F Ammar-Khodja

A Benabdallah

M González-Burgos

L De Teresa
email: deteresa@matem.unam.mx

The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials

Keywords: Parabolic systems, Boundary Controllability, Biorthogonal families, Kalman Rank condition

This paper is devoted to prove the controllability to trajectories of a system of n one-dimensional parabolic equations when the control is exerted on a part of the boundary by means of m controls. We give a general Kalman condition (necessary and sufficient) and also present a construction and sharp estimates of a biothorgonal family in L 2 (0, T ; C) to {t j e -Λ k t }.

Statement of the problem. Main results

This work is devoted to the study of the controllability properties of the following parabolic system      y t = y xx + Ay in Q = (0, π) × (0, T ), y(0, •) = Bv, y(π, •) = 0 on (0, T ),

y(•, 0) = y 0 in (0, π), (1)
where T > 0 is given, A ∈ L(C n ) and B ∈ L(C m ; C n ) are two given matrices and y 0 ∈ H -1 (0, π; C n ) is the initial datum. In system [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF], v ∈ L 2 (0, T ; C m ) is a control function (to be determined) which acts on the system by means of the Dirichlet boundary condition at point x = 0. The aim of this work is to give a necessary and sufficient condition for the exact controllability to trajectories of system [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF]. Let us remark that, for every v ∈ L 2 (0, T ; C m ) and y 0 ∈ H -1 (0, π; C n ), system (1) possesses a unique solution (defined by transposition; see Section 2) which satisfies

y ∈ L 2 (Q; C n ) ∩ C 0 ([0, T ]; H -1 (0, π; C n ))
and depends continuously on the data v and y 0 , i.e., there exists a constant C = C(T ) > 0 such that y L 2 (Q;C n ) + y C 0 ([0,T ];H -1 (0,π;C n )) ≤ C y 0 H -1 (0,π;C n ) + v L 2 (0,T ;C m ) .

It will be said that system [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF] is approximately controllable in H -1 (0, π; C n ) at time T if for every y 0 , y d ∈ H -1 (0, π; C n ) and for every ε > 0, there exists a control v ∈ L 2 (0, T ; C m ) such that the solution y to (1) satisfies y(•, T )y d H -1 (0,π;C n ) ≤ ε.

Also, it will be said that system [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF] is exactly controllable to trajectories at time T if, for every initial datum y 0 ∈ H -1 (0, π; C n ) and every trajectory y ∈ L 2 (Q; C n ) ∩ C 0 ([0, T ]; H -1 (0, π; C n )) of system (1) (i.e., a solution to (1) associated to fixed v ∈ L 2 (0, T ; C m ) and y 0 ∈ H -1 (0, π; C n )), there exists a control v ∈ L 2 (0, T ; C m ) such that the corresponding solution y of (1) satisfies

y(•, T ) = y(•, T ) in H -1 (0, π; C n ).
Thanks to the linear character of system [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF], this last property is equivalent to the null controllability at time T . That is, for every y 0 ∈ H -1 (0, π; C n ) there exists a control v ∈ L 2 (0, T ; C m ) such that the solution y to (1) satisfies

y(•, T ) = 0 in H -1 (0, π; C n ).
It is interesting to point out that we want to control the system [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF], which has n equations, by means of the control v, which has m components. Of course, the most interesting case is the case in which the number of controls is less than the number of equations: m < n.

Nowadays, the controllability properties of system (1) are well known in the scalar case, i.e., in the case n = 1 (see for instance [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF][START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF][START_REF] Russell | Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions[END_REF][START_REF] Fabre | Approximate controllability of the semilinear heat equation[END_REF][START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF][START_REF] Fursikov | Controllability of Evolution Equations[END_REF][START_REF] Fernández-Cara | The cost of approximate controllability for heat equations: the linear case[END_REF]). Thus, when n = 1 and B ≡ 0, system [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF] are exactly controllable to trajectories, null controllable and approximately controllable in H -1 (0, π; C n ) at time T (see for instance [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF][START_REF] Fursikov | Controllability of Evolution Equations[END_REF]). In fact, the boundary controllability results for system (1) can be easily obtained from the corresponding distributed controllability results and vice versa. As it is proved in [START_REF] Fernández-Cara | On the boundary controllability of non-scalar parabolic systems[END_REF][START_REF] Fernández-Cara | Boundary controllability of parabolic coupled equations[END_REF], the situation is quite different when n ≥ 2. More details will be given below.

On the other hand, controllability of linear ordinary differential systems is well-known. In particular we have at our disposal the famous Kalman rank condition (see for example [START_REF] Kalman | Topics in Mathematical Control Theory[END_REF]Chapter 2,p. 35]), that is to say, if n, m ∈ N with n, m ≥ 1 and A ∈ L(C n ) and B ∈ L(C m ; C n ), then the linear ordinary differential system Y ′ = AY + Bu is controllable at time T > 0 if and only if

rank [A | B] = rank [A n-1 B | A n-2 B | • • • | B] = n. (2)
In the framework of distributed controllability, an extension of this algebraic condition to a class of coupled second order parabolic equations has been obtained in [START_REF] Ammar-Khodja | A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems[END_REF] and [START_REF] Ammar-Khodja | A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems[END_REF]. Let us describe the Kalman condition and the result of controllability proved in [START_REF] Ammar-Khodja | A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems[END_REF]. Let R be a scalar second order elliptic selfadjoint operator. Let us also consider the matrices D = diag (d 1 , d 2 , ..., d n ) ∈ L(R n ) (where d i > 0 for every i :

1 ≤ i ≤ n), A ∈ L(R n ) and B ∈ L (R m ; R n ).
Let Ω be a bounded and regular domain of R d and ω ⊂⊂ Ω a nonempty open subset. Denoting by L the operator given by L := DR + A, the authors define the Kalman operator associated with (L, B) by the matrix operator

K := [L | B] : D(K) ⊂ L 2 (Ω) nm → L 2 (Ω) n , with D(K) := {u ∈ L 2 (Ω) nm : Ku ∈ L 2 (Ω) n }, where [L | B] = L n-1 B | L n-2 B | ... | LB | B .
They prove that the following system ∂ t y = (DR + A)y + Bv1 ω in Ω T = Ω × (0, T ),

y = 0 on Σ T = ∂Ω × (0, T ), y(•, 0) = y 0 (•) in Ω, (3)
is null controllable if and only if Ker (K * ) = {0} .

Let us point out that when D = I d , this last condition is equivalent to the algebraic Kalman condition [START_REF] Khodja | Controllability to the trajectories of phasefield models by one control force[END_REF]. In fact, in [START_REF] Ammar-Khodja | A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems[END_REF] the authors study this case and consider an operator L and coupling matrices A and B which depend on t.

In [START_REF] Fernández-Cara | On the boundary controllability of non-scalar parabolic systems[END_REF][START_REF] Fernández-Cara | Boundary controllability of parabolic coupled equations[END_REF], the authors study the controllability properties of system [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF] when n = 2 and m = 1. They prove that, unlike of system (3) with D = I d , the algebraic Kalman rank condition (2) is not a sufficient condition for the null controllability of system [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF]. They also exhibit an additional condition which is equivalent to the exact controllability to trajectories of system (1) (n = 2 and m = 1): Denote by µ 1 and µ 2 the eigenvalues of A * . Then, [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF] is exactly controllable to trajectories at any time T if and only if rank [A | B] = 2 and

µ 1 -µ 2 = k 2 -l 2 , ∀k, l ∈ N with k = l.
This work is an extension of both [START_REF] Ammar-Khodja | A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems[END_REF] and [START_REF] Fernández-Cara | Boundary controllability of parabolic coupled equations[END_REF]. For n, m ∈ N * , we give a suitable extension of the finite-dimensional Kalman rank condition. We show that the exact controllability to trajectories for system [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF] is equivalent to this Kalman condition (see Theorem 1.1).

In the last ten years, the study of the controllability properties of coupled parabolic systems has had an increasing interest (see for instance [START_REF] De Teresa | Insensitizing controls for a semilinear heat equation[END_REF], [START_REF] Khodja | Controllability to the trajectories of phasefield models by one control force[END_REF] [8], [START_REF] Ammar-Khodja | Null controllability of some systems of parabolic type by one control force[END_REF], [START_REF] Ammar-Khodja | Null controllability of some reaction-diffusion systems with one control force[END_REF], [START_REF] González-Burgos | Controllability results for some nonlinear coupled parabolic systems by one control force[END_REF], [START_REF] Guerrero | Null controllability of some systems of two parabolic equations with one control force[END_REF], [START_REF] Ammar-Khodja | A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems[END_REF], [START_REF] Ammar-Khodja | A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems[END_REF], [START_REF] González-Burgos | Controllability results for cascade systems of m coupled parabolic PDEs by one control force[END_REF], [START_REF] Kavian | Unique continuation principle for systems of parabolic equations[END_REF] and [START_REF] Coron | Null controllability of a parabolic system with a cubic coupling term[END_REF]). In these papers, almost all the results have been established for 2 × 2 systems where the distributed control is exerted on one equation (n = 2 and m = 1). The most general results in this context seem to be those in [START_REF] González-Burgos | Controllability results for cascade systems of m coupled parabolic PDEs by one control force[END_REF], [START_REF] Ammar-Khodja | A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems[END_REF] and [START_REF] Ammar-Khodja | A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems[END_REF]. In [START_REF] González-Burgos | Controllability results for cascade systems of m coupled parabolic PDEs by one control force[END_REF], the authors study a cascade parabolic system of n equations (n ≥ 2) controlled with one single distributed control.

To our knowledge, the unique works that study the boundary controllability problem for general coupled parabolic systems are [START_REF] Fernández-Cara | On the boundary controllability of non-scalar parabolic systems[END_REF] and [START_REF] Fernández-Cara | Boundary controllability of parabolic coupled equations[END_REF]. It is also worth mentioning the paper [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF] where a boundary controllability result for a particular hyperbolic coupled system is proved.

In this work we will use the following notation:

Given z ∈ L(C N , C M ), N, M ≥ 1, z * ∈ L(C M , C N ) stands for the conjugate transpose of z. If N = M = 1, i.e., if z = a + bi ∈ C, then z * = z = a -bi is the complex conjugate of z.
Let us now precise our controllability result.

It is well known that the operator -∂ xx on (0, π) with homogenous Dirichlet boundary conditions admits a sequence of eigenvalues and normalized eigenfunctions given by

λ k = k 2 , φ k (x) = 2 π sin kx, k ≥ 1, x ∈ (0, π) (5)
which is a Hilbert basis of L 2 (0, π). Thus, if y ∈ L 2 (0, π; C n ) there exists a unique sequence

{y k } k≥1 ⊂ C n such that y = k≥1 y k φ k . Let L : D(L) ⊂ L 2 (0, π; C n ) → L 2 (0, π; C n ) the unbounded linear operator defined by L = I d ∂ xx + A, D(L) = H 2 (0, π; C n ) ∩ H 1 0 (0, π; C n ).
Its adjoint operator is given by

L * = I d ∂ xx + A * , D(L * ) = H 2 (0, π; C n ) ∩ H 1 0 (0, π; C n ).
Then for any y = k≥1 y k φ k ∈ D(L) = D(L * ), we have

Ly = k≥1 [(-λ k I d + A) y k ] φ k , L * y = k≥1 [(-λ p I d + A * ) y k ] φ k .
In what follows, we set:

L k = -λ k I d + A ∈ L(C n ) and L * k = -λ k I d + A * ∈ L(C n ), ∀k ≥ 1. (6)
For k ≥ 1, let us introduce the matrices

B k =    B . . . B    ∈ L(C m ; C nk ), L k =      L 1 0 • • • 0 0 L 2 • • • 0 . . . • • • . . . . . . 0 • • • 0 L k      ∈ L(C nk ), (7)
and let us write the Kalman matrix associated with the pair (L k , B k ):

K k = [L k | B k ] = [B k | L k B k | L 2 k B k | • • • | L nk-2 k B k | L nk-1 k B k ] ∈ L(C mnk , C nk ).
The main result of this work is the following characterization of the exact controllability to trajectories at time T of sytem (1):

Theorem 1.1. Let us fix A ∈ L(C n ) and B ∈ L(C m ; C n ). Then, system (
K k = nk, ∀k ≥ 1. ( 8
)
Remark 1.1.

1. Actually, condition (8) only has to be checked for a frequency. In Corollary 3.3, we will show that there exists a positive integer k 0 , only depending on A, such that rank K k0 = nk 0 if and only if rank K k = nk for every k ≥ 1. 2. Note that the algebraic Kalman condition, rank [A | B] = n, corresponds to k = 1 and then, it is a necessary condition for the exact controllability to trajectories of system (1). 3. We will see that when B ∈ C n , i.e., when m = 1 (one control force), condition ( 8) is equivalent to the algebraic Kalman condition, rank [A | B] = n and

µ i -µ j = λ k -λ l , ∀(k, i), (l, j) ∈ N × {1, 2, ..., p} with (k, i) = (l, j),
where {µ l } 1≤l≤p ⊂ C is the set of distinct eigenvalues of A * . In this sense, Theorem 1.1 generalizes the results obtained in [START_REF] Fernández-Cara | On the boundary controllability of non-scalar parabolic systems[END_REF] and [START_REF] Fernández-Cara | Boundary controllability of parabolic coupled equations[END_REF]. 4. We will also see that if rank B = n (and therefore m ≥ n), then the pair (A, B) fulfills condition (8) and system (1) is exactly controllable to trajectories at time T . This boundary controllability result has been obtained in [START_REF] Fernández-Cara | Boundary controllability of parabolic coupled equations[END_REF] in the N -dimensional case. 5. From Theorem 1.1 we can conclude that, unlike the scalar case, n = 1, the distributed controllability property of parabolic systems in not equivalent to the boundary control property: the Kalman rank condition is a necessary condition for the controllability of both systems but is not a sufficient condition for the boundary controllability of system [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF]. This shows that there is an important difference between the controllability properties for scalar parabolic problems and coupled parabolic systems.

The sufficient part of Theorem 1.1 is proved through a moment problem. This method has been successfully used to prove the boundary controllability problem for the scalar heat equation (see [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF]). Let us briefly remember this method in the case of the scalar heat equation.

Let us fix y 0 ∈ H -1 (0, π). Then, there exists v ∈ L 2 (0, T ) such that the solution to

     y t -y xx = 0 in Q, y(0, •) = v, y(π, •) = 0 on (0, T ), y(•, 0) = y 0 in (0, π), satisfies y(•, T ) = 0 in (0, π) if and only if v ∈ L 2 (0, T ) satisfies -y 0 , e -λ k T φ k H -1 (0,π),H 1 0 (0,π) = T 0 v(t)e -λ k (T -t) ∂ x φ k (0) dt, ∀k ≥ 1.
In the previous equality λ k and φ k are given in [START_REF] Ammar-Khodja | A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems[END_REF].

Using the Fourier decomposition of y 0 , y 0 = k≥1 y 0,k φ k , it will be sufficient to find a control v ∈ L 2 (0, T ) such that

k 2 π T 0 e -λ k (T -t) v(t) dt = -e -λ k T y 0,k ∀k ≥ 1.
This problem is called a moment problem.

Let us recall that a family

{p k } k≥1 ⊂ L 2 (0, T ) is biorthogonal to {e -λ k t } k≥1 if it satisfies T 0 e -λ k t p l (t) = δ kl , ∀(k, l) : k, l ≥ 1.
In [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF] and [START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF], the authors solve the previous moment problem by proving the existence of a biorthogonal family {p k } k≥1 to {e -λ k t } k≥1 which satisfies the additional property: for every ǫ > 0 there exists a constant C(ǫ, T ) > 0 such that

p k L 2 (0,T ) ≤ C(ǫ, T )e ǫλ k .
In fact, the control v is obtained as a linear combination of {p k } k≥1 and the previous bounds are used to prove that this combination converges in L 2 (0, T ).

In this paper we will follow the previous technique for proving the sufficient part of Theorem 1.1. In the case considered here, we have two difficulties: first, the control v may act only on some of the n equations of the system (in general m < n); second, the spectrum of the operator L * = ∂ xx I d + A * may be complex with eigenvalues of multiplicity greater than 1. This leads to a moment problem associated to families as {t j e -Λ k t } k≥1,0≤j≤η-1 with η ≥ 1 a positive integer.

In [START_REF] Fernández-Cara | Boundary controllability of parabolic coupled equations[END_REF], the authors considered the case η = 2 and proved the existence of a suitable family {q k , q k : k ≥ 1} biorthogonal to {e -Λ k t , te -Λ k t : k ≥ 1}. We extend this result to any η ≥ 1 and to a large class of sequences {Λ k } k≥1 ⊂ C. To our knowledge, this construction produces a new and interesting result by itself and it is our second main result.

Let us fix η > 1, a positive integer, and let us consider a sequence of complex numbers Λ = {Λ k } k≥1 ⊂ C + = {λ ∈ C : ℜλ > 0}. Throughout this work we will use the notation:

e k,j (t) = t j e -Λ k t , ∀t > 0, with k ≥ 1 and j : 0 ≤ j ≤ η -1.
Given T ∈ (0, ∞], we define

A(Λ, η, T ) = span {e k,j : k ≥ 1, 0 ≤ j ≤ η -1} L 2 (0,T ;C) .

Let us recall that the family {ϕ

k,j } k≥1,0≤j≤η-1 ⊂ A(Λ, η, T ) is biorthogonal to {e k,j } k≥1,0≤j≤η-1 if the equalities (e k,j , ϕ l,i ) L 2 (0,T ;C) := T 0 t j e -Λ k t ϕ * l,i (t) dt = δ kl δ ij , ∀(k, j), (l, i) : k, l ≥ 1, 0 ≤ i, j ≤ η -1, (9)
holds.

Our second main result is the following:

Theorem 1.2. Let us fix η ≥ 1, a positive integer, and

T ∈ (0, ∞]. Assume that {Λ k } k≥1 is a sequence of complex numbers such that,      ℜΛ k ≥ δ|Λ k |, |Λ k -Λ l | ≥ ρ|k -l|, ∀k, l ≥ 1, k≥1 1 |Λ k | < ∞, (10)
for two positive constants δ and ρ. Then, there exists a family {ϕ k,j } k≥1,0≤j≤η-1 ⊂ A(Λ, η, T ) biorthogonal to {e k,j } k≥1,0≤j≤η-1 such that, for every ε > 0, there exists C(ε, T ) > 0 for which

ϕ k,j L 2 (0,T ;C) ≤ C(ε, T )e εℜΛ k , ∀(k, j) : k ≥ 1, 0 ≤ j ≤ η -1. ( 11
)
The plan of the paper is the following: In Section 2, we will address some preliminary results. In Section 3 we will study the Kalman condition (8) in some interesting cases. Section 4 concerns one of the main results, the construction and estimates of a biorthogonal family (proof of Theorem 1.2). Section 5 is devoted to the proof of Theorem 1.1. Finally, in Section 6 we give some comments and open problems.

Preliminary results

We devote this section to recalling some known results that will be used below. We begin by recalling some results for system [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF]. First, we introduce the concept of solution by transposition to system (1). To this end, let us consider the linear backward in time problem

     -ϕ t -ϕ xx = A * ϕ + g in Q, ϕ(0, •) = 0, ϕ(π, •) = 0 in (0, T ), ϕ(• , T ) = 0 in (0, 1), (12)
where

g ∈ L 2 (Q; C n ) is given. It is well known that, for every g ∈ L 2 (Q; C n ), this problem has a unique strong solution ϕ ∈ L 2 (0, T ; H 2 (0, π; C n )) ∩ C 0 ([0, T ]; H 1 0 (0, π; C n )),
which depends continuously on g, i.e., there exists a constant C = C(T ) > 0 such that

ϕ L 2 (0,T ;H 2 (0,π;C n )) + ϕ C 0 ([0,T ];H 1 0 (0,π;C n )) ≤ C g L 2 (Q;C n ) .
Thanks to the previous properties, we can introduce the following definition:

Definition 2.1. Let y 0 ∈ H -1 (0, π; C n ) and v ∈ L 2 (0, T ; C m ) be given. It will be said that y ∈ L 2 (Q; C n ) is a solution by transposition to (1) if, for each g ∈ L 2 (Q; C n ), one has Q (y , g) C n dx dt = y 0 , ϕ(•, 0) + T 0 (v(t) , B * ϕ x (0, t)) C m dt,
where ϕ is the solution to (12) associated to g and • , • , (•, •) C n and (•, •) C m stands for, resp., the usual duality pairing between H -1 (0, π; C n ) and H 1 0 (0, π; C n ) and the scalar products in C n and C m .

We can now establish the existence and uniqueness of solution to system (1). One has: Proposition 2.2. Assume that y 0 ∈ H -1 (0, π; C n ) and v ∈ L 2 (0, T ; C m ) are given. Then problem (1) admits a unique solution by transposition y that satisfies:

         y ∈ L 2 (Q; C n ) ∩ C 0 ([0, T ]; H -1 (0, π; C n )), y t ∈ L 2 (0, T ; D(-∆; C n ) ′ ), y t -y xx = Ay in L 2 (0, T ; D(-∆; C n ) ′ ), y(• , 0) = y 0 in H -1 (0, π; C n ) and y L 2 (Q;C n ) + y t L 2 (0,T ;D(-∆;C n ) ′ ) ≤ C y 0 H -1 (0,π;C n ) + v L 2 (0,T ;C m ) , for a positive constant C = C(T ).
This result can be proved using standard arguments. Anyway, for a detailed proof of the result, see for instance [START_REF] Fernández-Cara | Boundary controllability of parabolic coupled equations[END_REF].

As it is well-known, the controllability properties of system (1) are equivalent to appropriate properties of the following adjoint system:

     -ϕ t = ϕ xx + A * ϕ in Q, ϕ(0, •) = 0, ϕ(π, •) = 0 on (0, T ), ϕ(•, T ) = ϕ 0 in (0, π), (13)
where ϕ 0 ∈ H 1 0 (0, π; C n ). Let us observe that, for every ϕ 0 ∈ H 1 0 (0, π; C n ), system (13) admits a unique solution ϕ ∈ L 2 (0, T ;

H 2 (0, π; C n )) ∩ C 0 ([0, T ]; H 1 0 (0, π; C n ))
and, for a positive constant C = C(T ), one has

ϕ L 2 (0,T ;H 2 (0,π;C n )) + ϕ C 0 ([0,T ];H 1 0 (0,π;C n )) ≤ C ϕ 0 H 1 0 (0,π;C n ) .
The solutions y to problem (1) and ϕ to the adjoint system [START_REF] Fernández-Cara | On the boundary controllability of non-scalar parabolic systems[END_REF] are related by means of the following result Proposition 2.3. Let y 0 ∈ H -1 (0, π; C n ), ϕ 0 ∈ H 1 0 (0, π; C n ) and v ∈ L 2 (0, T ; C m ) be given. Let y and ϕ be, resp., the solution to (1) associated to y 0 and v and the solution to the adjoint system (13) associated to ϕ 0 . Then:

y(•, t), ϕ(•, t) -y 0 , ϕ(•, 0) = t 0 (v(s) , B * ϕ x (0, s)) C m ds ∀t ∈ [0, T ]. ( 14
)
This proposition is an easy consequence of Proposition 2.2 and the details are left to the reader. As said above, the controllability properties of (1) can be characterized in terms of appropriate properties of the solutions to [START_REF] Fernández-Cara | On the boundary controllability of non-scalar parabolic systems[END_REF]. More precisely, we have: Proposition 2.4. The following properties are equivalent:

1. There exists a positive constant C such that, for any y 0 ∈ H -1 (0, π; C n ), there exists a control v ∈ L 2 (0, T ; C m ) such that

v 2 L 2 (0,T ;C m ) ≤ C y 0 2 H -1 (0,π;C n )
and the associated solution to (1) satisfies y(•, T ) = 0 in H -1 (0, π; C n ). 2. There exists a positive constant C such that, for any trajectory y ∈ C 0 ([0, T ];

H -1 (0, π; C n ))
of (1) and any

y 0 ∈ H -1 (0, π; C n ), there exists a control v ∈ L 2 (0, T ; C m ) such that v -v 2 L 2 (0,T ;C m ) ≤ C y 0 -y(• , 0) 2 H -1 (0,π;C n )
and the associated solution y to (1) satisfies y(• , T ) = y(• , T ) in H -1 (0, π; C n ). 3. There exists a positive constant C such that the observability inequality

ϕ(•, 0) 2 H 1 0 (0,π;C n ) ≤ C T 0 |B * ϕ x (0, t)| 2 dt ( 15
)
holds for every ϕ 0 ∈ H 1 0 (0, π; C n ). In [START_REF] Fernández-Cara | The cost of approximate controllability for heat equations: the linear case[END_REF], ϕ is the solution to the adjoint system (13) associated to ϕ 0 .

Again, this result is well known and is a consequence of formula [START_REF] Fernández-Cara | Boundary controllability of parabolic coupled equations[END_REF]. For its proof, see for instance [START_REF] Fernández-Cara | Boundary controllability of parabolic coupled equations[END_REF].

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 [START_REF] Fernández-Cara | On the boundary controllability of non-scalar parabolic systems[END_REF]. More precisely, (1) is approximately controllable if and only if the following unique continuation property holds: "Let ϕ 0 ∈ H 1 0 (0, π; C n ) be given and let ϕ be the associated adjoint state. Then, if B * ϕ x (0, t) = 0 on (0, T ), one has ϕ ≡ 0 on Q."

The Kalman condition

We will devote this section to showing some properties related to the Kalman condition [START_REF] Bodart | Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient[END_REF]. To be precise, we will give equivalent conditions to [START_REF] Bodart | Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient[END_REF] in two important cases, m = 1 and m ≥ n, and will clarify Remark 1.1.

Throughout this work we will use the following notation: Notation: Let us denote by {µ l } 1≤l≤p ⊂ C the set of distinct eigenvalues of A * . For l : 1 ≤ l ≤ p, we denote by n l the geometric multiplicity of µ l and assume that we have

n 1 ≥ n l , 2 ≤ l ≤ p.
The sequence {v l,j } 1≤j≤n l will denote a basis of eigenvectors of A * associated to µ l , i.e., a basis of the eigenspace associated to µ l . To each eigenvector v l,j we associate its Jordan chain (of dimension τ l,j ) and the corresponding set of generalized eigenvectors {v i l,j } 1≤i≤τ l,j defined by:

A * v i l,j = µ l v i l + v i+1 l , 1 ≤ i < τ l,j , A * v τ l,j l,j = µ l v τ l,j l,j (so that v τ l,j l,j = v l,j ).
We will first present an equivalent condition to the Kalman rank condition that will be used later. To this end, let us consider A ∈ L(C N ) and B ∈ L(C M , C N ) two matrices, (N and M are positive integers). Let {θ l } 1≤l≤ p ⊂ C be the set of distinct eigenvalues of A * . For l : 1 ≤ l ≤ p, we denote by m l the geometric multiplicity of θ l . The sequence {w l,j } 1≤j≤m l will denote a basis of eigenvectors of A * associated to θ l , i.e., a basis of the eigenspace associated to θ l . With this notation, one has: Proposition 3.1. Under the previous notations for the pair (A, B), the following conditions are equivalent:

1. rank [A | B] = rank [B | AB | A 2 B | • • • | A N -1 B] = N . 2. rank [B * w l,1 | B * w l,2 | • • • | B * w l,m l ] = m l , for every l, with 1 ≤ l ≤ p.
Proof: We will deduce the proof from the Hautus test which is an equivalent condition to the Kalman rank condition. Indeed, it is well known (for instance, see [START_REF] Tucsnak | Observation and Control of Operator Semigroups, Advanced Texts[END_REF], page 15) that rank

[A | B] = N if and only if rank A * -θ l I d B * = N, ∀l : 1 ≤ l ≤ p. ( 16
)
Let us assume that the Kalman rank condition holds and let us proof that the set

{B * w l,j } 1≤j≤m l ⊂ R M
is linearly independent for every l : 1 ≤ l ≤ p. To this end, let us suppose that for

{α j } 1≤j≤m l ⊂ C one has m l j=1 α j B * w l,j = 0.
In particular, w = m l j=1 α j w l,j ∈ C n is an eigenvector of A * associated to θ l and is a solution to the linear system

A * -θ l I d B * w = 0.
Using [START_REF] Fursikov | Controllability of Evolution Equations[END_REF], we conclude w = m l j=1 α j w l,j ≡ 0, i.e., α j ≡ 0, for every j : 1 ≤ j ≤ m l . Let us now assume that the set {B * w l,j } 1≤j≤m l ⊂ C M is linearly independent for every l : 1 ≤ l ≤ p and let us proof that (A, B) fulfills condition [START_REF] Fursikov | Controllability of Evolution Equations[END_REF]. Thus, we consider w ∈ C N a solution to the previous linear system. In particular, B * w = 0 and w is an eigenvector of A * associated to θ l . As a consequence, w can be written as w = m l j=1 α j w l,j , with α j ∈ C. Evidently, the equality B * w = 0 implies α j = 0 for every j whence w ≡ 0. This finalizes the proof.

Our next task will be to clarify the first point in Remark 1.1. Before let us prove the following result: Proposition 3.2. Let A ∈ L(C n ) be given and let us denote by {µ l } 1≤l≤p ⊂ C the set of distinct eigenvalues of A * . Then, there exists an integer k 0 = k 0 (A) ∈ N, only depending on A, such that,

µ i -µ j = λ k -λ l , (17)
for every k > k 0 , l ≥ 1, k = l, and i, j : 1 ≤ i, j ≤ p.

Proof: First, let us observe that λ k is given by ( 5) and then, for ρ 0 = 1 and K 0 = 1, one has

|λ k -λ l | ≥ ρ 0 |k 2 -l 2 |, ∀k, l ≥ K 0 .
Let us consider

k 0 = max K 0 , 1 ρ 0 max 1≤i,j≤p |µ i -µ j | + 1 ,
and let us take k > k 0 , l ≥ 1, with k = l, and i, j,

with 1 ≤ i, j ≤ p. Then, if µ i -µ j ∈ R, we can conclude the result. If µ i -µ j ∈ R and, for instance, k > l, µ i -µ j ≤ |µ i -µ j | ≤ ρ 0 k 0 < ρ 0 (k + l) ≤ ρ 0 (k + l)(k -l) = ρ 0 (k 2 -l 2 ) ≤ λ k -λ l .
Finally, if µ iµ j ∈ R and k < l, one has

µ j -µ i ≤ |µ i -µ j | ≤ ρ 0 k 0 < ρ 0 (k + l) ≤ ρ 0 (k + l)(l -k) ≤ λ l -λ k .
We have thus the proof.

By means of the previous result we can establish an equivalent condition to [START_REF] Bodart | Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient[END_REF]. Thus, one has:

Corollary 3.3. Let A ∈ L(C n ) and B ∈ L(C m ; C n ) be
given and let us consider k 0 ≥ 1 provided by Proposition 3.2. Then, the three following conditions are equivalent

1. rank K k = nk for every k ≥ 1. 2. rank K k = nk for every k : 1 ≤ k ≤ k 0 . 3. rank K k0 = nk 0 .
Proof: Of course, condition 1 implies condition 2 and this one implies condition 3.

Let us now prove that condition 3 implies condition 1 and, to this end, let us take k 0 such that (17) holds. In order to prove the result, let us denote by σ(L * k ) the set of eigenvalues of the matrix L * k ∈ L(C nk ). From the definition of L k (see ( 6) and ( 7)) we get

σ(L k ) = {-λ l + µ i : 1 ≤ l ≤ k, 1 ≤ i ≤ p}, (remember that {µ i } 1≤i≤p ⊂ C is the set of distinct eigenvalues of A * ).
Let us start with the case k = k 0 + 1. By contradiction, let us suppose that rank

K k0+1 < n(k 0 + 1). It is clear that σ(L * k0+1 ) = σ(L * k0 ) ∪ {-λ k0+1 + µ i : 1 ≤ i ≤ p}.
Using the Hautus criterium [START_REF] Fursikov | Controllability of Evolution Equations[END_REF] for the couple (L * k0+1 , B k0+1 ) and taking into account that rank K k0 = nk 0 we deduce that there exists an eigenvector

V ∈ C n(k0+1) of L * k0+1 associated to -λ k0+1 + µ i , with i : 1 ≤ i ≤ p, such that B * k0+1 V = 0. ( 18
)
From Proposition 3.2, we deduce that σ(L * k0 ) ∩ {-λ k0+1 + µ i : 1 ≤ i ≤ p} = ∅ and therefore, the vector

V = (V j ) 1≤j≤k0+1 ∈ C n(k0+1) is an eigenvector of L * k0+1 associated to -λ k0+1 + µ i if and only if V j = 0, ∀j = k 0 + 1, and V k0+1 = v,
where v ∈ C n is an eigenvector of A * associated to µ i . Thus, condition [START_REF] González-Burgos | Controllability results for some nonlinear coupled parabolic systems by one control force[END_REF] implies that v belongs to the kernel of B * . Using again the Hautus test [START_REF] Fursikov | Controllability of Evolution Equations[END_REF], this time applied to (A, B), we infer that rank

[A | B] = rank K 1 < n. But this last inequality contradicts condition 2.
The general case k ≥ k 0 can be obtained combining an induction argument and the previous reasoning.

Let us now prove the case k < k 0 . Actually, we can prove that if rank

K k = nk, then rank K k-1 = n(k -1). Indeed, by contradiction, if rank K k-1 < n(k -1), using again Proposi- tion 3.1, there exists an eigenvector V ∈ C n(k-1) of the matrix L * k-1 associated to θ ∈ C such that B * k-1 V = 0.
It is easy to check that θ is also an eigenvalue of L * k which has as associated eigenvector

V = V 0 ∈ C nk and B * k V = 0.
This contradicts the previous assumption and ends the proof.

In the next result we will study the Kalman condition (8) in the particular case m = 1. Thus, one has

Proposition 3.4. Let us fix A ∈ L(C n ) and B ∈ C n (m = 1)
. Then, the following conditions are equivalent

1. rank K k = nk for every k ≥ 1. 2. rank [A | B] = n and µ i -µ j = λ k -λ l , ∀(k, i), (l, j) ∈ N × {1, 2, ..., p} with (k, i) = (l, j). Proof: Let us assume that rank K k = nk for every k ≥ 1. It is then clear that rank [A | B] = n.
By contradiction, assume that there exist k, l ≥ 1, with k > l, and i, j : 1 ≤ i, j ≤ p such that

-λ k + µ i = -λ l + µ j := θ. Then, θ ∈ σ(K * k ). Let us take w 1 ∈ C n and w 2 ∈ C n eigenvectors of A * associated, resp., to µ i and µ j . Then, V 1 = (V 1,ℓ ) 1≤ℓ≤k , V 2 = (V 2,ℓ ) 1≤ℓ≤k ∈ C nk , with V 1,k = w 1 and V 1,ℓ = 0, ∀ℓ = k, V 2,l = w 2 and V 2,ℓ = 0, ∀ℓ = l, are two independent eigenvectors of K * k associated to θ. Using Proposition 3.1 we deduce that the set {B * k V 1 , B * k V 1 } ⊂ C is linearly independent (m = 1)
. This is evidently absurd. So, we have condition 2.

On the other hand, let us assume that (A, B) satisfies condition 2. We deduce that (17) holds with k 0 ≡ 1. Applying directly Corollary 3.3 with k 0 = 1 we obtain rank K k = nk for every k ≥ 1. This ends the proof. Remark 3.1. As said in Remark 1.1 and in view of this last result, we deduce that Theorem 1.1 generalizes the controllability result for system (1) stated in [START_REF] Fernández-Cara | On the boundary controllability of non-scalar parabolic systems[END_REF] and [START_REF] Fernández-Cara | Boundary controllability of parabolic coupled equations[END_REF] Proof: Let us fix k ≥ 1. In order to prove (8), we will use Proposition 3.1 for the pair (L k , B k ) (see [START_REF] Ammar-Khodja | Boundary controllability of coupled parabolic systems[END_REF] for the definition of these matrices). Therefore, let us consider θ ∈ σ(L * k ), an eigenvalue of L * k . Associated to θ, let us consider the set,

I(θ) = {(l, i) : 1 ≤ l ≤ k, 1 ≤ i ≤ p, θ = -λ l + µ i }.
Aided by this set we can obtain a basis of eigenvectors of L * k associated to θ:

EV (θ) = (l,i)∈I(θ) {V l i,j ∈ C kn : 1 ≤ j ≤ n i }, with V l i,j = (V l,ℓ i,j ) 1≤ℓ≤k and V l,ℓ i,j ∈ C n given by V l,ℓ i,j = δ l,ℓ v i,j . The set {B * k V : V ∈ EV (θ)
} can be writen as

{B * k V : V ∈ EV (θ)} = {B * v i,j : i such that, for l ≥ 1, (l, i) ∈ I(θ) and 1 ≤ j ≤ n i } ⊂ {B * v i,j : 1 ≤ i ≤ p, 1 ≤ j ≤ n i }.
Taking into account that rank B * = n and the set {v i,j : 1

≤ i ≤ p, 1 ≤ j ≤ n i } ⊂ C n is linearly independent, we deduce that {B * k V : V ∈ EV (θ)
} is also linearly independent. This finalizes the proof of the result.

Biorthogonal families: construction and estimates

This section will be devoted to proving Theorem 1.2. To this end, we will follow here the strategy based on the Laplace transform which explicitly construct the biorthogonal family from the fixed complex sequence. This strategy has been used for instance in [START_REF] Schwartz | Étude des sommes d'exponentielles, 2ième éd[END_REF] in order to construct a biorthogonal family to the set e -Λ k t k≥1 , where {Λ k } k≥1 is a real positive sequence satisfying suitable properties.

Throughout this section η ≥ 1 will denote a positive integer and

Λ = {Λ k } k≥1 ⊂ C + = {λ ∈ C : ℜλ > 0} is a complex sequence satisfying Λ k = Λ j , ∀k, j ∈ N with k = j. ( 19
)
We will obtain the proof of the Theorem 1.2 reasoning as follows:

1. First, we will prove the existence of a biorthogonal family {ϕ k,j } k≥1,0≤j≤η-1 ⊂ A(Λ, η, ∞) to t j e -Λ k t k≥1,0≤j≤η-1 (see Proposition 4.1). In view of other applications, we will consider more general Λ k than those satisfying condition [START_REF] Fabre | Approximate controllability of the semilinear heat equation[END_REF] in Theorem 1.2. In this proposition we will also give an estimate of the norm in L 2 (0, ∞; C) of ϕ k,j in terms of a Blaschke product associated to the sequence Λ (see ( 21)). 2. Secondly, we will use assumptions [START_REF] Fabre | Approximate controllability of the semilinear heat equation[END_REF] in order to get an estimate of the Blaschke product P k in (21) (see Proposition 4.5). 3. Finally, we will directly prove Theorem 1.2 when T = ∞ and will deduce the general case T ∈ (0, ∞) using a well known argument (see Corollary 4.6).

For T ∈ (0, ∞], let us recall that A(Λ, η, T ) is the space given by

A(Λ, η, T ) = span {t j e -Λ k t : k ≥ 1, 0 ≤ j ≤ η -1} L 2 (0,T ;C)
and is a closed subspace of L 2 (0, T ; C).

Let us also recall that the function e k,j is given by e k,j (t) = t j e -Λ k t , t ≥ 0, with (k, j) such that k ≥ 1 and 0 ≤ j ≤ η -1.

We will obtain the proof of Theorem 1.2 from several previous results. Let us start with the following one: [START_REF] González-Burgos | Controllability results for cascade systems of m coupled parabolic PDEs by one control force[END_REF] and the assumption

Proposition 4.1. Assume that Λ = {Λ k } k≥1 ⊂ C + satisfies
k≥1 ℜΛ k (1 + ℜΛ k ) 2 + (ℑΛ k ) 2 < ∞. (20)
Then, there exists a biorthogonal family {ϕ k,j } k≥1,0≤j≤η-1 ⊂ A(Λ, η, T ) to {e k,j } k≥1,0≤j≤η-1 such that

ϕ k,j L 2 (0,∞;C) ≤ C 1 + 1 ℜΛ k (2η-j)(η-j-1)+1 (ℜΛ k ) η(η-j) |1 + Λ k | 2η(η-j) P η(η-j) k , ( 21
)
where C is a positive constant, only depending on η, and P k is given by

P k := ℓ≥1 ℓ =k 1 + Λ k /Λ * ℓ 1 -Λ k /Λ ℓ .
To prove this result, we need some preliminary lemmata.

         W (λ) = W (λ, Λ) = k≥1 δ k 1 -λ/Λ k 1 + λ/Λ * k , λ ∈ C + δ k = Λ k Λ * k |Λ k -1| |Λ k + 1| Λ * k + 1 Λ * k -1 (δ k = 1 if Λ k = 1).
Then, W ∈ H ∞ (C + ), the space of bounded and holomorphic functions defined on C + , and satisfies

|W (λ)| < 1 for ℜλ > 0, |W (iτ )| = 1 a.e. in R and
W (λ 0 ) = 0 ⇐⇒ λ 0 = Λ k with k ≥ 1.
Moreover, Λ k is a simple root of W , for any k ≥ 1.

Proof: Let U be the unit ball of C and let us consider a sequence

{α k } k≥1 ⊂ U such that k≥1 (1 -|α k |) < ∞.
Then, it is well known (see for instance [START_REF] Rudin | Real and complex analysis[END_REF]) that the following function

C(z) = k≥1 |α k | α k α k -z 1 -α * k z , with z ∈ U,
is well defined and C ∈ H ∞ (U ). Moreover, C(e iθ ) = 1 for almost all θ ∈ (-π, π).

We will obtain the proof of the lemma from the previous properties of the function C. Indeed, it is not difficult to check that

h : z ∈ U → h(z) = 1 + z 1 -z ∈ C + is a bijective map. In addition, h is holomorphic in U and W (λ) = C(h -1 (λ)) with α k = h -1 (Λ k ). Observe that (1 -|α k |) = 1 -1 - 4ℜΛ k (1 + ℜΛ k ) 2 + (ℑΛ k ) 2 1/2 and k≥1 (1 -|α k |) < ∞ if
Φ(λ) = W (λ) (1 + λ) 2 η , for λ ∈ C + . (22)
Simple computations immediately show that Φ ∈ H 2 (C + ), the space of holomorphic functions on

C + such that +∞ -∞ |Φ(σ + iτ )| 2 dτ < ∞, ∀σ > 0, with norm Φ H 2 (C+) = +∞ -∞ |Φ (iτ )| 2 dτ 1/2
.

Using the properties of the function W , it is not difficult to check that, for a positive constant C η (only depending on η), one has

Φ H 2 (C+) ≤ C η .
It is well known that the Laplace transform is a homeomorphism

4 from L 2 (0, ∞; C) into H 2 (C + ). Therefore, there exists a nontrivial function ϕ ∈ L 2 (0, ∞; C) such that Φ(λ) = 1 2π ∞ 0 e -λt ϕ * (t) dt.
Observe that, thanks to Lemma 4.2, {Λ k } k≥1 are the zeros of Φ and have multiplicity η. In particular Φ (j) (Λ k ) = 0 for every k ≥ 1 and j : 0

≤ j ≤ η -1. Thus ∞ 0 t j e -Λ k t ϕ * (t) dt = (e k,j , ϕ) L 2 (0,∞;C) = 0, ∀(k, j) : k ≥ 1, 0 ≤ j ≤ η -1.
We have then proved that there exists ϕ ∈ L 2 (0, ∞; C), with ϕ ≡ 0, such that ϕ ∈ A(Λ, η, ∞) ⊥ . This finalizes the proof.

Starting from the function Φ defined in [START_REF] Kalman | Topics in Mathematical Control Theory[END_REF], we would like to construct a family of functions {Φ k,j } ⊂ H 2 (C + ) satisfying some additional conditions. These are given in the following result: Lemma 4.4. Assume that the sequence {Λ k } k≥1 satisfies [START_REF] González-Burgos | Controllability results for cascade systems of m coupled parabolic PDEs by one control force[END_REF] and [START_REF] Guerrero | Null controllability of some systems of two parabolic equations with one control force[END_REF]. Then, there exists a familly

{Φ k,j } k≥1,1≤j≤η ⊂ H 2 (C + ) such that Φ (ν) k,j (Λ l ) = (-1) ν δ kl δ jν , ∀(k, j), (l, ν) : k, l ≥ 1, 0 ≤ j, ν ≤ η -1, (23)
and

Φ k,j H 2 (C+) ≤ C H j (Λ k ) Φ (η) (Λ k ) η-j , ∀(k, j) : k ≥ 1, 0 ≤ j ≤ η -1, ( 24
)
where C is a positive constant, only depending on η, and Φ and H j (Λ k ) are respectively given by [START_REF] Kalman | Topics in Mathematical Control Theory[END_REF] and

H j (Λ k ) = 1 + 1 ℜΛ k (2η-j)(η-j-1)+1
. 4 For the space H 2 (C + ) and the properties of the Laplace transform, see for instance [29, pp. 19-20]).

Before presenting the proof of this lemma, let us complete the proof of Proposition 4.1.

Proof of Proposition 4.1: From Lemma 4.4 we deduce that Φ k,j ∈ H 2 (C + ) for every (k, j) : k ≥ 1 and 0 ≤ j ≤ η -1. Thus, using again the Laplace transform, for any k ≥ 1 and j : 0 ≤ j ≤ η -1, there exists a nontrivial function

ϕ k,j ∈ L 2 (0, ∞; C) such that Φ k,j (λ) = 1 2π ∞ 0 e -λt ϕ * k,j (t) dt, ∀λ ∈ C + , and ϕ k,j L 2 (0,∞;C) ≤ C Φ k,j H 2 (C+)
for a positive constant C. We also have

Φ (ν) k,j (λ) = (-1) ν 1 2π ∞ 0 t ν e -λt ϕ * k,j (t) dt, ∀λ ∈ C + , ∀ν ≥ 0.
Let us consider the projection operator Π Λ :

L 2 (0, ∞; C) → A(Λ, η, ∞). One has ∞ 0 t j e -Λ k t Π Λ ϕ * (t) dt = ∞ 0 t j e -Λ k t ϕ * (t) dt, ∀(k, j) : k ≥ 1, 0 ≤ j ≤ η-1, ∀ϕ ∈ L 2 (0, ∞; C).
Taking into account [START_REF] Kavian | Unique continuation principle for systems of parabolic equations[END_REF] and the two previous equalities, we deduce that the set {ϕ k,j } k≥1,0≤j≤η , with ϕ k,j = Π Λ ϕ k,j /2π, is a biorthogonal family associated to {t j e -Λ k } k≥1,0≤j≤η and

ϕ k,j L 2 (0,∞;C) ≤ C Φ k,j H 2 (C+)
for a positive constant C.

From [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF] and in order to prove [START_REF] Hansen | Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems[END_REF], let us calculate |Φ (η) (Λ k )|. First, the function Φ can be written as Φ(λ) = [f (λ)] η with f a holomorphic function on C + . Since Λ k is a simple zero of f , we get Φ (η)

(Λ k ) = η![f ′ (Λ k )] η , i.e., Φ (η) (Λ k ) = η! W ′ (Λ k ) 1 + Λ 2 k η ,
where W is given in Lemma 4.2. On the other hand, a simple calculation gives

W ′ (Λ k ) = -δ k -Λ * k 2Λ k ℜΛ k ℓ≥1 ℓ =k δ ℓ 1 -Λ k /Λ ℓ 1 + Λ k /Λ * ℓ ,
and therefore

|Φ (η) (Λ k )| = η!   1 2|1 + Λ k | 2 ℜΛ k ℓ =k 1 -Λ k /Λ ℓ 1 + Λ k /Λ * ℓ   η .
Finally, from [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF] we get [START_REF] Hansen | Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems[END_REF]. This ends the proof.

Proof of Lemma 4.4: Let us fix k ≥ 1 and j : 0 ≤ j ≤ η -1 and let us set

f k,j (λ) := Φ(λ) (λ -Λ k ) η-j , ∀λ ∈ C + , ( 25
)
where Φ is given by [START_REF] Kalman | Topics in Mathematical Control Theory[END_REF]. From the properties of the function Φ, we get

           f (ν) k,j (Λ l ) = 0, ∀l ≥ 1 with l = k, ∀ν : 0 ≤ ν ≤ η -1, f (ν) k,j (Λ k ) = 0, ∀ν : 0 ≤ ν ≤ j -1, f (j+ν) k,j (Λ k ) = (j + ν)! (η + ν)! Φ (η+ν) (Λ k ), ∀ν : ν ≥ 0. ( 26
)
In particular, f

(j) k,j (Λ k ) = j! η! Φ (η) (Λ k ) = 0 (
remember that Λ k is a zero of Φ of multiplicity η; see [START_REF] Kalman | Topics in Mathematical Control Theory[END_REF]).

We will obtain the proof after several steps.

Step 1: In this first step we will prove that there exists a polynomial function p = p k,j of degree ηj -1 such that the function defined by

Φ k,j (λ) := p(λ)f k,j (λ), ∀λ ∈ C + , (27)
satisfies [START_REF] Kavian | Unique continuation principle for systems of parabolic equations[END_REF].

Clearly, for any polynomial p we have

Φ (ν) k,j (Λ l ) = 0, ∀l : l = k, ∀ν : 0 ≤ ν ≤ η -1, Φ (ν) k,j (Λ k ) = 0, ∀ν : 0 ≤ ν < j ≤ η -1.
Thus, in order to get [START_REF] Kavian | Unique continuation principle for systems of parabolic equations[END_REF], we have to show that there is a polynomial p such that Φ

(j) k,j (Λ k ) = (-1) j and Φ (j+ν) k,j (Λ k ) = 0 for 1 ≤ ν ≤ η -j -1.
In view of ( 26)), these relations lead to

           p(Λ k ) = (-1) j f (j) k,j (Λ k ) = η! j! (-1) j Φ (η) (Λ k ) , ν-1 ℓ=0 a νℓ p (ℓ) (Λ k ) + p (ν) (Λ k ) = 0, ∀ν : 1 ≤ ν ≤ η -j -1, (28)
where

a νℓ = j + ν ℓ j + ν ν f (j+ν-ℓ) k,j (Λ k ) f (j) k,j (Λ k ) = ν!η! ℓ!(η + ν -ℓ)! Φ (η+ν-ℓ) (Λ k ) Φ (η) (Λ k ) . ( 29
)
for every ν, ℓ : 0 ≤ ℓ < ν ≤ ηj -1. These relations allow us to compute p (ν) (Λ k ) for 0 ≤ ν ≤ ηj -1 and thus

p(λ) = η-j-1 ν=0 p (ν) (Λ k ) ν! (λ -Λ k ) ν .
Evidently Φ k,j (λ) := p(λ)f k,j (λ) satisfies [START_REF] Kavian | Unique continuation principle for systems of parabolic equations[END_REF].

Step 2: Let us now prove some estimates of the polynomial p constructed in the previous step.

We can rewrite the identities in (28) as a linear system of the form AP = B with

A =         1 0 • • • 0 0 a 10 1 • • • 0 0 a 20 a 21 . . . 0 0 . . . . . . . . . . . . 0 a η-j-1,0 a η-j-1,1 • • • a η-j-1,η-j-2 1         ∈ L(C η-j ), B =       (-1) j η! j!Φ (η) (Λ k ) 0 . . . 0       ∈ C η-j and P = p (ν) (Λ k ) 0≤ν≤η-j-1 ∈ C η-j . Thus P = A -1 B and |P | C η-j ≤ A -1 |B| C η-j
, where • stands for the Hilbert-Schmidt norm of a matrix, i.e., if M ∈ L(C η-j ), then

M =   1≤r,s≤η-j |m rs | 2   1/2 .
Let us write A = I d -V . Then (see [START_REF] Gil | Localization and perturbation of zeros of entire functions[END_REF]Theorem 1.4.1, p. 6]):

A -1 ≤ 1 + V 2 + 1 η -j -1 (η-j-1)/2
.

Recall that V = I d -A. Taking into account the expression of the elements a νℓ of V (see ( 29)) we have:

1 + V 2 = 1 + 0≤ℓ<ν≤η-j-1 |a νℓ | 2 = 1 + 0≤ℓ<ν≤η-j-1 ν!η! ℓ!(η + ν -ℓ)! Φ (η+ν-ℓ) (Λ k ) Φ (η) (Λ k ) 2 = 0≤ℓ≤ν≤η-j-1 ν!η! ℓ!(η + ν -ℓ)! Φ (η+ν-ℓ) (Λ k ) Φ (η) (Λ k ) 2 .
Coming back to P , one has:

|P | C η-j ≤ A -1 |B| C η-j ≤ C   0≤ℓ≤ν≤η-j-1 Φ (η+ν-ℓ) (Λ k ) 2   η-j-1 2 1 Φ (η) (Λ k ) η-j , (30)
for a positive constant C which only depends on η.

Finally, let us estimate

|Φ (η+ν-ℓ) (Λ k )| for (ν, ℓ) : 0 ≤ ℓ ≤ ν ≤ η-j -1.
Since Φ is a holomorphic function on C + , we can write:

Φ (m) (Λ k ) = m! 2iπ |z-Λ k |=r Φ(z) (z -Λ k ) m+1 dz, ∀m ≥ 0 where r > 0 is such that {z ∈ C : |z -Λ k | = r} ⊂ C + .
Observe that we can take r = ℜΛ k /2. On the other hand, from the definition of Φ (see [START_REF] Kalman | Topics in Mathematical Control Theory[END_REF]) and the properties of W (Lemma 4.2), we deduce

|Φ (z)| ≤ 1 |1 + z| 2n ≤ 1, ∀z ∈ C + .
Thus:

Φ (m) (Λ k ) ≤ m! 2πr m+1 |z-λ k |=r dz = m! r m , ∀m ≥ 0, with r = ℜΛ k /2.
Going back to (30), we get

|P | C η-j ≤ C   0≤ℓ≤η-j-1 1 (ℜΛ k ) 2(η+ℓ)   (η-j-1)/2 1 Φ (η) (Λ k ) η-j ≤ C H 0j (Λ k ) Φ (η) (Λ k ) η-j , (31)
for a new positive constant C only depending on η and where H 0,j (Λ k ) given by

H 0j (Λ k ) = 1 + 1 ℜΛ k (2η-j-1)(η-j-1)
.

Step 3: We finalize the proof of Lemma 4.4 showing that the function

Φ k,j (λ) = p(λ)f k,j (λ) = Φ(λ) (λ -Λ k ) η-j η-j-1 ν=0 p (ν) (Λ k ) ν! (λ -Λ k ), ∀λ ∈ C + ,
(the function Φ is given by ( 22)) satisfies Φ k,j ∈ H 2 (C + ) and ( 24).

On the one hand, taking into account [START_REF] Tucsnak | Observation and Control of Operator Semigroups, Advanced Texts[END_REF], we can infer

|p(iτ )| ≤ C H 0j (Λ k ) Φ (η) (Λ k ) η-j η-j-1 ν=0 |iτ -Λ k | ν ≤ C H 0j (Λ k ) Φ (η) (Λ k ) η-j 1 + |iτ -Λ k | η-j-1 ,
with C a positive constant only depending on η. On the other hand, we can estimate,

                       Φ k,j 2
H 2 (C+) ≤ C 2 H 0j (Λ k ) 2 Φ (η) (Λ k ) 2(η-j) ∞ -∞ 1 + |iτ -Λ k | η-j-1 2 |iτ -Λ k | 2(η-j) |Φ(iτ )| 2 dτ ≤ ≤ C 2 H 0j (Λ k ) 2 Φ (η) (Λ k ) 2(η-j) 1 (ℜΛ k ) η-j + 1 ℜΛ k 2 Φ 2 H 2 (C+) ≤ C 2 H j (Λ k ) 2 Φ (η) (Λ k ) 2(η-j) ,
where H j is given in the statement of Lemma 4.4 and C is a new positive constant only depending on η. This last inequality shows that Φ k,j ∈ H 2 (C + ), inequality [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF] and finishes the proof of Lemma 4.4.

In Proposition 4.1 we have proved that, under assumptions ( 19) and ( 20) on the sequence Λ = {Λ k } k≥1 ⊂ C + , there exists a biorthogonal family {ϕ k,j } k≥1,0≤j≤η-1 ⊂ A(Λ, η, ∞) to the set t j e -Λ k t k≥1,0≤j≤η-1 (η ≥ 1 is fixed) which satisfies [START_REF] Hansen | Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems[END_REF]. Now, we will see that if we impose slightly stronger assumptions upon the sequence Λ (see assumptions in Theorem 1.2) we can estimate the infinite product P k given in the statement of Proposition 4.1. One has: Proposition 4.5. Let {Λ k } k≥1 be a sequence of complex numbers satisfying [START_REF] Fabre | Approximate controllability of the semilinear heat equation[END_REF]. Then, for every ε > 0 there exists a constant C(ε) > 0 such that

P k := ℓ≥1 ℓ =k 1 + Λ k /Λ * ℓ 1 -Λ k /Λ ℓ ≤ C(ε)e εℜΛ k , ∀k ≥ 1.
The proof of this result can be found for instance in [START_REF] Luxemburg | Entire functions and Müntz-Szász type approximation[END_REF], [START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF] and [START_REF] Fernández-Cara | Boundary controllability of parabolic coupled equations[END_REF] (see [START_REF] Luca | System controllability and Diophantine approximations[END_REF], p. 1730 in the last reference). See also [START_REF] Hansen | Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems[END_REF] where a slightly stronger inequality is proved under assumptions on {Λ k } k≥1 which, in particular, imply [START_REF] Fabre | Approximate controllability of the semilinear heat equation[END_REF].

Proof of Theorem 1.2: Let us start proving Theorem 1.2 in the case T = ∞.

First, if {Λ k } k≥1 satisfies [START_REF] Fabre | Approximate controllability of the semilinear heat equation[END_REF], then one has [START_REF] González-Burgos | Controllability results for cascade systems of m coupled parabolic PDEs by one control force[END_REF] and there exists a positive constant

C δ such that        ℜΛ k (1 + ℜΛ k ) 2 + (ℑΛ k ) 2 ≤ 1 ℜΛ k ≤ 1 δ 1 |Λ k | , ℜΛ k (1 + ℜΛ k ) 2 + (ℑΛ k ) 2 ≥ δ |Λ k | (1 + |Λ k |) 2 + |Λ k | 2 ≥ C δ 1 |Λ k | .
Therefore (20) holds and we can apply Proposition 4.1 to the sequence {Λ k } k≥1 deducing the existence of a family {ϕ k,j } k≥1,0≤j≤η-1 ⊂ A(Λ, η, ∞) biorthogonal to t j e -Λ k t k≥1,0≤j≤η-1 which satisfies [START_REF] Hansen | Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems[END_REF].

Secondly, taking ε > 0 and using that ℜΛ k → ∞ we infer that for a positive constant C(η, ε) one has

1 + 1 ℜΛ k (2η-j)(η-j-1)+1 (ℜΛ k ) η(η-j) |1 + Λ k | 2η(η-j) ≤ C 1 (η, ε)e εℜΛ k /2 ,
for any (k, j) with k ≥ 1 and 0 ≤ j ≤ η -1.

Finally, applying Lemma 4.5, with ε/(2η(ηj)), instead of ε and taking into account the previous inequality and ( 21) we obtain [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF]. This finalizes the proof in the case T = ∞.

Before continuing the proof of Theorem 1.2, let us present a consequence of the result proved for T = ∞: Corollary 4.6. Let us assume the assumptions of Theorem 1.2. Then, for any T ∈ (0, ∞) the restriction operator R T : A(Λ, η, ∞) → A(Λ, η, T ) defined by

R T ϕ = ϕ| (0,T ) , ∀ϕ ∈ A(Λ, η, ∞)
is a topological isomorphism. In particular, there exists a constant C(T ) > 0 such that

ϕ L 2 (0,∞;C) ≤ C(T ) R T ϕ L 2 (0,T ;C) , ∀ϕ ∈ A(Λ, η, ∞).
The proof of this result can be seen for instance in [START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF], [START_REF] Hansen | Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems[END_REF] or [START_REF] Fernández-Cara | Boundary controllability of parabolic coupled equations[END_REF].

Let us now go back to the proof of Theorem 1.2 and so, let us assume that T ∈ (0, ∞). If we apply the previous case, we deduce the existence of a family { ϕ k,j } k≥1,0≤j≤η-1 ⊂ A(Λ, η, ∞) biorthogonal to t j e -Λ k t k≥1,0≤j≤η-1 in L 2 (0, ∞; C) which satisfies [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF]. Let us set

ϕ k,j = R -1 T * ϕ k,j ∈ A(Λ, η, T ), ∀(k, j) : k ≥ 1, 0 ≤ j ≤ η -1.
From Corollary 4.6 and the properties of the family { ϕ k,j } k≥1,0≤j≤η-1 , it is clear that ϕ k,j satisfies [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF] for any (k, j).

On the other hand, with the notation e k,j (t) = t j e -Λ k t , we can write

δ kl δ ij = (e k,j , ϕ l,i ) L 2 (0,∞;C) = (R -1 T R T e k,j , ϕ l,i ) L 2 (0,∞;C) = (R T e k,j , R -1
T * ϕ l,i ) L 2 (0,T ;C) = (e k,j , ϕ l,i ) L 2 (0,T ;C) , ∀(k, j), (l, i), i.e., {ϕ k,j } k≥1,0≤j≤η-1 ⊂ A(Λ, η, T ) is a biorthogonal family to {e k,j } k≥1,0≤j≤η-1 in L 2 (0, T ; C) which also satisfies estimate [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF]. This ends the proof of Theorem 1.2. Remark 4.1. In Corollary 4.3 we have proved that, under assumption [START_REF] Guerrero | Null controllability of some systems of two parabolic equations with one control force[END_REF], A(Λ, η, ∞) is a closed proper subspace of L 2 (0, ∞; C). In fact, from the results proved in Proposition 4.1 it is clear that {e k,j : k ≥ 1, 1 ≤ j ≤ η} form a strongly independent set, i.e., each element e k,j of this set is outside the closure of the space spanned by the other functions of the set. This two results can be easily generalize to the case T ∈ (0, ∞): "Let us assume that hypotheses in Proposition 4.1 holds. Then A(Λ, η, T ) is a closed proper subspace of L 2 (0, T ; C). In addition, the set {e k,j : k ≥ 1, 0 ≤ j ≤ η -1} forms a strongly independent set.

Exact controllability to trajectories. Proof of Theorem 1.1

We will devote this section to prove Theorem 1.1. Using Proposition 2.4, we will just prove the null controllability at time T of the system. To this end, we will follow the method used by Fattorini and Russell in [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF] and [START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF] for the study of the null controllability of a scalar heat equation. By means of this method we will reduce the controllability problem for system (1) to a moment problem.

In subsection 5.1, we explain the moment method and derive the moment problems that must be satisfied by the components of the control v. We end this section with the proof of Theorem 1.1. All along this section we will assume that the coupling matrices A and B fulfill conditions [START_REF] Bodart | Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient[END_REF].

Recall that {µ l } 1≤l≤p ⊂ C is the set of distinct eigenvalues of A * . In this section we will use the notation introduced in Section 3 (see p. 8).

The moment problem

In this subsection we will see that, under assumption (8), the null controllability problem for system (1) is equivalent to a problem (the moment problem) for the unknown control v.

Let us now fix y 0 ∈ H -1 (0, π; C n ). Using formula [START_REF] Fernández-Cara | Boundary controllability of parabolic coupled equations[END_REF] for t = T , we deduce that, if ϕ is a solution of the adjoint system (13) corresponding to ϕ 0 ∈ H 1 0 (0, π; C n ), then the null controllability problem for system (1) is equivalent to the problem

     Find v ∈ L 2 (0, T ; C m ) such that -y 0 , ϕ(•, 0) = T 0 (v(t) , B * ϕ x (0, t)) C m dt, ∀ϕ 0 ∈ H 1 0 (0, π; C n ) . ( 32
)
If ϕ 0 ∈ H 1 0 (0, π; C n ) is given, then the corresponding solution to ( 13) is given by

ϕ(x, t) = k≥1 e (-λ k I d +A * )(T -t) φ k (x)ϕ 0,k , with ϕ 0,k = π 0 ϕ 0 (x)φ k (x) dx ∈ C n ,
(φ k given in ( 5)).

Let us fix k 0 ≥ 1 as in Proposition 3.2 and let us consider the finite-dimensional space

X 0 = {w : w = 1≤k≤k0 w k φ k with w k ∈ C n }.
In general, given y ∈ H -1 (0, π; C n ) (resp. y ∈ L 2 (0, π; C n )), we will use the notation

y k = y, φ k ∈ C n , (resp. y k = (y, φ k ) L 2 (0,π) ),
where •, • stands for the usual duality pairing between H -1 (0, π) and H 1 0 (0, π). With this notation, we consider

Y 0 = √ π k √ 2 y 0,k 1≤k≤k0 ∈ C nk0 , with y 0,k = y 0 , φ k ∈ C n , (33)
and

Φ 0 = k 2 π ϕ 0,k 1≤k≤k0 ∈ C nk0 . Then, it is not difficult to see,          B * ϕ x (0, t) = B * k0 e L * k 0 (T -t) Φ 0 + k>k0 k 2 π B * e (-λ k I d +A * )(T -t) ϕ 0,k , t ∈ (0, T ), -y 0 , ϕ(•, 0) = -(Y 0 , e L * k 0 T Φ 0 ) C nk 0 - k>k0 (y 0,k , e (-λ k I d +A * )T ϕ 0,k ) C n
with (B k0 , L k0 ) given by [START_REF] Ammar-Khodja | Boundary controllability of coupled parabolic systems[END_REF].

Taking first ϕ 0 arbitrary in X 0 and then ϕ 0 = aφ k , with a ∈ C n and k > k 0 , and using that {φ k } k≥1 is an orthonormal basis of L 2 (0, π), (32) transforms into the problem

             Find v ∈ L 2 (0, T ; C m ) such that T 0 (v(T -t) , B * k0 e L * k 0 t Φ 0 ) C m dt = F(Y 0 , Φ 0 ), ∀Φ 0 ∈ C nk0 , T 0 (v(T -t) , B * e (-λ k I d +A * )t a) C m dt = f k (y 0 , a), ∀a ∈ C n , ∀k > k 0 , (34)
where we have introduced the bilinear forms F :

C nk0 ×C nk0 → C and f k : H -1 (0, π; C n )×C n → C given by      F (Y 0 , Φ 0 ) = -(Y 0 , e L * k 0 T Φ 0 ) C nk 0 , ∀(Y 0 , Φ 0 ) ∈ C nk0 × C nk0 , f k (y 0 , a) = - 1 k π 2 (y 0,k , e (-λ k I d +A * )T a) C n , ∀(y 0 , a) ∈ H -1 (0, π; C n ) × C n . ( 35
)
So we have reduced our control problem to a vector moment problem. In order to analyze this moment problem, let us first introduce the Jordan structure of the matrix L * k0 : Let {γ ℓ } 1≤ℓ≤e p ⊂ C be the set of distinct eigenvalues of L * k0 . Following the notation that we have introduce for the matrix A * (see p. 8), for ℓ : 1 ≤ ℓ ≤ p, we denote by N ℓ the geometric multiplicity of γ ℓ and we assume that we have numbered the eigenvalues in such a way that

N 1 ≥ N ℓ , 2 ≤ ℓ ≤ p.
Also, the sequence {V ℓ,j } 1≤j≤N ℓ will denote a basis of eigenvectors of L * k0 associated to γ ℓ . To each eigenvector V ℓ,j we associate its Jordan chain (of dimension τ ℓ,j ) and the corresponding set of generalized eigenvectors {V i ℓ,j } 1≤i≤e τ ℓ,j defined by:

L * k0 V i ℓ,j = γ ℓ V i ℓ + V i+1 ℓ , 1 ≤ i < τ ℓ,j , L * k0 V e τ ℓ,j ℓ,j = γ ℓ V e τ ℓ,j ℓ,j
(so that V e τ ℓ,j ℓ,j = V ℓ,j ). In fact, the eigenvalues, eigenvectors and the Jordan canonical form of the matrix L * k0 is determined by the eigenvalues, eigenvectors and the Jordan canonical form of A * . Thus, γ ℓ , with 1 ≤ ℓ ≤ p, is an eigenvalue of L * k0 if and only if for k : 1 ≤ k ≤ k 0 and l : 1 ≤ l ≤ p, one has γ ℓ = -λ k + µ l . In this last case, the vector V = (V i ) 1≤i≤k0 ∈ C nk0 is an eigenvector of L * k0 associated to γ ℓ if and only if

V i = 0, ∀i = k, and V k = v l,j with j : 1 ≤ j ≤ n l .
On the other hand, it is also clear that the following properties hold:

p ≤ p ≤ pk 0 , N 1 = max{N ℓ : 1 ≤ ℓ ≤ p} ≥ n 1 = max{n l : 1 ≤ l ≤ p}. ( 36
)
With the previous notation, if t ∈ R, we can write:

             e tL * k 0 V ı ℓ, = e γ ℓ t e τ ℓ, -ı σ=0 t σ σ! V ı+σ ℓ, , ∀(ℓ, , ı) : 1 ≤ ℓ ≤ p, 1 ≤  ≤ N ℓ , 1 ≤ ı ≤ τ ℓ, , e tA * v i l,j = e µ l t τ l,j -i σ=0 t σ σ! v i+σ l,j , ∀(l, j, i) : 1 ≤ l ≤ p, 1 ≤ j ≤ n l , 1 ≤ i ≤ τ l,j . (37)
For the sake of simplicity, we will show the moment method under the assumption:

τ l,j = τ l , ∀l, j : 1 ≤ l ≤ p, 1 ≤ j ≤ n l ,
i.e., we will suppose that the Jordan's block associated to the eigenvalues of A * have the same dimension. Observe that this assumption also implies τ ℓ, = τ ℓ for every ℓ, , with 1 ≤ ℓ ≤ p and 1 ≤  ≤ N ℓ . Now, let us fix (ℓ, , ı) and (l,

j, i) with 1 ≤ ℓ ≤ p, 1 ≤  ≤ N ℓ , 1 ≤ ı ≤ τ ℓ,j ≡ τ ℓ , 1 ≤ l ≤ p, 1 ≤ j ≤ n l and 1 ≤ i ≤ τ l,j ≡ τ l ,
and let us take Φ 0 = V ı ℓ, and a = v i l,j in (34). So, from (37) and taking into account that {V ı ℓ, : 1

≤ ℓ ≤ p, 1 ≤  ≤ N ℓ , 1 ≤ ı ≤ τ ℓ } and {v i l,j : 1 ≤ l ≤ p, 1 ≤ j ≤ n l , 1 ≤ i ≤ τ l } are basis of C nk0 and C n (resp.), we deduce that (34) is equivalent to                              Find v ∈ L 2 (0, T ; C m ) such that e τ ℓ -ı σ=0 ( T 0 t σ σ! e γ * ℓ t v(t) dt , B * k0 V ı+σ ℓ, ) C m = F (Y 0 , V ı ℓ, ), ∀(ℓ, , ı) : 1 ≤ ℓ ≤ p, 1 ≤  ≤ N ℓ and 1 ≤ ı ≤ τ ℓ , τ l -i σ=0 ( T 0 t σ σ! e (-λ k +µ * l )t v(t) dt , B * v i+σ l,j ) C m = f k (y 0 , v i l,j ), ∀(k, l, j, i) : k > k 0 , 1 ≤ l ≤ p, 1 ≤ j ≤ n l and 1 ≤ i ≤ τ l , (38)
where

v(t) = v(T -t), for t ∈ [0, T ].
From Proposition 3.1 applied to C ≡ L k0 and D ≡ B k0 , we know that condition (8

) (with k = k 0 ) is equivalent to rank [B * k0 V ℓ,1 | B * k0 V ℓ,2 | • • • | B * k0 V ℓ,N ℓ ] = N ℓ , ∀ℓ : 1 ≤ ℓ ≤ p. ( 39
)
In particular, we infer that m ≥ N ℓ for every ℓ : 1 ≤ ℓ ≤ p and from (36) also m ≥ n l for all l : 1 ≤ l ≤ p. Thus, the set {B * k0 V 1,j } 1≤j≤N1 ⊂ C m is linearly independent. We complete the previous set with the vectors { V j } 1≤j≤m-N1 ⊂ C m in order to have a basis of C m :

B ≡ {B * k0 V 1,j } 1≤j≤N1 ∪ { V j } 1≤j≤m-N1
. We can associate with each vector B * k0 V ı ℓ, and B * v i l,j of C m , its coordinates in this basis:

             B * k0 V ı ℓ, = N1 q=1 α q ℓ,,ı B * k0 V 1,q + m-N1 q=1 α q ℓ,,ı V q , 1 ≤ ℓ ≤ p, 1 ≤  ≤ N ℓ , 1 ≤ ı ≤ τ ℓ , B * v i l,j = N1 q=1 β q l,j,i B * k0 V 1,q + m-N1 q=1 β q l,j,i V q , 1 ≤ l ≤ p, 1 ≤ j ≤ n l , 1 ≤ i ≤ τ l . (40)
Thus, coming back to (38), we obtain that this problem is equivalent to

                                                   Find v ∈ L 2 (0, T ; C m ) such that e τ ℓ -ı σ=0 N1 q=1 α q, * ℓ,,ı+σ ( T 0 t σ σ! e γ * ℓ t v(t) dt , B * k0 V 1,q ) C m + e τ ℓ -ı σ=0 m-N1 q=1 α q, * l,,ı+σ ( T 0 t σ σ! e γ * ℓ t v(t) dt , V q ) C m = F (Y 0 , V ı ℓ, ), ∀(ℓ, , ı) with 1 ≤ ℓ ≤ p, 1 ≤  ≤ N ℓ and 1 ≤ ı ≤ τ ℓ , τ l -i σ=0 N1 q=1 β q, * l,j,i+σ ( T 0 t σ σ! e (-λ k +µ * l )t v(t) dt , B * k0 V 1,q ) C m + τ l -i σ=0 m-N1 q=1 β q, * l,j,i+σ ( T 0 t σ σ! e (-λ k +µ * l )t v(t) dt , V q ) C m = f k (y 0 , v i l,j ), ∀(k, l, j, i) with k > k 0 , 1 ≤ l ≤ p, 1 ≤ j ≤ n l and 1 ≤ i ≤ τ l .
Let us now consider {Φ 1,q } 1≤q≤N1 ∪ { Φ 1,q } 1≤q≤m-N1 a biorthogonal basis associated to the previous basis B of C m . We will look for a control v given by

v(t) = N1 q=1 u q (t)Φ 1,q , (41)
with u q ∈ L 2 (0, T ; C) for q : 1 ≤ q ≤ N 1 .

Using the equalities (Φ 1,q , B * k0 V 1,i ) C m = δ qi and (Φ 1,q , V j ) C m = 0, valid for every (q, i, j) : 1 ≤ q, i ≤ N 1 and 1 ≤ j ≤ m -N 1 , we can rewrite the previous moment problem as

                             Find u q ∈ L 2 (0, T ; C), with 1 ≤ q ≤ N 1 , such that e τ ℓ -ı σ=0 N1 q=1 α q, * ℓ,,ı+σ T 0 t σ σ! e γ * ℓ t u q (t) dt = F (Y 0 , V ı ℓ, ), ∀(ℓ, , ı) with 1 ≤ ℓ ≤ p, 1 ≤  ≤ N ℓ and 1 ≤ ı ≤ τ ℓ , τ l -i σ=0 N1 q=1 β q, * l,j,i+σ T 0 t σ σ! e (-λ k +µ * l )t u q (t) dt = f k (y 0 , v i l,j ), ∀(k, l, j, i) with k > k 0 , 1 ≤ l ≤ p, 1 ≤ j ≤ n l and 1 ≤ i ≤ τ l . (42)
Our aim is to prove that the previous moment problem admits, for every q, a solution u q which lies in L 2 (0, T ; C). The previous reasoning shows that the corresponding control v(t) = v(Tt), with v given by (41), is in L 2 (0, T ; C m ) and solves the null controllability problem for system [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF].

For (ℓ, ν) and (l, σ) such that 1

≤ ℓ ≤ p, 0 ≤ ν ≤ τ ℓ -1, 1 ≤ l ≤ p and 0 ≤ σ ≤ τ l -1, let us set                  X ℓ,ν = T 0 t ν ν! e γ * ℓ t u q (t) dt 1≤q≤N1 , Y k l,σ = T 0 t σ σ! e (-λ k +µ * l )t u q (t) dt 1≤q≤N1 ∈ C N1 , F ℓ,ν (Y 0 ) = F (Y 0 , V e τ ℓ -ν ℓ,j ) 1≤j≤N ℓ ∈ C N ℓ , F k l,σ (y 0 ) = f k (y 0 , v τ l -σ l,j ) 1≤j≤n l ∈ C n l and A ℓ,ν = α q, * ℓ,j,e τ ℓ -ν 1≤j≤N ℓ 1≤q≤N1 ∈ L(C N1 ; C N ℓ ), B l,σ = β q, * l,j,τ l -σ 1≤j≤n l 1≤q≤N1 ∈ L(C N1 ; C n l ), (43)
where F (Y 0 , Φ 0 ) and f k (y 0 , a) are given in (35).

Step 1: Let us first consider the case ℓ = 1. Let us recall that α q ℓ,j,i 1≤q≤N1 are the coordinates of B * k0 V i ℓ,j with respect to the basis

B = {B * k0 V 1,j } 1≤j≤N1 ∪ { V j } 1≤j≤m-N1 of C m . In particular, B * k0 V e τ1 1, = N1 q=1 α q 1,,e τ1 B * k0 V 1,q + m-N1 q=1 α q 1,,e τ1 V q , ∀ : 1 ≤  ≤ N 1 .
Since V e τ ℓ ℓ,j = V ℓ,j for every ℓ, j, if ℓ = 1 we get α q 1,j,e τ1 = δ qj , ∀j, q : 1 ≤ j, q ≤ N 1 and α q 1,j,e τ1 = 0, ∀j, q : 1 ≤ j, q ≤ m -N 1 . From (42) and using the previous equalities (ℓ = 1) we obtain:

ı = τ 1 , T 0 e γ * 1 t u j (t) dt = F (Y 0 , V e τ1 1,j ), ı = τ 1 -1, N1 q=1 α q, * 1,j,e τ1-1 T 0 e γ * 1 t u q (t) dt + T 0 te γ * 1 t u j (t) dt = F (Y 0 , V e τ1- 1 1,j ), . . . . .
. ı = 1, e τ1-1 σ=0 N1 q=1 α q, * 1,j,1+σ T 0 t σ σ! e γ * 1 t u q (t) dt = F (Y 0 , V 1 1,j ).
Using (43), we infer that, for ℓ = 1, the first part of system (42) is equivalent to the linear algebraic system

           X 1,0 = F 1,0 (Y 0 ), A 1,1 X 1,0 + X 1,1 = F 1,1 (Y 0 ), . . . A 1,e τ1-1 X 1,0 + A 1,e τ1-2 X 1,1 + • • • + X 1,e τ1-1 = F 1,e τ1-1 (Y 0 ), or in matrix form:        I d 0 0 • • • 0 A 1,1 I d 0 • • • 0 A 1,2 A 1,1 I d • • • 0 . . . . . . . . . . . . . . . A 1,e τ1-1 A 1,e τ1-2 A 1,e τ1-3 • • • I d        =M1        X 1,0 X 1,1 X 1,2 . . . X 1,e τ1-1        =        F 1,0 (Y 0 ) F 1,1 (Y 0 ) F 1,2 (Y 0 ) . . . F 1,e τ1-1 (Y 0 )       
Clearly this system possesses a unique solution given by 

    X 1,0 X 1,1 . . . X 1,e τ1-1      = M -1 1      F 1,0 (Y 0 ) F 1,1 (Y 0 ) . . . F 1,e τ1-1 (Y 0 )      ≡      C 1,0 (Y 0 ) C 1,1 (Y 0 ) . . . C 1,e τ1-1 (Y 0 )      .
Observe that the matrix M 1 ∈ L(C e τ1N1 ) only depends on the coupling matrices A and B and can be computed independently of the initial datum y 0 .

In other words, if we write C 1,σ (y 0 ) = (c 1,σ,q (Y 0 )) 1≤q≤N1 ∈ C N1 (Y 0 given in (33)), we have established that the components of u q of the control v with respect to the set {Φ 1,q } 1≤q≤N1 must solve the following first family of moment problems:

T 0 t σ σ! e γ * 1 t u q (t) dt = c 1,σ,q (y 0 ), ∀(σ, q) : 0 ≤ σ ≤ τ 1 -1, 1 ≤ q ≤ N 1 .
Finally, taking into account the expressions of C 1,σ (y 0 ), F 1,σ (Y 0 ) (see ( 43)) and F (Y 0 , Φ 0 ) (see ( 35)), we deduce the existence of a positive constant C = C(A, B) for which

|c 1,σ,q (y 0 )| ≤ C e L * k 0 T L(C nk 0 ) y 0 H -1 (0,π;C n ) , ∀(σ, q) : 0 ≤ σ ≤ τ 1 -1, 1 ≤ q ≤ N 1 .
Step 2: Consider now the case 2 ≤ ℓ ≤ p. Following the same reasoning as before, we find that the first equation of system (42) can be equivalently rewritten as:

           A ℓ,0 X ℓ,0 = F ℓ,0 (Y 0 ), A ℓ,1 X ℓ,0 + A ℓ,0 X ℓ,1 = F ℓ,1 (Y 0 ), . . . A ℓ,e τ ℓ -1 X ℓ,0 + A ℓ,e τ ℓ -2 X ℓ,1 + • • • + A ℓ,0 X ℓ,e τ ℓ -1 = F ℓ,e τ ℓ -1 (Y 0 ). (44)
Let us show that this linear system (which has τ ℓ N 1 unknown and τ ℓ N ℓ ≤ τ ℓ N 1 equations) is compatible. Indeed, remember that A ℓ,0 ∈ L(C N1 ; C N ℓ ) (see (43)) and its components α q ℓ,j,e τ ℓ are the coordinates of the vectors B * k0 V e τ ℓ ℓ,j = B * k0 V ℓ,j with respect to the basis B (see (40)). Again condition (39) implies that the set {B * k0 V ℓ,j : 1 ≤ j ≤ N ℓ } ⊂ C m is linearly independent and, evidently, also rank A ℓ,0 = N ℓ . Then, there exists a permutation matrix P ℓ ∈ L(R N1 ) (only depending on A ℓ,0 , i.e., on ℓ, A and B) such that A ℓ,0 P ℓ = A ℓ,0 | D ℓ,0 with A ℓ,0 ∈ L(C N ℓ ) a squared matrix and rank A ℓ,0 = rank A ℓ,0 = N ℓ . For each σ with 0 ≤ σ ≤ τ ℓ -1, we also set A ℓ,σ P ℓ = A ℓ,σ | D ℓ,σ with A ℓ,σ ∈ L(C n ℓ ). If we look for a solution under the form X ℓ,σ = P ℓ X ℓ,σ 0 , the previous system transforms into:

             A ℓ,0 X ℓ,0 = F ℓ,0 (Y 0 ), A ℓ,1 X ℓ,0 + A ℓ,0 X ℓ,1 = F ℓ,1 (Y 0 ), . . . A ℓ,e τ ℓ -1 X ℓ,0 + A ℓ,e τ ℓ -2 X ℓ,1 + • • • + A ℓ,0 X l,e τ ℓ -1 = F ℓ,e τ ℓ -1 (Y 0 ).
This system has a unique solution which can be written as

X ℓ,σ = C ℓ,σ (y 0 ) (0 ≤ σ ≤ τ ℓ -1) with        C ℓ,0 (y 0 ) C ℓ,1 (y 0 ) . . . C ℓ,e τ ℓ -1 (y 0 )        = M -1 ℓ       F ℓ,0 (Y 0 ) F ℓ,1 (Y 0 ) . . . F ℓ,e τ ℓ -1 (Y 0 )       , (45)
where M ℓ ∈ L(C e τ ℓ N ℓ ) is the coefficient matrix of the linear system (which, once again, only depends on ℓ and the coupling matrices A and B) and F ℓ,σ (Y 0 ) ∈ C N ℓ and Y 0 are given in (43) and (33). This proves that the system (44) is compatible and, in fact, we have obtained a particular solution.

We can repeat the arguments shown in the case ℓ = 1 and deduce that the components of the previous solution of system (44) satisfy the family of moment problems

T 0 t σ σ! e γ * ℓ t u q (t) dt = c ℓ,σ,q (y 0 ), ∀(ℓ, σ, q) : 2 ≤ ℓ ≤ p, 0 ≤ σ ≤ τ ℓ -1, 1 ≤ q ≤ N 1 ,
where the coefficient c ℓ,σ,q (y 0 ) are given by (c ℓ,σ,q (y 0 )) 1≤q≤N1 = P ℓ C ℓ,σ (y 0 ) 0 .

Again, taking into account the expressions of F ℓ,σ (Y 0 ) and F (Y 0 , Φ 0 ) (see ( 43) and ( 35)), we deduce again the existence of a positive constant C = C(A, B) such that, for every (ℓ, σ, q) with 2 ≤ ℓ ≤ p, 0 ≤ σ ≤ τ ℓ -1 and 1 ≤ q ≤ N 1 , one has

|c ℓ,σ,q (y 0 )| ≤ C e L * k 0 T L(C nk 0 ) y 0 H -1 (0,π;C n ) . ( 46
)
Step 3: Now, we are going to obtain an infinite family of moment problems using the second identity in (42). We fix k > k 0 and l : 1 ≤ l ≤ p. Following the same ideas as before, we obtain that the second identity in (42) is equivalent to

             B l,0 Y k l,0 = F k l,0 (y 0 ), B l,1 Y k l,0 + B l,0 Y k l,1 = F k l,1 (y 0 ), . . . B l,τ l -1 Y k l,0 + B l,τ l -2 Y k l,1 + • • • + B l,0 Y k l,τ l -1 = F k l,τ l -1 (y 0 ), (47)
where B l,σ and F k l,σ given in (43). Again, we have a compatible system of dimension (τ l n l ) × (τ l N 1 ) (n l ≤ N 1 , see (36)). Indeed, from the Kalman rank condition (2) we deduce

rank [B * v l,1 | B * v l,2 | • • • | B * v l,n l ] = n l , ∀l : 1 ≤ l ≤ p,
and also rank B l,0 = n l . Therefore, for a permutation matrix Q l ∈ L(R N1 ) (only depending on l, A and B), we can write B l,0 Q l = B l,0 | D l,0 with B l,0 ∈ L(C n l ) and rank B l,0 = n l . We also

write B l,σ Q l = B l,σ | D l,σ , with B l,σ ∈ L(C n l ) (0 ≤ σ ≤ τ l -1)
, and we obtain a solution to (47)

as Y k l,σ = Q l Y k l,σ 0 with Y k l,σ ∈ C n l solution to              B l,0 Y k l,0 = F k l,0 (y 0 ), B l,1 Y k l,0 + B l,0 Y k l,1 = F k l,1 (y 0 ), . . . B l,τ l -1 Y k l,0 + B l,τ l -2 Y k l,1 + • • • + B l,0 Y k l,τ l -1 = F k l,τ l -1 (y 0 ),
If we denote by means of M l ∈ L(C τ l n l ) the previous coefficient matrix (which only depends on l, A and B), then we have managed to obtain a solution of the system (47) under the form

Y k l,σ = D k l,σ (y 0 ) with D k l,σ (y 0 ) = Q l D k l,σ (y 0 ) 0 and D k l,σ (y 0 ) 1≤σ≤τ l -1 (y 0 ) = M -1 l F k l,σ (y 0 ) 1≤σ≤τ l -1 . (48)
Finally, let us remark that, from the expressions of F k l,σ (y 0 ) and f k (y 0 , a) (see ( 43) and ( 35)), we have the existence of a positive constant C = C(A, B) such that the components d k 1,σ,q (y 0 ) (q : 1 ≤ q ≤ N 1 ) of D k l,σ (y 0 ) satisfy

|d k 1,σ,q (y 0 )| ≤ C k e (-λ k I d +A * )T L(C n ) |y 0,k | (49)
for every (k, l, σ, q) with k > k 0 , 1 ≤ l ≤ p, 0 ≤ σ ≤ τ l -1 and 1 ≤ q ≤ N 1 .

Summarizing, with the previous notation and assuming that τ l,j = τ l , for all l, j : 1 ≤ l ≤ p and 1 ≤ j ≤ n l , we have proved: Proposition 5.1. Assume that condition (8) holds and let us consider the integer k 0 provided by Proposition 3.2. Let us also fix y 0 ∈ H -1 (0, π; C n ). Then, for every (ℓ, l) with 1 ≤ ℓ ≤ p and 1 ≤ l ≤ p, there exist matrices M ℓ ∈ L(C e τ ℓ N ℓ ), M l ∈ L(C τ l n l ) and permutation matrices P ℓ , Q l ∈ L(R N1 ), which only depend on the coupling matrices A and B, such that, if for every q, with 1 ≤ q ≤ N 1 , the function u q ∈ L 2 (0, T ; C) satisfies the family of moment problems:

         T 0 t ν ν! e γ * ℓ t u q (t) dt = c ℓ,ν,q (y 0 ), ∀(ℓ, ν) : 1 ≤ ℓ ≤ p, 0 ≤ ν ≤ τ ℓ -1, T 0 t σ σ! e (-λ k +µ * l )t u q (t) dt = d k l,σ,q (y 0 ), ∀(k, l, σ) : k > k 0 , 1 ≤ l ≤ p, 0 ≤ σ ≤ τ l -1, (50)
then the control v given by v(t) = v(T -t) (t ∈ (0, T )), with v given by (41), is in L 2 (0, T ; C m ) and solves the null controllability problem for system [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF]. In (50) the coefficients c ℓ,ν,q and d k l,σ,q (y 0 ) are respectively given by (c ℓ,ν,q (y 0 )) 1≤q≤N1 = P ℓ C ℓ,ν (y 0 ) 0 , d k l,σ,q (y 0 ) 1≤q≤N1 = Q l D k l,σ (y 0 ) 0 and D k l,σ (y 0 ) and C ℓ,ν (y 0 ) by (45) and (48). Finally, there exists a positive constant C, only depending on A and B, such that (46) and (49) hold.

Proof of Theorem 1.1

In this subsection we are going to prove Theorem 1.1.

Necessary condition: Let us show that condition [START_REF] Bodart | Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient[END_REF] is necessary in order to get the exact controllability to the trajectories of system (1). To this end, we will use Proposition 2.4. To be precise, let us assume that, for k 0 ≥ 1, one has

rank K k0 = rank [B k0 | L k0 B k0 | L 2 k0 B k0 | • • • | L nk0-2 k0 B k0 | L nk0-1 k0 B k ] < nk 0 ,
and let us prove that the observability inequality ( 15) fails (L k and B k are given in ( 7)). Indeed, using the Kalman rank condition for ordinary differential systems, we deduce that the pair (L * k0 , B * k0 ) is not observable, i.e., there exists Φ 0 ∈ C nk0 with Φ 0 ≡ 0 such that the solution Φ to the system

-Φ ′ = L * k0 Φ in (0, T ), Φ(T ) = Φ 0 ∈ C nk0 , satisfies B * k0 Φ(t) = 0, ∀t ∈ (0, T ).
If now we do Φ 0 = k 2 π ϕ 0,k 1≤k≤k0 (with ϕ 0,k ∈ C n for every k : 1 ≤ k ≤ k 0 ) and we take

ϕ 0 (x) = k0 k=1 ϕ 0,k φ k (x),
then, ϕ 0 ∈ H 1 0 (0, π; C n ), ϕ 0 ≡ 0 and the corresponding solution ϕ to (13) satisfies

B * ϕ x (0, •) = B * k0 Φ(t) = 0 on (0, T ).
Evidently, this proves that inequality (15) fails.

Sufficient condition: Let us assume that the pair (A, B) satisfies condition [START_REF] Bodart | Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient[END_REF] and let us prove that system (1) is exactly controllable to trajectories at time T or, equivalently, is null controllable at time T (T ∈ (0, T ) is given). To this ends, let us fix y 0 ∈ H -1 (0, π; C n ).

As said before, we will follow the technique from [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF] and we will prove the result as a consequence of Proposition 5.1 and Theorem 1.2.

Case 1: With the notation of Section 3 and Subsection 5.1, let us first assume that the matrix A is such that τ l,j = τ l , for all l, j : 1 ≤ l ≤ p and 1 ≤ j ≤ n l .

Let us also take k 0 ≥ 1 provided by Proposition 3.2. Recall that {γ ℓ } 1≤ℓ≤e p ⊂ C and {µ l } 1≤l≤p ⊂ C are, resp., the set of distinct eigenvalues of the matrices L * k0 and A * . Let us fix µ > 0 such that ℜ(λ 1µ l + µ) > 0 for every l : 1 ≤ l ≤ p. With this notation, let us set

Λ ℓ = -γ ℓ + µ, for 1 ≤ ℓ ≤ p, Λ e p+(i-1)p+l = λ i+k0 -µ l + µ, for i ≥ 1, 1 ≤ l ≤ p.
Recall that, given ℓ : 1 ≤ ℓ ≤ p, one has γ ℓ = -λ k + µ l , with 1 ≤ k ≤ k 0 and 1 ≤ l ≤ p. Thanks to Proposition 3.2 we also have that the sequence Λ = {Λ k } k≥1 satisfies [START_REF] González-Burgos | Controllability results for cascade systems of m coupled parabolic PDEs by one control force[END_REF]. On the other hand, from the property satisfied by µ, we deduce that ℜΛ k > 0, i.e.,

{Λ k } k≥1 ⊂ C + .
Our next task will be to prove that the sequence {Λ k } k≥1 satisfies (10):

(a) ℜΛ k ≥ δ|Λ k | for a positive δ wich only depends on A. Indeed,

lim k→∞ ℜΛ k |Λ k | = 1,
whence we deduce the existence of K 0 ≥ 1 (only depending on

{µ l } 1≤l≤p ) such that ℜΛ k > 1 2 |Λ k | for every k ≥ K 0 . Taking δ = min 1 2 , ℜΛ k |Λ k | : 1 ≤ k ≤ K 0 ,
we deduce this first property.

(b) Let us recall that λ k = k 2 (see ( 5)). Therefore, the second property

k≥1 1 |Λ k | < ∞,
can be easily deduced.

(c) Finally, let us prove that there exists ρ > 0 (which only depends on A) such that |Λ k -Λ l | ≥ ρ|k -l| for every k > l ≥ 1. Let us first consider k ≥ p + 1 and l < k. Then, for some ν : 1 ≤ ν ≤ p, we can write k = p + (i -1)p + ν and Λ k = λ iµ ν + µ. Let us take,

M 0 = max 1≤ν,σ≤p |µ ν -µ σ | M 1 = max 1≤ν≤p,1≤ℓ≤e p |γ ℓ -µ ν | ≥ λ k0 , and          I 0 = max {0, [M 0 + 2(p -k 0 -1)]} + 1, I 1 = max 0, 1 2 p -2k 0 + (p -2k 0 ) 2 + 4(M 1 -k 0 + p -1) + 1, K 0 = p + I 0 p and K 1 = p + I 1 p.
Observe that M 1 ≥ k 2 0 and, thus, I 1 is well defined. It is also clear that the constants M 0 , M 1 , K 0 and K 1 only depend on the matrix A.

Then, if k ≥ K 0 and p < l < k, that is to say, if k ≥ K 0 and l = p + (j -1)p + σ with 1 ≤ j < i and σ : 1 ≤ σ ≤ p, we have i ≥ I 0 and so

|k -l| = (i -j)p + ν -σ ≤ (i -j)p + p -1 ≤ (i + j + 2k 0 )(i -j) -M 0 ≤ (i + k 0 ) 2 -(j + k 0 ) 2 -|µ σ -µ ν | ≤ |Λ k -Λ l |.
Let us now assume that k ≥ K 1 and 1 ≤ l ≤ p. In particular, i ≥ I 1 , Λ l = -γ l + µ and we can write

|k -l| = p + (i -1)p + ν -l ≤ p + ip -1 ≤ (i + k 0 ) 2 -M 1 ≤ (i + k 0 ) 2 -|γ l -µ ν | ≤ |Λ k -Λ l | Summarizing, if k ≥ K = max {K 0 , K 1 }, we have proved |Λ k -Λ l | ≥ |k -l|, ∀l : 1 ≤ l < k. Finally, let us set ρ = m 0 K -2 , with m 0 = min 1≤l<k<K |Λ k -Λ l |.
Thus, ρ > 0 (thanks to [START_REF] González-Burgos | Controllability results for cascade systems of m coupled parabolic PDEs by one control force[END_REF]), ρ only depends on the matrix A and |Λ k -Λ l | ≥ ρ|k -l| for every k, l with 1 ≤ l < k < K. This finishes the proof of condition [START_REF] Fabre | Approximate controllability of the semilinear heat equation[END_REF].

Let us take η = max {τ l , τ ℓ : 1 ≤ l ≤ p, 1 ≤ ℓ ≤ p} (we are following the notations introduced in Subsection 5.1). With η and the sequence {Λ k } k≥1 we can apply Theorem 1.2 and deduce the existence of a family F ≡ {ϕ k,j } k≥1,0≤j≤η-1 ⊂ L 2 (0, T ; C) biorthogonal to t j e -Λ k t k≥1,0≤j≤η-1 which satisfies [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF].

Our objective is to apply Proposition 5.1 and, in particular, to solve the family of moment problems (50). Given y 0 ∈ H -1 (0, π; C n ), we will take as control in system (1) the following function: v(t) = v(Tt) for every t ∈ (0, T ), with v given by (41) and u q (1 ≤ q ≤ N 1 ) defined on the interval (0, T ) by

u q (t) = e p ℓ=1 e τ ℓ -1 ν=0 c ℓ,ν,q (y 0 )e -µt ϕ ℓ,ν (t) + k>k0 p l=1 τ l -1 σ=0 d k l,σ,q (y 0 )e -µt ϕ e p+(k-k0-1)+l,σ (t),
where the coefficients c ℓ,ν,q (y 0 ) and d k l,σ,q (y 0 ) are provided by Proposition 5.1 and satisfy (46) and (49).

Using the orthogonality properties of the family F we deduce that v and u q solve the moment problems (50). Therefore, if we prove that u q ∈ L 2 (0, T ; C), we could apply Proposition 5.1 and conclude that the control v ∈ L 2 (0, T ; C m ) solves the null controllability problem for the coupled parabolic system (1).

Let us take ε > 0 (which will be chosen later). Using ( 11), ( 46) and (49) we get

                                   u q L 2 (0,T ;C) ≤ C(ε, T, A, B) e L * k 0 T L(C nk 0 ) y 0 H -1 (0,π;C n ) e p ℓ=1 e -εℜγ ℓ + C(ε, T, A, B) k>k0 p l=1 1 k e (-λ k I d +A * )T L(C n ) e εℜ(λ k -µ l ) |y 0,k | ≤ C(ε, T, A, B) y 0 H -1 (0,π;C n ) + k>k0 1 k e -(T -ε)λ k |y 0,k | ≤ C(ε, T, A, B) 1 + k>k0 1 k e -2(T -ε)λ k y 0 H -1 (0,π;C n ) ,
where C(ε, T, A, B) is a positive constant. Taking ε ∈ (0, T ), for example ε = T /2, we obtain that the series in the definition of u q converges absolutely in L 2 (0, T ; C). As a consequence u q ∈ L 2 (0, T ; C), for every q : 1 ≤ q ≤ N 1 , and for a positive constant C(T, A, B)

the control v satisfies v L 2 (0,T ;C m ) ≤ C y 0 H -1 (0,π;C n ) .
This proves the sufficient implication of Theorem 1.1 under the hypothesis τ l,j = τ l , for all l, j : 1 ≤ l ≤ p and 1 ≤ j ≤ n l .

Case 2: Let us now prove that system (1) is null controllable at time T in the general case. Thanks to Proposition 2.4, this null controllability property is equivalent to the observability inequality [START_REF] Fernández-Cara | The cost of approximate controllability for heat equations: the linear case[END_REF] for the solutions ϕ of the adjoint system [START_REF] Fernández-Cara | On the boundary controllability of non-scalar parabolic systems[END_REF]. Thus, let us show this observability inequality.

Following the notations introduced in Subsection 3 (see p. 8), we can write A * = P J * P -1 with P ∈ L(C n ) a regular matrix and J * the Jordan canonical form of A * which is given by

J * = diag (J 1 (µ 1 ), J 2 (µ 2 ), • • • , J p (µ p )) ∈ L(C n ), where J l (µ l ) = diag (J l,1 (µ l ), J l,2 (µ l ), • • • , J l,n l (µ l )) ∈ L(C m l ), 1 ≤ l ≤ p, m l = n l
j=1 τ l,j is the algebraic multiplicity of µ l and J l,j (µ l ) ∈ L(C τ l,j ) (1 ≤ j ≤ n l ) is the Jordan block associated to the eigenvector v l,j of A * , i.e.,

K k = [ L k | B k ] = [ B k | L k B k | L 2 k B k | • • • | L nk-2 k B k | L nk-1 k B k ] ∈ L(C mnk , C nk ).
where L i = -λ i I d + J. with P l : z ∈ C n → P l (z) ∈ C m l and P l,j : z ∈ C n → P l,j (z) ∈ C τ l,j (1 ≤ l ≤ p, 1 ≤ j ≤ n l ). Our next objective is to change the matrices J and B in order to get new matrices J and B such that the set {µ l } 1≤l≤p is also the set of distinct eigenvalues of J * (with the same geometric multiplicity n l ), with the property: "for every l : 1 ≤ l ≤ p, the Jordan blocks of J associated to µ l have the same dimension τ l " and for which the previous case could be applied.

To this end, let us take

τ l = max
V = (V i ) 1≤i≤k ∈ C nk , with V i ∈ C n , then Π k V = (ΠV i ) 1≤i≤k ∈ C b nk
). Indeed, the set of eigenvectors of the matrix L * k (resp. L * k ) can be easily constructed from the eigenvectors of J * (resp. J * ). On the other hand, it is also easy to check that v ∈ C n is an eigenvector of J * associated to µ l if and only if Πv ∈ C b n is an eigenvector of J * associated to µ l .

2.

Let us assume now that in system (1) A ∈ L(R n ) and B ∈ L(R m ; R n ). In this case, if y 0 ∈ H -1 (0, π; R n ) the null control for system (1) can be chosen in L 2 (0, T ; R m ). Indeed, if v ∈ L 2 (0, T ; C m ) is a control for which the solution y of system (1) satisfies y(•, T ) = 0 in (0, π), then ℜv also gives the null controllability result.

3.

For the sake of simplicity, we have presented our controllability result for the Laplacian operator -∂ xx with boundary Dirichlet conditions. It is possible to consider general second order self-adjoint differential operators R given by (Ry)(x) = (p(x)y ′ (x)) ′ + q(x)y(x), x ∈ (0, π),

where p ∈ C 2 (0, π) , q ∈ C 0 (0, π) and for a positive constant c 1 one has 0 < c 1 ≤ p(x), x ∈ (0, π).

In this case it is well known that the operator R with homogeneous boundary conditions has a sequence of eigenvalues {λ k } k≥1 and eigenfunctions {φ k } k≥1 such that

λ k = (k + α) 2 + O(1), |φ ′ k (0)| = c 2 λ k + O(1), for k → ∞,
with c 2 a positive constant (for instance, see [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF]). The same proof of Theorem 1.1 given in this work can be easily adapted to the operator R to give the same result.

  for n = 2 and m = 1. Let us now analyze condition (8) when m ≥ n (at least the same number of controls than equations) and the matrix B ∈ L(C m ; C n ) satisfies rank B = n. One has: Proposition 3.5. Let us fix A ∈ L(C n ) and B ∈ L(C m ; C n ) such that rank B = n. Then, the pair (A, B) satisfies condition (8) for any k ≥ 1.

Lemma 4 . 2 .

 42 Under the assumptions of Proposition 4.1, let us consider the Blaschke product associated to Λ, W : C + → C, defined by:

2 H 1 0 1 T 0 |

 2110 C τ l,j ).Let us set B = P * B. If we perform the change of variables ϕ = P ψ, with ϕ the solution to (13) associated to ϕ 0 ∈ H 1 0 (0, π; C n ), then the observability inequality (15) is equivalent to the existence of a positive constant C 1 such thatψ(•, 0) (0,π;C n ) ≤ C B * ψ x (0, t)| 2 dt,(51)for everyψ 0 ∈ H 1 0 (0, π; C n ), with ψ the solution to      -ψ t = ψ xx + J * ψ in Q, ψ(0, •) = 0, ψ(1, •) = 0 on (0, T ), ψ(•, T ) = ψ 0 in (0, π).(52)Let us fix a positive integer k. With the new matrices (J, B), we can introduce ( L k , B k ) as in[START_REF] Ammar-Khodja | Boundary controllability of coupled parabolic systems[END_REF], i.e., (C m ; C nk ),L k = diag ( L 1 , • • • , L k ) ∈ L(C nk ) and

  One has: rank K k = nk, ∀k ≥ 1.(53) Indeed, if we set P k = diag (P, P,k) • • •, P ) ∈ L(C nk ), then P k is a regular matrix, B k = P * k B k , L k = P * k L k ( P * k ) -1 and K k = P * k K k .From[START_REF] Bodart | Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient[END_REF] we infer (53). From the Jordan canonical form of A * we can obtain a decomposition of C n as follows: if z ∈ C n , then

  1≤j≤n l τ l,j , m l = n l τ l , n = p l=1 m l and J * = diag ( J 1 (µ 1 ), J 2 (µ 2 ), • • • , J p (µ p )) ∈ L(C b n ), where                J l (µ l ) = diag ( J l,1 (µ l ), J l,2 (µ l ), • • • , J l,n l (µ l )) ∈ L(C b m l ), 1 ≤ l ≤ p,andJ l,j (µ l ) = C τ l,j ), 1 ≤ j ≤ n l .In particular the Jordan blocks of J * associated to each eigenvalue have the same dimension. Let us also introduce the operator Π :z ∈ C n → Πz ∈ C b n b m l and Π l,j : z ∈ C n → Π l,j z = 0 P l,j z ∈ C b τ l . Finally, if B = ( b 1 | • • • | b m ), let us set B = (Π b 1 | • • • | Π b m ).With the previous notation and using the pair ( J, B), we can also construct the corresponding matricesB k ∈ L(C m ; C b nk ), L k ∈ L(C b nk ) and K k ∈ L(C mb nk , C b nk) as above. Thus, if k ≥ 1 is given, one has the following properties:1. σ( L * k ) = σ( L * k ) = {-λ i + µ l : 1 ≤ i ≤ k, 1 ≤ l ≤ p}. Moreover, the geometric multiplicity of µ ∈ σ( L * k ) ≡ σ( L * k ) coincides. 2. V ∈ C nk is an eigenvector of L * k associated to µ if and only if Π k V ∈ C b nk is an eigenvector of L * k associated to µ (Π k : C nk → C bnk is the operator defined as follows: if

4 .Find v ∈ L 2 6 . 2 .

 4262 As in[START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF], one can consider a control that depends only on time     y t = y xx + Ay + Bf v in Q = (0, π) × (0, T ), y(0, •) = 0, y(π, •) = 0 on (0, T ), y(•, 0) = y 0 in (0, π),(55)where A ∈ L(C n ) and B ∈ L(C m ; C n ) are two given matrices, y 0 ∈ L 2 (0, π; C n ) is the initial datum and f ∈ L 2 (0, π; C) is a given function such that for every ε > 0inf k≥1 |f k | e ελ k > 0 (56)wheref k = (f, φ k ) L 2 (0,T ) ∈ C. In system (55), v ∈ L 2 (0, T ; C m) is a control function that, of course, only depends on time.In order to deal with this controllability problem we can reason as before. In this case the control problem is the following (0, T ; C m ) such that -(y 0 , ϕ(•, 0)) L 2 (0,π;C n ) = T 0 (v(t) , B * (f , ϕ(•, t)) L 2 (0,π) ) C m dt, ∀ϕ 0 ∈ L 2 (0, π; C n ) .An inspection of the prove of Theorem 1.1, shows that by using the same arguments, one hasTheorem Let us fix A ∈ L(C n ), B ∈ L(C m ; C n ) and f ∈ L 2 (0, π; C) satisfying (56). Then, system(55) is exactly controllable to trajectories at any time T if and only if the pair (L k , B k ) is controllable for all k ≥ 1, with (L k , B k ) defined in (7). 5. A natural question is what happens if we consider the situation where some controls act on x = 0 and x = π. Let      y t = y xx + Ay in Q = (0, π) × (0, T ), y(0, •) = B 1 v 1 , y(π, •) = B 2 v 2 on (0, T ), y(•, 0) = y 0 in (0, π),

  Corollary 4.3. Under the assumptions of Proposition 4.1, one has that A (Λ, η, ∞) is a proper closed subspace of L 2 (0, ∞; C).

	Proof: Since conditions (19) and (20) are assumed, it follows that W ∈ H ∞ (C + ) given by Lemma 4.2, satisfies W ≡ 0. Let us set

and only if

[START_REF] Guerrero | Null controllability of some systems of two parabolic equations with one control force[END_REF]

holds. Combining the previous properties we conclude that W ∈ H ∞ (C + ) if and only if condition (20) is fulfilled.

Finally, it can be easily checked that |W (λ)| < 1 for ℜλ > 0, |W (iτ )| = 1 for almost all τ ∈ R and the two last properties.

As a direct consequence of the previous result we deduce:

Acknowledgements

The last draft of this paper was done during the trimester "Control of Partial Differential Equations and Applications" at Institut Henri Poincaré, Paris, November 2010. The authors want to thanks IHP and the organizers of the trimester for the support.

 [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF]Supported by l'Agence Nationale de la Recherche under grant ANR-07-JCJC-0139-01 2 Supported by grant MTM2006-07932 of the D.G.E.S. (Spain). 3 Supported by CONACyT (Mexico).

3. rank K k = nk. Indeed, first condition (53) holds. Using Proposition 3.1, this last condition is equivalent to: dim span { B * k V : V is an eigenvector of L * k associated to µ} = geometric multiplicity of µ, for every µ ∈ σ( L * k ). From the two previous properties we can clearly deduce:

Therefore, using again Proposition 3.1, we conclude that rank K k = nk. 4. We can apply the previous step to system [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF] (with coupling matrices ( J, B) instead of (A, B)) and conclude that this system is exactly controllable to the trajectories at time T . Equivalently, there exists a positive constant C 1 such that the observability inequality

holds for every solution ψ of

with

Let us now finalize the proof of Theorem 1.1. If we fix ψ 0 ∈ H 1 0 (0, π; C n ) and we take ψ 0 = Πψ 0 , the corresponding solution to (54) is given by ψ = Πψ with ψ the solution to problem (52) associated to ψ 0 . The observability inequality (51) is now an easy consequence of the corresponding observability inequality established for the solutions to problem (54).

This ends the proof of Theorem 1.1.

The arguments given in the proof of Theorem 1.1 and Proposition 3.2 allow us to prove the following consequence:

Let us assume that the algebraic Kalman condition (2) holds. Then for any y 0 ∈ X ⊥ 0 there exists v ∈ L 2 (0, T ; C m ) such that the solution to (1) satisfies y(•, T ) = 0 in (0, π). The space X 0 is given by

where k 0 is provided in Proposition 3.2.

Further results and open problems

1. In this work we have dealt with the null controllability result for system [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF]. Taking into account the results in the paper, it is not difficult to prove the following approximate controllability result: Theorem 6.1. Let us fix A ∈ L(C n ) and B ∈ L(C m ; C n ). Then, system (1) is approximately controllable at any time T > 0 if and only if

are given matrices and y 0 ∈ H -1 (0, π; C n ) is the initial datum. In system (57), v 1 ∈ L 2 (0, T ; C m1 ), v 2 ∈ L 2 (0, T ; C m2 ) are the controls functions which act on the system by means of the Dirichlet boundary condition at points x = 0 and x = π.

We set

With the same notations and following the same ideas as below, one has:

57) is exactly controllable to trajectories at any time T if and only if

6. The boundary controllability problem for this kind of parabolic systems in higher dimension of space is widely open except of course in the case where rank B = n.

7.

Let us consider the system     

where

with d i > 0 for 1 ≤ i ≤ n. The null controllability problem for this system is widely open. When n = 2, m = 1 and A and B are given by

in [START_REF] Fernández-Cara | Boundary controllability of parabolic coupled equations[END_REF] it is proved that the approximate controllability of system (58) at time T > 0 holds if and only if

The null controllability problem is much more intricate; in [START_REF] Luca | System controllability and Diophantine approximations[END_REF] it is also showed that there are matrices D such that d 1 /d 2 ∈ Q (and then, system (58) is approximately controllable at any time T ) for which system (58) is not null controllable for any time T . In a forthcoming work [START_REF] Ammar-Khodja | Boundary controllability of coupled parabolic systems[END_REF], we show that we can define a Kalman condition for system (58) by replacing L k (defined in ( 6)) by -λ k D + A. The approximate controllability of this system is equivalent to the same Kalman rank condition [START_REF] Bodart | Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient[END_REF]. Nevertheless we cannot generalize the proof of exact controllability to trajectories. The main difference between D = I d and the previous case is that the eigenvalues of L * = D∂ xx + A * may not satisfy the separability condition in assumption (10) of Theorem 1.2.