of appropriate properties of the solutions to (10). More precisely, we have: Proposition 2.3. The following properties are equivalent: 1. There exists a positive constant Csuch that, for any y0∈H−1(0,1)2, there exists a control v∈L2(0, T)such that kvk2 L2(0,T )≤Cky0k2 H−1(0,1) (11) and the associated state satisfies (2). 2. There exists a positive constant Csuch that, for any trajectory by∈C0([0, T]; H−1(0,1)2) of (1) and any y0∈H−1(0,1)2, there exists a control v∈L2(0, T )such that kvk2 L2(0,T )≤Cky0−by(·,0)k2 H−1(0,1) (12) and the associated state satisfies y(·, T) = by(·, T)in H−1(0,1)2. 4 3. There exists a positive constant Csuch that the observability inequality kϕ(·,0)k2 H1 0(0,1) ≤CZT 0|B∗ϕx(0, t)|2dt (13) holds for every ϕ0∈H1 0(0,1)2. In (13),ϕis the adjoint state associated to ϕ0. Again, this result is well known. For completeness, the proof is presented in Appendix B, at the end of the paper. Remark 2.1. It is also well known that the approximate controllability of (1) can be characterized in terms of a property of the solutions to (10). More precisely, (1) is approximately controllable if and only if the following unique continuation property holds: “Let ϕ0∈H1 0(0,1)2be given and let ϕbe the associated adjoint state. Then, if B∗ϕx(0, t) = 0 on (0, T), one has ϕ≡0on Q.” 3 Biorthogonal families: construction, estimates and applications In this Section, some technical results are given. They will be used below to prove Theorem 1.1. Let us first present a fundamental lemma whose first part was essentially proved by Luxemburg and Korevaar in [20]. For the sake of completeness, we have included the proof below. As far as we know, the second part of this lemma is new. Lemma 3.1. Suppose that {Λn}n≥1is a sequence of complex numbers such that, for some δ, ρ > 0, one has:      <(Λn)≥δ|Λn|,|Λn−Λk| ≥ |n−k|ρ∀n, k ≥1, ∞ X n=1 1 |Λn|<∞.(14) Then, a) There exists a sequence {hn}biorthogonal to {e−Λnt}such that, for every ε > 0, one has khnkL2(0,∞)≤K(ε)eε<(Λn)∀n≥1.(15) b) There exists a sequence {qn,˜qn}biorthogonal to {e−Λnt, te−Λnt}such that, for every ε > 0, one has k(qn,˜qn)kL2(0,∞)≤K(ε)eε<(Λn)∀n≥1.(16) As a consequence, we also have: Lemma 3.2. Let us assume that (14) holds. Then: a) For every T > 0, there exists C(T)>0such that, for all m≥1and Aj∈C, one has: ZT 0| m X j=1 Aje−Λjt|2dt ≥C(T)Z∞ 0| m X j=1 Aje−Λjt|2dt. b) For every T > 0, there exists C(T)>0such that, for all m≥1and Aj, Bj∈C, one has: ZT 0| m X j=1 (Aj+tBj)e−Λjt|2dt ≥C(T)Z∞ 0| m X j=1 (Aj+tBj)e−Λjt|2dt. 5 Let us introduce the (closed) spaces ET= [e−Λjt:j≥1]L2(0,T ), E∞= [e−Λjt:j≥1]L2(0,∞), FT= [e−Λjt, te−Λjt:j≥1]L2(0,T ), F∞= [e−Λjt, te−Λjt:j≥1]L2(0,∞), spanned by the functions e−Λjt(and te−Λjt) in L2(0, T) and L2(0,∞), respectively. Let us also introduce the canonical mappings Γ : E∞7→ ETand e Γ : F∞7→ FT, with Γv=v|(0,T )∀v∈E∞and e Γw=w|(0,T )∀w∈F∞. A trivial consequence of Lemma 3.2 is the following: Lemma 3.3. Let us assume that (14) holds. Then: a) For every T > 0,Γ : E∞7→ ETis an isomorphism. In particular, there exists C(T)>0 such that kvkL2(0,∞)≤C(T)kΓvkL2(0,T )∀v∈E∞. b) For every T > 0,e Γ : F∞7→ FTis an isomorphism. In particular, there exists C(T)>0 such that kwkL2(0,∞)≤C(T)ke ΓwkL2(0,T )∀w∈F∞. These lemmas are crucial for the proof of the main result in this Section, that is the following: Proposition 3.4. Let us assume that (14) holds. Then: a) For every T > 0, there exists C(T)>0such that ZT 0|X j≥1 Aje−Λjt|2dt ≥C(T)X j≥1 |Aj|2 |Λj|e−<(Λj)T,(17) whenever the sum in the left hand side makes sense. b) For every T > 0, there exists C(T)>0such that ZT 0|X j (Aj+tBj)e−Λjt|2dt ≥C(T)X j≥1 |Aj|2+|Bj|2 |Λj|e−<(Λj)T,(18) whenever the sum in the left hand side makes sense. We will first give the proof of Proposition 3.4 assuming that Lemmas 3.1, 3.2 and 3.3 hold true. Then, we will present the proofs of these lemmas. Proof of Proposition 3.4: Let us prove part b). Part a) is simpler and can be established in a similar way. Let us take qkand eqkas in Lemma 3.1 b) and let us assume that the left hand side of (18) is meaningful, i.e. X j (Aj+tBj)e−Λjt∈L2(0, T). In view of Lemma 3.3 b), we also have Pj(Aj+tBj)e−Λjt∈L2(0,∞) and Z∞ 0|X j (Aj+tBj)e−Λjt|2dt ≤C(T)ZT 0|X j (Aj+tBj)e−Λjt|2dt. (19) 6 For all k≥1, we have                Z∞ 0|X j (Aj+tBj)e−Λjt|2dt ≥1 kqkk2Z∞ 0X j (Aj+tBj)e−Λjtqk(t)dt 2 =(Ake−Λkt, qk)2 kqkk2=|Ak|2 kqkk2. Consequently,  Z∞ 0|X j (Aj+tBj)e−Λjt|2dt  X k 1 |Λk|!≥X k 1 |Λk||Ak|2 kqkk2. Let us fix ε > 0. Then this inequality together with (16) imply Z∞ 0|X j (Aj+tBj)e−Λjt|2dt ≥C K(ε)2X k 1 |Λk||Ak|2e−2ε<(Λk). Taking ε=T/2, we see that Z∞ 0|X j (Aj+tBj)e−Λjt|2dt ≥C(T)X k 1 |Λk||Ak|2e−<(Λk)T(20) for some C(T)>0. Proceeding as before, but using eqkinstead of qk, we also get Z∞ 0|X j (Aj+tBj)e−Λjt|2dt ≥C(T)X k 1 |Λk||Bk|2e−<(Λk)T.(21) Now, combining (20) and (21), we find that Z∞ 0|X j (Aj+tBj)e−Λjt|2dt ≥C(T)X j≥1 |Aj|2+|Bj|2 |Λj|e−<(Λj)T. Finally, from (19), we get (18). Let us now present the proofs of Lemmas 3.1 and 3.2. Proof of Lemma 3.1: In this proof, k · k will stand for the norm in L2(0,+∞). Part a) can be deduced from the proof given in [9] (see also [8]); it can also be deduced from part b). However, for clarity and completeness, we will include here the proof. Thus, let us set pn(t) = e−Λntand let us introduce the space En= [pk:k6=n]L2(0,∞), that is to say, the closed span in L2(0,∞) of the functions pkwith k6=n. Thanks to M¨untz’s Theorem (see [22], p. 24), pn6∈ Enand there exists a unique rn∈Ensuch that kpn−rnk= dist (pn, En). Of course, rnis characterized by rn∈Enand (pn−rn)⊥En. 7 Let us choose hn=pn−rn kpn−rnk2. It is then clear that (hn, pk) = δkn for all kand n, i.e. the sequence {hn}is biorthogonal to{e−Λnt}. Let us now prove the inequalities (15) or, equivalently, let us estimate kpn−rnkfrom below. For each m∈Nlet us denote by rm nthe projection of pnover Em n= [pk:k6=n, 1≤k≤m]L2(0,∞). Then rm n→rnin L2(0,∞) as m→ ∞ and kpn−rnk= limm→∞ kpn−rm nk. We also have kpn−rm nk2= (e−Λnt, e−Λnt−rm n) = Z∞ 0 e−Λnte−Λnt−rm n(t)dt = Φ(Λn), where Φ is given by Φ(Λ) = Z∞ 0 e−Λte−Λnt−rm n(t)dt ∀Λ∈Cwith <(Λ) ≥0. Observe that Φ depends on nand m; however, in order to simplify the notation, from now on we will not indicate explicitly this dependence. Since rm n∈Em n, we can write Φ(Λ) = 1 Λ + Λn− m X j=1,j6=n am j Λ + Λj =g(Λ) (Λ + Λn)Qm j=1,j6=n(Λ + Λj) for some am j∈C. Here, gis a polynomial of degree ≤m−1. The orthogonality properties of rm n imply that Φ(Λj) = 0 for all jwith 1 ≤j≤mand j6=n. As a consequence, this is also satisfied by gand we have g(Λ) = K m Y j=1,j6=n (Λ −Λj) (22) for some K∈C. On the other hand, we also have g(Λ) = m Y j=1,j6=n (Λ + Λj)−(Λ + Λn) m X j=1, j6=n am j m Y i=1, i6=n,j (Λ + Λi) , whence g(−Λn) = m Y j=1,j6=n (Λj−Λn). This and (22) together imply that K= m Y j=1,j6=n Λn−Λj Λn+ Λj and Φ(Λ) = 1 Λ + Λn m Y j=1,j6=n (Λn−Λj)(Λ −Λj) (Λn+ Λj)(Λ + Λj). In particular, we see that Φ(Λn) = 1 2<(Λn) m Y j=1,j6=n 1−Λn Λj 2 1 + Λn Λj 2.(23) 8 Taking limits as m→ ∞ in (23), we get kpn−rnk=Pn, where Pn=1 2<(Λn)1/2∞ Y j=1,j6=n 1−Λn Λj 1 + Λn Λj .(24) Following the ideas in [9] and [20], it can be proved that, for every ε > 0, there exists C(ε)>0 such that Pn≥C(ε)e−ε<(Λn).(25) For completeness, we give a proof below. From these inequalities, taking into account the definition of hn, we directly obtain (15). This ends the proof of part a). Let us now prove (25). Let us fix  > 0. From (14), there exists N0()∈Nsuch that X j≥N0() 1 |Λj|≤. Thus, using the inequality 1 + x≤ex,x∈R, we can estimate the denominator of (24) as follows: ∞ Y j=1,j6=n1 + Λn Λj≤ ∞ Y j=1,j6=n1 + |Λn| |Λj|= N0()−1 Y j=1,1 + |Λn| |Λj|∞ Y j=N0()1 + |Λn| |Λj| ≤ N0()−1 Y j=1,1 + |Λn| c∞ Y j=N0() e |Λn| |Λj|≤1 + |Λn| cN0()−1 e|Λn| ≤C1()e2|Λn|∀n∈N, (26) for a positive constant C1(). In the previous inequality we have used that, for some constant c > 0, one has |Λj| ≥ c > 0 for every j∈N. Let us now work on the numerator of (24). We introduce S1(n) = {j:|Λj| ≤ 1 2|Λn|}, S2(n) = {j6=n:1 2|Λn|<|Λj| ≤ 2|Λn|} and S3(n) = {j:|Λj|>2|Λn|}. Then Y j∈S1(n)1−Λn Λj≥Y j∈S1(n)|Λn| |Λj|−1≥1∀n∈N.(27) On the other hand, Y j∈S3(n)1−Λn Λj≥Y j∈S3(n)1−|Λn| |Λj|≥Y j∈S3(n) e−2|Λn| |Λj|=e−2|Λn|Pj∈S3(n)1 |Λj|.(28) In this inequality we have used that e−2x≤1−xif x∈[0,1/2]. Using (14), we deduce that there exists N1()∈Nsuch that, if n≥N1(), one has X j∈S3(n) 1 |Λj|≤. From (28) and the previous inequality we deduce that, if n≥N1(), then Y j∈S3(n)1−Λn Λj≥e−2|Λn|. 9 and |ebm,j|=(ekm,eqj)L2(0,∞)≤ kekmkL2(0,∞)keqjkL2(0,∞)≤Cεeε<(Λj) for all m≥1 and all 1 ≤j≤N(m). We also have kΓekmkL2(0,T )→0 (43) and kekmkL2(0,∞)= 1 ∀m≥1.(44) Let 0 < ε < T/3 be given and let us introduce the set Uε={z∈C:<(z)>3ε, |=(z)|<δ−2−1−1/2ε}, where δ > 0 is given in (14). Using assumption (14), we have |=(Λj)| ≤ δ−2−11/2<(Λj) for all j≥1. Therefore, if z∈Uε, one has: |e−Λjz|=e=(Λj)=(z)−<(Λj)<(z)≤e−(<(z)−ε)<(Λj)∀j≥1. Observe that, thanks to (14), we have limj→∞ <(Λj) = ∞. Consequently, |ekm(z)| ≤ N(m) X j=1 |eam,j|+|z||ebm,j||e−Λjz| ≤ Cε N(m) X j=1 e−<(Λj)(<(z)−2ε)(1 + |z|) =Cεe−<(Λ1)(<(z)−2ε)(1 + |z|) N(m) X j=1 e−[<(Λj)−<(Λ1)](<(z)−2ε) ≤Cεe−<(Λ1)(<(z)−2ε)(1 + |z|) ∞ X j=1 e−ε[<(Λj)−<(Λ1)] ≡e Cεe−<(Λ1)(<(z)−2ε)(1 + |z|) for all z∈Uε. We deduce that the holomorphic function ekm(z) is uniformly bounded in Uε. Therefore, there exist a subsequence (still denoted by ekm) and a holomorphic function ekin Uεsuch that ekm→ek uniformly on the compacts of Uε. In particular, ekm(t)→ek(t) for all t∈(3ε, ∞) and |ekm(t)| ≤ e Cεe−<(Λ1)(t−2ε)(1 + t)∀t∈(3ε, ∞). Using Lebesgue’s Theorem, we also deduce that ekm→ekin L2(3ε, ∞) (strongly). In view of (43), ek(t) = 0 for all t∈(3ε, T). Since ekis holomorphic, we must have ek≡0 in Uε, whence Z∞ T|ekm(t)|2dt →0. But this and (43) imply kekmkL2(0,∞)→0, which contradicts (44). This ends the proof of Lemma 3.2. 4 Proof of Theorem 1.1 We will devote this Section to the proof of Theorem 1.1. First of all, observe that the Kalman’s rank condition (6) is a necessary condition for the controllability of system (1). Indeed, if B= 0, it is clear that (1) is not null controllable at time T. Therefore, let us assume that B6≡ 0 and rank [B|AB]=1, i. e. AB =αB for some α∈R. 16 Let us choose e B∈R2such that det e P= det [B|e B]6= 0. Then it is not difficult to see that e P−1B=e1=1 0and e P−1Ae P=e C:= α β1 0β2 for some β1, β2∈R. The change of variables z=e P−1yleads to the following reformulation of (1):    zt−zxx =e Cz in Q , z(0,·) = e1v, z(1,·) = 0 in (0, T), z(·,0) = e P−1y0in (0,1) . But it is clear that this system is neither approximately nor null controllable, since the second component of zis independent of v. Consequently, this is also the case for (1) and (6) is certainly a necessary condition. Henceforth, it will be assumed that (6) is satisfied. Let us take P= [B|AB] and let us introduce ey=P−1y. Arguing as above, we obtain P−1B=e1and P−1AP =e A:= 0a1 1a2, where a1and a2are the coefficients of the characteristic polynomial of A: pA(µ) = µ2−a2µ−a1. The eigenvalues of Aand e Aare the same. The system satisfied by eyis   eyt−eyxx =e Aeyin Q , ey(0,·) = e1v, ey(1,·) = 0 in (0, T), ey(·,0) = P−1y0in (0,1) , Therefore, the previous change of variables reduces the situation to the case where A=0a1 1a2and B=1 0.(45) For simplicity, it will be assumed in the rest of the proof that Aand Bare given by (45). In view of Proposition 2.3, we just have to see whether or not the solutions to the adjoint system (10) satisfy the observability inequality (13). In order to deal with this inequality, we are going to reformulate the original control problem in a more simple way. Thus, let Fbe a fundamental matrix of the linear ordinary differential system ξt=Aξ. Let us introduce wand ψ, with y=F(t)wand ϕ=F(t)∗ψ, where yis the solution to (1) (i.e. the state associated to y0and v) and ϕis the solution to (10) (i.e. the adjoint state associated to ϕ0). Then the functions wand ψrespectively satisfy    wt−wxx = 0 in Q , w(0,·) = e B(t)v(t), w(1,·) = 0 in (0, T), w(·,0) = y0in (0,1) (where e B(t) = F(t)−1e1for all t) and      ψt+ψxx = 0 in Q , ψ(0,·) = 0, ψ(1,·) = 0 in (0, T), ψ(·, T) = ψ0in (0,1) . (46) 17 Obviously, (1) is null controllable at time Tif and only if this is the case for the system satisfied by w. Clearly, this property is also equivalent to the fact that the solutions to (46), where the final data ψ0belong to H1 0(0,1)2, satisfy the observability inequality kψ(·,0)k2 H1 0(0,1) ≤CZT 0|e B∗(t)ψx(0, t)|2dt. (47) We can now distinguish three different cases, depending on the spectrum of A: Case 1: Ahas two different real eigenvalues. Let µ1and µ2denote the eigenvalues of A, with µ1< µ2, i. e. a2 2+ 4a1>0 and µ1=a2 2−pa2 2+ 4a1 2, µ2=a2 2+pa2 2+ 4a1 2. We can then choose Msuch that Fis given by F(t) = "−µ2eµ1t−µ1eµ2t eµ1teµ2t#. Consequently, it can be assumed that e B(t) = F(t)−1e1=1 µ2−µ1 −e−µ1t e−µ2t! and (47) reads kψ(·,0)k2 H1 0(0,1) ≤C |µ2−µ1|2ZT 0|e−µ1t∂xψ1(0, t)−e−µ2t∂xψ2(0, t)|2dt. (48) The eigenvalues and eigenfunctions of the (one-dimensional) Dirichlet Laplacian in (0,1) are λj=π2j2, θj(x) = sin(πjx), j = 1,2,.... Hence, if ψ0∈L2(0,1)2is given, the associated solution ψis ψ(x, t) = X j≥1aj bje−λj(T−t)sin(πjx), where the aj,bjare the Fourier coefficients of the components of ψ0. Replacing this expression in (48) and performing the change of variables t→T−t, we readily see that (48) is equivalent to X j≥1 λja2 j+b2 je−2λjT≤CZT 0|X j≥1 jaje−µ1(T−t)−bje−µ2(T−t)e−λjt|2dt. (49) for every aj, bjsuch that Pj≥1j2a2 j+b2 j<∞. Case 1.1: there exists j0, k0∈Nwith j06=k0such that µ2−µ1=λj0−λk0. Setting (ak0=j0e−µ2Tand aj= 0,∀j6=k0, bj0=k0e−µ1Tand bj= 0,∀j6=j0, we see that the observability inequality (49) fails. Therefore, system (1) is not null controllable at time Tin this case. 18 Remark 4.1. Actually, in view of this argument, when (7) is not satisfied, (1) is not approximately controllable, since the solutions to the adjoint system (10) do not necessarily satisfy the related unique continuation property; see Remark 2.1. Case 1.2: µ2−µ16=λj−λkfor all j, k ≥1. Let us show that in this case (49) holds and, consequently, (1) is null controllable. Let us introduce the sequence {Λj}, where {Λj:j≥1}={λk:k≥1}∪{λk+µ2−µ1:k≥1} and the indexes are fixed in such a way that Λj≤Λj+1 for all j. Observe that Λ1=π2>0, since µ2−µ1>0. Denoting by [x] the integer part of x, we see that, whenever i≥i0:= µ2−µ1 2π2+1 2+ 1, one has λi−1+µ2−µ1< λi< λi+µ2−µ1< λi+1. Therefore, Λ2k−1=λkand Λ2k=λk+µ2−µ1∀k≥i0. Let us check that the sequence {Λj}satisfies the assumptions of Lemma 3.1. Clearly, if k≤2i0−2, Λk+1 −Λk≥ρ1>0, where ρ1only depends on µ2−µ1. On the other hand, if k≥2i0−1, it is not difficult to check that Λk+1 −Λk≥min{µ2−µ1,(2i0+ 1)π2−µ2+µ1}=ρ2>0 and, again, ρ2only depends on µ2−µ1. Hence, condition (14) is fulfilled by taking ρ= min{ρ1, ρ2}. Finally, X n≥1 1 Λn <∞. Thus, we can apply Proposition 3.4 to the sequence {Λj}with Ck=(jaje−µ1Tif Λk=λj+µ2−µ1, −jbje−µ2Tif Λk=λj. Observe that, if k≥i0, |C2k|2 Λ2k =k2a2 ke−2µ1T π2k2+µ2−µ1≥C|ak|2e−2µ1T,|C2k−1|2 Λ2k−1 =1 π2|bk|2e−2µ2T, where C=C(µ1, µ2) is a positive constant. Consequently, for 1 ≤k≤2(i0−1) we have |Ck|2 Λk≥       1 π2(i0−1)2+µ2−µ1|aj|2e−2µ1Tif Λk=λj+µ2−µ1, 1 π2(i0−1)2|bj|2e−2µ2Tif Λk=λj. Let us now prove the observability inequality (49). The following holds: I=ZT 0|X j≥1 jaje−µ1(T−t)−bje−µ2(T−t)e−λjt|2dt =ZT 0 e2µ2t|X j≥1 jaje−µ1Te−(λj+µ2−µ1)t−X j≥1 jbje−µ2Te−λjt|2dt ≥min{1, e2µ2T}ZT 0|X j≥1 jaje−µ1Te−(λj+µ2−µ1)t−X j≥1 jbje−µ2Te−λjt|2dt = min{1, e2µ2T}ZT 0|X k≥1 Cke−Λkt|2dt. 19 In view of (17), taking into account the properties of the sequence {Ck}k≥1, we get (Cis a positive constant which depends on µ1and µ2): I≥C(T) min{1, e2µ2T}X k≥1 |Ck|2 Λk e−ΛkT ≥CC(T) min{1, e2µ2T}X j≥1a2 je−(µ1+µ2)T+b2 je−2µ2Te−λjT ≥CC(T) min{1, e−2µ2T}X j≥1a2 j+b2 je−λjT≥e C0 Tmin{1, e−2µ2T}X j≥1 λja2 j+b2 je−2λjT, where e C0 T>0 is such that 0<e C0 T≤1 λj C(T)eλjT∀j≥1.(50) This proves (49) and concludes the proof in this first case. Case 2: Ahas two complex eigenvalues. In this case, a2 2+ 4a1<0, µ1=α+iβ and µ2=α−iβ, where α=a2/2 and β=p−(a2 2+ 4a1)/2. We can choose Msuch that F(t) = eαt "−(αcos βt +βsin βt)−(αsin βt −βcos βt) cos βt sin βt #. Consequently, e B(t) = F(t)−1e1=e−αt β −sin βt cos βt !. Again, (47) and (49) are equivalent. Now, we consider the complex sequence {Λk}, with Λ2k−1=λk=π2k2,Λ2k=λk−2iβ =π2k2+ 2iβ ∀k≥1. The assumptions in Lemma 3.1 are again fulfilled. Indeed, one has <(Λ2k−1) = λk=|Λ2k−1|and <(Λ2k) = π2k2≥δ(π4k4+ 4β2)1/2=δ|Λ2k| for some δ∈(0,1) (which depends on β). On the other hand,    |Λ2k−Λ2n|=|Λ2k−1−Λ2n−1|=π2|k2−n2| ≥ 3π2|k−n| =3π2 2|2k−2n|=3π2 2|(2k−1) −(2n−1)| and (|Λ2k−1−Λ2n|2=π4|k2−n2|2+ 4β2≥9π4|k−n|2+ 4β2 ≥min{9π4/8,2β2}|2k−1−2n|2 for every k, n ≥1. Finally, X n≥1 1 |Λn|<∞. As a consequence, we can apply Proposition 3.4 a), with C2k−1=kake−µ1Tand C2k=−kbke−µ2T∀k≥1, 20 which satisfies |C2k−1|2 |Λ2k−1|≥Ce−2αT |ak|2and |C2k|2 |Λ2k|≥Ce−2αT |bk|2 for a positive constant C=C(β). This gives: I=ZT 0|X j≥1 jaje−µ1(T−t)−bje−µ2(T−t)e−λjt|2dt ≥min{1, e2αT }ZT 0|X k≥1 Cke−Λkt|2dt. If we now apply (17) to these Λkand Ck, we deduce that                    I≥C(T) min{1, e2αT }X k≥1 |Ck|2 |Λk|e−<(Λk)T ≥CC(T) min{1, e−2αT }X j≥1a2 j+b2 je−λjT ≥Ce C(T) min{1, e−2αT }X j≥1 λja2 j+b2 je−2λjT, where e C(T) is a positive constant satisfying (50). This proves (49) in this case. Case 3: Ahas a double real eigenvalue. We denote by µthe eigenvalue of A. One has µ=a2/2∈Rand we can assume that F(t) = eµt "−µ1−µ(t−T) 1t−T#and e B(t) = e−µt −(t−T) 1,∀t∈[0, T]. The observability inequality (47) is now equivalent to prove that X j≥1 λja2 j+b2 je−2λjT≤CZT 0|X j≥1 jeµt (taj+bj)e−λjt|2dt for all ajand bjsuch that Pj≥1j2a2 j+b2 j<∞. But this inequality can be readily obtained from (18) by applying Proposition 3.4 to the sequences {λj},{aj}and {bj}and taking into account (50). This ends the proof of Theorem 1.1. Combining Proposition 2.3 and Theorem 1.1, we deduce the following: In the conditions of Theorem 1.1, there exists a positive constant C, only depending on T, such that the observability inequality (13) is satisfied by the solutions to (10) if and only if conditions (6) and (7) hold. 5 Further results and open problems 5.1 Some changes in Theorem 1.1 Obviously, the statement of Theorem 1.1 is valid if, instead of (1), we consider the controlled problem      yt−yxx =Ay in Q, y(0,·)=0, y(1,·) = Bv in (0, T), y(·,0) = y0in (0,1). 21 Again, A∈L(R2) and B∈R2are given, y0∈H−1(0,1)2and v∈L2(0, T) is a control function to be determined. On the other hand, system (1) can be posed in a general spatial interval (0, `) with ` > 0. In this case, the additional condition (7) must be changed by `2 π2(µ1−µ2)6=j2−k2∀k, j ∈Nwith k6=j. (51) Finally, it is possible to identify the natural numbers n≥1 that can be expressed in the form n=j2−k2with k, j ∈Nand j > k ≥1. It is not difficult to see that, given n∈N, there exist j, k ∈Nwith j > k ≥1 such that n=j2−k2if and only if n= 4(m+ 1) or n= 2m+ 1 for some m≥1. Thus, the controllability result for the coupled parabolic system (1) in the spatial interval (0, `) (with ` > 0) reads as follow: Theorem 5.1. Let ` > 0,A∈L(R2)and B∈R2be given and let us denote by µ1and µ2the eigenvalues of A. Then, system (1) is exactly controllable to the trajectories at time Tif and only if rank [B|AB]=2, and (`/π)2(µ1−µ2)is not an integer of the form 4(m+ 1) or 2m+ 1 for some m≥1. 5.2 Approximate controllability As a consequence of the result stated at the end of Section 4 and the arguments in the proof of Theorem 1.1, the conditions (6) and (7) are also equivalent to the approximate controllability at time Tof system (1). To be precise, one has the following result, that we state without proof: Theorem 5.2. Let A∈L(R2)and B∈R2be given and let us denote by µ1and µ2the eigenvalues of A. Then, system (1) is approximately controllable in H−1(0,1)2at time Tif and only if (6) and (7) hold. 5.3 The case of mcontrol forces Theorem 1.1 can be generalized to the case in which mcontrol forces, with m≥2, appear in system (1), i. e. to the case B∈L(Rm;R2) and v∈L2(0, T)m. There are two possible situations: •rank B≤1: it is then easy to check that the controllability properties of system (1) are determined by (6) and (7), as in Theorems 1.1 and 5.2. •rank B= 2: then (6) is automatically satisfied. Let us see that system (1) is exactly controllable to the trajectories independently of (7). In fact, we will deduce this property as a consequence of a similar (and well known) result for scalar parabolic problems. For convenience, this will be proved in a more general framework. Thus, let us assume that N≥1, Ω ⊂RNis a bounded connected open set with boundary ∂Ω of class C2and γis a nonempty relative open subset of ∂Ω. For n, m ≥2, we consider the controlled system      yt−∆y=Ay in Q= Ω ×(0, T), y=Bv1γon Σ = ∂Ω×(0, T ), y(·,0) = y0in Ω, (52) where A∈L(Rn), B∈L(Rm;Rn) and y0∈H−1(Ω)nare given and 1γis the characteristic function on γ. In (52), y= (y1, . . . , yn)∗is the state and v∈L2(Σ)mis the control function. As in Section 2, it can be shown that, for every y0∈H−1(Ω)nand v∈L2(Σ)m, the linear system (52) possesses exactly one solution (defined by transposition) y∈L2(Q)n∩C0([0, T]; H−1(Ω)n). Our main assumption reads as follows: rank B=n. (53) 22 Then, one has: Theorem 5.3. In the previous conditions, if (53) holds, then (52) is exactly controllable to the trajectories and approximately controllable in H−1(Ω)nat time T > 0. For the proof, we will use the global Carleman inequality given in the following result, by Fursikov and Imanuvilov [12]: Theorem 5.4. There exist a positive function α0∈C2(Ω) and two positive constants σ0and C0(only depending on Ωand γ) such that, for every s≥s0=σ0T+T2and every z∈ L2(0, T;H1 0(Ω)) with zt±∆z∈L2(Q), the following (global Carleman) estimate holds: I(z)≤C0 ZZQ e−2sα |zt±∆z|2dx dt +sZZγ×(0,T ) e−2sαρ ∂z ∂n 2 dΓdt!. Here, I(z)and the functions αand ρare given as follows: I(z) = ZZQ (sρ)−1e−2sα |zt|2+|∆z|2+ (sρ)2|∇z|2+ (sρ)4|z|2dx dt, α(x, t) = α0(x) t(T−t)∀(x, t)∈Q, ρ(t)=(t(T−t))−1∀t∈(0, T). Let us now present the main ideas of the proof of Theorem 5.3. Let us consider the adjoint problem (−ϕt−∆ϕ=A∗ϕin Q, ϕ= 0 on Σ, ϕ(·, T) = ϕ0in Ω,(54) where ϕ0∈H1 0(Ω)n. If ϕis the (strong) solution to (54) associated to ϕ0∈H1 0(Ω)n, it is possible to apply Theorem 5.4 to each component of ϕand deduce that I(ϕ) := n X i=1 I(ϕi)≤C1 ZZQ e−2sα|ϕ|2dx dt +sZZγ×(0,T ) e−2sαρ ∂ϕ ∂n  2 dΓdt! for all s≥s0=σ0T+T2, where C1is a new constant which depends on n, Ω, γand A. If we now take s3≥2−5T6C1, then C1≤(sρ)3/2 and we can write I(ϕ)≤C2sZZγ×(0,T ) e−2sαρ ∂ϕ ∂n  2 dΓdt for all s≥s1=σ1T+T2, where C2= 2C1and σ1= max{σ0,2−5/3C1/3 1}. Taking into account (53), we get the following global Carleman estimates for the solutions to (54): I(ϕ)≤C1sZZγ×(0,T ) e−2sαρB∗∂ϕ ∂n  2 dΓdt ∀s≥σ1T+T2,(55) where C3is a positive constant depending on n, Ω, γ,Aand B. The Carleman inequality (55) leads to a unique continuation property for the solutions to (54): “If ϕ∈C0([0, T]; H1 0(Ω)n) is a solution to (54) and B∗∂ϕ ∂n = 0 on γ×(0, T), then ϕ≡0.” This is equivalent to the approximate controllability of (52) in H−1(Ω)nat time T. We turn now to the exact controllability to trajectories. As above, this property is equivalent to the observability of the adjoint problem (54), i.,e. to the following property: there exists a positive constant C > 0 such that kϕk2 H1 0(Ω) ≤CZZγ×(0,T )B∗∂ϕ ∂n  2 dΓdt ∀ϕ0∈H1 0(Ω)n. 23 But, combining the global Carleman inequality (55) and the energy inequality satisfied by the solutions to (54), it is easy to show that this is true; see for instance [11] for a detailed presentation of the argument in the case of a similar scalar problem. This ends the proof. 5.4 Possible generalizations to n×ncoupled systems It would be interesting to generalize the results presented in this work to the case of a n×n coupled system (with n≥3), controlled by mboundary control forces (m≥1). To be precise, let us consider the system (1) with A∈L(Rn), B∈L(Rm;Rn) and y0∈ H−1(0,1)n. Then, it can be proved that the Kalman’s rank condition rank [A|B] = n is a necessary condition for the approximate controllability and also for the exact controllability to the trajectories; for a proof, see for instance [3] and [4]. On the other hand, as in the case n= 2, there are some necessary conditions which arise in the study of the controllability properties. Thus, let us assume that n≥3 and m= 1, i.e., B∈Rn. Let us also suppose that there exist j0, k0≥1 with j06=k0and two eigenvalues µand eµof Asuch that π−2(µ−eµ) = j2 0−k2 0.(56) Then, (1) is neither null nor approximately controllable in H−1(Ω)nat time T. Indeed, there must exist P∈L(Cn) (with det P6= 0) and J∈L(Cn−2), two matrices, such that A=P  µ0 0 0eµ0 0 0 J P−1. If ϕ0∈H1 0(Ω)n, then the solution ϕto the adjoint problem (10) satisfies B∗ϕx(0, t) = πX j≥1 B∗j(P∗)−1   e(−πj2+µ)(T−t)0 0 0e(−πj2+eµ)(T−t)0 0 0 e(−πj2Id.+J∗)(T−t)  P∗aj, where Id.is the identity matrix in L(Cn−2) and the aj∈Rnare the Fourier coefficients aj=Z1 0 ϕ0(x) sin(πjx)dx. Let us set B∗(P∗)−1= (β1, β2,e β∗) and P∗aj= (α1 j, α2 j,eα∗ j)∗, with e β, eαj∈Cn−2. Then B∗ϕx(0, t) = πX j≥1 jβ1e(−πj2+µ)(T−t)α1 j+β2e(−πj2+eµ)(T−t)α2 j+e β∗e(−πj2Id.+J∗)(T−t)eαj. Choosing αj= 0, for every j≥1, α1 j= 0, for every j6=j0,α2 j= 0, for every j6=k0, and α1 j0and α2 k0such that j0β1α1 j0=−k0β2α2 k0and α1 j02+α2 k026= 0 and taking into account the equality (56) we deduce that ϕ6≡ 0 in Qand nevertheless B∗ϕx(0,·)≡ 0 on (0, T). Therefore, the function ϕdoes not satisfy the unique continuation property, nor the observability inequality (13). Summarizing, we have proved that the opposite to (56) is a necessary condition for the controllability of (1) when n≥3 and m= 1. 24 5.5 An example In this paper, up to now, we have assumed that all the diffusion coefficients in the considered systems are the same. In this Section we give a simple example which shows that, when the diffusion matrix is not Id., the situation can be much more complex and, again, results valid for distributed controls are no longer valid for boundary controls; see [14] and [4]. This gives an idea of the, in some sense, unnatural difficulties that arise when we try to control a non-scalar system from the boundary. We will be concerned with the following cascade system, where ν > 0:    yt−Dyxx =Ay in Q, y(0,·) = Bv, y(1,·) = 0 in (0, T), y(·,0) = y0in (0,1) , (57) where D=ν0 0 1 , A =0 0 1 0 and B=1 0. We address the following approximate controllability question: Let ε > 0, y0∈H−1(0,1)2and y1∈H−1(0,1)2be given; then, does there exist v∈L2(0, T) such that the corresponding solution to (57) satisfies ky(·, T)−y1kH−1≤ε? In the present situation, the adjoint system is    −ϕt−D∆ϕ=A∗ϕin Q, ϕ(0,·) = ϕ(1,·) = 0 in (0, T), ϕ(., T) = ϕ0in (0,1) (58) and the previous controllability property is equivalent to the following: B∗ϕx|x=0 = 0 in L2(0, T) implies ϕ≡0 in Q. (59) We then have: Theorem 5.5. Suppose that ν6= 1. Then (57) is approximately controllable at time T > 0if and only if √ν6∈ Q. Proof: The proof is given of two parts. In the first part we prove the unique continuation property when √ν6∈ Q. In the second one, we give a counter-example to (59) when ν6= 1 and √ν∈Q. In what follows, λjdenotes the j-th eigenvalue of the Dirichlet Laplacian in (0,1) and wjis the associated eigenfunction of norm 1 in L2(0,1). That is, λj=π2j2and wj(x)≡sin(πjx) for all j≥1. First Part: Let ϕ0∈H−1(0,1)2be given. Since ν6= 1, we have the following expression for the solution to (58): ϕ(x, t) = X j≥1    aj−bj (ν−1)λje−νλj(T−t)+bj (ν−1)λj e−λj(T−t) bj (ν−1)λj e−λj(T−t)    wj(x), with aj bj=Z1 0 ϕ0(x) sin(πjx)dx ∈R2. Then, B∗ϕx(0, t) = X j≥1 (jπ)aj−bj (ν−1)λje−νλj(T−t)+bj (ν−1)λj e−λj(T−t). 25