Partial null controllability of parabolic linear systems

This paper is devoted to the partial null controllability issue of parabolic linear systems with n equations. Given a bounded domain in R N, we study the effect of m localized controls in a nonempty open subset 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 2x2 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, (1.1) 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 [4]). For example, in [11], 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.

System (1.1) 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 (1.1). The first of them is about cascade systems (see [20]). The authors prove the null controllability of System (1.1) 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 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 . . . A similar result on parabolic systems with cascade coupling matrices can be found in [1]. The null controllability of parabolic 3 × 3 linear systems with space/time dependent coefficients and non cascade structure is studied in [8] and [23] (see also [20]).
If A ∈ L(R n ) and B ∈ L(R m , R n ) (the constant case), it has been proved in [3] that System (1.1) is null controllable on the time interval 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 (1. 6) In [2] the authors prove first that, if there exists t 0 ∈ [0, T ] such that rank [A|B](t 0 ) = n, (1.7) then System (1.1) is null controllable on the time interval (0, T ). Secondly that System (1.1) 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. (1.8) 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 [25].
Our first result is the following: is equivalent to the Π p -null/approximate controllability on the time interval (0, T ) of System (1.1).
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 [2,20] where the authors consider a time-dependent second order elliptic operator L(t) given by and the uniform elliptic condition: there exists a 0 > 0 such that 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 in Ω, (1.12) 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).
, 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 (1.12) satisfies y(T ) ≡ 0 in Ω.
in Ω with Dirichlet boundary conditions and for all k, l ∈ N * , If the function α satisfies |α kl | C 1 e −C2|k−l| for all k, l ∈ N * , (1.13) for two positive constants C 1 > 0 and C 2 > b − a, then System (1.12) is Π 1 -null controllable for any open set ω ⊂ Ω.
We will not prove item (1) in Theorem 1.3, because it is a direct consequence of Theorem 1.2.
Concerning item (2), we remark that Condition (1.13) is equivalent to the existence of two constants C 1 > 0, C 2 > π such that, for all p ∈ N, 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 (1.14) 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 (5.28)), what seems true only for coupling terms α of the form (1.14). Thus α is not zero on the boundary.
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 [17]), one can prove that System (1.15) is null controllable when f satisfies Remark 5. Consider the same system as System (1.12) except that the control is now on the boundary, that is where y 0 , z 0 ∈ H −1 (0, π). In Theorem 5.1, we provide an explicit coupling function α for which the 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 (1.12) can be Π 1 -null controllable or not. But this system is Π 1 -approximately controllable for all α ∈ L ∞ (Ω): , 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 (1.12) satisfies This result is a direct consequence of the unique continuation property and existence/unicity of solutions for a single heat equation. Indeed System (1.12) is Π 1 -approximately controllable (see Proposition 2.1) if and only if for all φ 0 ∈ L 2 (Ω) the solution to the adjoint system If we assume that, for an initial data φ 0 ∈ L 2 (Ω), the solution to System (1.18) satisfies φ ≡ 0 in ω × (0, T ), then using Mizohata uniqueness Theorem in [24], 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 [5].
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 *      In [6], the authors have proved that the system 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 [10]. Consider now It is also proved in [6] that: If T > T 0 (α), then System (1.19) is null controllable at time T and if T < T 0 (α), then System (1.19) 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 [4]).
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 (1.12) in Section 5.3.

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.

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 in Ω, 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 .., r} (e j is the j-th element of the canonical basis of R n ).
The proof of this result uses a Carleman estimate (see [17]) and can be found in [2] or [20].

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 [4]). We detail below the observability for the Π p -controllability.
satisfies the observability inequality

Adjoint system
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 (1.1) at time t ∈ [0, T ]. For all t ∈ [0, T ], let us consider the operators S t and L t defined as follows

In view of the definition in (2.5) of S T and L
This is equivalent to That means In other words Corollary 2.1. Let us suppose that for all ϕ 0 ∈ L 2 (Ω) p , the solution ϕ to the adjoint System (2.3) satisfies the observability inequality (2.4). Then for all initial condition The proof is classical and will be omitted (estimate (2.11) can be obtained directly following the method developed in [16]).

Partial null controllability with constant coupling matrices
Let us consider the system in Ω, Let the natural number s be defined by 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.
in Ω, If P is constant, we have

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.
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 Proof of Theorem 1.1. Let us remark that Thus, for all l ∈ {s, s + 1, ..., n} and i ∈ {0, ..., s − 1}, there exist α l,i such that We first prove in (a) that condition (1.9) is sufficient, and then in (b) that this condition is necessary.
(a) Sufficiency part: Let us assume first that condition (1.9) holds. Then, using (3.7), we have Let be y 0 ∈ L 2 (Ω) n . We will study the Π p -null controllability of System (3.1) 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 (3.1). 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 ], 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 (3.10) Using (3.9) and (3.10), we can write 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 (3.8), P 11 is invertible and thus P (T ) also. Furthermore, since Let us suppose first that T * = 0. Since P (t) is an element of 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 (3.1) if and only if w := P (t) −1 y is the solution to System (3.3) where C, D are given by for all t ∈ [0, T ]. Using (3.6) and (3.10), we obtain where Then If now T * = 0, let y be the solution in W (0, T * ) n to System (3.1) 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 (3.1) 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 (3.1) 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 then, for this control u, y is the solution in W (0, T ) n to System (3.1). Moreover y satisfies Π s y(T ) ≡ 0 in Ω.  Thus, equation (3.15) yields Since rank [C|D] = rank [A|B] = s, we proceed as in Case 1 forward deduce that System (3.3) is Π s -null controllable, that is there exists a control u ∈ L 2 (Q T ) such that the solution w to System (3.3) satisfies Π s w(T ) ≡ 0 in Ω.
Moreover the matrix Q can be rewritten where Q 22 ∈ L(R n−p ). Thus 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.
Step 1. Let us suppose first that k 11 = ... = k 1s = 0 and rank We want to prove that, for some initial condition y 0 ∈ L 2 (Ω) n , a control u ∈ L 2 (Q T ) cannot be found such that the solution to System (3.1) satisfies y 1 (T ) ≡ 0 in Ω. Let us consider the matrix P ∈ L(R n ) defined by P := (B|...|A s−1 B|e 1 |e s+2 |...|e n ). with C 11 defined in (3.13). Then C can be rewritten as where C 12 ∈ L(R n−s , R s ) and C 22 ∈ L(R n−s ). Furthermore and with the Definition (3.17) of P we get 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 (3.3) satisfying w s+1 (T ) ≡ 0 in Ω. Therefore we apply Proposition 2.1 and prove that the observability inequality (2.4) 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 (3.3) 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 in Ω (3.19) satisfies the observability inequality But for all ϕ 0 s+1 ≡ 0 in Ω, the inequality (3.20) 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 [18]), we have clearly (ϕ s+1 (0), ..., ϕ n (0)) ≡ 0 in Ω.  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 (3.3) with w := P −1 y, y being a solution to System (3.1). We remark that if we denote by (k ij ) := [C|D], we havek 11 = ... =k 1s = 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 (3.3) satisfies Thus, with the definition of Q, for all control u in L 2 (Q T ) the solution y to System (3.1) satisfies Suppose Π p y(T ) ≡ 0 in Ω, then w 1 (T ) ≡ 0 in Ω and this contradicts (3.21).
As a consequence of Proposition 2.1, the Π p -null controllability implies the Π p -approximate controllability of System (3.3). If now Condition (1.9) is not satisfied, as for the Π p -null controllability, we can find a solution to System (3.19) such that φ 1 ≡ 0 in ω × (0, T ) and φ ≡ 0 in Q T and we conclude again with Proposition 2.1.
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 [2]. 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 We first prove in (a) that condition (1.9) is sufficient, and then in (b) that this condition is necessary. (a) Sufficiency part: Let us suppose first that (1.9) is satisfied. Let be y 0 ∈ L 2 (Ω) n . We will prove that we need only r forces to control System (3.1). 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, (3.24) whereB = (b l1 |b l2 | · · · |b lr ) ∈ L(R r , R n ). Using (1.9) and (3.23), 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 and for 1 i j r the matrices C ij ∈ L(R sj , R si ) are given by whereC 11 is defined in (3.30). Then C can be written as whereC 12 ∈ L(R s , R n−s ) andC 22 ∈ L(R n−s ). Furthermore, the matrix D can be written where D 1 ∈ L(R m , R s ). Using (3.34), we get 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 (3.3) satisfying w s+1 (T ) ≡ 0 in Ω. Therefore we apply Proposition 2.1 and prove that the observability inequality (2.4) 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 (3.3) 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 satisfies the observability inequality But for all ϕ 0 s+1 ≡ 0 in Ω, the inequality (3.37) 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 2.1, the Π p -null controllability implies the Π papproximate controllability of System (3.24). If Condition (1.9) is not satisfied, as for the Π p -null controllability, we can find a solution to System (3.36) 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 2.1.
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 (3.24) if and only if w := P (t) −1 y is the solution to System (3.3) where C, D are given by for all t ∈ [0, T ]. Using (4.2) and (4.5), we obtain and for 1 i j r, the matrices C ij ∈ C 0 ([0, T ]; ∈ L(R sj , R si )) are given here by   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.

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 in Ω, 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 (5.1) 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 (5.1) satisfies y(T ) ≡ 0 in Ω.

Example of controllability
In this subsection, we will provide an example of Π 1 -null controllability for System (5.1) with the help of the method of moments initially developed in [14]. 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 where φ 0 ∈ L 2 (0, π). For an initial data φ 0 ∈ L 2 (0, π) in adjoint System (5.2), we get with the notation q T := ω × (0, T ). Since (w k ) k 1 spans L 2 (0, π), System (5.1) 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 (5.2) satisfies the following equality where (φ k , ψ k ) is the solution to adjoint System (5.2) 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 (5.2): for all t ∈ [0, T ]. If we write: 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 ), (5.5)

Example of controllability
where, for all k, l ∈ N * , α kl is defined in (2). In (5.5) we implicitly used the convention: 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 −t) . With these expressions of φ k and ψ k , the equality (5.4) reads for all k 1 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,[7]) There exists f ∈ L 2 (0, π) such that Supp f ⊂ ω and for a constant β, one has inf where, for all k ∈ N * , f k := π 0 f w k dx.
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 (5.6), the Π 1 -null controllability of System (5.1) is reduced to find a solution γ ∈ L 2 (0, T ) to the following problem of moments: The function γ(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 * , With the Condition (1.13) on α, there exists a positive constant C T which do not depend on k such that for all k ∈ N * ∞ l=1 e −l 2 T +C2l C T e −C2k (5.10) and (5.11) Combining the three last inequalities (5.9)-(5.11), for all k ∈ N * l 1 where C T is a positive constant independent of k. Let ε ∈ (0, 1). Then, with Lemma 5.1, (5.8) and (5.12), there exists a positive constant C T,ε independent of k such that for all k ∈ N * Thus, using Lemma 5.2, for ε small enough and a positive constant C T,ε

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 where y is the solution to (5.1). We can find in [9] the argument relating the null/approximate controllability and this kind of functional. Using the Fenchel-Rockafellar theory (see [13] 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 (qT ) + ε 2 ϕ 0 2 L 2 (QT ) + y(T ; y 0 , 0), ϕ 0 L 2 (Ω) , where ϕ is the solution to the backward System (5.35). 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 (5.35) with the initial data ϕ(T ) = ϕ 0,ε . A simple computation leads to ∇J ε (ϕ 0 ) = Λϕ 0 + εϕ 0 + y(T ; y 0 , 0), with the Gramiam operator Λ defined as follows where w is the solution to the following backward and forward systems  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 [9] can be adapted to prove that where y ε is the solution to System (5.1) for the control u ε . System (5.1) 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) and (3)  Systems (5.1) and (5.35)-(5.36) 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 [9]) 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 fully-discretized 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 (5.1) and Gramiam Λ (used to the minimisation of F ǫ ), we refer to Section 3 in [9] and in [19, p. 37] respectively. The results are depicted Figure 1    As mentioned in the introduction of the present article (see Theorem 1.3), in both situations (a) and (b), System (5.1) is Π 1 -approximately controllable and we observe indeed in Figure 1 and 2 that the norm of the numerical solution to System (5.1) 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