A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems

Differential Equations &Applications Volume 1, Number 3 (2009), 427–457 A GENERALIZATION OF THE KALMAN RANK CONDITION FOR TIME–DEPENDENT COUPLED LINEAR PARABOLIC SYSTEMS F. AMMAR KHODJA,A.BENABDALLAH,C.DUPAIX AND M. GONZ ´ ALEZ-BURGOS (Communicated by M. Kirane) Abstract. In this paper we present a generalization of the Kalman rank condition for linear ordinary differential systems to the case of systems of ncoupled parabolic equations (posed in the time interval (0,T)with T>0) where the coupling matrices Aand Bdepend on the time variable t. To be precise, we will prove that the Kalman rank condition rank [A|B](t0)=n, with t0∈[0,T],isasufficient condition (but not necessary) for obtaining the exact controllability to the trajectories of the considered parabolic system. In the case of analytic matrices Aand B (and, in particular, constant matrices), we will see that the Kalman rank condition characterizes the controllability properties of the system. When the matrices Aand Bare constant and condition rank [A|B]=nholds, we will be able to state a Carleman inequality for the corresponding adjoint problem. 1. Statement of the problem. Main results Let Ω⊂RNbe a bounded connected open set with boundary ∂ Ωof class C2. Let ω ⊂Ωbe a nonempty open subset and assume T>0. Let us consider the timedependent second order elliptic operator L(t)y(x,t)=− N ∑ i,j=1 ∂ ∂ xiaij(x,t) ∂ y ∂ xj (x,t)+ N ∑ i=1 bi(x,t) ∂ y ∂ xi (x,t) +c(x,t)y(x,t),(1) where aij ∈W1,∞(Q),bi,c∈L∞(Q),1⩽i,j⩽N,(Q=Ω×(0,T)), aij(x,t)=aji(x,t)a.e. in Q,(2) Mathematics subject classification (2000): 93B05, 93B07, 35K05, 35K55, 35R30. Keywords and phrases: Kalman condition, control, observability, Carleman estimates, parabolic systems. The authors F. Ammar Khodja, A. Benabdallah, and C. Dupaix are partially supported by grant ANR-07-JCJC-0139- 01 (Agence Nationale de la recherche, France). The author M. Gonz´alez-Burgos is supported by D.G.E.S. (Spain), grant MTM2006-07932. c    ,Zagreb Paper DEA-01-24 427 428 F. AMMAR KHODJA,A.BENABDALLAH,C.DUPAIX AND M. GONZ ´ ALEZ-BURGOS and the coefficients aij satisfy the uniform elliptic condition N ∑ i,j=1 aij(x,t) ξ i ξ j⩾a0| ξ |2,∀ ξ ∈RN,a.e. in Q,(3) for a positive constant a0. For n⩾2andm⩾1 given, we consider the linear parabolic system  ∂ ty+L(t)y=A(t)y+B(t)v1 ω in Q=Ω×(0,T), y=0onΣ= ∂ Ω×(0,T),(4) where A(·)∈CM−1([0,T];L(Rn)) and B(·)∈CM([0,T];L(Rm,Rn)) (5) for an integer M⩾n.In(4),y=(yi)1⩽i⩽nis the state, v∈L2(Q)mis the control and 1 ω denotes the characteristic function of the open subset ω . Let us observe that, for every y0∈L2(Ω)nand v∈L2(Q)m, system (4) admits a unique solution y∈L2(0,T;H1 0(Ω)n)∩C0([0,T];L2(Ω)n)which satisfies y(·,0)=y0in Ω. The main goal of this paper is to analyze the controllability properties of system (4) when mdistributed control forces are exerted on the system. Let us fixT0,T1∈[0,T]with T0<T1. It will be said that (4) is approximately controllable in L2(Ω)non the time interval (T0,T1)if, for any y0,yd∈L2(Ω)nand any ε >0, there exists a control function v∈L2(Q)msuch that the solution y∈ C0([T0,T1];L2(Ω)n)to (4) corresponding to the initial condition y(·,T0)=y0satisfies ||y(·,T1)−yd||L2⩽ ε . On the other hand, it will be said that (4) is exactly controllable to the trajectories on (T0,T1)if for every trajectory y∗∈C0([0,T];L2(Ω)n)of (4) (i.e., a solution to (4) corresponding to v≡0) and y0∈L2(Ω)n, there exists a control v∈L2(Q)msuch that the solution yto (4) corresponding to the initial condition y(·,T0)=y0satisfies y(·,T1)=y∗(·,T1)in Ω. It will be said that system (4) is null controllable on the interval (T0,T1)when the previous condition is satisfied for the trajectory y∗≡0. Let us remark that for linear systems as (4), the exact controllability to the trajectories and the null controllability are equivalent concepts. There are few results on null controllability of system (4) when n>1 and most of them are proved for n=2andB=(1,0)∗(m=1). In [18], [5] and [13] the authors consider linear and nonlinear systems of two heat equations, one of them being forward and the other one backward in time, and show the null controllability of the considered systems. In [1] and [2], the authors give a null controllability result for a phase-field system and for reaction-diffusion systems (two nonlinear heat equations) when one distributed control force is exerted on the system. The results in [1] and [2] have been generalized in [11] in two directions: on the one hand, there are not restrictions on TIME-DEPENDENT COUPLED LINEAR PARABOLIC SYSTEMS 429 the dimension N, and on the other hand, the authors consider nonlinearities which depend on the gradient of the state. When n>2, we point out [12], [16], [3] and [4]. In [12] the authors provide a null controllability result for a general cascade parabolic system. In [16] the author gives a necessary and sufficient condition for the approximate controllability of a parabolic system with diagonalizable diffusion matrix. Finally, in [3] and [4] it is also proved a necessary and sufficient condition for the exact controllability to the trajectories of the autonomous system  ∂ ty−DRy =Ay +Bv1 ω in Q=Ω×(0,T), y=0onΣ= ∂ Ω×(0,T),y(·,0)=y0in Ω,(6) where Ris a self adjoint elliptic second order operator given by Ry(x)= N ∑ i,j=1 ∂ ∂ xirij(x) ∂ y ∂ xj (x)+c(x)y(x), with rij ∈W1,∞(Ω),∑i,jrij(x) ξ i ξ j⩾a0| ξ |2,forevery ξ ∈RN(a0>0), and rij =rji in Ω(1⩽i,j⩽N), c∈L∞(Ω),D=P−1diag(d1,....,dn)P(with detP=0anddi>0, for every i,1⩽i⩽n)andA∈L(Rn)and B∈L(Rm,Rn)are constant matrices. The controllability properties of ordinary differential systems are, nowadays, well known. To be precise, let us consider the system x=A(t)x+B(t)uon (0,T),(7) where A∈Cn−2([0,T];L(Rn)) and B∈Cn−1([0,T];L(Rm,Rn)) are given and uis a control. Let us define ⎧ ⎨ ⎩ B0(t)=B(t), Bi(t)=A(t)Bi−1(t)−d dt Bi−1(t),(8) (1⩽i⩽n−1) and denote by [A|B]∈C0([0,T];L(Rnm;Rn)) the matrix function given by: [A|B](t)=(B0(t)|B1(t)|···|Bn−1(t)). Let us remark that when Aand Bare constant matrices, [A|B]∈L(Rnm,Rn)is the matrix given by [A|B]=B|AB|A2B|···|An−1B.(9) With this notation, one has the following result. THEOREM 1.1. (Silverman-Meadows [17]). Under the previous assumptions, the following holds: 1. If there exists t0∈[0,T]such that rank [A|B](t0)=n, then system (7) is completely controllable on (0,T), i.e., for every x0,xd∈RNthere exists a control u ∈L2(0,T)m such that the solution x(·)to (7) corresponding to the initial condition x(0)=x0satis- fies x(T)=xd. 430 F. AMMAR KHODJA,A.BENABDALLAH,C.DUPAIX AND M. GONZ ´ ALEZ-BURGOS 2.System (7) is totally controllable on (0,T), i.e., system (7) is completely controllable on every subinterval (T0,T1)of (0,T)(0⩽T0<T1⩽T)if and only if there exists E , a dense subset of (0,T), such that rank [A|B](t)=n for every t ∈E. In the particular case in which Aand Bare constant matrices, the concepts of complete and total controllability on (0,T)for system (7) coincide. Thus, the exact controllability of system (7) is equivalent to the well-known Kalman’s rank condition (e.g., see [15]) rank[A|B]=n.(10) The objective of the present paper is to extend the controllability results stated in Theorem 1.1 for ordinary differential systems to the case in which the parabolic system (4) is considered. On the other hand, all along this paper we will assume that the operator L(t)is given by (1) and satisfies (2)-(3). Under assumption (5), if 1 ⩽p⩽M, we can define Kp(t)=(B0(t)|B1(t)|··· |Bp−1(t)) ∈C1([0,T];L(Rmp;Rn)),(11) where Bi(t)is given by (8), for 0 ⩽i⩽p−1. With this notation, the first result of our work reads as follows. THEOREM 1.2. Assume that the matrices A and B satisfy (5). Then, the following holds: 1.If there exist t0∈[0,T]and p ∈{1,...,M}such that rank Kp(t0)=n,(12) then system (4) is exactly controllable to the trajectories on the time interval (0,T). 2.System (4) is exactly controllable to the trajectories on every interval (T0,T1)with 0⩽T0<T1⩽T if and only if there exists E a dense subset of (0,T)such that rank [A|B](t)=n for every t ∈E, (or equivalently, rank Kp(t)=n for all p ∈{n,...,M} and t ∈E). Actually, we will show that, under assumption (12), if we write B(t)=b1(t)|b2(t)|···|bm(t),bi(·)∈CM([0,T])n,1⩽i⩽m, then, there exist r∈{1,...,m}(the total number of effective controls), a set {bli:1⩽ i⩽r}(the effective controls) and a non degenerate closed interval [T0,T1]⊆[0,T]such that system (4) is exactly controllable to the trajectories on the time interval (T0,T1) (and therefore, on (0,T)) when we exert on the system the rcontrol forces  B(t)=(0|···|0|bl1(t)|0|···|0|bl2(t)|0|···|0|blr(t)|0|···|0). In fact, we will see that, on the time interval (T0,T1), system (4) is equivalent to a cascade parabolic system. We will deduce the null controllability properties of (4) from the corresponding results for cascade parabolic systems stated in [12]. TIME-DEPENDENT COUPLED LINEAR PARABOLIC SYSTEMS 431 REMARK 1.1. Assume that Aand Bsatisfy (5) and let us fixaninteger p∈ {n,..,M}.Atfirst sight it could seem that condition (12) is weaker that the following condition: there exists t1∈[0,T]such that rank Kn(t1)=n. But,infact,wewillsee that both conditions are equivalent. In Section 4 (see Corollary 4.2) it will be seen, max t∈[0,T]rank Kn(t)= max t∈[0,T]rank Kp(t), for every p⩾n. A similar result has been proved by J.-M. Coron in [6] (see Proposition 1.19, p. 11): “If A(·)∈C∞([0,T];L(Rn)) and B(·)∈C∞([0,T];L(Rm,Rn)) and there exists t0∈[0,T]such that (12) holds, then there exists ε >0such that rank Kn(t)=nfor every t ∈([0,T]∩(t0− ε ,t0+ ε ))\{t0}”. As in the case of finite-dimensional linear systems, it is interesting to point out that the existence of t0∈[0,T]satisfying (12) is not a necessary condition to have the exact controllability to the trajectories on (0,T)of system (4). Following [6], we will give an example of matrices A∈C∞([0,T];L(Rn)) and B∈C∞([0,T];L(Rm,Rn)) such that rank Kp(t)<n,∀p⩾1and∀t∈[0,T], and system (4) is exactly controllable to the trajectories on (0,T)(see Section 5). Nevertheless, when Aand Bare analytic functions on [0,T], we will show that condition (12) is a necessary and sufficient condition for the exact controllability to trajectories of system (4). One has the following result. THEOREM 1.3. Let us suppose that A and B are analytic on [0,T]. Then, system (4) is exactly controllable to the trajectories on (0,T)if and only if there exist t0∈[0,T]and an integer p such that (12) holds. As in the case of system (7) and as a consequence of Theorem 1.2 (or Theorem 1.3), when Aand Bare constant matrices we obtain that the Kalman rank condition (10) is a necessary and sufficient condition for the exact controllability to the trajectories of system (4) on (0,T). Thus, one has the following result. THEOREM 1.4. Assume A ∈L(Rn)and B ∈L(Rm,Rn). Then, system (4) is exactly controllable to the trajectories on (0,T)if and only if (10) holds. Let us remark that a similar result to Theorem 1.4 has been already proved in [3] and [4] for system (6). To be precise, in [3] and [4], a necessary and sufficient condition for the null controllability on the time interval (0,T)of the system (6) is stated (D=P−1diag(d1,....,dn)P,A∈L(Rn)and B∈L(Rm,Rn)are constant matrices and detP=0anddi>0, for every i,1⩽i⩽n). When D≡Id we obtain system (4) for L(t)≡−Rand the condition given in [3] is equivalent to (10). So as to state our next result, let us consider a trajectory of system (4) y∗∈ L2(0,T;H1 0(Ω)n)∩C0([0,T];L2(Ω)n). Thus, it is possible to characterize the initial data y0that can be exactly driven to y∗(·,T)when Aand Bare constant matrices and condition (10) is not fulfilled. One has the following result. 432 F. AMMAR KHODJA,A.BENABDALLAH,C.DUPAIX AND M. GONZ ´ ALEZ-BURGOS THEOREM 1.5. Assume A ∈L(Rn),B∈L(Rm,Rn)and rank[A|B]=<n. Let X ⊂Rnbe the linear space generated by the columns of [A|B]. Then, given y0∈ L2(Ω)nand y∗∈L2(0,T;H1 0(Ω)n)∩C0([0,T];L2(Ω)n), a trajectory of (4), there exists a control v ∈L2(Q)msuch that the solution to (4) corresponding to the initial condition y(·,0)=y0in Ωsatisfies y(·,T)=y∗(·,T)in Ω, if and only if y0−y∗(·,0)∈L2(Ω;X). In order to study the controllability properties of system (4), we will consider the corresponding adjoint problem − ∂ t ϕ +L∗(t) ϕ =A∗(t) ϕ +F0+∇·Fin Q, ϕ =0onΣ, ϕ (·,T)= ϕ 0in Ω,(13) where F0=(F1 0,...,Fn 0)∗∈L2(Q)n,F=(F1|F2|...|Fn)∈L2(Q;L(Rn;RN)), ϕ 0∈ L2(Ω)nand where, by means of ∇·F, we are denoting the column vector ∇·F= (∇·F1,∇·F2,...,∇·Fn)∗. The operator L∗(t), the adjoint operator to L(t),is given by L∗(t)y(x,t)=− N ∑ i,j=1 ∂ ∂ xiaij(x,t) ∂ y ∂ xj (x,t)− N ∑ i=1 ∂ (biy) ∂ xi (x,t) +c(x,t)y(x,t). It is by now well known that the exact controllability on (0,T)to the trajectories of system (4) is equivalent to the existence of a positive constant Csuch that, for every ϕ 0∈L2(Ω)n, the solution ϕ ∈C0([0,T];L2(Ω)n)to the adjoint system (13) (corresponding to F0=0andF=0) satisfies the observability inequality || ϕ (·,0)||2 L2(Ω)⩽C ω ×(0,T)|B∗(t) ϕ (x,t)|2,(14) where we have omitted, as we do along all this paper when no confusion is possible, the Lebesgue measure dxdt. Under hypothesis (12), we will prove inequality (14) as a consequence of a global Carleman inequality for the solutions to (13) in the effective control interval (T0,T1). In the particular case in which Aand Bare constant matrices we will see that (T0,T1)≡(0,T)and will obtain THEOREM 1.6. There exist a positive function α 0∈C2(Ω)(only depending on Ωand ω )such that, if A ∈L(Rn)and B ∈L(Rm,Rn)satisfy (10), then there exist two positive constants C0and σ 0(only depending on Ω, ω ,(aij)1⩽i,j⩽N,n,m,Aand B) and integers ⩾3,1⩾0and 2⩾2(only depending on n, m, A and B)such that, for every ϕ 0∈L2(Q)n, the solution ϕ to (13) satisfies I(3, ϕ )⩽ C1s ω ×(0,T) e−2s αγ (t)|B∗ ϕ |2 +s1Q e−2s αγ (t)1|F0|2+s2Q e−2s αγ (t)2|F|2,(15) TIME-DEPENDENT COUPLED LINEAR PARABOLIC SYSTEMS 433 ∀s⩾s0= σ 0T+T2+T2||c||2/3 ∞+T2||b||2 ∞. In inequality (15), α (x,t), γ (t)and I(d,z)are respectively given by: α (x,t)≡ α 0(x)/t(T−t), γ (t)≡(t(T−t))−1and I(d,z)≡sd−2Q e−2s αγ (t)d−2|∇z|2+sdQ e−2s αγ (t)d|z|2. We will prove Theorem 1.6 from the corresponding global Carleman inequality satisfied by the solutions to scalar parabolic equations with a right hand side in L2(0,T;H−1(Ω)) (see e.g. [9] and [14]). The rest of the work is organized as follows. In Section 2 we will recall some known results on controllability of cascade parabolic systems which will be used later. Section 3 will be devoted to proving Theorems 1.4, 1.5 and 1.6, i.e., we will study in this section the case of constant matrices A∈L(Rn)and B∈L(Rm,Rn). We will deal with the general case in Section 4. We will devote Section 5 to give some remarks and additional results. We will finalize the work with an appendix where we will give a sketch of the proofs of the results stated in Section 2. 2. Some known results As said above, we will obtain the proofs of Theorems 1.2, 1.4 and 1.6 as a consequence of the results on controllability for cascade systems stated in [12]. For the sake of completeness, let us recall these results in the precise way they are going to be used. Let us consider the controlled system  ∂ tw+L(t)w=C(t)w+Du1 ω in Q, w=0onΣ,w(·,0)=w0in Ω,(16) where u=(u1,u2,...,ur)∗, with 1 ⩽r⩽n, and the coupling and control matrices C and Dsatisfy: C∈C0([0,T];L(Rn)) is given by C(t)=⎛ ⎜ ⎜ ⎜ ⎝ C11(t)C12(t)··· C1r(t) 0C22(t)··· C2r(t) . . .. . ..... . . 00··· Crr(t) ⎞ ⎟ ⎟ ⎟ ⎠,(17) with Cii(t)=⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ α i 11(t) α i 12(t) α i 13(t)... α i 1,si(t) 1 α 22(t) α i 23(t)... α i 2,si(t) 01 α i 33(t)... α i 3,si(t) . . .. . ........ . . 00... 1 α i si,si(t) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , si∈N,∑r i=1si=n,andD∈L(Rr,Rn)such that D=(eS1|eS2|···|eSr)with Si= 1+∑i−1 j=1sj,1⩽i⩽r(ejis the j-th element of the canonical basis of Rn). Observe that (16) is a cascade system and, by means of D, we are exerting rdistributed controls. Therefore, we can apply the controllability results stated in [12] and obtain 434 F. AMMAR KHODJA,A.BENABDALLAH,C.DUPAIX AND M. GONZ ´ ALEZ-BURGOS THEOREM 2.1. Under the previous assumptions, given w0∈L2(Ω)n, there exists a control u ∈L2(Q)rsuch that the corresponding solution w to (16) satisfies w(x,T)= 0in Ω. This controllability result is a consequence of an appropriate global Carleman inequality satisfied by the solutions to the adjoint problem to (16): − ∂ t ψ +L∗(t) ψ =C∗(t) ψ +G0+∇·Gin Q, ψ =0onΣ, ψ (·,T)= ψ 0in Ω,(18) where G0=(G1 0,...,Gn 0)∗∈L2(Q)n,G=(G1|G2|...|Gn)∈L2(Q;L(Rn;RN)) and ψ 0∈L2(Ω)n. We recall that ∇·Gis the column vector given by ∇·G=(∇·G1,∇· G2,...,∇·Gn)∗. Under the previous hypotheses, one has the following result. THEOREM 2.2. There exist two positive constants  C0and  σ 0(only depending on Ω, ω ,(aij)1⩽i,j⩽N, n, C and D)and integers j⩾3,1 k⩾0and 2 k⩾2, with 1⩽j⩽r and 1⩽k⩽n, (only depending on (si)1⩽i⩽r, n and r)such that, for every ψ 0∈L2(Ω)n, the solution ψ to (18) satisfies n ∑ i=1 I(3, ψ i)⩽ C0r ∑ j=1 sj ω ×(0,T) e−2s αγ (t)j| ψ Sj|2 + n ∑ k=1s1 kQ e−2s αγ (t)1 k|Gk 0|2+s2 kQ e−2s αγ (t)2 k|Gk|2,(19) for every s ⩾ s0= σ 0T+T2+T2||c||2/3 ∞+T2||b||2 ∞. In the previous inequality, α (x,t), γ (t)and I(d,z)are as in Theorem 1.6. Theorem 2.2 is proved in [12] but, for the sake of completness, we will include a sketch of the proof in an appendix at the end of the paper. 3. The case in which Aand Bare constant We will devote this section to prove Theorems 1.4, 1.5 and 1.6. As said above, we will show that, by means of an appropriate change of variables, system (4) is equivalent on (0,T)to a cascade parabolic system as system (16). We will deduce the proofs of Theorems 1.4 and 1.6 from Theorems 2.1 and 2.2. We first present the proofs when B∈Rn(m=1, one control force) and then, the general case B∈L(Rm,Rn). 3.1. One control force All along this subsection, we will assume that A∈L(Rn)and B∈Rn. Therefore, [A|B]∈L(Rn). TIME-DEPENDENT COUPLED LINEAR PARABOLIC SYSTEMS 435 Proof of Theorem 1.4. Firstly, observe that, as an easy consequence of Theorem 1.5, (10) is a necessary condition to obtain the exact controllability to trajectories of system (4) on the interval (0,T). On the other hand, let us assume that condition (10) holds. Then, we have that the matrix P=[A|B]∈L(Rn)satisfies detP=0. Also, we can readily show the equalities AP =AB|A2B|···|AnB=PC Pe1=B(20) where e1=(1,0,...,0)∗,Cis the matrix C=⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 00 0 ... −an 10 0 ... −an−1 01 0 ... −an−2 . . .. . ........ . . 00... 1−a1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (21) and ai∈R,1⩽i⩽n,arethecoefficients of the characteristic polynomial of A. Indeed, if p( λ )= λ n+a1 λ n−1+a2 λ n−2+···+an−1 λ +anis the characteristic polynomial of Athen, as a consequence of the Cayley-Hamilton theorem, we have p(A)=0, that is to say, An=−a1An−1−a2An−2−···−an−1A−anId, hence AnB=−a1An−1B−a2An−2B−···−an−1AB −anB. Thus, P−1AP =Cwith Cgiven by (21). If yis the solution to (4) associated to the condition y(·,0)=y0∈L2(Ω)n,then w=P−1yis the solution to  ∂ tw+L(t)w=Cw+e1v1 ω in Q, w=0onΣ,w(x,0)=P−1y0in Ω,(22) with Cgiven by (21). Clearly, system (4) is exactly controllable to the trajectories on the time interval (0,T)if and only if system (4) is null controllable on (0,T).Since detP=0, system (4) is null controllable on (0,T)if and only if system (22) also is null controllable on (0,T). To finalize, the null controllability of system (22) can be obtained as a consequence of Theorem 2.1 applied to C,D=e1,m=1, r=1, S1=1ands1=n. REMARK 3.1. Observe that condition (10) is independent of ω . Thus, if condition (10) is not satisfied, then system (4) is not exactly controllable to the trajectories on the time interval (0,T)even if we take ω ≡Ω. On the other hand, we have deduced the exact controllability to the trajectories of system (4) showing that, under assumption (10), system (4) is equivalent to system (22) (Cis given by (21)) on the time interval (0,T). Observe that [C|e1]≡Id. The pair (C,e1)is the so-called Brunovsky’s form of (A,B). 442 F. AMMAR KHODJA,A.BENABDALLAH,C.DUPAIX AND M. GONZ ´ ALEZ-BURGOS LEMMA 4.1. Assume that max{rankKp(t):t∈[0,T]}=⩽n,for an integer p ∈ {n,...,M}. Then, there exist T0,T1∈[0,T], with T0<T1,r∈{1,...,m}and sequences {sj}1⩽j⩽r⊂{1,2,...,n}, with ∑r i=1sj=, and {lj}1⩽j⩽r⊂{1,2,...,m}such that, for every t ∈[T0,T1],theset B(t)= r  j=1 {blj 0(t),blj 1(t),...,blj sj−1(t)}, is linearly independent, generates the columns of Kp(t)and satisfies blj sj(t)= j ∑ k=1 θ lj,lk sj,0(t)blk 0(t)+ θ lj,lk sj,1(t)blk 1(t)+···+ θ lj,lk sj,sk−1(t)blk sk−1(t),(30) for every t ∈[T0,T1]and j, 1⩽j⩽r, where θ lj,lk sj,0, θ lj,lk sj,1, ..., θ lj,lk sj,sk−1∈C1([T0,T1]). Proof. Since Kp∈C1([0,T];L(Rmp;Rn)), there exists a non degenerate closed interval I0⊆[0,T]such that rankKp(t)=, ∀t∈I0.(31) The proof will be made in the following steps. Step 1.We define l1=min{j∈{1,...,m}:∃t0∈I0s.t. bj(t0)(= bj 0(t0)) =0}, and rl1=1. Observe that, if 1 ⩽k⩽l1−1, bk(t)≡0inI0and, therefore, K1 p(t)|···|Kl1−1 p(t)≡0inI0. Let  I0⊆I0be a non degenerate closed interval such that bl1 0(t)=0, forall t∈ I0. We will show the statement of the lemma for l1, s1=max t∈ I0 rank K1 p(t)|···|Kl1 p(t)≡max t∈ I0 rankKl1 p(t) (which satisfies 1 ⩽s1⩽n), a non degenerate closed interval Irl1(to be determined) and rl1instead of m,,[T0,T1]and r. Firstly, let us fixI1 0⊆ I0such that rank K1 p(t)|···|Kl1 p(t)≡rankKl1 p(t)=s1,∀t∈I1 0.(32) On the other hand, let us consider the set {bl1 0(t),bl1 1(t)}with t∈I1 0. Thus: 1. if {bl1 0(t),bl1 1(t)}is linearly dependent for every t∈I1 0, then, we deduce the existence of a function θ l1 10 ∈CM−1(I1 0)such that bl1 1(t)= θ l1 10(t)bl1 0(t),∀t∈I1 0. TIME-DEPENDENT COUPLED LINEAR PARABOLIC SYSTEMS 443 In this case, it is easy to show that, for every k,1⩽k⩽p−1, one has bl1 k(t)= θ l1 k0(t)bl1 0(t),∀t∈I1 0, for a function θ l1 k0∈CM−k(I1 0). From (32), we deduce s1≡1 and, therefore, for every t∈Irl1≡I1 0, we obtain that the set {bl1 0(t)}is l.i., generates the columns of K1 p(t)|··· |Kl1 p(t)and satisfies (30) for j=rl1=1; 2. if {bl1 0(t1 0),bl1 1(t1 0)}, with t1 0∈I1 0, is linearly independent, then, there exists a new non degenerate interval I1 1⊆I1 0such that {bl1 0(t),bl1 1(t)}is linearly independent for every t∈I1 1. In this last case, we consider the set {bl1 0(t),bl1 1(t),bl1 2(t)}with t∈I1 1.Iftheset {bl1 0(t),bl1 1(t),bl1 2(t)}is linearly dependent for every t∈I1 1, there exist two functions θ l1 20, θ l1 21 ∈CM−2(I1 1)such that bl1 2(t)= θ l1 20(t)bl1 0(t)+ θ l1 21(t)bl1 1(t),∀t∈I1 1. Again, we readily check that bl1 k(t)= θ l1 k0(t)bl1 0(t)+ θ l1 k1(t)bl1 1(t),∀t∈I1 1,∀k:2⩽k⩽p−1, with θ l1 k0, θ l1 k1∈CM−k(I1 1). In this case, s1≡2 and again we obtain the result for Irl1=I1 1. On the other hand, if for t1 1∈I1 1the set {bl1 0(t1 1),bl1 1(t1 1),bl1 2(t1 1)}is linearly independent, we can continue the previous process until we find  s1∈{1,..., p}and a non degenerate interval Irl1⊆I0such that for all t∈Irl1the set {bl1 0(t),bl1 1(t),...,bl1  s1−1(t)} is linearly independent and generates the columns of Kl1 p(t). From (32), it is clear that  s1=s1and, so, we have (30) for j=1. Step 2.Assume that, for l1⩽i⩽m−1, we have constructed ri∈{1,...,i}, a non degenerate closed interval Iri⊆Iri−1⊆I0,sj∈{1,...,n}and lj∈{1,...,m} (1 ⩽j⩽ri) such that, if we define Blj(t)={blj 0(t),blj 1(t),...,blj sj−1(t)}, one has that for every t∈Irithe set ri  j=1 Blj(t) is linearly independent, generates the columns of K1 p(t)|···|Ki p(t)and satisfies (30) for 1 ⩽j⩽ri. We argue as above and we consider the set Di+1={q∈{i+1,...,m}:∃t0∈Iris.t. {bq(t0)}∪ ri  j=1 Blj(t0)is l.i.}. 444 F. AMMAR KHODJA,A.BENABDALLAH,C.DUPAIX AND M. GONZ ´ ALEZ-BURGOS 1. If Di+1=/0, then, for every q,i+1⩽q⩽m, there exist coefficients θ q,lj 00 (t), θ q,lj 01 (t), ..., θ q,lj 0,sj−1(t)(1⩽j⩽ri) such that, for each t∈Iri, one has bq 0(t)= ri ∑ j=1 θ q,lj 00 (t)blj 0(t)+ θ q,lj 01 (t)blj 1(t)+···+ θ q,lj 0,sj−1(t)blj sj−1(t).(33) It is not difficult to check that the previous coefficients satisfy θ q,lj 00 , θ q,lj 01 ,..., θ q,lj 0,sj−1∈CM+1− S0(Iri)⊂C1(Iri), where  S0=max1⩽j⩽risj(1⩽ S0⩽n). From equality (33) and taking into account (29), we infer that the set {bq 1(t)}∪ ri  j=1 Blj(t) is linearly dependent for every t∈Iri. So, we deduce that bq 1(t)can be written as bq 1(t)= ri ∑ j=1 θ q,lj 10 (t)blj 0(t)+ θ q,lj 11 (t)blj 1(t)+···+ θ q,lj 1,sj−1(t)blj sj−1(t), for new coefficients θ q,lj 10 , θ q,lj 11 ,..., θ q,lj 1,sj−1∈CM+1− S1(Iri)⊂C1(Iri)(1⩽j⩽ri), where  S1=max{2,sj:1⩽j⩽ri}. By induction, we readily obtain that the set {bq l(t)}∪ ri  j=1 Blj(t)(34) is linearly dependent for every l:0⩽l⩽p−1andt∈Iriand bp l(t)can be written as bq l(t)= ri ∑ j=1 θ q,lj l0(t)blj 0(t)+ θ q,lj l1(t)blj 1(t)+···+ θ q,lj l,sj−1(t)blj sj−1(t), for new coefficients θ q,lj l0, θ q,lj l1,..., θ q,lj l,sj−1∈CM+1− Sl(Iri)(1⩽j⩽ri) with  Sl=max{l+ 1,sj:1⩽j⩽ri}. In this case, we set rl=ri,foreveryi+1⩽l⩽mand the lemma is proved if we take r=rmand [T0,T1]≡Irm. 2. If Di+1=/0,weset l=minDi+1,rj=riif i+1⩽j⩽l−1, rl=ri+1andlrl=l. For q:1⩽q⩽l−1, we can argue as above and obtain that for every t∈Irq(= Iri),the set (34) is linearly dependent for every l:0⩽q⩽p−1. So,forevery t∈Irl−1,theset ∪rl−1 j=1Blj(t)is l.i., generates the columns of the matrix K1 p(t)|···|Kl−1 p(t)and (30) holds for 1 ⩽j⩽rl−1. Now, we concentrate on the case q=l. Our next goal is to construct a new non degenerate interval Irl⊆Iriand slsuch that, for every t∈Irl,theset∪rl j=1Blj(t)is l.i., generates the columns of the matrix K1 p(t)|···|Kl p(t) TIME-DEPENDENT COUPLED LINEAR PARABOLIC SYSTEMS 445 and (30) holds for 1 ⩽j⩽rl. Taking into account the definition of l, we deduce that there exists a new non degenerate interval Irl 0⊆Irisuch that the set {bl(t)}∪ri j=1Blj(t)is l.i. for every t∈Irl 0. We now consider the set {bl 0(t),bl 1(t)}∪ri j=1Blj(t)with t∈Irl 0.If{bl 0(t),bl 1(t)}∪ ri j=1Blj(t)is l.d. for every t∈Irl 0, there exist functions θ l,l 10 , θ l,lj 10 , θ l,lj 11 ,..., θ l,lj 1,sj−1∈ CM+1− S1(Irl 0)(1 ⩽j⩽ri), where  S1=max{2,sj:1⩽j⩽ri}, such that, for each t∈Irl 0, one has bl 1(t)= θ l,l 10 (t)bl 0(t)+ ri ∑ j=1 θ l,lj 10 (t)blj 0(t)+ θ l,lj 11 (t)blj 1(t)+···+ θ l,lj 1,sj−1(t)blj sj−1(t). From this equality and following the argument above, we infer that bl l(t), with 1 ⩽ l⩽p−1andt∈Irl 0, can be written as a linear combination of {bl 0(t)}∪ri j=1Blj(t). Thus, we obtain the previous property for the set ∪rl j=1Blj(t)by taking srl=1and Irl=Irl 0⊆Iri. On the other hand, if there exists trl 0∈Irl 0such that the set {bl 0(trl 0),bl 1(trl 0)}∪ ri j=1Blj(trl 0)is l.i., then we can obtain a new non degenerate interval Irl 1⊂Irl 0such that the previous set goes on being a l.i. set for every t∈I l 1. As in the previous step, we can continue the method until we obtain srl∈{1,...,n}and a non degenerate interval Irl⊂Irisuch that the set rl j=1Blj(t)is l.i. for every t∈Irl, generates the columns of the matrix K1 p(t)|K2 p(t)|···|Kl p(t)and satisfies (30) for 1 ⩽j⩽rl. Finally, from the reasoning by induction, we deduce the existence of r(= rm)and a non degenerate closed interval [T0,T1]⊆I0such that, for every t∈[T0,T1],theset r j=1Blj(t)is l.i., generates the columns of the matrix K1 p(t)|K2 p(t)|···|Km p(t)and satisfies (30) for 1 ⩽j⩽r. From (31), we also conclude that ∑r i=1sj=. This concludes the proof of Lemma 4.1.  As a consequence of Lemma 4.1 we deduce. COROLLARY 4.2. Under the assumptions of Lemma 4.1,letbe a set defined by =max{rankKp(t):t∈[0,T]}with p ∈{n,..,M}an integer. Then, max{rankKn(t): t∈[0,T]}=and there exist T0,T1∈[0,T], with T0<T1, such that rank Kn(t)=, ∀t∈[T0,T1]. We are now ready to prove Theorem 1.2. Proof of Theorem 1.2.All along this proof we will assume that Aand Bsatisfy (5). 1. Let us suppose that condition (12) is satisfied for t0∈[0,T]and p∈{1,...,M}.If p<n, in particular rank Kn(t0)=nand we can apply Lemma 4.1 with =n.If 446 F. AMMAR KHODJA,A.BENABDALLAH,C.DUPAIX AND M. GONZ ´ ALEZ-BURGOS p∈{n,...,M}we can directly apply Lemma 4.1 also with =nand obtain in both cases the existence of an interval [T0,T1]⊆[0,T](T0<T1), r⩽m,lj∈{1,2,...,m} and sj∈{1,2,...,n},sj⩽p,(1⩽j⩽r) with ∑r j=1sj=nsuch that if P(t)=bl1 0|bl1 1|···|bl1 s1−1|bl2 0|bl2 1|···|bl2 s2−1|···|blr 0|blr 1|···|blr sr−1(t), then, detP(t)=0foreveryt∈[T0,T1]. In addition, P∈C1([0,T];L(Rn)) and, thanks to the properties of the columns of P(t)stated in Lemma 4.1, if 1 ⩽j⩽rand t∈ [T0,T1]one has (30) for θ lj,li sj,0, θ lj,li sj,1, ..., θ lj,li sj,si−1∈C1([T0,T1]). Therefore, A(t)P(t)−P(t)=P(t)C(t)and P(t)eSi=bli 0(t),1⩽i⩽r, with Si=1+∑i−1 j=1sj,1⩽i⩽r,C(t)given by (17) and Cii ∈C0([T0,T1];L(Rsi)) and Cij ∈C0([T0,T1];L(Rsj;Rsi)),1⩽i⩽j⩽r,givenby Cii = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 00 0 ... θ li,li si,0 10 0 ... θ li,li si,1 01 0 ... θ li,li si,2 . . .. . ........ . . 00... 1 θ li,li si,si−1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ,Cij = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 00... 0 θ lj,li sj,0 00... 0 θ lj,li sj,1 00... 0 θ lj,li sj,2 . . .. . ........ . . 00... 0 θ lj,li si,si−1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . As done in Section 3, we will prove that system (4) is in fact exactly controllable to trajectories on the interval [T0,T1]⊆[0,T]when we exert rcontrol forces. Obviously, this fact implies that system (4) is exactly controllable to the trajectories on the interval (0,T). To this end, we consider the system  ∂ ty+L(t)y=A(t)y+ B(t)v1 ω in  Q=Ω×(T0,T1), y=0on Σ= ∂ Ω×(T0,T1),y(x,T0)=y0in Ω,(35) with  B(t)=(0|···|0|bl1 0(t)|0|···|0|bl2 0(t)|0|···|0|blr 0(t)|0|···)and y0∈L2(Ω)n. Let us see that system (35) is null controllable on [T0,T1](which clearly implies that system (4) is exactly controllable to the trajectories on the interval (0,T)). If we do w=P(t)−1yand v=(v1,...,vm)∗∈L2(Ω)m, the null controllability result on the interval [T0,T1]for system (35) is equivalent to the null controllability on the interval [T0,T1]of system ⎧ ⎪ ⎨ ⎪ ⎩ ∂ tw+L(t)w=C(t)w+ r ∑ j=1 eSjvlj1 ω in  Q, w=0on Σ,w(x,T0)=P−1(T0)y0in Ω, where C(t)is given by (17). Once again, the null controllability property of the previous system is deduced from Theorem 2.1. TIME-DEPENDENT COUPLED LINEAR PARABOLIC SYSTEMS 447 2. Let us fixp∈{n,...,M}and T0,T1∈[0,T]with T0<T1. Let us assume that there exists a dense set Ein [0,T]such that rank Kp(t)=nfor every t∈E. Then, one has max{rank Kp(t):t∈[T0,T1]}=n. If we repeat the arguments developed in the previous point, we deduce that system (4) is exactly controllable to the trajectories on the interval [T0,T1]. Now, let us assume that system (4) is exactly controllable to the trajectories on every non degenerate interval [T0,T1]of [0,T]. We will prove the existence of a dense subset E⊆[0,T]for which rank Kn(t)=nfor every t∈E. Evidently, this fact implies that rank Kp(t)=nfor every p∈{n,...,M}and t∈E. By contradiction, let us also assume that for a non degenerate interval [ τ 0, τ 1]⊆ [0,T]one has max{rank Kn(t):t∈[ τ 0, τ 1]}=<n. Thus, applying Lemma 4.1 in the interval [ τ 0, τ 1], we deduce the existence of a new non degenerate interval [ τ 0, τ 1]⊆ [ τ 0, τ 1],r∈{1,...,m}and sequences {sj}1⩽j⩽r⊂{1,2,...,n}, with ∑r i=1sj=,and {lj}1⩽j⩽r⊂{1,2,...,m}such that, for every t∈[ τ 0, τ 1],theset B(t)= r  j=1 {blj 0(t),blj 1(t),...,blj sj−1(t)}, is linearly independent, generates the columns of Kn(t)and satisfies (30) in [ τ 0, τ 1].In particular, if we fix τ ∈( τ 0, τ 1),thesetB( τ )is linearly independent and generates the columns of Kn( τ ). Let us now take a set {p+1,..., pn}such that B( τ )∪{p+1,..., pn} is a basis of Rn. Using the continuity of Kn(t)in [0,T], we infer that the set B(t)∪ {p+1,..., pn}is a basis of Rnand B(t)generates the columns of Kn(t)in a new non degenerate subinterval [ τ 0, τ 1]of [ τ 0, τ 1].Fort∈[ τ 0, τ 1],letP(t)be the matrix whose columns are the elements of B(t)∪{p+1,..., pn}.Ifweset  B(t)=P−1(t)B(t)and C(t)=P−1(t)A(t)P(t),∀t∈[ τ 0, τ 1], then, from the previous properties, we get  B(t)= B1(t) 0in [ τ 0, τ 1],and C(t)=C11(t)C12(t) 0C22(t),t∈[ τ 0, τ 1] with  B1∈C0([ τ 0, τ 1];L(Rm;R)),C11 ∈C0([ τ 0, τ 1];L(R)), C12 ∈C0([ τ 0, τ 1];L(Rn−,R)),C22 ∈C0([ τ 0, τ 1];L(Rn−)), and are such that one has rank[C11 | B1]=. Again, reasoning as in the proof of Theorem 1.5, we obtain that system (4) is not null controllable on the time interval [ τ 0, τ 1]since on this time interval this system is equivalent to the decoupled system ⎧ ⎪ ⎨ ⎪ ⎩ ∂ tw+L(t)w=C11(t)C12(t) 0C22(t)w+ B1(t) 01 ω vin Ω×( τ 0, τ 1), w=0on ∂ Ω×( τ 0, τ 1). (36) 448 F. AMMAR KHODJA,A.BENABDALLAH,C.DUPAIX AND M. GONZ ´ ALEZ-BURGOS This contradicts our assumption and finalizes the proof of Theorem 1.2.  REMARK 4.1. As said above, under condition (12), system (4) is exactly controllable to the trajectories on (0,T)when we exert on the system r⩽meffective control forces, the controls ∑r j=1blj(·)vlj. Observe that, in fact, theses control forces are exerted on the system in the effective time interval [T0,T1]. As a consequence of Lemma 4.1, we can establish a Carleman estimate for the solutions of the adjoint problem (13) in the time interval (T0,T1). The following result holds true. COROLLARY 4.3. Let us assume that A and B satisfy hypothesis (5). Then, there exist a positive function α 0∈C2(Ω)(only depending on Ωand ω )such that, if (12) is fulfilled, there exist a time interval (T0,T1)⊆(0,T), two positive constants C0= C0(Ω, ω ,(aij)1⩽i,j⩽N,n,m,A(·),B(·)) and σ 0= σ 0(Ω, ω ,(aij)1⩽i,j⩽N,n,m,A(·),B(·)) and integers ⩾3,1⩾0and 2⩾2(only depending on n, m, A(·)and B(·)) such that, for every ϕ 0∈L2(Q)n, the solution ϕ to (13) satisfies I(3, ϕ )⩽ C1s ω ×(T0,T1) e−2s α  γ (t)|B∗ ϕ |2 +s1Ω×(T0,T1) e−2s α  γ (t)1|F0|2+s2Ω×(T0,T1) e−2s α  γ (t)2|F|2, for every s ⩾s0= σ 0 T+ T2+ T2||c||2/3 ∞+ T2||b||2 ∞with  T=T1−T0. In the previous inequality  α (x,t), γ (t)and I(d,z)are respectively given by:  α (x,t)≡ α 0(x)/(t−T0)(T1−t), γ (t)≡((t−T0)(T1−t))−1, and I(d,z)≡sd−2Ω×(T0,T1) e−2s α  γ (t)d−2|∇z|2+sdΩ×(T0,T1) e−2s α  γ (t)d|z|2. Proof. Let us consider the adjoint problem (13) and let us assume that condition (12) holds. As in the proof of Theorem 1.2 1, we can apply Lemma 4.1 with =nand obtain an interval [T0,T1]⊆[0,T](T0<T1)andasetB(t)such that B(t) is a basis of Rnfor every t∈(T0,T1)and satisfies (30). We now consider the matrix P(t)∈C1([T0,T1];L(Rn)) whose columns are the elements of B(t).If ϕ is the solution to (13) corresponding to ϕ 0∈L2(Ω)nand we perform the change ψ =P(t)∗ ϕ in (T0,T1), then, from the properties satisfied by the basis B(t)(see (30)), it is not difficult to check that ψ solves − ∂ t ψ +L(t) ψ =C∗(t) ψ +G0+∇·Gin Ω×(T0,T1), ψ =0on ∂ Ω×(T0,T1), ψ (x,T1)=P(T1)∗ ϕ (x,T1)in Ω, TIME-DEPENDENT COUPLED LINEAR PARABOLIC SYSTEMS 449 where C(t)is the matrix given by (17), G0=P∗(·)F0and G=FP(·). If we apply to ψ Theorem 2.2 in the interval (T0,T1)we deduce the existence of α 0∈C2(Ω), only depending on Ωand ω , two positive constants  C0= C0(Ω, ω ,(aij)1⩽i,j⩽N,n,m,A(·),B(·)) and  σ 0= σ 0(Ω, ω ,(aij)1⩽i,j⩽N,n,m,A(·),B(·)) and integers j⩾3, 1 k⩾0and2 k⩾2, with 1 ⩽j⩽rand 1 ⩽k⩽n, (only depending on n,m,A(·)and B(·))insuchaway that ψ satisfies the inequality n ∑ i=1I(3, ψ i)⩽ C0r ∑ j=1 sj ω ×(T0,T1) e−2s α  γ (t)j| ψ Sj|2 + n ∑ k=1s1 kΩ×(T0,T1) e−2s α  γ (t)1 k|Gk 0|2+s2 kΩ×(T0,T1) e−2s α  γ (t)2 k|Gk|2, for every s⩾ s0= σ 0T+T2+T2||c||2/3 ∞+T2||b||2 ∞,whererand siare provided by Theorem 2.2 and Si=1+∑i−1 j=1sj,(1⩽i⩽r). Finally, the proof can be obtained if we take =max1⩽j⩽rj,1=max1⩽k⩽n1 k and 2=max1⩽k⩽n2 k, we replace ϕ =(P∗(t))−1 ψ in the previous inequality and we take into account the equality e∗ Si ψ Sj=(blj)∗(t) ϕ .Thisfinalizes the proof.  We will finish this section proving Theorem 1.3 when Aand Bare analytic in [0,T]. Proof of Theorem 1.3.It is clear that if there exist t0∈[0,T]and p⩾1 such that condition (12) holds then system (4) is exactly controllable to the trajectories on the interval [0,T]. Let us see the necessary part. By contradiction, assume that for every t0∈[0,T] one has rank Kp(t0)<n,∀p⩾1. If we fixt0∈[0,T], the previous condition implies the existence of ξ ∈Rn\{0}such that ξ ∗Bi(t0)=0,∀i⩾0.(37) Under this last condition we will see that system (4) is equivalent (on the time interval [0,T]) to an appropriate system which contains, at least, one equation which cannot be controlled. Therefore, we will deduce that system (4) is not null controllable on [0,T]. Let us consider a fundamental matrix F(t)associated to the ordinary differential system x=A(t)xon [0,T]. It is well known that F(t)is nonsingular for all t∈[0,T], is analytic on [0,T]and satisfies d dt F(t)−1=−F(t)−1A(t),∀t∈[0,T]. It is not difficult to check the equality (see (8)) di dtiF(t)−1B(t)=(−1)iF(t)−1Bi(t),∀t∈[0,T],∀i⩾0. 450 F. AMMAR KHODJA,A.BENABDALLAH,C.DUPAIX AND M. GONZ ´ ALEZ-BURGOS We now take the function g(t)= ξ ∗F(t0)F(t)−1B(t)(analytic on [0,T]). The previous equality, together with condition (37), provides us the property dig dti(t0)=0,∀i⩾0, whence g(t)=0,i.e., ξ ∗F(t0)F(t)−1B(t)=0, for every t∈[0,T]. Finally, we consider a nonsingular matrix P∈L(Rn)whose first row is given by ξ ∗. Thus, if system (4) is null controllable on [0,T], performing the change w=PF(t0)F(t)−1yand taking  B(t)=PF(t0)F(t)−1B(t), the system  ∂ tw+L(t)w= B(t)v1 ω in Q, w=0onΣ,(38) is also null controllable on the time interval [0,T]. Nevertheless, the previous properties show  B(t)=0  B(t), with  B(·)∈C∞([0,T];L(Rm;Rn−1)) (in fact, analytic on [0,T]). We readily deduce that system (38) is not null controllable on [0,T]which contradicts the assumptions. This ends the proof.  REMARK 4.2. It is interesting to remark that, when Aand Bare analytic on [0,T], the previous proof shows that a necessary and sufficient condition for the exact controllability to the trajectories of system (4) on [0,T]is: for every t ∈[0,T]there exists p ∈Nsuch that rank Kp(t)=n. On the other hand, in the proof of Theorem 1.2 we have shown that if for t0∈ [0,T]and p⩾1wehaverankKp(t0)=n, then there exists a non degenerate interval [T0,T1]⊆[0,T]such that rank Kn(t)=nfor every t∈[T0,T1]. In particular, when A and Bare analytic, we deduce the existence of a finite set Fsuch that rank Kn(t)=n for every t∈[0,T]\F. Therefore, a necessary and sufficient condition for the exact controllability to the trajectories of system (4) on [0,T]is the following one: there exists a finite set Fsuch that rank Kn(t)=n for every t ∈[0,T]\F. 5. Further results and comments We will finalize this work doing some remarks and establishing some additional results. 1. All along this work we have assumed that Aand Bare matrix functions which satisfy (5) for M⩾n. It is not difficult to see that Theorem 1.2 is still valid under the following assumptions: A∈WM−1,∞(0,T;L(Rn)) and B∈WM,∞(0,T;L(Rm,Rn)). 2. In the case of linear ordinary differential systems as (7) it is well known that the condition (10) is not necessary in order to obtain the null controllability result on the interval [0,T]. Let us show that for the system (4) condition (10) neither is a necessary TIME-DEPENDENT COUPLED LINEAR PARABOLIC SYSTEMS 451 condition for the exact controllability to the trajectories on the interval [0,T]. Indeed, following [6], let us set n=2andm=1andletustake A(t)≡0andB(t)=b1(t) b2(t),∀t∈(0,T), that is to say, let us take the system ⎧ ⎨ ⎩ ∂ ty+L(t)y=b1(t) b2(t)v1 ω in Q, y=0inΣ,y(·,0)=y0in Ω, (39) with y0∈L2(Ω)2,b1∈C∞ 0(0,T/2)and b2∈C∞ 0(T/2,T)such that for a positive constant β and nonempty time intervals ( τ 0, τ 1)⊂⊂ (0,T/2)and ( τ  0, τ  1)⊂⊂ (T/2,T) one has |b1(t)|⩾ β >0,∀t∈( τ 0, τ 1)and |b2(t)|⩾ β >0,∀t∈( τ  0, τ  1). Evidently, Aand Bsatisfy (5) and rank Kp(t)<2foreveryt∈[0,T]and p⩾1. Let us show that system (39) is exactly controllable to the trajectories or equivalently, null controllable on the interval [0,T]: Let us denote Y=(Y1,Y2)∗∈L2(0,T;H1 0(Ω)2) the solution to (39) corresponding to y0∈L2(Ω)2and v≡0. Firstly, thanks to the assumptions on the operator Land the function b1, it is not difficult to prove that there exists u1∈L2(Ω×( τ 0, τ 1)) such that the solution w1∈L2( τ 0, τ 1;H1 0(Ω)) to the scalar equation  ∂ tw1+L(t)w1=b1(t)u11 ω in Ω×( τ 0, τ 1) w1=0on ∂ Ω×( τ 0, τ 1),w1(·, τ 0)=Y1(·, τ 0) satisfies w1(·, τ 1)=0inΩ. On the other hand, we can reason in a similar way and deduce the existence of u2∈L2(Ω×( τ  0, τ  1)) such that the solution w2∈L2( τ  0, τ  1;H1 0(Ω)) to  ∂ tw2+L(t)w2=b2(t)u21 ω in Ω×( τ  0, τ  1) w2=0on ∂ Ω×( τ  0, τ  1),w2(·, τ  0)=Y2(·, τ  0) satisfies w2(·, τ  1)=0inΩ. Finally, let us set y1(x,t)=⎧ ⎨ ⎩ Y1(x,t)if t∈[0, τ 0], w1(x,t)if t∈[ τ 0, τ 1], 0ift∈[ τ 1,T], y2(x,t)=⎧ ⎨ ⎩ Y2(x,t)if t∈[0, τ  0], w2(x,t)if t∈[ τ  0, τ  1], 0ift∈[ τ  1,T], y=(y1,y2)∗and v(x,t)=v1(x,t)if t∈[0,T/2], v2(x,t)if t∈[T/2,T], where v1and v2are, respectively, the extensions by 0 of u1and u2to the whole interval [0,T]. Now, it is easy to check that yis the solution to system (39) corresponding to vand satisfies y(·,T)=0inΩ. In conclusion, condition (10) is not necessary to