Ammar Farid
email: farid.ammar-khodja@univ-fcomte.fr

Franz Ammar Khodja
email: franz.chouly@univ-fcomte.fr

Michel Chouly

Duprez
email: mduprez@math.cnrs.fr.

F Chouly

M Duprez
email: mduprez@math.cnrs.fr

PARTIAL NULL CONTROLLABILITY OF PARABOLIC LINEAR SYSTEMS

Keywords: 2010 Mathematics Subject Classification. Primary: 93B05, 93B07; Secondary: 93C20, 93C05, 35K40 Controllability, Observability, Kalman condition, Moment Method, Parabolic Systems

This paper is devoted to the partial null controllability issue of parabolic linear systems with n equations. Given a bounded domain Ω in R N (N ∈ N * ), we study the effect of m localized controls in a nonempty open subset ω only controlling p components of the solution (p, m n). The first main result of this paper is a necessary and sufficient condition when the coupling and control matrices are constant. The second result provides, in a first step, a sufficient condition of partial null controllability when the matrices only depend on time. In a second step, through an example of partially controlled 2 × 2 parabolic system, we will provide positive and negative results on partial null controllability when the coefficients are space dependent.

Introduction and main results.

Let Ω be a bounded domain in R N (N ∈ N * ) with a C 2 -class boundary ∂Ω, ω be a nonempty open subset of Ω and T > 0. Let p, m, n ∈ N * such that p, m n. We consider in this paper the following system of n parabolic linear equations      ∂ t y = ∆y + Ay + B1 ω u in Q T := Ω × (0, T ), y = 0 on Σ T := ∂Ω × (0, T ),

y(0) = y 0 in Ω, (1)
where y 0 ∈ L 2 (Ω) n is the initial data, u ∈ L 2 (Q T ) m is the control and

A ∈ L ∞ (Q T ; L(R n )) and B ∈ L ∞ (Q T ; L(R m , R n )).
In many fields such as chemistry, physics or biology it appeared relevant to study the controllability of such a system (see [START_REF] Khodja | Recent results on the controllability of linear coupled parabolic problems: a survey[END_REF]). For example, in [START_REF] Chakrabarty | Distributed parameters deterministic model for treatment of brain tumors using Galerkin finite element method[END_REF], the authors study a system of three semilinear heat equations which is a model coming from a mathematical description of the growth of brain tumors. The unknowns are the drug concentration, the density of tumors cells and the density of wealthy cells and the aim is to control only two of them with one control. This practical issue motivates the introduction of the partial null controllability.

For an initial condition y(0) = y 0 ∈ L 2 (Ω) n and a control u ∈ L 2 (Q T ) m , it is well-known that System (1) admits a unique solution in W (0, T ) n , where W (0, T ) := {y ∈ L 2 (0, T ; H 1 0 (Ω)), ∂ t y ∈ L 2 (0, T ; H -1 (Ω))}, with H -1 (Ω) := H 1 0 (Ω) and the following estimate holds (see [START_REF] Lions | Problèmes aux Limites Non Homogènes et Applications[END_REF])

y L 2 (0,T ;H 1 0 (Ω) n ) + y C 0 ([0,T ];L 2 (Ω) n ) C( y 0 L 2 (Ω) n + u L 2 (Q T ) m ), (2)
where C does not depend on time. We denote by y(•; y 0 , u) the solution to System (1) determined by the couple (y 0 , u).

Let us consider Π p the projection matrix of L(R n ) given by Π p := (I p 0 p,n-p ) (I p is the identity matrix of L(R p ) and 0 p,n-p the null matrix of L(R n-p , R p )), that is, Π p : R n → R p , (y 1 , ..., y n ) → (y 1 , ..., y p ). System (1) is said to be • Π p -approximately controllable on the time interval (0, T ), if for all real number ε > 0 and y 0 , y T ∈ L 2 (Ω) n there exists a control u ∈ L 2 (Q T ) m such that Π p y(T ; y 0 , u) -Π p y T L 2 (Ω) p ε. • Π p -null controllable on the time interval (0, T ), if for all initial condition y 0 ∈ L 2 (Ω) n , there exists a control u ∈ L 2 (Q T ) m such that Π p y(T ; y 0 , u) ≡ 0 in Ω.

Before stating our main results, let us recall the few known results about the (full) null controllability of System [START_REF] Alabau-Boussouira | A hierarchic multi-level energy method for the control of bidiagonal and mixed n-coupled cascade systems of PDE's by a reduced number of controls[END_REF]. The first of them is about cascade systems (see [START_REF] González-Burgos | Controllability results for cascade systems of m coupled parabolic PDEs by one control force[END_REF]). The authors prove the null controllability of System [START_REF] Alabau-Boussouira | A hierarchic multi-level energy method for the control of bidiagonal and mixed n-coupled cascade systems of PDE's by a reduced number of controls[END_REF] with the control matrix B := e 1 (the first vector of the canonical basis of R n ) and a coupling matrix A of the form

A :=        α 1,1 α 1,2 α 1,3 • • • α 1,n α 2,1 α 2,2 α 2,3 • • • α 2,n 0 α 3,2 α 3,3 • • • α 3,n . . . . . . . . . . . . . . . 0 0 • • • α n,n-1 α n,n        , (3)
where the coefficients α i,j are elements of L ∞ (Q T ) for all i, j ∈ {1, ..., n} and satisfy for a positive constant C and a nonempty open set ω 0 of ω α i+1,i C in ω 0 or -α i+1,i C in ω 0 for all i ∈ {1, ..., n -1}.

A similar result on parabolic systems with cascade coupling matrices can be found in [START_REF] Alabau-Boussouira | A hierarchic multi-level energy method for the control of bidiagonal and mixed n-coupled cascade systems of PDE's by a reduced number of controls[END_REF].

The null controllability of parabolic 3 × 3 linear systems with space/time dependent coefficients and non cascade structure is studied in [START_REF] Benabdallah | Controllability to trajectories for some parabolic systems of three and two equations by one control force[END_REF] and [START_REF] Mauffrey | On the null controllability of a 3 x 3 parabolic system with non-constant coefficients by one or two control forces[END_REF] (see also [START_REF] González-Burgos | Controllability results for cascade systems of m coupled parabolic PDEs by one control force[END_REF]).

If A ∈ L(R n ) and B ∈ L(R m , R n ) (the constant case), it has been proved in [START_REF] Khodja | A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems[END_REF] that System (1) is null controllable on the time interval (0, T ) if and only if the following condition holds rank [A|B] = n, (4) where [A|B], the so-called Kalman matrix, is defined as [A|B] := (B|AB|...|A n-1 B).

(

) 5
For time dependent coupling and control matrices, we need some additional regularity. More precisely, we need to suppose that A ∈ C n-1 ([0, T ]; L(R n )) and B ∈ C n ([0, T ]; L(R m ; R n )). In this case, the associated Kalman matrix is defined as follows. Let us define B 0 (t) := B(t), B i (t) := A(t)B i-1 (t) -∂ t B i-1 (t) for all i ∈ {1, ...

In [START_REF] Khodja | A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems[END_REF] the authors prove first that, if there exists t 0 ∈ [0, T ] such that rank [A|B](t 0 ) = n,

then System (1) is null controllable on the time interval (0, T ). Secondly that System (1) is null controllable on every interval (T 0 , T 1 ) with 0 T 0 < T 1 T if and only if there exists a dense subset E of (0, T ) such that rank [A|B](t) = n for every t ∈ E.

In the present paper, the controls are acting on several equations but on one subset ω of Ω. Concerning the case where the control domains are not identical, we refer to [START_REF] Olive | Null-controllability for some linear parabolic systems with controls acting on different parts of the domain and its boundary[END_REF].

Our first result is the following:

Theorem 1.1. Assume that the coupling and control matrices are constant in space and time, i. e., A ∈ L(R n ) and B ∈ L(R m , R n ). The condition

rank Π p [A|B] = p (9)
is equivalent to the Π p -null/approximate controllability on the time interval (0, T ) of System [START_REF] Alabau-Boussouira | A hierarchic multi-level energy method for the control of bidiagonal and mixed n-coupled cascade systems of PDE's by a reduced number of controls[END_REF].

The Condition [START_REF] Boyer | Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients[END_REF] for Π p -null controllability reduces to Condition (4) whenever p = n. A second result concerns the non-autonomous case: Theorem 1.2. Assume that

A ∈ C n-1 ([0, T ]; L(R n )) and B ∈ C n ([0, T ]; L(R m ; R n )). If rank Π p [A|B](T ) = p, (10)
then System (1) is Π p -null/approximately controllable on the time interval (0, T ).

In Theorems 1.1 and 1.2, we control the p first components of the solution y. If we want to control some other components a permutation of lines leads to the same situation.

Remark 1.

1. When the components of the matrices A and B are analytic functions on the time interval [0, T ], Condition [START_REF] Benabdallah | Controllability to trajectories for some parabolic systems of three and two equations by one control force[END_REF] is necessary for the null controllability of System (1) (see Th. 1.3 in [START_REF] Khodja | A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems[END_REF]). Under the same assumption, the proof of this result can be adapted to show that the following condition there exists t 0 ∈ [0, T ] such that : rank Π p [A|B](t 0 ) = p, is necessary to the Π p -null controllability of System (1).

2. As told before, under Condition [START_REF] Benabdallah | Controllability to trajectories for some parabolic systems of three and two equations by one control force[END_REF], System (1) is null controllable. But unlike the case where all the components are controlled, the Π p -null controllability at a time t 0 smaller than T does not imply this property on the time interval (0, T ). This roughly explains Condition [START_REF] Chakrabarty | Distributed parameters deterministic model for treatment of brain tumors using Galerkin finite element method[END_REF]. Furthermore this condition can not be necessary under the assumptions of Theorem 1.2 (for a counterexample we refer to [START_REF] Khodja | A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems[END_REF]).

Remark 2. In the proofs of Theorems 1.1 and 1.2, we will use a result of null controllability for cascade systems (see Section 2) proved in [START_REF] Khodja | A generalization of the Kalman rank condition for time-dependent coupled 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] where the authors consider a time-dependent second order elliptic operator L(t) given by

L(t)y(x, t) = - N i,j=1 ∂ ∂x i α i,j (x, t) ∂y ∂x j (x, t) + N i=1 b i (x, t) ∂y ∂x i (x, t) + c(x, t)y(x, t), (11)
with coefficients α i,j , b i , c satisfying α i,j ∈ W 1 ∞ (Q T ), b i , c ∈ L ∞ (Q T ) 1 i, j N, α i,j (x, t) = α j,i (x, t) ∀(x, t) ∈ Q T , 1 i, j N
and the uniform elliptic condition: there exists a 0 > 0 such that

N i,j=1 α i,j (x, t)ξ i ξ j a 0 |ξ| 2 , ∀(x, t) ∈ Q T .
Theorems 1.1 and 1.2 remain true if we replace -∆ by an operator L(t) in System [START_REF] Alabau-Boussouira | A hierarchic multi-level energy method for the control of bidiagonal and mixed n-coupled cascade systems of PDE's by a reduced number of controls[END_REF]. Now the following question arises: what happens in the case of space and time dependent coefficients ? As it will be shown in the following example, the answer seems to be much more tricky. Let us now consider the following parabolic system of two equations

         ∂ t y = ∆y + αz + 1 ω u in Q T , ∂ t z = ∆z in Q T , y = z = 0 on Σ T , y(0) = y 0 , z(0) = z 0 in Ω, (12)
for given initial data y 0 , z 0 ∈ L 2 (Ω), a control u ∈ L 2 (Q T ) and where the coefficient

α ∈ L ∞ (Ω). Theorem 1.3. (1) Assume that α ∈ C 1 ([0, T ]). Then System (12) is Π 1 -null con- trollable for any open set ω ⊂ Ω ⊂ R N (N ∈ N * ), that is for all initial conditions y 0 , z 0 ∈ L 2 (Ω), there exists a control u ∈ L 2 (Q T ) such that the solution (y, z) to System (12) satisfies y(T ) ≡ 0 in Ω. (2) Let Ω := (a, b) ⊂ R (a, b ∈ R), α ∈ L ∞ (Ω), (w k ) k 1 be the L 2 -normalized eigenfunctions of -∆ in Ω with Dirichlet boundary conditions and for all k, l ∈ N * , α kl := Ω α(x)w k (x)w l (x) dx.
If the function α satisfies

|α kl | C 1 e -C2|k-l| for all k, l ∈ N * , (13)
for two positive constants C 1 > 0 and C 2 > b -a, then System ( 12) is Π 1 -null controllable for any open set ω ⊂ Ω.

(3) Assume that Ω := (0, 2π) and ω ⊂ (π, 2π). Let us consider α ∈ L ∞ (0, 2π) defined by

α(x) := ∞ j=1 1 j 2 cos(15jx) for all x ∈ (0, 2π).
Then System [START_REF] Ekeland | Analyse Convexe et Problèmes Variationnels[END_REF] is not Π 1 -null controllable. More precisely, there exists k 1 ∈ {1, ..., 7} such that for the initial condition (y 0 , z 0 ) = (0, sin(k 1 x)) and any control u ∈ L 2 (Q T ) the solution y to System [START_REF] Ekeland | Analyse Convexe et Problèmes Variationnels[END_REF] is not identically equal to zero at time T .

We will not prove item (1) in Theorem 1.3, because it is a direct consequence of Theorem 1.2.

Remark 3. Suppose that Ω := (0, π). Consider α ∈ L ∞ (0, π) and the real sequence (α p ) p∈N such that for all x ∈ (0, π)

α(x) := ∞ p=0 α p cos(px).
Concerning item (2), we remark that Condition ( 13) is equivalent to the existence of two constants C 1 > 0, C 2 > π such that, for all p ∈ N,

|α p | C 1 e -C2p .
As it will be shown, the proof of item (3) in Theorem 1.3 can be adapted in order to get the same conclusion for any α ∈ H k (0, 2π) (k ∈ N * ) defined by α(x) := ∞ j=1 1 j k+1 cos((2k + 13)jx) for all x ∈ (0, 2π).

(

) 14
These given functions α belong to H k (0, π) but not to D((-∆) k/2 ). Indeed, in the proof of the third item in Theorem 1.3, we use the fact that the matrix (α kl ) k,l∈N * is sparse (see (103)), what seems true only for coupling terms α of the form [START_REF] Fernández-Cara | Boundary controllability of parabolic coupled equations[END_REF]. Thus α is not zero on the boundary.

Remark 4. From Theorem 1.3, one can deduce some new results concerning the null controllability of the heat equation with a right-hand side. Consider the system

     ∂ t y = ∆y + f + 1 ω u in (0, π) × (0, T ), y(0) = y(π) = 0 on (0, T ), y(0) = y 0 in (0, π), (15)
where y 0 ∈ L 2 (0, π) is the initial data and f, u ∈ L 2 (Q T ) are the right-hand side and the control, respectively. Using the Carleman inequality (see [START_REF] Fursikov | Controllability of Evolution Equations[END_REF]), one can prove that System (15) is null controllable when f satisfies

e C T -t f ∈ L 2 (Q T ), (16)
for a positive constant C. For more general right-hand sides it was rather open.

The second and third points of Theorem 1.3 provide some positive and negative null controllability results for System [START_REF] Fernández-Cara | The cost of approximate controllability for heat equations: the linear case[END_REF] with right-hand side f which does not fulfil Condition [START_REF] Fursikov | Controllability of Evolution Equations[END_REF].

Remark 5. Consider the same system as System [START_REF] Ekeland | Analyse Convexe et Problèmes Variationnels[END_REF] except that the control is now on the boundary, that is

         ∂ t y = ∆y + αz in (0, π) × (0, T ), ∂ t z = ∆z in (0, π) × (0, T ), y(0, t) = v(t), y(π, t) = z(0, t) = z(π, t) = 0 on (0, T ), y(x, 0) = y 0 (x), z(x, 0) = z 0 (x) in (0, π), (17)
where y 0 , z 0 ∈ H -1 (0, π). In Theorem 5.4, we provide an explicit coupling function α for which the Π 1 -null controllability of System (17) does not hold. Moreover one can adapt the proof of the second point in Theorem 1.3 to prove the Π 1 -null controllability of System [START_REF] Ghidaglia | Some backward uniqueness results[END_REF] under Condition [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF].

If the coupling matrix depends on space, the notions of Π 1 -null and approximate controllability are not necessarily equivalent. Indeed, according to the choice of the coupling function α ∈ L ∞ (Ω), System (12) can be Π 1 -null controllable or not. But this system is Π 1 -approximately controllable for all α ∈ L ∞ (Ω):

Theorem 1.4. Let α ∈ L ∞ (Q T ). Then System (12) is Π 1 -approximately control- lable for any open set ω ⊂ Ω ⊂ R N (N ∈ N * )
, that is for all y 0 , y T , z 0 ∈ L 2 (Ω) and all ε > 0, there exists a control u ∈ L 2 (Q T ) such that the solution (y, z) to System [START_REF] Ekeland | Analyse Convexe et Problèmes Variationnels[END_REF] satisfies

y(T ) -y T L 2 (Ω) ε.
This result is a direct consequence of the unique continuation property and existence/unicity of solutions for a single heat equation. Indeed System (12) is Π 1approximately controllable (see Proposition 1) if and only if for all φ 0 ∈ L 2 (Ω) the solution to the adjoint system

         -∂ t φ = ∆φ in Q T , -∂ t ψ = ∆ψ + αφ in Q T , φ = ψ = 0 on Σ T , φ(T ) = φ 0 , ψ(T ) = 0 in Ω (18) satisfies φ ≡ 0 in ω × (0, T ) ⇒ (φ, ψ) ≡ 0 in Q T .
If we assume that, for an initial data φ 0 ∈ L 2 (Ω), the solution to System (18) satisfies φ ≡ 0 in ω × (0, T ), then using Mizohata uniqueness Theorem in [START_REF] Mizohata | Unicité du prolongement des solutions pour quelques opérateurs différentiels paraboliques[END_REF], φ ≡ 0 in Q T and consequently ψ ≡ 0 in Q T . For another example of parabolic systems for which these notions are not equivalent we refer for instance to [START_REF] Khodja | Minimal time of controllability of two parabolic equations with disjoint control and coupling domains[END_REF]. Remark 6. The quantity α kl , which appears in the second item of Theorem 1.3, has already been considered in some controllability studies for parabolic systems.

Let us define for all

k ∈ N *      I 1,k (α) := a 0 α(x)w k (x) 2 dx, I k (α) := α kk .
In [START_REF] Khodja | New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence[END_REF], the authors have proved that the system

         ∂ t y = ∆y + αz in (0, π) × (0, T ), ∂ t z = ∆z + 1 ω u in (0, π) × (0, T ), y(0, t) = y(π, t) = z(0, t) = z(π, t) = 0 on (0, T ), y(x, 0) = y 0 (x), z(x, 0) = z 0 (x) in (0, π), (19)
is approximately controllable if and only if

|I k (α)| + |I 1,k (α)| = 0 for all k ∈ N * .
A similar result has been obtained for the boundary approximate controllability in [START_REF] Boyer | Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients[END_REF]. Consider now

T 0 (α) := lim sup k→∞ -log(min{|I k | , |I 1,k |}) k 2 .
It is also proved in [START_REF] Khodja | New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence[END_REF] that: If T > T 0 (α), then System [START_REF] González-Burgos | Controllability results for cascade systems of m coupled parabolic PDEs by one control force[END_REF] is null controllable at time T and if T < T 0 (α), then System ( 19) is not null controllable at time T . As in the present paper, we observe a difference between the approximate and null controllability, in contrast with the scalar case (see [START_REF] Khodja | Recent results on the controllability of linear coupled parabolic problems: a survey[END_REF]).

In this paper, the sections are organized as follows. We start with some preliminary results on the null controllability for the cascade systems and on the dual concept associated to the Π p -null controllability. Theorem 1.1 is proved in a first step with one force i.e. B ∈ R n in Section 3.1 and in a second step with m forces in Section 3.2. Section 4 is devoted to proving Theorem 1.2. We consider the situations of the second and third items of Theorem 1.3 in Section 5.1 and 5.2 respectively. This paper ends with some numerical illustrations of Π 1 -null controllability and non Π 1 -null controllability of System (12) in Section 5.3.

2.

Preliminaries. In this section, we recall a known result about cascade systems and provide a characterization of the Π p -controllability through the corresponding adjoint system.

2.1. Cascade systems. Some theorems of this paper use the following result of null controllability for the following cascade system of n equations controlled by r distributed functions

     ∂ t w = ∆w + Cw + D1 ω u in Q T , w = 0 on Σ T , w(0) = w 0 in Ω, (20)
where

w 0 ∈ L 2 (Ω) n , u = (u 1 , ..., u r ) ∈ L 2 (Q T ) r
, with r ∈ {1, ..., n}, and the coupling and control matrices C ∈ C 0 ([0, T ]; L(R n )) and D ∈ L(R r , R n ) are given by

C(t) :=      C 11 (t) C 12 (t) • • • C 1r (t) 0 C 22 (t) • • • C 2r (t) . . . . . . . . . . . . 0 0 • • • C rr (t)      (21)
with

C ii (t) :=        α i 11 (t) α i 12 (t) α i 13 (t) • • • α i 1,si (t) 1 α i 22 (t) α i 23 (t) • • • α i 2,si (t) 0 1 α i 33 (t) • • • α i 3,si (t) . . . . . . . . . . . . . . . 0 0 • • • 1 α i si,si (t)        , s i ∈ N,
S i = 1 + i-1
j=1 s j , i ∈ {2, ..., r} (e j is the j-th element of the canonical basis of R n ).

Theorem 2.1. System (20) is null controllable on the time interval (0, T ), i.e. for all w 0 ∈ L 2 (Ω) n there exists u ∈ L 2 (Ω) r such that the solution w in W (0, T ) n to System (20) satisfies w(T ) ≡ 0 in Ω.

The proof of this result uses a Carleman estimate (see [START_REF] Fursikov | Controllability of Evolution Equations[END_REF]) and can be found in [START_REF] Khodja | A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems[END_REF] or [START_REF] González-Burgos | Controllability results for cascade systems of m coupled parabolic PDEs by one control force[END_REF].

2.2. Partial null controllability of a parabolic linear system by m forces and adjoint system. It is nowadays well-known that the controllability has a dual concept called observability (see for instance [START_REF] Khodja | Recent results on the controllability of linear coupled parabolic problems: a survey[END_REF]). We detail below the observability for the Π p -controllability.

Proposition 1.

1. System ( 1) is Π p -null controllable on the time interval (0, T ) if and only if there exists a constant C obs > 0 such that for all initial data

ϕ 0 = (ϕ 0 1 , ..., ϕ 0 p ) ∈ L 2 (Ω) p the solution ϕ ∈ W (0, T ) n to the adjoint system      -∂ t ϕ = ∆ϕ + A * ϕ in Q T , ϕ = 0 on Σ T , ϕ(•, T ) = Π * p ϕ 0 = (ϕ 0 1 , ..., ϕ 0 p , 0, ..., 0) in Ω ( 22
)
satisfies the observability inequality

ϕ(0) 2 L 2 (Ω) n C obs T 0 B * ϕ 2 L 2 (ω) m dt. (23)
2. System (1) is Π p -approximately controllable on the time interval (0, T ) if and only if for all ϕ 0 ∈ L 2 (Ω) p the solution ϕ to System [START_REF] Mauffrey | On the null controllability of a 3 x 3 parabolic system with non-constant coefficients by one or two control forces[END_REF] satisfies

B * ϕ ≡ 0 in (0, T ) × ω ⇒ ϕ ≡ 0 in Q T .
Proof. For all y 0 ∈ L 2 (Ω) n , and u ∈ L 2 (Q T ) m , we denote by y(t; y 0 , u) the solution to System (1) at time t ∈ [0, T ]. For all t ∈ [0, T ], let us consider the operators S t and L t defined as follows

S t : L 2 (Ω) n → L 2 (Ω) n y 0 → y(t; y 0 , 0) and L t : L 2 (Q T ) m → L 2 (Ω) n u → y(t; 0, u). (24)
1. System (1) is Π p -null controllable on the time interval (0, T ) if and only if

∀y 0 ∈ L 2 (Ω) n , ∃u ∈ L 2 (Q T ) m such that Π p L T u = -Π p S T y 0 . (25)
Problem (25) admits a solution if and only if

Im Π p S T ⊂ Im Π p L T . (26)
The inclusion ( 26) is equivalent to (see [START_REF] Coron | Control and Nonlinearity[END_REF], Lemma 2.48 p. 58)

∃C > 0 such that ∀ϕ 0 ∈ L 2 (Ω) p , S * T Π * p ϕ 0 2 L 2 (Ω) n C L * T Π * p ϕ 0 2 L 2 (Q T ) m . (27)
We note that

S * T Π * p : L 2 (Ω) p → L 2 (Ω) n ϕ 0 → ϕ(0) and L * T Π * p : L 2 (Ω) p → L 2 (Q T ) m ϕ 0 → 1 ω B * ϕ,
where ϕ ∈ W (0, T ) n is the solution to System [START_REF] Mauffrey | On the null controllability of a 3 x 3 parabolic system with non-constant coefficients by one or two control forces[END_REF]. Indeed, for all

y 0 ∈ L 2 (Ω) n , u ∈ L 2 (Q T ) m and ϕ 0 ∈ L 2 (Ω) p Π p S T y 0 , ϕ 0 L 2 (Ω) p = y(T ; y 0 , 0), ϕ(T ) L 2 (Ω) n = T 0 ∂ t y(s; y 0 , 0), ϕ(s) L 2 (Ω) n ds + T 0 y(s; y 0 , 0), ∂ t ϕ(s) L 2 (Ω) n ds + y 0 , ϕ(0) L 2 (Ω) n = y 0 , ϕ(0) L 2 (Ω) n (28) and Π p L T u, ϕ 0 L 2 (Ω) p = y(T ; 0, u), ϕ(T ) L 2 (Ω) n = T 0 ∂ t y(s; 0, u), ϕ(s) L 2 (Ω) n ds + T 0 y(s; 0, u), ∂ t ϕ(s) L 2 (Ω) n ds = 1 ω Bu, ϕ L 2 (Q T ) n = u, 1 ω B * ϕ L 2 (Q T ) m . (29)
The inequality (27) combined with (28)-(29) lead to the conclusion. 2. In view of the definition in (24) of S T and L T , System (1) is Π p -approximately controllable on the time interval (0, T ) if and only if

∀(y 0 , y T ) ∈ L 2 (Ω) n × L 2 (Ω) p , ∀ε > 0, ∃u ∈ L 2 (Q T ) m such that Π p L T u + Π p S T y 0 -y T L 2 (Ω) p ε.
This is equivalent to

∀ε > 0, ∀z T ∈ L 2 (Ω) p , ∃u ∈ L 2 (Q T ) m such that Π p L T u -z T L 2 (Ω) p ε. That means Π p L T (L 2 (Q T ) m ) = L 2 (Ω) p .
In other words ker

L * T Π * p = {0}. Thus System (1) is Π p -approximately controllable on the time interval (0, T ) if and only if for all ϕ 0 ∈ L 2 (Ω) p L * T Π * p ϕ 0 = 1 ω B * ϕ ≡ 0 in Q T ⇒ ϕ ≡ 0 in Q T .
Corollary 1. Let us suppose that for all ϕ 0 ∈ L 2 (Ω) p , the solution ϕ to the adjoint System [START_REF] Mauffrey | On the null controllability of a 3 x 3 parabolic system with non-constant coefficients by one or two control forces[END_REF] satisfies the observability inequality [START_REF] Mizohata | Unicité du prolongement des solutions pour quelques opérateurs différentiels paraboliques[END_REF]. Then for all initial condition y 0 ∈ L 2 (Ω) n , there exists a control u ∈ L 2 (q T ) m (q T := ω × (0, T )) such that the solution y to System (1) satisfies Π p y(T ) ≡ 0 in Ω and

u L 2 (q T ) m C obs y 0 L 2 (Ω) n . ( 30
)
The proof is classical and will be omitted (estimate (30) can be obtained directly following the method developed in [START_REF] Fernández-Cara | The cost of approximate controllability for heat equations: the linear case[END_REF]).

3. Partial null controllability with constant coupling matrices. Let us consider the system

     ∂ t y = ∆y + Ay + B1 ω u in Q T , y = 0 on Σ T , y(0) = y 0 in Ω, (31)
where

y 0 ∈ L 2 (Ω) n , u ∈ L 2 (Q T ) m , A ∈ L(R n ) and B ∈ L(R m ; R n ).
Let the natural number s be defined by

s := rank [A|B] ( 32
)
and X ⊂ R n be the linear space spanned by the columns of [A|B].

In this section, we prove Theorem 1.1 in two steps. In subsection 3.1, we begin by studying the case where B ∈ R n and the general case is considered in subsection 3.2.

All along this section, we will use the lemma below which proof is straightforward.

Lemma 3.1. Let be y 0 ∈ L 2 (Ω) n , u ∈ L 2 (Q T ) m and P ∈ C 1 ([0, T ], L(R n )) such that P (t) is invertible for all t ∈ [0, T ].
Then the change of variable w = P -1 (t)y transforms System (31) into the equivalent system

     ∂ t w = ∆w + C(t)w + D(t)1 ω u in Q T , w = 0 on Σ T , w(0) = w 0 in Ω, ( 33
)
with w 0 := P -1 (0)y 0 , C(t) := -P -1 (t)∂ t P (t)+P -1 (t)AP (t) and D(t) := P -1 (t)B. Moreover Π p y(T ) ≡ 0 in Ω ⇔ Π p P (T )w(T ) ≡ 0 in Ω. If P is constant, we have [C|D] = P -1 [A|B].
3.1. One control force. In this subsection, we suppose that

A ∈ L(R n ), B ∈ R n and denote by [A|B] =: (k ij ) 1 i,j n and s := rank [A|B].
We begin with the following observation.

Lemma 3.2. {B, ..., A s-1 B} is a basis of X.

Proof. If s = rank [A|B] = 1, then the conclusion of the lemma is clearly true, since B = 0. Let s 2. Suppose to the contrary that {B, ..., A s-1 B} is not a basis of X, that is for some i ∈ {0, ..., s -2} the family {B, ..., A i B} is linearly independent and

A i+1 B ∈ span(B, ..., A i B), that is A i+1 B = i k=0 α k A k B with α 0 , ..., α i ∈ R.

Multiplying by A this expression, we deduce that

A i+2 B ∈ span(AB, ..., A i+1 B) = span(B, ..., A i B). Thus, by induction, A l B ∈ span(B, ..., A i B) for all l ∈ {i + 1, ..., n -1}. Then rank (B|AB|...|A n-1 B) = rank (B|AB|...|A i B) = i + 1 < s, contradicting with (32). Proof of Theorem 1.1. Let us remark that rank Π p [A|B] = dim Π p [A|B](R n ) rank [A|B] = s. ( 34
) Lemma 3.2 yields rank (B|AB|...|A s-1 B) = rank [A|B] = s. (35)
Thus, for all l ∈ {s, s + 1, ..., n} and i ∈ {0, ..., s -1}, there exist α l,i such that

A l B = s-1 i=0 α l,i A i B. (36)
Since, for all l ∈ {s, ..., n},

Π p A l B = s-1 i=0 α l,i Π p A i B, then rank Π p (B|AB|...|A s-1 B) = rank Π p [A|B]. ( 37
)
We first prove in (a) that condition ( 9) is sufficient, and then in (b) that this condition is necessary. (a) Sufficiency part: Let us assume first that condition (9) holds. Then, using (37), we have rank Π p (B|AB|...|A s-1 B) = p.

(38) Let be y 0 ∈ L 2 (Ω) n . We will study the Π p -null controllability of System (31) according to the values of p and s. Case 1 : p = s. The idea is to find an appropriate change of variable P to the solution y to System (31). More precisely, we would like the new variable w := P -1 y to be the solution to a cascade system and then, apply Theorem 2.1. So let us define, for all t ∈ [0, T ],

P (t) := (B|AB|...|A s-1 B|P s+1 (t)|...|P n (t)), (39)
where, for all l ∈ {s + 1, ..., n}, P l (t) is the solution in C 1 ([0, T ]) n to the system of ordinary differential equations

∂ t P l (t) = AP l (t) in [0, T ], P l (T ) = e l . (40)
Using (39) and (40), we can write

P (T ) = P 11 0 P 21 I n-s , (41)
where

P 11 := Π p (B|AB|...|A s-1 B) ∈ L(R s ), P 21 ∈ L(R s , R n-s
) and I n-s is the identity matrix of size n -s. Using (38), P 11 is invertible and thus P (T ) also. Furthermore, since

P (t) ∈ C 1 ([0, T ], L(R n )) continuous in time on the time interval [0, T ], there exists T * ∈ [0, T ) such that P (t) is invertible for all t ∈ [T * , T ].
Let us suppose first that T * = 0. Since P (t) ∈ C 1 ([0, T ], L(R n )) and invertible, in view of Lemma 3.1: for a fixed control u ∈ L 2 (Q T ), y is the solution to System (31) if and only if w := P (t) -1 y is the solution to System (33) where C, D are given by C(t) := -P -1 (t)∂ t P (t) + P -1 (t)AP (t) and D(t) := P -1 (t)B, for all t ∈ [0, T ]. Using (36) and (40), we obtain

   -∂ t P (t) + AP (t) = (AB|...|A s B|0|...|0) = P (t) C 11 0 0 0 in [0, T ], P (t)e 1 = B in [0, T ], (42)
where

C 11 :=        0 0 0 . . . α s,0 1 0 0 . . . α s,1 0 1 0 . . . α s,2 . . . . . . . . . . . . . . . 0 0 . . . 1 α s,s-1        ∈ L(R s ). ( 43
)
Then

C(t) = C 11 0 0 0 and D(t) = e 1 . (44)
Using Theorem 2.1, there exists u ∈ L 2 (Q T ) such that the solution to System (33) satisfies w 1 (T ) ≡ ... ≡ w s (T ) ≡ 0 in Ω. Moreover, using (41), we have Π s y(T ) = (y 1 (T ), ..., y s (T )) = P 11 (w 1 (T ), ..., w s (T )) ≡ 0 in Ω.

If now T * = 0, let y be the solution in W (0, T * ) n to System (31) with the initial condition y(0) = y 0 in Ω and the control u ≡ 0 in Ω × (0, T * ). We use the same argument as above to prove that System (31) is Π s -null controllable on the time interval [T * , T ]. Let v be a control in L 2 (Ω × (T * , T )) such that the solution z in W (T * , T ) n to System (31) with the initial condition z(T * ) = y(T * ) in Ω and the control v satisfies Π s z(T ) ≡ 0 in Ω. Thus if we define y and u as follows

(y, u) := (y, 0) if t ∈ [0, T * ], (z, v) if t ∈ [T * , T ],
then, for this control u, y is the solution in W (0, T ) n to System (31). Moreover y satisfies Π s y(T ) ≡ 0 in Ω. Case 2 : p < s. In order to use Case 1, we would like to apply an appropriate change of variable Q to the solution y to System (31). If we denote by [A|B] =: (k ij ) ij , equalities (35) and (38) can be rewritten rank

   k 11 • • • k 1s . . . . . . k n1 • • • k ns    = s and rank    k 11 • • • k 1s . . . . . . k p1 • • • k ps    = p.
Then there exist distinct natural numbers λ p+1 , ..., λ s such that we have {λ p+1 , ..., λ s } ⊂ {p + 1, ..., n} and rank

          k 11 • • • k 1s . . . . . . k p1 • • • k ps k λp+11 • • • k λp+1s . . . . . . k λs1 • • • k λss           = s. ( 45
)
Let Q be the matrix defined by

Q := (e 1 |...|e p |e λp+1 |...|e λn ) t ,
where {λ s+1 , ..., λ n } := {p + 1, ..., n}\{λ p+1 , ..., λ s }. Q is invertible, so taking w := P -1 y with P := Q -1 , for a fixed control u in L 2 (Q T ), y is solution to System (31) if and only if w is solution to System (33) where

w 0 := Qy 0 , C := QAQ -1 ∈ L(R n ) and D := QB ∈ L(R; R n ). Moreover there holds [C|D] = Q[A|B].
Thus, equation (45

) yields rank Π s [C|D] = rank Π s Q[A|B] = rank           k 11 • • • k 1n . . . . . . k p1 • • • k pn k λp+11 • • • k λp+1n . . . . . . k λs1 • • • k λsn           = s.
Since rank [C|D] = rank [A|B] = s, we proceed as in Case 1 forward deduce that System (33) is Π s -null controllable, that is there exists a control u ∈ L 2 (Q T ) such that the solution w to System (33) satisfies

Π s w(T ) ≡ 0 in Ω.
Moreover the matrix Q can be rewritten

Q = I p 0 0 Q 22 ,
where We suppose now that (9) is not satisfied: there exist p ∈ {1, ..., p} and β i for all i ∈ {1, ..., p}\{p} such that k pj = p i=1,i =p β i k ij for all j ∈ {1, ..., s}. The idea is to find a change of variable w := Qy that allows to handle more easily our system. We will achieve this in three steps starting from the simplest situation.

Q 22 ∈ L(R n-p ).
Step 1. Let us suppose first that

k 11 = ... = k 1s = 0 and rank    k 21 • • • k 2s . . . . . . k s+1,1 • • • k s+1,s    = s. ( 46
)
We want to prove that, for some initial condition

y 0 ∈ L 2 (Ω) n , a control u ∈ L 2 (Q T ) cannot
C = C 11 C 12 0 C 22 , (48)
where C 12 ∈ L(R n-s , R s ) and C 22 ∈ L(R n-s ). Furthermore

D = P -1 B = P -1 P e 1 = e 1 .
and with the Definition (47) of P we get

y 1 (T ) = w s+1 (T ) in Ω.
Thus we need only to prove that there exists w 0 ∈ L 2 (Ω) n such that we cannot find a control u ∈ L 2 (Q T ) with the corresponding solution w to System (33) satisfying w s+1 (T ) ≡ 0 in Ω. Therefore we apply Proposition 1 and prove that the observability inequality [START_REF] Mizohata | Unicité du prolongement des solutions pour quelques opérateurs différentiels paraboliques[END_REF] can not be satisfied. More precisely, for all w 0 ∈ L 2 (Ω) n , there exists a control u ∈ L 2 (Q T ) such that the solution to System (33) satisfies w s+1 (T ) ≡ 0 in Ω, if and only if there exists C obs > 0 such that for all ϕ 0 s+1 ∈ L 2 (Ω) the solution to the adjoint system

       -∂ t ϕ = ∆ϕ + C * 11 0 C * 12 C * 22 ϕ in Q T , ϕ = 0 on Σ T , ϕ (
But for all ϕ 0 s+1 ≡ 0 in Ω, the inequality (50) is not satisfied. Indeed, we remark first that, since ϕ 1 (T ) = ... = ϕ s (T ) = 0 in Ω, we have ϕ 1 = ... = ϕ s = 0 in Q T , so that ω×(0,T ) ϕ 2 1 dx = 0, while, if we choose ϕ 0 s+1 ≡ 0 in Ω, using the results on backward uniqueness for this type of parabolic system (see [START_REF] Ghidaglia | Some backward uniqueness results[END_REF]), we have clearly (ϕ s+1 (0), ..., ϕ n (0)) ≡ 0 in Ω.

Step 2. Let us suppose only that k 11 = ... = k 1s = 0. Since rank (B|...|A s-1 B) = s, there exists distinct λ 1 , ..., λ s ∈ {2, ..., n} such that rank

   k λ1,1 • • • k λ1,s . . . . . . k λs,1 • • • k λs,s    = s.
Let us consider the following matrix

Q := (e 1 |e λ1 |...|e λn-1 ) t ,
where {λ s+1 , ..., λ n-1 } = {2, ..., n}\{λ 1 , ..., λ s }. Thus, for P := Q -1 , again, for a fixed control u ∈ L 2 (Q T ), y is a solution to System (31) if and only if w := P -1 y is a solution to System (33) where C, D are given by C := QAQ -1 and D := QB. Moreover, we have

[C|D] = Q[A|B].
If we note ( kij ) ij := [C|D], this implies k11 = ... = k1s = 0 and rank

   k21 • • • k2s . . . . . . ks+1,1 • • • ks+1,s    = rank    k λ11 • • • k λ1s . . . . . . k λs,1 • • • k λs,s    = s.
Proceeding as in Step 1 for w, there exists an initial condition w 0 such that for all control u in L 2 (Q T ) the solution w to System (33) satisfies w 1 (T ) ≡ 0 in Ω. Thus, for the initial condition y 0 := Q -1 w 0 , for all control u in L 2 (Q T ), the solution y to System (31) satisfies

y 1 (T ) = w 1 (T ) ≡ 0 in Ω.
Step 3. Without loss of generality, we can suppose that there exists β i for all i ∈ {2, ..., p} such that k 1j = p i=2 β i k ij for all j ∈ {1, ..., s} (otherwise a permutation of lines leads to this case). Let us define the following matrix

Q := (e 1 - p i=2 β i e i )|e 2 |...|e n t .
Thus, for P := Q -1 , again, for a fixed initial condition y 0 ∈ L 2 (Ω) n and a control u ∈ L 2 (Q T ), consider System (33) with w := P -1 y, y being a solution to System (31). We remark that if we denote by ( kij ) := [C|D], we have k11 = ... = k1s = 0. Applying step 2 to w, there exists an initial condition w 0 such that for all control u in L 2 (Q T ) the solution w to System (33) satisfies

w 1 (T ) ≡ 0 in Ω. ( 51
)
Thus, with the definition of Q, for all control u in L 2 (Q T ) the solution y to System (31) satisfies

w 1 (T ) = y 1 (T ) - p i=2 β i y i (T ) in Ω.
Suppose Π p y(T ) ≡ 0 in Ω, then w 1 (T ) ≡ 0 in Ω and this contradicts (51).

As a consequence of Proposition 1, the Π p -null controllability implies the Π papproximate controllability of System (33). If now Condition ( 9) is not satisfied, as for the Π p -null controllability, we can find a solution to System (49) such that φ 1 ≡ 0 in ω × (0, T ) and φ ≡ 0 in Q T and we conclude again with Proposition 1.

m-control forces.

In this subsection, we will suppose that A ∈ L(R n ) and B ∈ L(R m , R n ). We denote by B =: (b 1 |...|b m ). To prove Theorem 1.1, we will use the following lemma which can be found in [START_REF] Khodja | A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems[END_REF]. Lemma 3.3. There exist r ∈ {1, ..., s} and sequences {l j } 1 j r ⊂ {1, ..., m} and {s j } 1 j r ⊂ {1, ..., n} with r j=1 s j = s, such that

B := r j=1
{b lj , Ab lj , ..., A sj -1 b lj } is a basis of X. Moreover, for every 1 j r, there exist α i k,sj ∈ R for 1 i j and 1 k s j such that

A sj b lj = j i=1 α i 1,sj b li + α i 2,sj Ab li + ... + α i si,sj A si-1 b li . ( 52
)
Proof of Theorem 1.1. Consider the basis B of X given by Lemma 3.3. Note that

rank Π p [A|B] = dim Π p [A|B](R n ) rank [A|B] = s.
If M is the matrix whose columns are the elements of B, i. (53) Indeed, relationship (52) yields

Π p A sj b lj = j i=1 α i 1,sj Π p b li + α i 2,sj Π p Ab li + ... + α i si,sj Π p A si-1 b li .
We first prove in (a) that condition ( 9) is sufficient, and then in (b) that this condition is necessary. (a) Sufficiency part: Let us suppose first that ( 9) is satisfied. Let be y 0 ∈ L 2 (Ω) n . We will prove that we need only r forces to control System (31). More precisely, we will study the Π p -null controllability of the system     

∂ t y = ∆y + Ay + B1 ω v in Q T , y = 0 on Σ T , y(0) = y 0 in Ω, (54)
|P s+1 (t)|...|P n (t)) ∈ L(R n ), (56)
where for all l ∈ {s + 1, ..., n}, P l is solution in C 1 ([0, T ]) n to the system of ordinary differential equations

∂ t P l (t) = AP l (t) in [0, T ], P l (T ) = e l . (57)
Using ( 56) and (57) we have We suppose first that T * = 0. Since P is invertible and continuous on [0, T ], for a fixed control v ∈ L 2 (Q T ) r , y is the solution to System (54) if and only if w := P (t) -1 y is the solution to System (33) where C, D are given by C(t) := -P -1 (t)∂ t P (t) + P -1 (t)AP (t) and D(t) := P -1 (t) B, for all t ∈ [0, T ]. Using (52) and (57), we obtain

P (T ) = P 11 0 P 21 I n-s , (58)
         -∂ t P (t) + AP (t) = (Ab l1
P (t)e Si = b li in [0, T ], (59) where
S i = 1 + i-1 j=1 s j for i ∈ {1, ..., r}, C11 :=      C 11 C 12 • • • C 1r 0 C 22 • • • C 2r . . . . . . . . . . . . 0 0 • • • C rr      ∈ L(R s ) (60)
and for 1 i j r the matrices C ij ∈ L(R sj , R si ) are given by where C11 is defined in (60). Then C can be written as

C ii :=        0 0 0 . . . α i 1,si 1 0 0 . . . α i 2,si 0
C = C11 C12 0 C22 , (64)
where C12 ∈ L(R s , R n-s ) and C22 ∈ L(R n-s ). Furthermore, the matrix D can be written

D = D 1 0 ,
where D 1 ∈ L(R m , R s ). Using (63), we get

y 1 (T ) = w s+1 (T ) in Ω.
Thus, we need only to prove that there exists w 0 ∈ L 2 (Ω) n such that we cannot find a control u ∈ L 2 (Q T ) m with the corresponding solution w to System (33) satisfying w s+1 (T ) ≡ 0 in Ω. Therefore we apply Proposition 1 and prove that the observability inequality [START_REF] Mizohata | Unicité du prolongement des solutions pour quelques opérateurs différentiels paraboliques[END_REF] can not be satisfied. More precisely, for all w 0 ∈ L 2 (Ω) n , there exists a control u ∈ L 2 (Q T ) m such that the solution w to System (33) satisfies w s+1 (T ) ≡ 0 in Ω, if and only if there exists C obs > 0 such that for all ϕ 0 s+1 ∈ L 2 (Ω) the solution to the adjoint system

       -∂ t ϕ = ∆ϕ + C * 11 0 C * 12 C * 22 ϕ in Q T , ϕ = 0 on Σ T , ϕ(T ) = (0, ..., 0, ϕ 0 s+1 , 0, ..., 0) t = e s+1 ϕ 0 s+1 in Ω (65)
satisfies the observability inequality

Ω ϕ(0) 2 dx C obs ω×(0,T ) (D * 1 (ϕ 1 , ..., ϕ s ) t ) 2 dx dt. ( 66
)
But for all ϕ 0 s+1 ≡ 0 in Ω, the inequality (66) is not satisfied. Indeed, we remark first that, since ϕ 1 (T ) = ... = ϕ s (T ) = 0 in Ω, we have ϕ 1 = ... = ϕ s = 0 in Q T . Furthermore, if we choose ϕ 0 s+1 ≡ 0 in Ω, as previously, we get (ϕ s+1 (0), ..., ϕ n (0)) ≡ 0 in Ω.

We recall that, as a consequence of Proposition 1, the Π p -null controllability implies the Π p -approximate controllability of System (54). If Condition ( 9) is not satisfied, as for the Π p -null controllability, we can find a solution to System (65) such that D * 1 (φ 1 , ..., φ s ) t ≡ 0 in ω × (0, T ) and φ ≡ 0 in Q T and we conclude again with Proposition 1.

4. Partial null controllability with time dependent matrices. We recall that

[A|B](•) = (B 0 (•)|...|B n-1 (•)) (see (6)). Since A(t) ∈ C n-1 ([0, T ]; L(R n )) and B(t) ∈ C n ([0, T ]; L(R m ; R n ))
, we remark that the matrix [A|B] is well defined and is an element of C 1 ([0, T ], L(R mn , R n ). We will use the notation

B i =: (b i 1 |...|b i m
) for all i ∈ {0, ..., n -1}. To prove Theorem 1.2, we will use the following lemma of [START_REF] González-Burgos | Controllability results for cascade systems of m coupled parabolic PDEs by one control force[END_REF] Lemma 4.1. Assume that max{rank [A|B](t) : t ∈ [0, T ]} = s n. Then there exist T 0 , T 1 ∈ [0, T ], with T 0 < T 1 , r ∈ {1, ..., m} and sequences (s j ) 1 j r ⊂ {1, ..., n}, with r i=1 s j = s, and (l j ) 1 j r ⊂ {1, ..., m} such that, for every t ∈ [T 0 , T 1 ], the set

B(t) = r j=1 {b lj 0 (t), b lj 1 (t), ..., b lj sj -1 (t)}, ( 67
)
is linearly independent, spans the columns of [A|B](t) and satisfies

b lj sj (t) = j k=1 θ lj ,l k sj ,0 (t)b l k 0 (t) + θ lj ,l k sj ,1 (t)b l k 1 (t) + ... + θ lj ,l k sj ,s k -1 (t)b l k s k -1 (t) , (68)
for every t ∈ [T 0 , T 1 ] and j ∈ {1, ..., r}, where Proof of Theorem 1.2. Let y 0 ∈ L 2 (Ω) n and s be the rank of the matrix [A|B](T ). As in the proof of the controllability by one force with constant matrices, let X being the linear space spanned by the columns of the matrix [A|B](T ). We consider B = B(t) the basis of X defined in (67).

As in the constant case, we will prove that we need only r forces to control System (1) that is we study the partial null controllability of System (54) with the coupling matrix

A(t) ∈ C n-1 ([0, T ]; L(R n )) and the control matrix B(t) = (B l1 (t)|B l2 (t)| • • • |B lr (t)) ∈ C n ([0, T ]; L(R r , R n )).
If we define M as the matrix whose columns are the elements of B(t), i.e. for all t ∈ [0, T ]

M (t) = (m ij (t)) 1 i n,1 j s := b l1 0 (t)
Π p M (T ) = rank Π p [A|B](T ) = p. ( 69
)
Indeed, using (68),

Π p b lj sj (t) = j k=1 θ lj ,l k sj ,0 (t)Π p b l k 0 (t) + θ lj ,l k sj ,1 (t)Π p b l k 1 (t) + ... + θ lj ,l k sj ,s k -1 (t)Π p b l k s k -1 (t) .
Case 1 : p = s. As in the constant case, we want to apply a change of variable P to the solution y to System (54). Let us define for all t ∈ [0, T ] the following matrix

P (t) := (b l1 0 (t)|b l1 1 (t)|...|b l1 s1-1 (t)| ...|b lr 0 (t)|b lr 1 (t)|...|b lr sr-1 (t)|P s+1 (t)|...|P n (t)) ∈ L(R n ), (70)
where for all i ∈ {s + 1, ..., n}, P l is solution in C 1 ([0, T ]) n to the system of ordinary differential equations

∂ t P l (t) = AP l (t) in [0, T ], P l (T ) = e l . (71)
Using ( 70) and (71), P (T ) can be rewritten

P (T ) = P 11 0 P 21 I n-s , ( 72
)
where 69), P 11 , and thus P (T ), are invertible. Furthermore, since P is continuous on [0, T ], there exists a

P 11 := Π p (b l1 0 (T )|b l1 1 (T )|...|b l1 s1-1 (T )|...|b lr 0 (T )|b lr 1 (T )|...|b lr sr-1 (T )) ∈ L(R s ) and P 21 ∈ L(R n-s , R s ). Using (
T * ∈ [0, T ) such that P (t) is invertible for all t ∈ [T * , T ].
As previously it is sufficient to prove the result for T * = 0. Since

P (t) ∈ C 1 ([0, T ], L(R n ))
and is invertible on the time interval [0, T ], again, for a fixed control v ∈ L 2 (Q T ) r , y is the solution to System (54) if and only if w := P (t) -1 y is the solution to System (33) where C, D are given by C(t) := -P -1 (t)∂ t P (t) + P -1 (t)AP (t) and D(t) := P -1 (t) B, for all t ∈ [0, T ]. Using (68) and (71), we obtain

         -∂ t P (t) + AP (t) = (b l1 1 (t)
P (t)e Si = b li 0 in [0, T ], (73) where
S i = 1 + i-1 j=1 s j for 1 i r, C11 :=      C 11 C 12 • • • C 1r 0 C 22 • • • C 2r . . . . . . . . . . . . 0 0 • • • C rr      ∈ L(R s ), (74)
and for 1 i j r, the matrices

C ij ∈ C 0 ([0, T ]; ∈ L(R sj , R si ))
are given here by

C ii =         0 0 0 . . . θ li,li si,0 1 0 0 . . . θ li,li si,1 0 1 0 . . . θ li,li si,2 . . . . . . . . . . . . . . . 0 0 . . . 1 θ li,li si,si-1        
and

C ij =          0 0 0 . . . θ lj ,
Using Theorem 2.1, there exists v ∈ L 2 (Q T ) r such that the solution to System (33) satisfies w 1 (T ) = ... = w s (T ) ≡ 0 in Ω. Moreover, the equality (72) leads to Π s y(T ) = (y 1 (T ), ..., y s (T )) t = P 11 (w 1 (T ), ..., w s (T )) t ≡ 0 in Ω.

Case 2 : p < s. The same method as in the constant case leads to the conclusion (see § 3.1).

The π p -approximate controllability can proved also as in the constant case.

5. Partial null controllability for a space dependent coupling matrix. All along this section, the dimension N will be equal to 1, more precisely Ω := (0, π) with the exception of the proof of the third point in Theorem 1.3 and the numerical illustration in Section 5.3 where Ω := (0, 2π). We recall that the eigenvalues of -∆ in Ω with Dirichlet boundary conditions are given by µ k := k 2 for all k 1 and we will denote by (w k ) k 1 the associated L 2 -normalized eigenfunctions. Let us consider the following parabolic system of two equations

         ∂ t y = ∆y + αz + 1 ω u in Q T , ∂ t z = ∆z in Q T , y = z = 0 on Σ T , y(0) = y 0 , z(0) = z 0 in Ω, (76)
where y 0 , z 0 ∈ L 2 (Ω) are the initial data, u ∈ L 2 (Q T ) is the control and the coupling coefficient α is in L ∞ (Ω). We recall that System (76) is Π 1 -null controllable if for all y 0 , z 0 ∈ L 2 (Ω), we can find a control u ∈ L 2 (Q T ) such that the solution (y, z) ∈ W (0, T ) 2 to System (76) satisfies y(T ) ≡ 0 in Ω.

Example of controllability.

In this subsection, we will provide an example of Π 1 -null controllability for System (76) with the help of the method of moments initially developed in [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF]. As already mentioned, we suppose that Ω := (0, π), but the argument of Section 5.1 can be adapted for any open bounded interval of R.

Let us introduce the adjoint system associated to our control problem

         -∂ t φ = ∆φ in (0, π) × (0, T ), -∂ t ψ = ∆ψ + αφ in (0, π) × (0, T ), φ(0) = φ(π) = ψ(0) = ψ(π) = 0 on (0, T ), φ(T ) = φ 0 , ψ(T ) = 0 in (0, π), (77)
where φ 0 ∈ L 2 (0, π). For an initial data φ 0 ∈ L 2 (0, π) in adjoint System (77), we get

π 0 φ 0 y(T ) dx - π 0 φ(0)y 0 dx - π 0 ψ(0)z 0 dx = q T φu dx dt, (78)
with the notation q T := ω × (0, T ). Since (w k ) k 1 spans L 2 (0, π), System (76) is Π 1 -null controllable if and only if there exists u ∈ L 2 (q T ) such that, for all k ∈ N * , the solution to System (77) satisfies the following equality

- π 0 φ k (0)y 0 dx - π 0 ψ k (0)z 0 dx = q T φ k u dx dt, (79)
where (φ k , ψ k ) is the solution to adjoint System (77) for the initial data φ 0 := w k . Let k ∈ N * . With the initial condition φ 0 := w k is associated the solution (φ k , ψ k ) to adjoint System (77):

φ k (t) = e -k 2 (T -t) w k in (0, π)
for all t ∈ [0, T ]. If we write:

ψ k (x, t) := l 1
ψ kl (t)w l (x) for all (x, t) ∈ (0, π) × (0, T ), then a simple computation leads to the formula

ψ kl (t) = e -k 2 (T -t) -e -l 2 (T -t) -k 2 + l 2 α kl for all l 1, t ∈ (0, T ), (80)
where, for all k, l ∈ N * , α kl is defined in [START_REF] Khodja | A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems[END_REF]. In (80) we implicitly used the convention: -t) . With these expressions of φ k and ψ k , the equality (79) reads for all k 1

if l = k the term (e -k 2 (T -t) -e -l 2 (T -t) )/(-k 2 + l 2 ) is replaced by (T -t)e -k 2 (T
-e -k 2 T y 0 k - l 1 e -k 2 T -e -l 2 T -k 2 + l 2 α kl z 0 l = q T e -k 2 (T -t) w k (x)u(t, x) dx dt. (81)
In the proof of Theorem 1.3, we will look for a control u expressed as u(x, t) = f (x)γ(t) with γ(t) = k 1 γ k q k (t) and (q k ) k 1 a family biorthogonal to (e -k 2 t ) k 1 . Thus, we will need the two following lemma Lemma 5.1. (see Lemma 5.1, [START_REF] Khodja | New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence[END_REF]) There exists f ∈ L 2 (0, π) such that Supp f ⊂ ω and for a constant β, one has

inf k 1 f k k 3 = β > 0, where, for all k ∈ N * , f k := π 0 f w k dx. Lemma 5.2. (see Corollary 3.2, [13]) There exists a sequence (q k ) k 1 ⊂ L 2 (0, T ) biorthogonal to (e -k 2 t ) k 1 , that is q k , e -l 2 t
L 2 (0,T ) = δ kl . Moreover, for all ε > 0, there exists C T,ε > 0, independent of k, such that

q k L 2 (0,T ) C T,ε e (π+ε)k , ∀k 1. ( 82
)
Remark 7. When Ω := (a, b) with a, b ∈ R, the inequality (82) of Lemma 5.2 is replaced by

q k L 2 (0,T ) C T,ε e (b-a+ε)k , ∀k 1.
Proof of the second point in Theorem 1.3. As mentioned above, let us look for the control u of the form u(x, t) = f (x)γ(t), where f is as in Lemma 5.1. Since f k = 0 for all k ∈ N * , using (81), the Π 1 -null controllability of System (76) is reduced to find a solution γ ∈ L 2 (0, T ) to the following problem of moments:

T 0 γ(T -t)e -k 2 t dt = f -1 k   -e -k 2 T y 0 k - l 1 e -k 2 T -e -l 2 T -k 2 + l 2 α kl z 0 l   := M k ∀k 0.
(83) The function γ(t) := k 1 M k q k (T -t) is a solution to this problem of moments. We need only to prove that γ ∈ L 2 (0, T ). Using the convexity of the exponential function, we get for all k ∈ N * ,

l 1 e -k 2 T -e -l 2 T -k 2 + l 2 |α kl | = k l=1 e -k 2 T -e -l 2 T -k 2 + l 2 |α kl | + ∞ l=k+1 e -k 2 T -e -l 2 T -k 2 + l 2 |α kl | k l=1 T e -l 2 T |α kl | + ∞ l=k+1 T e -k 2 T |α kl | =: A 1,k + A 2,k . (84)
With the Condition (13) on α, there exists a positive constant C T which do not depend on k such that for all k ∈ N *

A 1,k C 1 T k l=1 e -l 2 T e -C2(k-l) C 1 T e -C2k ∞ l=1 e -l 2 T +C2l C T e -C2k (85)
and

A 2,k C 1 T e -k 2 T ∞ l=k+1 e -C2(l-k) C 1 T e -k 2 T ∞ j=0 (e -C2 ) j C 1 T e -k 2 T 1 1 -e -C2 .
(86)

Combining the three last inequalities (84)-( 86), for all k

∈ N * l 1 e -k 2 T -e -l 2 T -k 2 + l 2 |α kl | C T e -C2k , (87)
where C T is a positive constant independent of k. Let ε ∈ (0, 1). Then, with Lemma 5.1, (83) and (87), there exists a positive constant C T,ε independent of k such that for all k ∈ N *

|M k | β -1 k 3 e -k 2 T y 0 L 2 (0,π) + C T e -C2k z 0 L 2 (0,π) C T,ε e -C2(1-ε)k ( y 0 L 2 (0,π) + z 0 L 2 (0,π) ).
Thus, using Lemma 5.2, for ε small enough and a positive constant C T,ε

γ L 2 (0,T ) C T,ε ( k∈N * e -[C2(1-ε)-π+ε]k )( y 0 L 2 (0,π) + z 0 L 2 (0,π) ) < ∞.

5.2.

Example of non controllability. In this subsection, to provide an example of non Π 1 -null controllability of System (76), we will first study the boundary controllability of the following parabolic system of two equations

         ∂ t y = ∆y + αz in Q T := (0, π) × (0, T ), ∂ t z = ∆z in Q T , y(0, t) = v(t), y(π, t) = z(0, t) = z(π, t) = 0 on (0, T ), y(x, 0) = y 0 (x), z(x, 0) = z 0 (x)
in Ω := (0, π), (88) where y 0 , z 0 ∈ H -1 (0, π) are the initial data, v ∈ L 2 (0, T ) is the boundary control and α ∈ L ∞ (0, π). For any given y 0 , z 0 ∈ H -1 (0, π) and v ∈ L 2 (0, T ), System (88) has a unique solution in L 2 (Q T ) 2 ∩ C 0 ([0, T ]; H -1 (Ω) 2 ) (defined by transposition; see [START_REF] Fernández-Cara | Boundary controllability of parabolic coupled equations[END_REF]).

As in Section 5.1, for an initial data (y 0 , z 0 ) ∈ H -1 (0, π) 2 we can find a control v ∈ L 2 (0, T ) such that the solution to (88) satisfies y(T ) ≡ 0 in (0, π) if and only if for all φ 0 ∈ H 1 0 (0, π) the solution to System (77) verifies the equality

-y 0 , φ(0) H -1 ,H 1 0 -z 0 , ψ(0) H -1 ,H 1 0 = T 0 v(t)φ x (0, t) dt, (89)
where the duality bracket •, • H -1 ,H 1 0 is defined as f, g H -1 ,H 1 0 := f (g) for all f ∈ H -1 (0, π) and all g ∈ H 1 0 (0, π). The used strategy here is inspired from [START_REF] Guerrero | Remarks on non controllability of the heat equation with memory[END_REF]. The idea involves constructing particular initial data for adjoint System (77): Lemma 5.3. Let m, G ∈ N * . For all M ∈ N\{0, 1}, there exists φ 0,M ∈ L 2 (0, π) given by φ 0,M = m i=1 φ 0,M GM +i w GM +i , with φ 0,M GM +1 , ..., φ 0,M GM +m ∈ R, such that the solution (φ M , ψ M ) to adjoint System (77) with

φ 0 = φ 0,M satisfies T 0 (φ M ) x (0, t) 2 dt 1/2 γ 1 M (2m-5)/2 , ( 90
)
where γ 1 does not depend on M . Morover for an increasing sequence (M j ) j∈N ⊂ N\{0, 1} and a k 1 ∈ {1, ..., m}, we have |φ 0,j GMj +k1 | = 1 for all G ∈ N * and j ∈ N. To study the controllability of System (88) we will use the fact that for fixed m, G ∈ N * , the quantity in the left-side hand in (90) converge to zero when M goes to infinity.

Proof. We remark first that

A M := T 0 (φ M ) x (0, t) 2 dt = T 0 GM +m k=GM +1 ke -k 2 (T -t) φ 0,M k 2 dt. (91)
We can rewrite A M as follows:

A M = T 0 m j=1 (GM + j)e -(G 2 M 2 +2GM j+j 2 )(T -t) φ 0,M GM +j 2 dt = T 0 e -2G 2 M 2 (T -t) g M (t) dt, (92)
where, for all t ∈ [0, T ], g M (t

) := f M (t) 2 with f M (t) := m j=1 (GM + j)e -(2GM j+j 2 )(T -t) φ 0,M GM +j .
Let (φ 0,M GM +1 , φ 0,M GM +2 , ..., φ 0,M GM +m ) be a nontrivial solution of the following homogeneous linear system of m -1 equations with m unknowns

f (l) M (T ) = m j=1
(GM + j)(2GM j + j 2 ) l φ 0,M GM +j = 0, for all l ∈ {0, ..., m -2}. (93)

Using Leibniz formula

g (l) M = l k=0 l k f (k) M f (l-k) M
we deduce that

g (l)
M (T ) = 0, for all l ∈ {0, ..., 2m -4}.

Using (94), after 2m -3 integrations by part in (92), we obtain

A M = -g M (0)e -2G 2 M 2 T 2G 2 M 2 + T 0 e -2G 2 M 2 (T -t) (-2G 2 M 2 ) g (1)
M (t)dt = 2m-4 l=0 g (l) M (0)e -2G 2 M 2 T (-2G 2 M 2 ) l+1 + T 0 e -2G 2 M 2 (T -t) (-2G 2 M 2 ) 2m-3 g (2m-3) M (t) dt.
By linearity, in (93) we can choose φ 0,M GM +1 , ..., φ 0,M GM +m such that sup i∈{1,...,m}

|φ 0,M GM +i | = 1. (95)
Thus, for all l ∈ N and all t ∈ [0, T ], the following estimate holds

|g (l) M (t)| = l k=0 l k f (k) M (t)f (l-k) M (t) l k=0 l k m j=1 (GM + j)(2GM j + j 2 ) k e -(2GM j+j 2 )(T -t) φ 0,M GM +j × m j=1 (GM + j)(2GM j + j 2 ) l-k e -(2GM j+j 2 )(T -t) φ 0,M GM +j (GM + m) 2 m 2 l k=0 l k (2GM m + m 2 ) l CM l+2 ,
where C does not depend on M . Then, since sup i∈{1,...,m}

|φ 0,M GM +i | = 1, there exist C, τ > 0 such that A M e -2G 2 M 2 T 2m-4 l=0 g (l) M ∞ (2G 2 M 2 ) l+1 + T g (2m-3) M ∞ (2G 2 M 2 ) 2m-3 e -τ M 2 ∞ l=0 C M l + C M 2m-5 CM -2 e -τ M 2 1 1 -M -2 + C M 2m-5 .
Thus there exists γ 1 > 0 such that we have the estimate

A M γ 1 M 2m-5
, where γ 1 does not depend on M . Using (104), for all M 2, there exists k 1 (M ) ∈ {1, ..., 7}, such that |φ 0,M 15M +k1(M ) | = 1. Thus there exists an increasing sequence (M j ) j∈N * such that |φ 0,Mj 15Mj +k1 | = 1 for a k 1 ∈ {1, ..., m} independent of j. Theorem 5.4. Let T > 0 and α be the function of L ∞ (0, π) defined by

α(x) := ∞ j=1 1 j 2 cos(15jx) for all x ∈ (0, π). (96)
Then there exists k 1 ∈ {1, .., 7} such that for (y 0 , z 0 ) := (0, w k1 ) and all control v ∈ L 2 (0, T ), the solution to System (88) verifies y(T ) ≡ 0 in (0, π).

Proof. To understand why the number 15 appears in the definition (96) of the function α, we will consider for all x ∈ (0, π)

α(x) := ∞ j=1 1 j 2 cos(Gjx) for all x ∈ (0, π), (97)
where G ∈ N * . We recall that for an initial condition (y 0 , z 0 ) ∈ L 2 (0, π) 2 and a control v ∈ L 2 (0, T ), the solution to System (96) satisfies y(T ) ≡ 0 in (0, π) if and only if for all φ 0 ∈ L 2 (0, π), we have the equality

-y 0 , φ(0) H -1 ,H 1 0 -z 0 , ψ(0) H -1 ,H 1 0 = T 0 v(t)φ x (0, t) dt, (98)
where (φ, ψ) is the solution to the adjoint System (77). Let us consider the sequences (M j ) j∈N * and (φ 0,Mj ) j∈N , k 1 defined in Lemma 5.3 and (φ Mj , ψ Mj ) the solution to

         -∂ t φ Mj = ∆φ Mj in (0, π) × (0, T ), -∂ t ψ Mj = ∆ψ Mj + αφ Mj in (0, π) × (0, T ), φ Mj (0) = φ Mj (π) = ψ Mj (0) = ψ Mj (π) = 0 on (0, T ), φ Mj (T ) = φ 0,Mj , ψ Mj (T ) = 0 in (0, π).
The goal is to prove that for the initial data (y 0 , z 0 ) := (0, w k1 ) and φ 0,Mj for j large enough, the equality (98) does not holds. Using Lemma 5.3, we have

T 0 v(t)(φ Mj ) x (0, t) dt γ 1 v L 2 (q T ) M j (2m-5)/2 . (99)
Since y 0 = 0, we obtain y 0 , φ Mj (0)

H -1 ,H 1 0 = 0. ( 100
)
Let us now estimate the term z 0 , ψ Mj (0) H -1 ,H 1 0 in the equality (98). We recall that the expression of α is given in (97). Then, the function α is of the form α(x) = ∞ p=0 α p cos(px) for all x ∈ (0, π), with

α p := 1 i 2 if p = Gi with i ∈ N * , 0 otherwise. ( 101
)
From the definition of α kl in (2), there holds for all k, l ∈ N * α kl = 1 π (α |k-l| -α k+l ). Let k ∈ {1, ..., m} and l ∈ {GM j + 1, ..., GM j + m}. We have k + l ∈ {GM j + 2, ..., GM j + 2m}.

Thus if we choose

G 2m + 1, (102) using (101), we obtain α k+l = 0 and

α |k-l| =    1 M j 2 if |k -l| = GM j , 0
otherwise. So that we have the following submatrix of (α kl ) 1 k,l GM +m :

(α kl ) 1 k m,GMj +1 l GMj +m = 1 πM j 2 I R m . (103)
According to Lemma 5.3, there exists k 1 ∈ {1, ..., m} such that |φ 0,Mj

GMj +k1 | = 1. (104)
Furthermore, since k 1 ∈ {1, ..., m},

|e -k 2 1 T -e -(GMj +k1) 2 T | |e -m 2 T -e -G 2 Mj 2 T | (105)
and

(GM j + k 1 ) 2 -k 2 1 (GM j + m) 2 -1. (106) Since z 0 = w k1 , the equality (103) leads to π 0 z 0 ψ Mj (0) dx = 7 s=1 e -k 2 1 T -e -(GMj +s)2T -k 2 1 + (GM j + s) 2 α k1,GMj +s φ 0,Mj GMj +s = e -k 2 1 T -e -(GMj +k1) 2 T -k 2 1 + (GM j + k 1 ) 2 1 πM j 2 .
Then using (105) and (106) for all j ∈ N * z 0 , ψ Mj (0)

H -1 ,H 1 0 = π 0 z 0 ψ Mj (0) dx γ 2 M j 4 , (107)
where γ 2 does not depend on j. Combining (99) and (107), we obtain a contradiction with equality (98). Thus, for this initial condition y 0 and z 0 , we can not find a control v ∈ L 2 (0, T ) such that the solution (y, z) to system (96) satisfies y(T ) ≡ 0 in (0, π).

Proof of the third point in Theorem 1.3. Using Theorem 5.4, for the initial data (p 0 , q 0 ) := (0, w k1 ) ∈ L 2 (0, π) 2 and all control v ∈ L 2 (0, T ), the solution (p, q) ∈ W (0, T ) 2 (defined by transposition) to the system

         ∂ t p = ∆p + αq in (0, π) × (0, T ), ∂ t q = ∆q in (0, π) × (0, T ), p(π, •) = v, p(0, •) = q(0, •) = q(π, •) = 0 on (0, T ), p(•, 0) = p 0 , q(•, 0) = q 0 in (0, π) (108)
satisfies p(T ) ≡ 0 in (0, π). Consider now p 0 , q 0 ∈ L 2 (0, 2π) defined by p 0 (x) = 0 and q 0 (x) = 2 π sin(k 1 x) for all x ∈ (0, 2π).

Remark that (p 0|(0,π) , q 0|(0,π) ) = (p 0 , q 0 ). Let ω ⊂ (0, π). Suppose now that the system For given (y 0 , z 0

) : (0, 2π) → R 2 , u : (0, 2π) × (0, T ) → R, Find (y, z) : (0, 2π) × (0, T ) → R 2 such that          ∂ t y = ∆y + αz + 1 ω u in (0, 2π) × (0, T ), ∂ t z = ∆z in (0, 2π) × (0, T ), y(0, •) = y(2π, •) = z(0, •) = z(2π, •) = 0 on (0, T ), y(•, 0) = y 0 , z(•, 0) = z 0 in (0, 2π) (109)
is Π 1 -null controllable, more particularly for the initial conditions y(0) = p 0 and z(0) = q 0 in (0, 2π), there exists a control u in L 2 ((0, 2π) × (0, T )) such that the solution (y, z) to System (109) satisfies y(T ) ≡ 0 in (0, 2π). We remark now that (p, q) := (y| (0,π) , z| (0,π) ) is a solution of (108) with (p(0), q(0)) = (p 0 , q 0 ) in (0, π), v(t) = y(π, t) in (0, T ) and satisfying p(T ) ≡ 0 in (0, π). This contradicts that for any control v ∈ L 2 (0, T ) the solution (p, q) to System (108) can not be identically equal to zero at time T.

Numerical illustration.

In this section, we illustrate numerically the results obtained previously in Sections 5.1 and 5.2. We adapt the HUM method to our control problem. For all penalty parameter ε > 0, we compute the control that minimizes the penalized HUM functional F ε given by

F ε (u) := 1 2 u 2 L 2 (ω×(0,T )) + 1 2ε y(T ; y 0 , u) 2 L 2 (Ω) ,
where y is the solution to (76). We can find in [START_REF] Boyer | On the penalised HUM approach and its applications to the numerical approximation of null-controls for parabolic problems, in CANUM 2012[END_REF] the argument relating the null/approximate controllability and this kind of functional. Using the Fenchel-Rockafellar theory (see [START_REF] Ekeland | Analyse Convexe et Problèmes Variationnels[END_REF] p. 59) we know that the minimum of F ε is equal to the opposite of the minimum of J ε , the so-called dual functional, defined for all ϕ 0 ∈ L 2 (Ω) by

J ε (ϕ 0 ) := 1 2 ϕ 2 L 2 (q T ) + ε 2 ϕ 0 2 L 2 (Q T ) + y(T ; y 0 , 0), ϕ 0 L 2 (Ω) ,
where ϕ is the solution to the backward System (110). Moreover the minimizers u ε and ϕ 0,ε of the functionals F ε and J ε respectively, are related through the equality u ε = 1 ω ϕ ε , where ϕ ε is the solution to the backward System (110) with the initial data ϕ(T ) = ϕ 0,ε . A simple computation leads to (111)

Then the minimizer u ε of F ε will be computed with the help of the minimizer ϕ 0,ε of J ε which is the solution to the linear problem (Λ + ε)ϕ 0,ε = -y(T ; y 0 , 0).

Remark 8. The proof of Theorem 1.7 in [START_REF] Boyer | On the penalised HUM approach and its applications to the numerical approximation of null-controls for parabolic problems, in CANUM 2012[END_REF] can be adapted to prove that (i) System (76) is Π 1 -null controllable if and only if sup System (76) with T = 0.005, Ω := (0, 2π), ω := (0, π) and y 0 := 100 sin(x) has been considered. We take the two expressions below for the coupling coefficient α that correspond respectively to Cases (1)-( 2 Systems (76) and ( 110)-(111) are discretized with backward Euler time-marching scheme (time step δt = 1/400) and standard piecewise linear Lagrange finite elements on a uniform mesh of size h successively equal to 2π/50, 2π/100, 2π/200 and 2π/300. We follow the methodology of F. Boyer (see [START_REF] Boyer | On the penalised HUM approach and its applications to the numerical approximation of null-controls for parabolic problems, in CANUM 2012[END_REF]) that introduces a penalty parameter ε = φ(h) := h 4 . We denote by E h , U h and L 2 δt (0, T ; U h ) the fullydiscretized spaces associated to L 2 (Ω), L 2 (ω) and L 2 (q T ). F h,δt ε is the discretization of F ε and (y h,δt ε , z h,δt ε , u h,δt ε ) the solution to the corresponding fully-discrete problem of minimisation. For more details on the fully-discretization of System (76) and Gramiam Λ (used to the minimisation of F ), we refer to Section 3 in [START_REF] Boyer | On the penalised HUM approach and its applications to the numerical approximation of null-controls for parabolic problems, in CANUM 2012[END_REF] and in [18, p. 37] respectively. The results are depicted Figure 1 and 2. As mentioned in the introduction of the present article (see Theorem 1.3), in both situations (a) and (b), System (76) is Π 1 -approximately controllable and we observe indeed in Figure 1 and 2 that the norm of the numerical solution to System (76) at time T (--) is decreasing when reducing the penality parameter ε = h 4 .

In Figure 1, the minimal value of the functional F h,δt ε (-• -) as well as the L 2 -norm of the control u h,δt ε (--) remain roughly constant whatever is the value of h (and ε = h 4 ). This appears in agreement with the results (1)-( 2) of Theorem 1.3, that state the Π 1 -null controllability of System (76) in Case (a) of a constant coupling coefficient α (see Remark 8 (i)). Furthermore the convergence to the null target is approximately of order 2 (slope of 2.27). This is in agreement with the convergence rate established in [8, Proposition 2.2], which should be h 2 for ε = h 4 (this result should be in fact slightly adapted to consider Π 1 -null controllability).

At the opposite, in Figure 2, the minimal value of the functional F h,δt ε as well as the L 2 -norm of the control u h,δt ε are strongly increasing whenever h (and ε) become smaller. This coincides with point (3) of Theorem 1.3: for the chosen value of the coupling coefficient α in Case (b), no Π 1 -null controllability of System (76) is expected. Moreover, convergence to the null target is quite slow, with a slope of approximately 8.34e -2.

  , n -1} and denote by [A|B](•) ∈ C 1 ([0, T ]; L(R nm ; R n )) the matrix function given by [A|B](•) := (B 0 (•)|B 1 (•)|...|B n-1 (•)).

r

  i=1 s i = n and D := (e S1 |...|e Sr ) with S 1 = 1 and

  Thus Π p y(T ) = Π p Qy(T ) = Π p w(T ) ≡ 0 in Ω. (b) Necessary part: Let us denote by [A|B] =: (k ij ) ij .

  be found such that the solution to System (31) satisfies y 1 (T ) ≡ 0 in Ω. Let us consider the matrix P ∈ L(R n ) defined byP := (B|...|A s-1 B|e 1 |e s+2 |...|e n ). (47)Using the assumption (46), P is invertible. Thus, in view of Lemma 3.1, for a fixed control u ∈ L 2 (Q T ), y is a solution to System (31) if and only if w := P -1 y is a solution to System (33) where C, D are given by C := P -1 AP and D := P -1 B. Using (36) we remark that A(B|AB|...|BA s-1 ) = (B|AB|...|BA s-1 ) C 11 0 , with C 11 defined in (43). Then C can be rewritten as

  T ) = (0, ..., 0, ϕ 0 s+1 , 0, ..., 0) t = e s+1 ϕ 0

  e. M = (m ij ) ij := b l1 |Ab l1 |...|A s1-1 b l1 |...|b lr |Ab lr |...|A sr-1 b lr , we can remark that rank Π p M = rank Π p [A|B].

  where B = (b l1 |b l2 | • • • |b lr ) ∈ L(R r , R n ). Using (9) and (53), we have rank Π p (b l1 |Ab l1 |...|A s1-1 b l1 |...|b lr |Ab lr |...|A sr-1 b lr ) = p. (55) Case 1 : p = s. As in the case of one control force, we want to apply a change of variable P to the solution y to System (54). Let us define for all t ∈ [0, T ] the following matrix P (t) := (b l1 |Ab l1 |...|A s1-1 b l1 |...|b lr |Ab lr |...|A sr-1 b lr

where P 11 :

 11 = Π s (b l1 |Ab l1 |...|A s1-1 b l1 |...|b lr |Ab lr |...|A sr-1 b lr ) ∈ L(R s ) and P 21 ∈ L(R n-s , R s ). From (55), P 11 and thus P (T ) are invertible. Furthermore, since P is continuous on [0, T ], there exists a T * ∈ [0, T ) such that P (t) is invertible for all t ∈ [T * , T ].

  |A 2 b l1 |...|A s1 b l1 |...|Ab lr |A 2 b lr |...|A sr b lr |0|...|0),

  t) = (e S1 |...|e Sr ). (61) Using Theorem 2.1, there exists v ∈ L 2 (Q T ) r such that the solution to System (33) satisfies w 1 (T ) = ... = w s (T ) ≡ 0 in Ω. Moreover, using (58), we have Π s y(T ) = (y 1 (T ), ..., y s (T )) = P 11 (w 1 (T ), ..., w s (T )) ≡ 0 in Ω. If now T * = 0, we conclude as in the proof of Theorem 1.1 with one force (see § 3.1). Case 2 : p < s. The proof is a direct adaptation of the proof of Theorem 1.1 with one force, it is possible to find a change of variable in order to get back to the situation of Case 1 (see § 3.1). (b) Necessary part: If (9) is not satisfied, there exist p ∈ {1, ..., p} and, for all i ∈ {1, ..., p}\{p}, scalars β i such that m pj = p i=1,i =p β i m ij for all j ∈ {1, ..., s}. As previously, without loss of generality, we can suppose that m 11 = ... = m 1s = 0 and rank permutation of lines leads to this case). Let us consider the matrix P defined by P := (b l1 |Ab l1 |...|A s1-1 b l1 |...|b lr |Ab lr |...|A sr-1 b lr |e 1 |e s+2 |...|e n ). (63) Relationship ensures (62) that P is invertible. Thus, again, for a fixed control u ∈ L 2 (Q T ) m , y is the solution to System (31) if and only if w := P -1 y is the solution to System (33) where C, D are given by C := P -1 AP and D := P -1 B. Using (52), we remark that A(b l1 |Ab l1 |...|A s1-1 b l1 |...|b lr |Ab lr |...|A sr-1 b lr ) = (Ab l1 |A 2 b l1 |...|A s1 b l1 |...|Ab lr |A 2 b lr |...|A sr b lr ) = P C11 0 ,

θLemma 4 . 2 .

 42 lj ,l k sj ,0 (t), θ lj ,l k sj ,1 (t), ..., θ lj ,l k sj ,s k -1 (t) ∈ C 1 ([T 0 , T 1 ]). With exactly the same argument for the proof of the previous lemma, we can obtain the If rank [A|B](T ) = s, then the conclusions of Lemma 4.1 hold true with T 1 = T .

  (e S1 |...|e Sr ).

∂

  ∇J ε (ϕ 0 ) = Λϕ 0 + εϕ 0 + y(T ; y 0 , 0), with the Gramiam operator Λ defined as followsΛ : L 2 (Ω) → L 2 (Ω), ϕ 0 → w(T ),where w is the solution to the following backward and forward systems t w = ∆w + 1 ω ϕ in Q T , w = 0, on Σ T , w(0) = 0 in Ω.

ε>0 inf L 2

 2 (ω×(0,T ))F ε < ∞, (ii) System (76) is Π 1 -approximately controllable if and only if y ε (T ) -→ ε→0 0,where y ε is the solution to System (76) for the control u ε .

1 p 2

 12 ) and (3) in Theorem 1.3:(a) α(x) = 1, (b) α(x) = p 0 cos(15px).

1 hFigure 1 .

 11 Figure 1. Minimal value of the functionalF h,δt in L 2 δt (0, T ; U h ), norm of the control u h,δt L 2δt (0,T ;U h ) , and distance to the target y h,δt (T ) E h in Case (a).

1 hFigure 2 .

 12 Figure 2. Minimal value of the functional F h,δt in L 2 δt (0, T ; U h ), norm of the control u h,δt L 2 δt (0,T ;U h ) , and distance to the target y h,δt (T ) E h in Case (b).

F. AMMAR KHODJA, F. CHOULY AND M. DUPREZ

Acknowledgements. The authors thank Assia Benabdallah and Franck Boyer for their interesting comments and suggestions. They thank as well the two referees for their remarks that helped to improve the paper.

This work was partially supported by Région de Franche-Comté (France).