Boundary null-controllability of two coupled parabolic equations : simultaneous condensation of eigenvalues and eigenfunctions

Let the matrix operator L = D$\partial$xx + q(x)A0, with D = diag(1, $\nu$), $\nu$ = 1, q $\in$ L $\infty$ (0, $\pi$), and A0 is a Jordan block of order 1. We analyze the boundary null controllability for system yt -- Ly = 0. When v \notin Q * + and q(x) = 1, x $\in$ (0, $\pi$) for instance, there exists a family of root vectors of (L * , D(L *)) forming a Riesz basis, moreover, F. Ammar Khodja, A.Benabdallah, M.Gonz\`alez-Burgos, L.Teresa, the authors show the existence of a minimal time of control depending on condensation of eigenvalues of (L * , D(L *)). But there exists q $\in$ L $\infty$ (0, $\pi$) such that the family of eigenfunctions of (L * , D(L *)) is complete but it is not a Riesz basis. In this framework new phenomena arise : simultaneous condensation of eigenvalues and eigenfunctions. We prove the existence of a minimal time T0 $\in$ [0, $\infty$] depending on the condensation of eigenvalues and associated eigenfunctions of (L * , D(L *)), such that the corresponding system is null controllable at any time T>T0 and is not if T


1.4)
Let us introduce the notion of null and approximate controllability for this kind of systems.
The null controllability of parabolic partial differential equations has been widely studied since the pioneering work of Fattorini and Russel [18]. From the works of Fursikov and Imanuvilov [20] and Lebeau and Robbiano [24], it was commonly admitted that, in the context of parabolic partial differential equations, there is no restriction on the final time T and no geometric restriction on the control domain (in case of internal or boundary control).
But recently the study of particular examples highlighted the existence of a positive minimal time for null controllability or a geometric condition on the control domain. Actually, such an example was already provided in the 70s in [14]. The more recent results concerning such a minimal time have been proved, for instance, in contexts of control of coupled parabolic equations [2], [3], [15] and more generally see [4].
The main goal of this article is to provide a complete answer to the null and approximate controllability issues for system (1.1). Let us first recall some known results about the controllability properties of scalar parabolic systems. The null controllability problem for scalar parabolic systems has been first considered in the one-dimensional case. Let us consider the following null controllability problem: Given y 0 ∈ H −1 (0, π), can we find a control v ∈ L 2 (0, T ) such that the corresponding solution y ∈ C([0, T ]; H −1 (0, π)) to    y t − ∂ xx y = 0 in Q T := (0, π) × (0, T ), y(0, ·) = v, y(π, ·) = 0 on (0, T ), y(·, 0) = y 0 in (0, π), (1.5) satisfies y(·, T ) = 0, in (0, π)? (1.6) Using the moment method, H. O. Fattorini and D. L. Russell gave a positive answer to the previous controllability question (see [17] and [18]). Let us briefly recall the main ideas of this moment method. 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 and the sequence {ϕ k } k≥1 is a Hilbert basis of L 2 (0, π). Given y 0 ∈ H −1 (0, π), there exists a control v ∈ L 2 (0, T ) such that the solution y to (1.5) satisfies (1.6) if and only if there exists v ∈ L 2 (0, T ) satisfying Using the Fourier decomposition of y 0 , y 0 = k≥1 y 0k ϕ k , this is equivalent to the existence of v ∈ L 2 (0, T ) such that This problem is called a moment problem. In [17] and [18], The authors solved the moment problem by proving the existence of a biorthogonal family {q k } k≥1 to {e −λ k t } k≥1 in L 2 (0, T ) which, in particular, satisfies the additional property: for every ǫ > 0 there exists a constant C(ǫ, T ) > 0 such that The control is obtained as a linear combination of and the previous bounds (1.8) are used to prove that this series converges in L 2 (0, T ) for any positive time T. In fact, in [17] and [18] the authors proved a general result on existence of a biorthogonal family to for a constant ρ > 0. This gap property in (1.9) for the sequence is crucial for obtaining property (1.8) and the null controllability result for system (1.1) for arbitrary small times T. In 1973, S. Dolecki addressed the pointwise controllability at time T of the one-dimensional heat equation (see [14]). That is to say: Given T > 0 and y 0 ∈ L 2 (0, π), can we find a control v ∈ L 2 (0, T ) such that the solution y ∈ C([0, T ]; L 2 (0, π)) of    y t − ∂ xx y = δ x0 v(t) in Q T := (0, π) × (0, T ), y(0, ·) = 0, y(π, ·) = 0 on (0, T ), y(·, 0) = y 0 in (0, π), (1.10) satisfies (1.6)? Here x 0 ∈ (0, π) is a given point and δ x0 is the Dirac distribution at this point x 0 . S. Dolecki exhibited a minimal time such that system (1.10) is not null controllable at time T if T < T 0 (x 0 ) and is null controllable at time T when T > T 0 (x 0 ). This minimal time depends on the point x 0 . To ourknowledge, this was the first result on null-controllability of parabolic problems where a minimal time of control appears.
Other recent works showed new phenomena in control of parabolic equation, in particular the presence of a minimal time. Examples, [2], [3], [5], [6], [7], [8], [15]. These phenomena are particularly present in the case of null-controllability of coupled parabolic equations. Their whole understanding is an open problem, even if some progress was made in [4].
In this work, we are interested in the following boundary control problem, given by (1.12) is the control and we want to control the complete system (n equations) by means of m controls exerted on the boundary condition at point x = 0. Observe that the most interesting (and difficult) case is the case m < n.
The first results of null controllability for system (1.12) was obtained in [19] in the case n = 2, m = 1 and D = Id, A ∈ M n (R). This result was generalized by [1] to the case n ≥ 2, m ≥ 1. In these two papers, the authors used the method of moments of Fattorini-Russell to give a necessary and sufficient condition of null controllability at any time T > 0 for system (1.12). In both cases, the sequence of eigenvalues Λ = {Λ k } k≥1 ⊂ R + of the matrix operator A = Id∂ xx + A with Dirichlet boundary conditions continue to satisfy the gap condition in (1.9). As in the scalar case (see system (1.5)), this gap property (together with appropriate properties for the coupling and control matrices A and B) provides the null controllability result for system (1.12) at any positive time.
In [2], the authors are interested in the extension of the previous null controllability results for system (1.12) to the case where D = Id, A ∈ M n (R), n > 1 and m < n. The main difference with the case D = Id lies in the behavior of the sequence of eigenvalues of the matrix operator A := D∂ xx + A. The operator −A admits a sequence of eigenvalues Λ = {Λ k } k≥1 which does not satisfy the gap condition appearing in (1.9) but the operator −A is diagonalizable, i.e., its eigenfunctions form a Riesz basis. As a consequence, the authors show the existence of a minimal time of control T 0 ∈ [0, +∞], depending to the so-called condensation index, c(Λ), of the sequence Λ of eigenvalues of the operator −A. To our knowledge, this condensation index has been introduced for the first time by V. Bernstein (see [11]) for increasing real sequences and later extended by J. R. Shackell (see [31]) to complex sequences. Roughly speaking, if we consider the complex sequence Λ = {Λ k } k≥1 ⊂ C, the condensation index of Λ, is a measure of the way how Λ n approaches Λ m for n = m.
In [3], the authors consider the case where D = Id, A(x) = q(x)A 0 , with q ∈ L ∞ (0, π) , n = 2 and m = 1. In this case the operator L * := −Id∂ xx + q(x)A * 0 admits a sequence of eigenvalues Λ = {Λ k } k≥1 which satisfy the gap condition appearing in (1.9) moreover using the eigenfunctions and the generalized eigenfunctions of the operator L * , we can construct a Riesz basis for the space L 2 (0, π; R 2 ) . As a consequence, the authors show the existence of a minimal time of control depending on q ∈ L ∞ (0, π) only. They also proved that for any τ ∈ [0, +∞], there exists q ∈ L ∞ (0, π) such that T 0 (q) = τ .
In this paper, we study the null controllability properties of system (1.12) to the case where D = Id and A(x) = q(x)A 0 , q ∈ L ∞ (0, π), n = 2 and m = 1. This situation may seem like a simple perturbation of previous cases ( in [2], [3]). It is not true, it contains a new phenomenon : simultaneous condensation of eigenvalues and eigenfunctions. This phenomenon was excluded from all previous cases because of the assumption that the familly of eigenfunctions of the operator (L * , D(L * )) form a Riesz basis for the Hilbert space where the system is posed. This condensation of eigenfunctions can compensate the condensation of eigenvalues and the minimum control time is affected. In [2] as in [3] the authors proved the controllability by using the moments method but this method does not take account for phenomena eigenfunctions condensation because it is insensitive to the second part of moments problem. Under appropriate assumptions on D and q ∈ L ∞ (0, π), the operator L * = −D∂ xx + q(x)A * 0 admits a sequence of eigenvalues Λ = {Λ k } k≥1 which does not satisfy the gap condition appearing in (1.9) however the sequence of associated eigenfunctions is complete but is not a Riesz basis (see Proposition 2.3) for some ν / ∈ Q * + and q ∈ L ∞ (0, π) . As a consequence, we will see that a minimal time of control T 0 ∈ [0, +∞], depends simultaneously of condensation of eigenvalues and associated eigenfunctions of (L * , D(L * )). To this end, we will use a technique developed by A. Benabdallah, F. Boyer, M. Morancey in [10].
The plan of the paper is the following one: In Section 1, we address some know results about the controllability of parabolic system and we give the main result of this work. In Section 2 we study the null-controllability of system (1.1) when √ ν / ∈ Q * + , √ ν > 1 . Section 3 is devoted to null-controllability of system (1.1) when √ ν ∈ Q * + . Finally, in the Appendix we first recall the proof of the equivalence between observability and null-controllability and we give then the proof of our main result in other situations.
Let us present our first boundary control results, when √ ν / ∈ Q * + .

System
Adopt the following labelling of the elements of σ(L * ) where, for any k ∈ N * : (1.19) and On the other hand, On the other hand, Let us now present our second boundary control results when √ ν = i0 j0 ∈ Q * + , where i 0 and j 0 are co-prime ( i 0 ∧ j 0 = 1 ).
For a proof of the previous results see for instance [32] or [19].
The controllability of system (1.1) can be characterized in terms of appropriate properties of the solutions to the adjoint problem (1.24). More precisely, we have Proposition 1.5. The following properties are equivalent: 1. There exists a positive constant C > 0 such that, for any and the associated state satisfies 2. There exists a positive constant C such that the observability inequality holds for every θ 0 ∈ H 1 0 (0, π; R 2 ). In (1.28), θ is the adjoint state associated with θ 0 . This result is well known. For completeness, the proof is presented in Appendix A, at the end of the paper.

Remark 1.2.
It is also well known that the approximate controllability of (1.1) can be characterized in terms of a property of the solutions to (1.24). More precisely, system (1.1) is approximately controllable if and only if the following unique continuation property holds: "Let θ 0 ∈ H 1 0 (0, π; R 2 ) be given and let θ be the associated adjoint state.
Fattorini gave an interesting characterization of the approximate controllability under a general abstract framework. In his paper [16], he proved that, under some reasonable assumptions, the only observation of the eigenfunctions completely characterizes the approximate controllability. Actually, this theorem has been proved for bounded observation operators but G. Olive in [26], give a generalization to the case of relatively bounded observation operators as follows: Theorem 1.6 (G. Olive [26]). Let H and U be some complex Hilbert spaces. Assume that A : D(A) ⊂ H → H generates a strongly continuous semigroup S(t) on H, has a compact resolvent, and the system of root vectors of its adjoint A * is complete in H. Let C : D(C) ⊂ H → U be relatively bounded with respect to A. Then, we have the property ∀z 0 ∈ D(A), (CS(t)z 0 = 0 f or a.e t ∈ (0, +∞)) =⇒ z 0 = 0, (1.29) if and only if Applying Theorem 1.6 to operators A generates a strongly continuous semigroup on L 2 (0, π; R 2 ), has a compact resolvent and the system of root vectors of A * is complete in L 2 (0, π; R 2 ) (see Lemma 2.2) . On the other hand, the operator C is relatively bounded with respect to A. Thus This previous Theorem justifies the second point of the Theorem 1.1 and will be used to prove the approximate controllability of system (1.1).

Some preliminary results
Let us consider the vectorial operator with domain D(L) = H 2 (0, π; R 2 ) ∩ H 1 0 (0, π; R 2 ) and also its adjoint L * . Proposition 2.1. Let A 0 be given by (1.2) and consider the operator L given by (1.13) and its adjoint L * . Assume that √ ν / ∈ Q, then, where ψ k is the unique solution of problem: Moreover, an explicit expression of ψ k is given by : Remark 2.1. Consider the problem where λ ∈ R * + . The general solution to (2.3) is given by consequently, On the other hand, suppose that we can write are obtained as usual by Proof of Proposition 2.1. Let us assume √ ν / ∈ Q, given k ≥ 1 . First, L * can be written Let λ be an eigenvalue of L * and y = (y 1 , y 2 ) T an associated eigenfunction. Thus y is a solution of problem: If y 1 ≡ 0, then, λ = νk 2 is an eigenvalue of L * and taking y 2 = ϕ k , we obtain Φ * 2,k as associated eigenfunction of L * . Now assume that y 1 ≡ 0, then λ = k 2 and y 1 = ϕ k is a (normalized) solution to the first o.d.e and inserting this expression in the second equation, we get for y 2 : This proves that Φ * 1,k , is the second eigenfunction of L * , associated to k 2 . Moreover (2.1) (resp. (2.2)) can be deduce from (2.4) (resp. (2.6)).
is not a Riesz basis for L 2 (0, π; R 2 ) for some ν / ∈ Q * + and q ∈ L ∞ (0, π). Proof of Proposition 2.3 . Lemma 2.4. For a sequence {f k } k≥1 in Hilbert space H the following conditions are equivalent: the space of square summable scalar sequences.
Proof of Lemma 2.4 . See for instance [12], Theorem 3.6.6, page 66. The determinant of, the Gram matrix associated to the normalized vectors of B * is equal to Remark that, there exists a constant C(ν, q L ∞ ) > 0 such that, Lemma 2.5. For any σ ∈ (0, ∞), there exist an irrational number ν > 0 and a sequence of rational numbers {k p , j p } p≥0 such that k p and j p are co-prime positive integers, the sequences {k p } p≥0 and {j p } p≥0 are strictly increasing and In particular, we deduce the existence of a positive constant C > 0 such that Proof of Lemma 2.5 . See [2], Lemma 6.22, page 47.
Corollary 2.6. Thanks to Lemma 2.5, we can extract a subsequence (k p + j p ) p≥0 of even numbers only or odd numbers only.

By choosing the subsequence of even numbers with
.
Proof of Corollary 2.6 Let us show the first point of Lemma (2.5). Let us fix σ > 0 and √ ν > 0. We have Thanks to formula (2.8), 2.2 Proof of Theorem 1.1 : In this subsection, our objective is to prove that system (1.
In the sequel we shall study only the case √ ν > 1 and we will include the proof of case √ ν < 1 in Appendix B. To this end, we are going to use a technique developed by A. Benabdallah, F. Boyer, M. Morancey in [10].
If y is the solution of System (1.1) associated with y 0 ∈ H −1 (0, π; R 2 ) and u ∈ L 2 (0, T ), then it can be checked that , the corresponding solution to the adjoint problem (1.24) is given by , the null controllability problem for system (1.1) amounts to: (2.12) So, the null controllability property at time T for system (1.1) is equivalent to find u ∈ L 2 (0, T ) such that: We are going to give some results that will be crucial, to solve (2.13), in the proof of positive nullcontrollability. One has:. Proposition 2.7. Let us define 1. If √ ν > 1, then the function I is injective.

Moreover
Corollary 2.8. Let us consider the notations of Proposition 2.7.

If
is a infinite set.

If
There exists a sequence of integers (n k ) k∈N * strictly increasing, such that i n k +1 − i n k > 1. Actually, let us take for instance Finally, reasoning as before, we prove that J is a injective function also.

Positive null controllability result
Let us recall that is a injective fonction , where for any k ∈ N * i k is the nearest integer to √ νk, moreover I = N * \ I(N * ) = { i k : k ≥ 1} is a infinite set (see Corollary 2.8). Thanks to Proposition 2.7 we can reformulate (2.13). We say that the null controllability property at time T for system (1.1) is equivalent to find u ∈ L 2 (0, T ) such that: (2.16) Proposition 2.9. Let us introduce the (closed) space E T ⊂ L 2 (0, T ) given by . Then 1. There exists a family {q k } k≥1 ⊂ E T such that (2.17) 2. If T > T 0 = max T 1 , T 2 (see (1.19)) then we infer that an explicit solution u of moment problem (2.16) given by We deduce that u is an absolutely convergent series in L 2 (0, T ) and thus u ∈ L 2 (0, T ). This will prove the null controllability of system (1.1) at time T when T > T 0 = max T 1 , T 2 .
Proof of Proposition 2.9. Let us start by recalling classical properties of the Laplace transform (see for instance [30], pp. [19][20]. Let H 2 (C + ) the space of holomorphic functions Φ on C + such that Then the Laplace transform is an isomorphism. Let us fix k ≥ 1 and consider the function where α k , β k and γ k are constants to be determined and Notice that for any k ≥ 1, This implies J k ∈ H 2 (C + ), moreover So, using that the Laplace transform is a isomorphism from L 2 (0, ∞; C) into H 2 (C + ), we infer the existence of a nontrivial functionq k ∈ L 2 (0, ∞; C) such that Now, we can choose α k , β k and γ k such that we obtain The Parseval equality gives Using for instance [2] (Lemma 4.2 page 26), we deduce that, by denoting q k = ℜ q k|(0,T ) , there exists C > 0 such that thanks to equalities (2.22), we obtain , ∀k ≥ 1: thus the control u defines an element of and Let us give first some properties of L k (see (2.19)).

Let us work with
. (2.29) • Let us fix ε > 0, there exists N (ε) such that Using the inequality 1 + x ≤ e x , x ∈ R, we can estimate the denominator of (2.29) as follows: for a positive constant C 1 (ε).
• Let us now work on the numerator of (2.29). Let us recall that (see Proposition 2.7 and Corollary 2.8) Consequently, for every ε > 0 there exists a constant C 2 (ε) > 0 such that : ∀j ≥ 1 with j = k, One has Putting together both inequalities above, we have proved the existence of a positive constant Similarly, we show that |L k (i 2 k )| ≥ C 4 (ε)e εi 2 k , and |L k ( i 2 k )| ≥ C 5 (ε)e ε i 2 k , ∀k ≥ 1.

The negative null controllability result
Let us prove that if 0 < T < T 0 , then system (1.1) is not null controllable at time T . We argue by contradiction. In particular, we assume that T 0 > 0, otherwise there is nothing to prove. By Proposition 1.5, system (1.1) is null-controllable at time T if and only if there exists C > 0 such that any solution θ of the adjoint problem (1.24) satisfies the observability inequality: Let us recall that • Let us suppose T 0 = T 1 , and work with the particular solutions of the adjoint problem (1.24) associated with initial data θ 0 k = ψ 1,k , where ψ 1,k is given by (1.17). With this choice, the solution θ k of the adjoint problem is given by The observability inequality reads as From the definition of T 1 , there exists a subsequence {k n } n≥1 such that : in this case, for every ε > 0, there exits a positive integer n ε ≥ 1 such that This proves that the observability inequality does not hold • Let us suppose now, T 0 = T 2 . Let us fix k ≥ 1 and work with the particular solutions associated with initial data θ 0 k = ψ 1,i k − ψ 2,k , with this choice, the solution θ k of adjoint problem is given by We have and, From the definition of T 2 , there exists an increasing unbounded subsequence (k n ) n≥1 such that in this case, for every ε > 0, there exits a positive integer n ε ≥ 1 such that thus, This proves that the observability inequality does not hold and finishes the proof of the negative null controllability result of (1.1).

Some preliminary results
In this subsection we will give some properties which will be used below. We are interested in studying the spectrum of the operators L * . Let us define the sets  Observe that Λ 1 , Λ 2 and Λ 3 are disjoint sets. : j ∈ N * \ j 0 N * } and Λ 3 := {l 2 : l ≥ 1}. Thus in the sequel, the controllability of the system (1.1) will be studied in only when √ ν = i0 j0 ∈ Q * + and i 0 > 1. The case i 0 = 1 is much easier and it is left to the reader.
In the sequel, for the proof of Theorem 1.2, we shall use the following notations and T 0 (ν, q) becomes Proposition 3.1. Let A 0 be given by (1.2) and consider the operator (L, D(L)) given by (1.13) and its adjoint L * .
1. The spectrum of L * is given by where, ψ k is the unique solution of problem: moreover, an explicit expression of ψ k is given by : with (see remark 2.1 for instance)

10)
and ψ l is the unique solution of problem: an explicit expression of ψ l is given by : (3.12) Moreover (see remark 2.1 for instance) In the next result we are going to give some properties of ψ k (3.8) and ψ l (3.11). This properties will be used later and will be crucial in the proof of Theorem 1.2. 1. Let us fix q ∈ L ∞ (0, π) and take k ∈ N * \ i 0 N * . Then, one has : (3.14) In addition, there exists a constant C > 0 such that 1. Let us fix k ∈ N * \ i 0 N * , the expression (3.14) can be deduced from (2.5) and (2.6). Moreover, there exists a constant C > 0 such that where, for each k, n k is the nearest integer of kj0 2. The properties (3.16) can be deduced from (3.11) . This finalizes the proof.
Thus, choosing z 1 = ϕ i0l and inserting this expression in the second equation, we get : (3.18) A necessary and sufficient condition for the previous nonhomogeneous Sturm-Liouville problem to have a solution is that With this value of c, (3.18) has a continuum of solutions. This proves that, for k = i 0 l, Φ 1,l is a generalized eigenfunction of L * associated to λ = i 2 0 l 2 . -Otherwise, if k = i 0 l, then (3. 19) and Φ 1,k is a eigenfunction of L * associated to λ = k 2 with k = i 0 l.
Thus, using Theorem 1.7, system (1.1) is not approximately controllable at time T . Sufficient condition: Let us suppose that condition (1.22) hold. The set of the eigenvectors associated with the eigenvalue νk 2 of L * is generated by Φ * 2,k (see Proposition 2.1). In this case, we remark that for all k ∈ N * Moreover, the set of the eigenvectors associated with the eigenvalue k 2 , k ∈ N * \ i 0 N * , of L * is generated by Φ * 1,k (see Proposition 2.1). In this case, we remark that for all k ∈ N * \ i 0 N * We conclude with the help of Theorem 1.7.

Positive null controllability result
In this subsection, our objective is to prove that system (1.1) is exactly controllable to zero at time T if T > T 0 ∈ [0, ∞) (see (1.23)). To this end, for y 0 ∈ H −1 (0, π; R 2 ) we will reformulate the null controllability problem as a moment problem. Let us first observe that condition (1.22) is a necessary condition for having the null controllability property of system (1.1) at time T > 0. Using Proposition 1.3 and Proposition 1.4 we deduce that the control u ∈ L 2 (0, T ) drives the solution of (1.1) to zero at time T if and only if u ∈ L 2 (0, T ) satisfies where θ is the solution to the adjoint problem (1.24) associated with θ 0 . Since B * is complete in H 1 0 (0, π; R 2 ), the null controllability property at time T for system (1.1) is equivalent to find u ∈ L 2 (0, T ) such that where θ 2,j is the solution of adjoint problem (1.24) associated with θ 0 = Φ * 2,j , and θ 1,k (resp. θ 1,l ) is the solution of adjoint problem(1.24) associated with θ 0 = Φ * 1,k (resp. θ 0 = Φ * 1,k ) . Developing the equality (3.20), one has: 1. If we take θ 0 ≡ Φ * 2,j , the solution of the adjoint problem is θ 2,j (·, t) = e −νj 2 (T −t) Φ * 2,j and we obtain, for a positive constant C independent of α ∈ Λ 1 and y 0 .
2. If we take θ 0 ≡ Φ * 1,k , the solution of the adjoint problem is θ 1,k (·, t) = e −k 2 (T −t) Φ * 1,k and we obtain , Using the properties of the function ψ ′ k stated in Proposition 3.2, one has | M (y 0 , β)| ≤ C y 0 for a new positive constant C independent of β ∈ Λ 2 and y 0 .

The negative null controllability result
Let us prove that if 0 < T < T 0 , then system (1.1) is not null controllable at time T . We argue by contradiction. Let us first remark that (3.34) • Let us suppose T 0 = T 0,1 . Let us work with the particular solutions θ k associated with initial data .
With this choice, the solution θ k of (1.24) is given by From the definition of T 0,1 (see (3.33)), then there exists an subsequence (k n ) n≥1 ⊂ N * \ i 0 N * such that: thus θ kn (·, 0) 2 This proves that the observability inequality (3.34) does not hold.
With this choice, the solution θ l of (1.24) is given by thus, the observability inequality (3.34) becomes We can choose a l = I(i 2 0 l 2 ) and b l = −I(i 2 0 l 2 )ψ ′ l (0)/ϕ ′ j0l (0), we obtain for a new constant C > 0 not depending on l : From the definition of T 0,2 (see (3.33)), there exists a subsequence {l n } n≥1 such that : Assume 0 < T 0,2 < +∞. In this case, for every ε > 0, there exits a positive integer n ε ≥ 1 such that Taking for instance ε = This proves that the observability inequality (3.34) does not hold and finishes the proof of the negative null controllability result of (1.1).
Let y be the state associated to y 0 and u. Thanks to (1.4), and y ε (·, T ) ⇀ y(·, T ) weakly in H −1 (0, π; R 2 ) Consequently, using (A.7), we see that we have found a control u satisfying (1.26) such that the associated state satisfies (1.27). This ends the proof.
is a injective fonction, where for any k ∈ N * j k is the nearest integer to k √ ν moreover J = N * \ J(N * ) = { j k : k ≥ 1} is a infinite set. Thanks to Proposition 2.7 we can reformulate (2.13). We say that the null controllability property at time T for system (1.1) is equivalent to find u ∈ L 2 (0, T ) such that:              We deduce that u is an absolutely convergent series in L 2 (0, T ) and thus u ∈ L 2 (0, T ). This will prove the null controllability of system (1.1) at time T when T > T 0 .
Proof of Proposition B.1.