Pierre Gaillard
email: pierre.gaillard@u-bourgogne.fr

Families of solutions to the KPI equation and the structure of their rational representations of order N

Keywords: numbers : 33Q55, 37K10, 47.10A-, 47.35.Fg, 47.54.Bd

come

Families of solutions to the KPI equation and the structure of their rational representations of order N

Introduction

We consider the Kadomtsev-Petviashvili equation (KP) which can be written in the form (4u t -6uu x + u xxx ) x -3u yy = 0,

subscripts x, y and t denoting partial derivatives. Kadomtsev and Petviashvili [START_REF] Kadomtsev | On the stability of solitary waves in weakly dispersing media[END_REF] first proposed that equation in 1970. That equation is considered as a model, for example, for surface and internal water waves [START_REF] Ablowitz | On the evolution of packets of water waves[END_REF], and in nonlinear optics [START_REF] Pelinovsky | Self-focusing of plane dark solitons in nonlinear defocusing media[END_REF]. In 1974 Dryuma showed how the KP equation could be written in Lax form [START_REF] Dryuma | On analytical solutions of the two-dimensional Kortewegde Vries equation[END_REF]. Zakharov extended the inverse scattering transform (IST) to this KPI equation, and obtained several exact solutions.

The first rational solutions were found by Manakov, Zakharov, Bordag and Matveev published in november 1977 [START_REF] Manakov | Twodimensional solitons of the Kadomtsev-Petviashvili equation and their interaction[END_REF] and at the same time by Krichever published in january 1978 [START_REF] Krichever | Rational solutions of the Kadomtcev-Petviashvili equation and integrable systems of n particules on a line[END_REF].

From the end of the seventies, a lot of methods have been carried out to solve that equation. Krichever constructed for the first time in 1976 published in [START_REF] Krichever | Novikov Holomorphic bundles over riemann surfaces and the kadomtsev-petviashvili equation[END_REF] the solutions to KPI given in terms of Riemann theta functions in the frame of algebraic geometry. Practically at the same time, a more general study in this frame was realized by Dubrovin [START_REF]Dubrovin Theta functions and non-linear equations[END_REF].

Various researches were conducted and more general rational solutions of the KPI equation were obtained. Among many works, one can quote of the studies of Krichever in 1978 [START_REF] Krichever | Elliptic solutions of the KP equation and integrable systems of particles[END_REF], Satsuma and Ablowitz in 1979 [START_REF] Satsuma | Two-dimensional lumps in nonlinear dispersive systems[END_REF], Matveev in 1979 [START_REF] Matveev | Darboux transformation and explicit solutions of the Kadomtcev-Petviaschvily equation depending on functional parameters[END_REF], Freeman and Nimmo in 1983 [START_REF] Freeman | Nimmo Rational solutions of the KdV equation in wronskian form[END_REF][START_REF] Freeman | The use of Bäcklund transformations in obtaining N-soliton solutions in wronskian form[END_REF], Matveev in 1987 [START_REF] Matveev | Salle New families of the explicit solutions of the Kadomtcev-Petviaschvily equation and their application to Johnson equation[END_REF], Pelinovsky and Stepanyants in 1993 [START_REF] Pelinovsky | New multisolitons of the Kadomtsev-Petviashvili equation[END_REF], Pelinovsky in 1994 [START_REF] Pelinovsky | Rational solutions of the Kadomtsev-Petviashvili hierarchy and the dynamics of their poles. I. New form of a general rational solution[END_REF], Ablowitz, Villarroel, Chakravarty, Trubatch [START_REF] Ablowitz | Solutions to the time dependent schrödinger and the Kadomtsev-Petviashvili equations[END_REF][START_REF] Villarroel | On the discrete spectrum of the nonstationary Schrdinger equation and multipole lumps of the Kadomtsev-Petviashvili I equation[END_REF][START_REF] Ablowitz | A novel class of solution of the non-stationary Schrödinger and the KP equations[END_REF] in 1997-2000, Biondini and Kodama [START_REF] Biondini | On a family of solutions of the KadomtsevPetviashvili equation which also satisfy the Toda lattice hierarchy[END_REF][START_REF] Kodama | Young diagrams and N solitons solutions to the KP equation[END_REF][START_REF] Biondini | Line Soliton Interactions of the Kadomtsev-Petviashvili Equation[END_REF] in 2003-2007.

In the following, we recall the results of the author about the representations of solutions to the KPI equation. We have expressed the solutions in terms of Fredholm determinants of order 2N depending on 2N -1 parameters. We have also given another representation in terms of wronskians of order 2N with 2N -1 parameters. These representations allow to obtain an infinite hierarchy of solutions to the KPI equation, depending on 2N -1 real parameters . We have used these results to build rational solutions to the KPI equation, making a parameter to 0 tend to 0. Rational solutions of order N depending on 2N -2 parameters without the presence of a limit have been constucted. These families depending on 2N -2 parameters for the N -th order can be written as a ratio of two polynomials of x, y and t of degree 2N (N + 1). We prove that the maximum of the modulus of those solutions at order N is equal to 2(2N + 1) 2 . That provides an effective method to build an infinite hierarchy of rational solutions of order N depending on 2N -2 real parameters. We present here the representations of their modulus in the plane of the coordinates (x, y) according to the 2N -2 real parameters a i and b 1 and time t for N between 2 and 6.

Rational solutions to KPI equation of order

N depending on 2N -2 parameters 2.1 Families of rational solutions of order N depending on 2N -2 parameters

We need to define some notations. First one defines real numbers λ j such that -1 < λ ν < 1, ν = 1, . . . , 2N depending on a parameter ǫ which will be intended to tend towards 0; they can be written as

λ j = 1 -2ǫ 2 j 2 , λ N +j = -λ j , 1 ≤ j ≤ N, (2)
The terms κ ν , δ ν , γ ν , τ ν and x r,ν are functions of λ ν , 1 ≤ ν ≤ 2N ; they are defined by the formulas :

κ j = 2 1 -λ 2 j , δ j = κ j λ j , γ j = 1-λj 1+λj ,; x r,j = (r -1) ln γj -i γj +i , r = 1, 3, τ j = -12iλ 2 j 1 -λ 2 j -4i(1 -λ 2 j ) 1 -λ 2 j , κ N +j = κ j , δ N +j = -δ j , γ N +j = γ -1 j , x r,N +j = -x r,j , , τ N +j = τ j j = 1, . . . , N. (3)
e ν 1 ≤ ν ≤ 2N are defined in the following way :

e j = 2i 1/2 M -1 k=1 a k (je) 2 k+1 -i 1/2 M -1 k=1 b K (je) 2 k+1 , e N +j = 2i 1/2 M -1 k=1 a k (je) 2 k+1 + i 1/2 M -1 k=1 b k (je) 2 k+1 , 1 ≤ j ≤ N, a k , b k ∈ R, 1 ≤ k ≤ N -1. (4)
ǫ ν , 1 ≤ ν ≤ 2N are real numbers defined by :

ǫ j = 1, ǫ N +j = 0 1 ≤ j ≤ N. (5)
Let I be the unit matrix and D r = (d jk ) 1≤j,k≤2N the matrix defined by :

d νµ = (-1) ǫν η =µ γ η + γ ν γ η -γ µ exp(iκ ν x -2δ ν y + τ ν t + x r,ν + e ν ). (6)
Then we recall the following result

where n(x, y, t) = det(I + D 3 (x, y, t)),

d(x, y, t) = det(I + D 1 (x, y, t)), (8)
and D r = (d jk ) 1≤j,k≤2N the matrix

d νµ = (-1) ǫν η =µ γ η + γ ν γ η -γ µ exp(iκ ν x -2δ ν y + τ ν t + x r,ν + e ν ). ( 10
)
is a solution to the KPI equation ( 1), dependent on 2N -

1 parameters a k , b k , 1 ≤ k ≤ N -1 and ǫ.
We recall a second result on the solutions to KPI equation obtained recently by the author in terms of wronskians. We need to define the following notations : [START_REF] Matveev | Darboux transformation and explicit solutions of the Kadomtcev-Petviaschvily equation depending on functional parameters[END_REF] with the arguments

φ r,ν = sin Θ r,ν , 1 ≤ ν ≤ N, φ r,ν = cos Θ r,ν , N + 1 ≤ ν ≤ 2N, r = 1, 3,
Θ r,ν = κν x 2 + iδ ν y -i xr,ν 2 -i τν 2 t + γ ν w -i eν 2 , 1 ≤ ν ≤ 2N. ( 12
)
We denote W r (w) the wronskian of the functions φ r,1 , . . . , φ r,2N defined by

W r (w) = det[(∂ µ-1 w φ r,ν ) ν, µ∈[1,...,2N ] ]. ( 13
)
We consider the matrix D r = (d νµ ) ν, µ∈[1,...,2N ] defined in [START_REF] Satsuma | Two-dimensional lumps in nonlinear dispersive systems[END_REF]. Then we have the following statement2

Theorem 2.2 The function v defined by

v(x, y, t) = -2 |W 3 (φ 3,1 , . . . , φ 3,2N )(0)| 2 (W 1 (φ 1,1 , . . . , φ 1,2N )(0)) 2 is a solution to the KPI equation depending on 2N -1 real parameters a k , b k 1 ≤ k ≤ N -1 and ǫ, with φ r ν defined by φ r,ν (w) = sin( κν x 2 + iδ ν y -i xr,ν 2 -i τν 2 t + γ ν w -i eν 2 ), 1 ≤ ν ≤ N, φ r,ν (w) = cos( κν x 2 + iδ ν y -i xr,ν 2 -i τν 2 t + γ ν w -i eν 2 ), N + 1 ≤ ν ≤ 2N, r = 1, 3,
κ ν , δ ν , x r,ν , γ ν , e ν being defined in(3), ( 2) and (4).

From those two preceding results, we construct rational solutions to the KPI equation as a quotient of two determinants. We use the following notations :

X ν = κ ν x 2 + iδ ν y -i x 3,ν 2 -i τ ν 2 t -i e ν 2 , Y ν = κ ν x 2 + iδ ν y -i x 1,ν 2 -i τ ν 2 t -i e ν 2 ,
for 1 ≤ ν ≤ 2N , with κ ν , δ ν , x r,ν defined in [START_REF] Pelinovsky | Self-focusing of plane dark solitons in nonlinear defocusing media[END_REF] and parameters e ν defined by (4). We define the following functions :

ϕ 4j+1,k = γ 4j-1 k sin X k , ϕ 4j+2,k = γ 4j k cos X k , ϕ 4j+3,k = -γ 4j+1 k sin X k , ϕ 4j+4,k = -γ 4j+2 k cos X k , (14)
for 1 ≤ k ≤ N , and

ϕ 4j+1,N +k = γ 2N -4j-2 k cos X N +k , ϕ 4j+2,N +k = -γ 2N -4j-3 k sin X N +k , ϕ 4j+3,N +k = -γ 2N -4j-4 k cos X N +k , ϕ 4j+4,N +k = γ 2N -4j-5 k sin X N +k , (15)
for 1 ≤ k ≤ N . We define the functions ψ j,k for 1 ≤ j ≤ 2N , 1 ≤ k ≤ 2N in the same way, the term X k is only replaced by Y k .

ψ 4j+1,k = γ 4j-1 k sin Y k , ψ 4j+2,k = γ 4j k cos Y k , ψ 4j+3,k = -γ 4j+1 k sin Y k , ψ 4j+4,k = -γ 4j+2 k cos Y k , (16)
for 1 ≤ k ≤ N , and

ψ 4j+1,N +k = γ 2N -4j-2 k cos Y N +k , ψ 4j+2,N +k = -γ 2N -4j-3 k sin Y N +k , ψ 4j+3,N +k = -γ 2N -4j-4 k cos Y N +k , ψ 4j+4,N +k = γ 2N -4j-5 k sin Y N +k , (17)
for 1 ≤ k ≤ N .

The following ratio q(x, t)

:= W 3 (0) W 1 (0)
can be written as

q(x, t) = ∆ 3 ∆ 1 = det(ϕ j,k ) j, k∈[1,2N ] det(ψ j,k ) j, k∈[1,2N ] . (18)
Then we get the following result3 :

Theorem 2.3 The function v defined by v(x, y, t) = -2 | det((n jk) j,k∈[1,2N ] )| 2 det((d jk) j,k∈[1,2N ] ) 2 (19)
is a rational solution to the KPI equation [START_REF] Kadomtsev | On the stability of solitary waves in weakly dispersing media[END_REF].

(4u t -6uu x + u xxx ) x -3u yy = 0,
where

n j1 = ϕ j,1 (x, y, t, 0), 1 ≤ j ≤ 2N n jk = ∂ 2k-2 ϕj,1 ∂ǫ 2k-2 (x, y, t, 0), n jN +1 = ϕ j,N +1 (x, y, t, 0), 1 ≤ j ≤ 2N n jN +k = ∂ 2k-2 ϕj,N+1 ∂ǫ 2k-2
(x, y, t, 0),

d j1 = ψ j,1 (x, y, t, 0), 1 ≤ j ≤ 2N d jk = ∂ 2k-2 ψj,1 ∂ǫ 2k-2 (x, y, t, 0), d jN +1 = ψ j,N +1 (x, y, t, 0), 1 ≤ j ≤ 2N d jN +k = ∂ 2k-2 ψj,N+1 ∂ǫ 2k-2 (x, y, t, 0), 2 ≤ k ≤ N, 1 ≤ j ≤ 2N (20)
The functions ϕ and ψ are defined in ( 14),( 15), ( 16), [START_REF] Ablowitz | Solutions to the time dependent schrödinger and the Kadomtsev-Petviashvili equations[END_REF].

. The structure of the solutions to the KPI equation is given by the following result4 :

Theorem 2.4 The function v defined by v(x, y, t) = -2 det((n jk) j,k∈[1,2N ] ) 2 (det((d jk) j,k∈[1,2N ] )) 2 (21)
is a rational solution to the KPI equation ( 1) quotient of two polynomials n(x, y, t) and d(x, y, t) depending on 2N -2 real parameters a j and b j , 1 ≤ j ≤ N -1. n and d are polynomials of degrees 2N (N + 1) in x, y and t.

The terms n jk and d jk are defined by [START_REF] Biondini | On a family of solutions of the KadomtsevPetviashvili equation which also satisfy the Toda lattice hierarchy[END_REF] and the functions ϕ and ψ are defined in ( 14), ( 15), ( 16), [START_REF] Ablowitz | Solutions to the time dependent schrödinger and the Kadomtsev-Petviashvili equations[END_REF].

3 The highest amplitude of the modulus of the N-order solution to the KPI equation equal to 2(2N + 1) 2

Another wronskian representation of solutions to KPI equation

We have recalled in the previous section a recent result obtained by the author (2.2).

We choose here to give another representation of the solutions to the KPI equation depending only on terms γ ν , 1 ≤ ν ≤ 2N . For this, we express the terms κ ν , δ ν , τ ν and x r,ν in function of γ ν , for 1 ≤ ν ≤ 2N and we obtain :

κ j = 4γj (1+γ 2 j ) , δ j = 4γj (1-γ 2 j ) (1+γ 2 j ) 2 , x r,j = (r -1) ln γj -i γj +i , τ j = -8iγj (3-2γ 2 j +3γ 4 j ) (1+γ 2 j ) 3 , 1 ≤ j ≤ N, κ j = 4γj (1+γ 2 j ) , δ j = - 4γj (1-γ 2 j ) (1+γ 2 j ) 2 , x r,j = (r -1) ln γj +i γj -i , τ j = -8iγj (3-2γ 2 j +3γ 4 j ) (1+γ 2 j ) 3 , N + 1 ≤ j ≤ 2N.
We have the following representation :

Theorem 3.1 The function v defined by v(x, y, t) = -2 det[(∂ µ-1 w φ3,ν (0)) ν, µ∈[1,...,2N ] ] 2 (det[(∂ µ-1 w φ1,ν (0)) ν, µ∈[1,...,2N ] ]) 2 (22)
is solution to the KPI equation ( 1).

The functions φr,ν are defined by φr,j (w) = sin

2γj (1+γ 2 j ) x + i 4γj (1-γ 2 j ) (1+γ 2 j ) 2 t -4 γj (3-2γ 2 j +3γ 4 j ) (1+γ 2 j ) 3 y -i (r-1) 2 ln γj -i γj +i + γ j w -i ej 2 , φr,N+j (w) = cos 2γj (1+γ 2 j ) x -i 4γj (1-γ 2 j ) (1+γ 2 j ) 2 t -4 γj (3-2γ 2 j +3γ 4 j ) (1+γ 2 j ) 3 + i (r-1) 2 ln γj -i γj +i + 1 γj w -i eN+j 2 , 1 ≤ j, ≤ N, with e j = 2i( 1/2 M -1 k=1 a k (je) 2 k-1 -i 1/2 M -1 k=1 b K (je) 2 k-1 ), e N +j = 2i( 1/2 M -1 k=1 a k (je) 2 k-1 + i 1/2 M -1 k=1 b k (je) 2 k-1 ), 1 ≤ j ≤ N, a k , b k ∈ R, 1 ≤ k ≤ N. (23)
Remark 3.1 In the formula [START_REF] Biondini | Line Soliton Interactions of the Kadomtsev-Petviashvili Equation[END_REF], the determinants det[(∂ µ-1 w f ν (0)) ν, µ∈[1,...,2N ] ] are the wronskians of the functions f 1 , . . . , f 2N evaluated in w = 0. In particular ∂ 0 w f ν means f ν .

The highest amplitude of the modulus of the solution to the KPI equation

There is any freedom to choose γ j in such a way that the conditions on λ j are checked. In order to get the more simple expressions in the determinants, we choose particular solutions in the previous families.

Here we choose γ j = jǫ for 1 ≤ j ≤ N as simple as possible in order to have the conditions on λ j checked.

Theorem 3.2

The function v 0 defined by

v 0 (x, y, t) = -2 | det((n jk ) j,k∈[1,2N ] )| 2 det((d jk ) j,k∈[1,2N ] ) 2 (aj =bj =0, 1≤j≤N -1) (24)
is the solution of order N solution to the KPI equation ( 1) whose highest amplitude in modulus is equal to 2(2N + 1) 2 .

Remark 3.2 In (24), the matrices (n jk ) j,k∈ [1,2N ] and (d jk ) j,k∈ [1,2N ] are defined in [START_REF] Biondini | On a family of solutions of the KadomtsevPetviashvili equation which also satisfy the Toda lattice hierarchy[END_REF].

4 Study of the patterns of the modulus of the rational solutions to the KPI equation in function of parameters and time

We have explicitly constructed rational solutions to the KPI equation of order N depending on 2N -2 parameters for 1 ≤ N ≤ 6.

In the following, we only give patterns of the modulus of the solutions in the plane (x, y) of coordinates in function of the parameters a i , and b i , for 1 ≤ i ≤ N -1 for 2 ≤ N ≤ 6, and time t.

Case N = 3

Figure 5. Solution of order 3 to KPI, on the left for t = 0; in the center for t = 0, 01; on the right for t = 0, 1; all the parameters are equal to 0.

Figure 6. Solution of order 3 to KPI, on the left for t = 0, 2; in the center for t = 10 2 ; on the right for t = 10 3 ; all the parameters are equal to 0.

Conclusion

From the previous representations of the solutions to the KPI equation given by the author in terms of Fredholm determinants of order 2N depending on 2N -1 real parameters and in terms of wronskians of order 2N depending on 2N -1 real parameters, we succeed in obtaining rational solutions to the KPI equation depending on 2N -2 real parameters. These solutions can be expressed in terms of a ratio of two polynomials of degree 2N (N + 1) in x, y and t. The maximum of the modulus of those solutions is equal to 2(2N + 1) 2 . That gives a new approach to find explicit solutions for higher orders and try to describe the structure of those rational solutions.

In the (x, y) plane of coordinates, different structures appear.

In the case N = 1, one obtains a peak which the height decreases very quickly as t increases.

For N = 2, the formation of three peaks is obtained when the parameters a 1 or b 1 are not equal to 0.

In the case N = 3, for a 1 = 0 or b 1 = 0 and the other parameters equal to zero, we obtain a triangle with 6 peaks; for a 2 = 0 or b 2 = 0, and other parameters equal to zero, we obtain a concentric rings of 5 peaks with a peak in the center.

For N = 4, for a 1 = 0 or b 1 = 0 and the other parameters equal to zero, we obtain a triangle with 10 peaks; for a 2 = 0 or b 2 = 0, and other parameters equal to zero, we obtain two concentric rings of 5 peaks on each of them; in the last case, when a 3 = 0 or b 3 = 0, and the other parameters equal to zero, we obtain one ring with 7 peaks.

In the case N = 5, for a given t, when one parameter grows and the other ones are equal to 0 we obtain triangular or rings or concentric rings. There are four types of patterns. For a 1 = 0 or b 1 = 0, and other parameters equal to zero, we obtain a triangle with 15 peaks. For a 2 = 0 or b 2 = 0, and other parameters equal to zero, we obtain three concentric rings of 5 peaks on each of them. For a 3 = 0 or b 3 = 0, and other parameters equal to zero, we obtain two concentric rings of 7 peaks on each of them with a central peak; in the last case, when a 4 = 0 or b 4 = 0, and other parameters equal to zero, we obtain one ring with 9 peaks with the lump L 3 in the center.

In the last case studied N = 6, for a given t, when one parameter grows and the other ones are equal to 0 we obtain triangular, rings or concentric rings. There are five types of patterns. For a 1 = 0 or b 1 = 0, and other parameters equal to zero, we obtain a triangle with 21 peaks. For a 2 = 0 or b 2 = 0, and other parameters equal to zero, we obtain three concentric rings of 5, 10, 5 peaks respectively. For a 3 = 0 or b 3 = 0, and other parameters equal to zero, we obtain three concentric rings of 7 peaks on each of them. In the case where a 4 = 0 or b 4 = 0, and other parameters equal to zero, we obtain two rings with 9 peaks with the lump L 4 in the center. In the last case where a 5 = 0 or b 5 = 0, and other parameters equal to zero, we obtain one ring with 11 peaks with the lump L 4 in the center.

It will be relevant to go on this study for higher orders to try to understand the structure of those rational solutions.

4. 1 Case N = 1 Figure 1 .

 111 Figure 1. Solution of order 1 to KPI, on the left for t = 0; in the center for t = 10 4 ; on the right for t = 10 8 .

Figure 2 .

 2 Figure 2. Solution of order 2 to KPI for t = 0, on the left a 1 = 0, b 1 = 0, ; in the center a 1 = 10, b 1 = 0, ; on the right a 1 = 10, b 1 = 10.

Figure 3 .

 3 Figure 3. Solution of order 2 to KPI for t = 0, on the left a 1 = 2 , b 1 = 0; in the center a 1 = 10 4 , b 1 = 0; on the right for t = 10, a 1 = , b 1 = 0.

Figure 4 .

 4 Figure 4. Solution of order 2 to KPI, on the left for t = 5, a 1 = 0, b 1 = 0; in the center for t = 10, a 1 = 0, b 1 = 0; on the right for t = 100, a 1 = 0, b 1 = 0.

Figure 7 .

 7 Figure 7. Solution of order 3 to KPI, on the left for a 1 = 10 3 ; in the center for b 1 = 10 3 ; on the right for a 2 = 10 6 ; here t = 0.

Figure 8 .

 8 Figure 8. Solution of order 3 to KPI, on the left for t = 0, b 2 = 10 6 ; in the center for t = 0, 01, a 1 = 10 3 all the other parameters are equal to 0; on the right for t = 0, 1, b 1 = 10 3 all the parameters are equal to 0.

4. 4 Case N = 4 Figure 9 .

 449 Figure 9. Solution of order 4 to KPI, on the left for t = 0; in the center for t = 0, 01; on the right for t = 0, 1; all the parameters to equal to 0.

Figure 10 .

 10 Figure10. Solution of order 4 to KPI, on the left for t = 0, 2; in the center for t = 10; on the right for t = 50; all the parameters to equal to 0.

Figure 11 .

 11 Figure 11. Solution of order 4 to KPI for t = 0, on the left for a 1 = 10 3 ; in

Figure 12 .

 12 Figure 12. Solution of order 4 to KPI for t = 0, on the left for b 2 = 10 6 ; in the center for a 3 = 10 9 ; on the right for b 3 = 10 9 ; all the other parameters to equal to 0.

4. 5 Case N = 5 Figure 13 .

 5513 Figure[START_REF] Freeman | The use of Bäcklund transformations in obtaining N-soliton solutions in wronskian form[END_REF]. Solution of order 5 to KPI, on the left for t = 0; in the center for t = 0, 01; on the right for t = 0, 1; all parameters equal to 0.

Figure 14 .

 14 Figure 14. Solution of order 5 to KPI, on the left for t = 0, 2; in the center for t = 20; on the right for t = 50; all parameters equal to 0.

Figure 15 .

 15 Figure 15. Solution of order 5 to KPI for t = 0, on the left for a 1 = 10 4 ; in the center for b 1 = 10 4 ; on the right for a 2 = 10 6 ; all other parameters equal to 0.

Figure 16 .

 16 Figure 16. Solution of order 5 to KPI for t = 0, on the left for b 2 = 10 6 ; in the center for a 3 = 10 8 ; on the right for b 3 = 10 8 ; all other parameters equal to 0.

Figure 17 .

 17 Figure 17. Solution of order 5 to KPI for t = 0, on the left for a 4 = 10 8 ; in the center for b 4 = 10 8 ; on the right for b 4 = 10 8 , sight on top; all other parameters equal to 0.

4. 6 Case N = 6 Figure 18 .

 6618 Figure 18. Solution of order 6 to KPI, on the left for t = 0; in the center for t = 0, 01; on the right for t = 0, 1; all parameters equal to 0.

Figure 19 .

 19 Figure[START_REF] Ablowitz | A novel class of solution of the non-stationary Schrödinger and the KP equations[END_REF]. Solution of order 6 to KPI, on the left for t = 0, 2; in the center for t = 3; on the right for t = 10; all parameters equal to 0.

Figure 20 .

 20 Figure 20. Solution of order 6 to KPI, on the left for a 1 = 10 3 ; in the center for b 1 = 10 3 ; on the right for b 1 = 10 6 ; all parameters equal to 0 and t = 0.

Figure 21 .

 21 Figure 21. Solution of order 6 to KPI, on the left for a 2 = 10 6 , sight on top; in the center for b 2 = 10 6 ; on the right for b 2 = 10 6 , sight on top; all parameters equal to 0 and t = 0.

Figure 22 .

 22 Figure 22. Solution of order 6 to KPI, on the left for a 3 = 10 8 ; in the center for a 3 = 10 7 ; on the right for a 3 = 10 9 ; all parameters equal to 0 and t = 0.

Figure 23 .

 23 Figure 23. Solution of order 6 to KPI, on the left for b 4 = 10 9 ; in the center for a 5 = 10 9 ; on the right for b 5 = 10 9 ; all parameters equal to 0 and t = 0.

The proof of this result has been given by the author[START_REF] Gaillard | Rational solutions to the KPI equation and multi rogue waves[END_REF][START_REF] Gaillard | Fredholm and Wronskian representations of solutions to the KPI equation and multi-rogue waves[END_REF]

The proof of this result has been given[START_REF] Gaillard | Rational solutions to the KPI equation and multi rogue waves[END_REF][START_REF] Gaillard | Fredholm and Wronskian representations of solutions to the KPI equation and multi-rogue waves[END_REF].

This result has been proved by the author and submitted to a review.

This result has been also proved by the author and submitted to a review.


Pierre Lissy
email: lissy@ceremade.dauphine.fr.

Ionel Rovent ¸a
email: ionelroventa@yahoo.com

Optimal approximation of internal controls for a wave-type problem with fractional Laplacian using finite-difference method

Keywords: fractional Laplacian, hyperbolic equations, control approximation, moment problem, biorthogonal families. Mathematical subject codes: 35R11, 93B05, 30E05, 65M06

We consider a finite-difference semi-discrete scheme for the approximation of internal controls of a one-dimensional evolution problem of hyperbolic type involving the spectral fractional Laplacian. The continuous problem is controllable in arbitrary small time. However, the high frequency numerical spurious oscillations lead to a loss of the uniform (with respect to the mesh size) controllability property of the semi-discrete model in the natural setting. For all initial data in the natural energy space, if we filter the high frequencies of these initial data in an optimal way, we restore the uniform controllability property in arbitrary small time. The proof is mainly based on a (non-classic) moment method.

Introduction 1.Motivation

In many realistic models coming from different branches of sciences (e.g. physics, chemistry, engineering), fractional Laplacians naturally arise to describe various phenomena.

More specifically, since the seminal paper by Richardson on turbulence diffusion [START_REF] Richardson | Atmospheric diffusion shown on a distance-neighbour graph[END_REF], systems with weaker/strongly diffusivity (subdiffusion, superdiffusion) modelled by fractional equations are frequently used and are usually related with Brownian diffusion and the deviation of the statistics of fluctuation from the Gaussian law, see e.g. [START_REF] Dubkov | Levy superflight diffusion: an introduction[END_REF][START_REF] Metzler | The random walk's guide to anomalous diffusion: A fractional dynamics approach[END_REF][START_REF] Sokolov | Fractional kinetics[END_REF] and the references therein.

Notably, examples of fractional parabolic dynamical system in the subdiffusive regime include (but Let us mention that many definitions of the fractional Laplacians co-exist (see e.g. [START_REF] Servadei | On the spectrum of two different fractional operators[END_REF]). However, they basically have the same behaviour at high frequency, meaning that choosing one definition instead of another should not change the controllability properties of the system under study, even if it may cause some additional technical difficulties. Here, for the sake of simplicity, we will consider the spectral definition of the fractional Laplace operator, which has the great advantage to enjoy explicit eigenfunctions and eigenvalues.

On the other hand, the approximation of controls of PDEs is a very popular direction of research and has attracted a lot of attention these last twenty years, due to the necessity of understanding how to achieve effective implementation of controls problems. This study has been initialized in [START_REF] Glowinski | A numerical approach to the exact boundary controllability of the wave equation (I). Dirichlet controls: Description of the numerical methods[END_REF] for the boundary controls of the wave equation and in [START_REF] Leon | Boundary controllability of the finite-difference space semi-discretizations of the beam equation[END_REF] concerning the boundary controls beam equation (these equations appear to be limiting cases of the present study, in a sense explained in details afterwards). However, to our knowledge, the question of the approximation of controls in the case of fractional wave equations has never been investigated so far, despite its interest from a theoretical and applicative point of view. To conclude, let us mention that a numerical study of the one-dimensional fractional heat equation with finite element method has been performed in [START_REF] Biccari | Controllability of a one-dimensional fractional heat equation: theoretical and numerical aspects[END_REF].

The aim of the present paper is to study the semi-discrete approximation of the controls of a one-dimensional evolution problem of hyperbolic type involving a family of spectral fractional Laplacian. Note that a more realistic framework would be to consider a fully discretized (i.e. in space and time) version of the problem. However, in the case of conservative systems, the study of a semi-discrete version is sufficient to derive also a result for a fully discretized version of the system under study, involving some natural CFL-like condition on both time and space discretization steps (for more details see [START_REF] Ervedoza | On the observability of time-discrete conservative linear systems[END_REF] and notably Section 4.2 of this article).

Mathematically speaking, our model can be seen as an intermediate case between the cases of the wave equation and the beam equation. We consider the simplest numerical approximation of such an equations as a finite difference semi-discrete scheme for the approximation of internal controls. It is well-known that, due to the high frequency numerical spurious oscillations, the uniform (with respect to the mesh-size) controllability property of the semi-discrete model fails in the natural setting. Consequently, the convergence of the approximate internal controls corresponding to an initial data in the energy space cannot be guaranteed.

Many possibilities have been proposed to overcome this difficulty: a Tychonov regularization of the HUM cost functional (see [START_REF] Glowinski | A numerical approach to the exact boundary controllability of the wave equation (I). Dirichlet controls: Description of the numerical methods[END_REF][START_REF] Zuazua | Propagation, observation, and control of waves approximated by finite difference methods[END_REF]), a change of the numerical scheme (mixed finite elements [START_REF] Castro | Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method[END_REF], vanishing viscosity [START_REF] Bugariu | Approximation of the controls for the beam equation with vanishing viscosity[END_REF][START_REF] Micu | Uniform boundary controllability of a semidiscrete 1-D wave equation with vanishing viscosity[END_REF] and other type of finite difference schemes [START_REF] Münch | Famille de schemas implicites uniformement controlables pour l'equation des ondes 1-D. (French.English, French summary) [Family of implicit schemes uniformly controllable for the 1-D wave equation[END_REF]), the introduction of non-uniform meshes ( [START_REF] Ervedoza | Observability properties of a semi-discrete 1D wave equation derived from a mixed finite element method on nonuniform meshes[END_REF][START_REF] Ervedoza | Numerical meshes ensuring uniform observability of one-dimensional waves: construction and analysis[END_REF]), an approximation of discrete controls [START_REF] Cîndea | An approximation method for exact controls of vibrating systems[END_REF] (which does not lead exactly the discrete solution to zero, but converges to an exact control of the continuous problem), and finally an appropriate filtering technique, introduced in [START_REF] Micu | Uniform boundary controllability of a semi-discrete 1-D wave equation[END_REF] and notably used in [START_REF] Cîndea | Boundary controllability for finite-differences semidiscretizations of a clamped beam equation[END_REF][START_REF] Micu | Approximation of the controls for the linear beam equation[END_REF][START_REF] Lissy | Optimal filtration for the approximation of boundary controls for the one-dimensional wave equation using finite-difference method[END_REF] in the context of wave or beam equation, which consists in relaxing the control requirement by controlling only the low-frequency part of the solution. This later approach will be considered in this paper.

Our approach consider a filtering technique which cut the frequencies of the initial conditions. Even if the initial condition is filtered, the control will excite all frequencies. This creates a lot of technical difficulties, because the spectral is not uniform with respect to the discretization step h (see Remark 1.1). For more general uniform controllability results by using filtered spaces and resolvent estimates, the interested reader is referred to [START_REF] Ervedoza | Spectral conditions for admissibility and observability of wave systems: applications to finite element schemes[END_REF][START_REF] Ervedoza | Observability in arbitrary small time for discrete conservative linear systems, Some Problems in Nonlinear Hyperbolic Equations[END_REF][START_REF] Miller | Resolvent conditions for the control of unitary groups and their approximations[END_REF]. Most of the above results treated the boundary controllability problems and we have seen that the negative results are due to the bad numerical approximation of the high eigenmodes. Since we are dealing with a fractional Laplacian problem of order α ∈ (1, 2), and the question of prescribing non-homogeneous boundary conditions for these kinds of fractional Laplace operators is a tricky question (see e.g. [START_REF] Yuan | Boundary Problems for Fractional Laplacians[END_REF]), we will not consider the case of boundary controls here, but rather the case when the control acts in the interior of the domain.

In the case α = 2 (i.e. the beam equation), the spurious numerical high eigenmodes do not play such an important role as in the case of boundary controls and the uniform controllability property holds automatically. For more details, see [START_REF] Bugariu | Approximation of the controls for the beam equation with vanishing viscosity[END_REF]Section 7].

The case α ∈ [1, 2) is more intricate. The best result in the case α = 1 (i.e. the wave equation) has recently been obtained by the authors in [START_REF] Lissy | Optimal filtration for the approximation of boundary controls for the one-dimensional wave equation using finite-difference method[END_REF] for boundary controls (the same result would hold for internal controls). It is notably proved an optimal result with respect to the range of filtration of the initial condition, together with precise (but non-optimal) estimates on the minimal time of uniform controllability. Even if the spirit of the present article is quite similar to [START_REF] Lissy | Optimal filtration for the approximation of boundary controls for the one-dimensional wave equation using finite-difference method[END_REF], we would like to emphasize that the present study is more intricate and is far from being a direct consequence of the computations made in [START_REF] Lissy | Optimal filtration for the approximation of boundary controls for the one-dimensional wave equation using finite-difference method[END_REF]. This is due to the following facts:

• Contrary to the corresponding product in [START_REF] Lissy | Optimal filtration for the approximation of boundary controls for the one-dimensional wave equation using finite-difference method[END_REF], it is not clear at all that the product R m (x) is uniformly bounded with respect to the discretization step h and x in the all range |x| (2/h) α . Notably, it can be proved numerically that the product S m is not bounded in the previous range. Hence, we had to understand what kind of estimates are needed, and to exhibit some very precise (and quite tricky to prove, see Section 2.2) estimates in this range, that depend both on x, h and the discrete eigenvalues (see Remark 2.2), and are difficult to handle. These estimates are likely to be non-optimal but are enough for our purpose.

• The use of the Paley-Wiener Theorem requires to compensate the explosion of the product R m by an adequate multiplier. Because of the bad behaviour of the product R m emphasized above, the construction given in [START_REF] Lissy | Optimal filtration for the approximation of boundary controls for the one-dimensional wave equation using finite-difference method[END_REF] is not sufficient to ensure the uniform boundedness of the discrete control with respect to h. Here, we use a new construction of the multiplier in the context of semi-discrete control problems. This construction is made of two parts.

The first part is similar to the multiplier is similar to the one given in [START_REF] Lissy | Optimal filtration for the approximation of boundary controls for the one-dimensional wave equation using finite-difference method[END_REF], and enables to compensate R m (x) for high |x|. The second part is similar to the construction used in [START_REF] Lissy | Construction of Gevrey functions with compact support using the Bray-Mandelbrojt iterative process and applications to the moment method in control theory[END_REF] in the context of continuous fractional control problems, and is used to compensate the bad behaviour of the product R m (x) for low |x| (this is the main difference with [START_REF] Lissy | Optimal filtration for the approximation of boundary controls for the one-dimensional wave equation using finite-difference method[END_REF]). However, the use we make of this multiplier is quite different from the continuous case. Notably, we must analyze very precisely in different ranges how it can compensate the product R m by obtaining at the end uniform estimates in h (see Section 2.3). To our knowledge, such a (quite complex) construction of a multiplier has never been considered in the context of semi-discrete problems, and might be of interest for a larger class of semi-discrete control problems. Note that for this later construction, it is crucial that α > 1.

• Contrary to [START_REF] Lissy | Optimal filtration for the approximation of boundary controls for the one-dimensional wave equation using finite-difference method[END_REF] (and many papers on this topic), we are able to work in the natural energy space. This is due to the use of a refinement of the moment method that has been introduced in [START_REF] Lagnese | Control of wave processes with distributed controls supported on a subregion[END_REF] and rediscovered recently in [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations by boundary controls[END_REF]. However, this construction requires to be much more careful about the estimation of the L 2 -norm of the discrete control introduced in (3.1). It is also likely that the use of this refined moment method is of wider interest and might be used in many other context, probably improving many results of the literature concerning technical restrictions on the space of initial conditions.

Let us mention some perspectives and open question related to the present work. Concerning the model under study, one very natural question is to ask if we can study non-uniform mesh grids with the same technique. This will clearly require a deeper understanding of the phenomena under study, since the present work mainly relies on the fact that we know exactly the eigenvectors and eigenvalues of our discrete operator.

To finish, a more long-term remaining open question consists in finding the sharp filtering scale in a general setting, which should depend both on the defect of uniform discrete spectral gap, the asymptotics of the discrete/continuous eigenvalues and properties of localization of the discrete eigenvectors. We hope that the present study can be of help to reach this distant goal.

Presentation of the problem

This paper is concerned with the internal null-controllability (which is equivalent to the exact internal controllability) for a finite-difference semi-discrete scheme of the one-dimensional fractional wave equation on the space interval (0, 1).

More precisely, let us consider the 1-D Laplacian ∆ with domain H 2 (0, 1) ∩ H 1 0 (0, 1), and state space L 2 (0, 1). It is well-known that -∆ : D(∆) → L 2 (0, 1) is a positive definite operator with compact resolvent, the k-th eigenvalue is λ k = k 2 π 2 , with associated normalized eigenvector e k (x) := √ 2 sin (kπx) .

Thanks to the continuous functional calculus for positive self-adjoint operators, one can define any positive power of -∆. We consider some s > 0. Then, we introduce

D((-∆ D ) s ) = {f ∈ L 2 (Ω) ∞ k=1 λ 2s k | < f, e k > | 2 L 2 (0,1) < ∞}. For f = ∞ k=1 < f, e k > e k ∈ D((-∆ D ) s ), we define (-∆ D ) s f (x) = ∞ k=1 λ s k < f, e k > e k .
Let us consider some T > 0 and some α ∈ (1, 2) (for α 2, the problem becomes trivial, see Remark 1.2). We consider 0 < a < b < 1. We are interested in the following controlled equation:

       u (t, x) + (-∆ D ) α u(t, x) = χ (a,b) (x)v(t, x) t ∈ (0, T ), x ∈ (0, 1), u(t, 0) = 0 t ∈ (0, T ), u(t, 1) = 0 t ∈ (0, T ), u(0, x) = u 0 (x), u (0, x) = u 1 (x)
x ∈ (0, 1),

where v ∈ L 2 ((0, T ) × (0, 1)) is a control, assumed to be localized in (a, b). Let us remark that we have existence and uniqueness of solutions of (1.1) as soon as the initial condition (u 0 , u 1 ) lies in D((-∆ D ) α/2 ) × L 2 (0, 1). More precisely, from now on we consider the following notations for the space of initial conditions

H = D((-∆ D ) α/2 ) × L 2 (0, 1) = (u 0 , u 1 ) = k∈Z * a k 1 k α i , -1 sin(kπx) | k∈Z * |a k | 2 < ∞ . (1.2)
It is well-known that for this kind of fractional wave equation (see [START_REF] Micu | On the controllability of a fractional order parabolic equation[END_REF]Section 6] for a result with a scalar control and an imposed profile, but the result can be easily extended to (1.1) using a technique similar to [START_REF] Lagnese | Control of wave processes with distributed controls supported on a subregion[END_REF]), null-controllability for initial conditions in the space H holds in arbitrary small time if and only if α > 1.

Let N ∈ N * and h = 1 N +1 . We introduce

U (t) :=     u(t, h) u(t, 2h) . . . u(t, N h)     , U 0 :=     u 0 (h) u 0 (2h) . . . u 0 (N h)     and U 1 :=     u 1 (h) u 1 (2h) . . . u 1 (N h)     .
For any T > 0, we consider the following semi-discrete space approximation of the fractional wave equation (1.1) by the following explicit finite-difference method, that we write directly in matrix form:

U + A α h U = B h (v h (t)) 1 ≤ j ≤ N, t ∈ (0, T ), U (0, •) = U 0 , U (0, •) = U 1 , (1.3)
where A h ∈ M N (R) is the discrete Laplacian given by

A h := 1 h 2           2
          and B h ∈ M N (R) is the discrete control operator given by B h (v 1 , . . . v N ) := (χ (a,b) (jh)v(j)) T 1 j N .
Note that A h is a symmetric positive matrix, so that by usual continuous functional calculus, it is very easy to define any of its positive powers. Moreover, since the eigenfunctions and eigenvalues of A h are known explicitly, it is also very easy to compute numerically such powers. To conclude, it is easy to prove that for any α ∈ (1, 2), the semi-discrete scheme given by (1.3) is consistant of order (at least) 2 with respect to equation (1.1) (see [START_REF] Szekeres | Finite difference approximation of space-fractional diffusion problems: the matrix transformation method[END_REF]Theorem 1]), so that it can be proved to be convergent and hence to be a reasonable approximation of (1.1).

Given T > 0, h > 0 and ((u 0 j , u 1 j )) 1≤j≤N ∈ C 2N , we study the existence of a control function v h ∈ L 2 (0, T ) such that the solution of the equation (1.3) verifies

u j (T ) = u j (T ) = 0 (j = 1, 2, ..., N ).
More precisely, our aim is to study the existence of a uniformly bounded sequence of controls (v h ) h>0 with respect to the mesh size h, by using the moment method introduced in [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF] for the boundary controllability of the heat equation. Note that here, contrary to what is done in general, the control is not a scalar control (i.e. it is not of the form f (x)g(t), with f some imposed profile in space in an adequate class, and g ∈ L 2 (0, T ) some scalar control function). For the continuous system (1.1), the control lives in an infinite dimensional subspace, whereas in the discrete case of (1.3), the control lives in a finite dimensional space of dimension N . This generalization of the moment method has been introduced for the internal control of the wave equation in [START_REF] Lagnese | Control of wave processes with distributed controls supported on a subregion[END_REF], used notably in the context of parabolic equations in [START_REF] Allonsius | Spectral analysis of discrete elliptic operators and applications in control theory[END_REF][START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations by boundary controls[END_REF] and studied in a more general setting in [START_REF] Ammar-Khodja | Quantitative Fattorini-Hautus test and minimal null control time for parabolic problems[END_REF]. It enables us to get rid of technical assumptions on the initial conditions, that can be taken in the natural energy space H, contrary to many previous results in the literature.

On the other hand, the choice of an appropriate approximation ((u 0 j , u 1 j )) 1≤j≤N ∈ C 2N for the initial datum (u 0 , u 1 ) ∈ H is very important in order to ensure the uniform boundedness of the sequence of controls (v h ) h>0 .

Let us introduce some additional notations. We consider L α h ∈ M 2N ×2N (R) given by

L α h :=   0 -I n A α h 0   ,
where

I n ∈ M n (R) is the identity matrix of size n.
The eigenvalues of the discretization matrix L α h are given by the family (i λ n (h)) 1≤|n|≤N , where

λ n (h) = sgn(n) 2 h sin |n|πh 2 α , 1 ≤ |n| ≤ N, (1.4)
and the corresponding eigenvectors are

Φ n h = 1 iλn ϕ n h -ϕ n h (1 ≤ |n| ≤ N ),
where

(ϕ n h ) 1≤|n|≤N =     sin(nπh) sin(2nπh) • • • sin(N nπh)     ∈ C N .
For the sake of simplicity, we will often write λ n instead of λ n (h) in what follows, when there is no ambiguity. Note that the eigenvalues of the matrix A α h are (λ 2 n ) 1≤n≤N and (Φ n h ) 1≤|n|≤N forms an orthonormal basis in C 2N (see [START_REF] Micu | Uniform boundary controllability of a semi-discrete 1-D wave equation[END_REF]Proposition 3.1]).

We conclude this preliminary section with some gap estimates which we believe that determine the real behavior of the sequence of discrete controls.

Remark 1.1. Let us consider α ≥ 1 and M > 0. We remark that the gap between the eigenvalues λ n (h) from (1.4) depends dramatically on the position of α. More precisely, it is easy to prove that, for any h > 0 sufficiently small, the following statements hold:

• if α = 1 then |λ n+1 -λ n | = Θ(1) 1 ≤ |n| ≤ δN, ∀δ ∈ (0, 1) Θ(h) N -M ≤ |n| ≤ N , • if α ∈ (1, 2) then |λ n+1 -λ n | =    Θ(1) 1 ≤ |n| ≤ M Θ( 1 h α-1 ) δN 2 ≤ |n| ≤ δN, ∀δ ∈ (0, 1) Θ(h 2-α ) N -M ≤ |n| ≤ N , • if α = 2 then |λ n+1 -λ n | =    Θ(1) 1 ≤ |n| ≤ M Θ( 1 h ) δN 2 ≤ |n| ≤ δN, ∀δ ∈ (0, 1) Θ(1) N -M ≤ |n| ≤ N , • if α > 2 then |λ n+1 -λ n | =    Θ(1) 1 ≤ |n| ≤ M Θ( 1 h α-1 ) δN 2 ≤ |n| ≤ δN, ∀δ ∈ (0, 1) Θ( 1 h α-2 ) N -M ≤ |n| ≤ N ,
where the Bachmann-Landau notation Big Theta has to be understood in the asymptotics h → 0.

Remark 1.2. Note that, our problem deals with the difficult case α ∈ (1, 2). Indeed, if α 2, we proceed as in [6, Section 7](see also [START_REF] Leon | Boundary controllability of the finite-difference space semi-discretizations of the beam equation[END_REF]), taking into account the uniform gap estimates given in Remark 1.1, we can use a generalization of Ingham's inequality [ [START_REF] Komornik | Fourier Series in Control Theory[END_REF], Theorem 4.6].

More precisely, if we have a family with the gap tending to infinity when h goes to zero and another finite family (depending on T) with gap O(1), then we can use an Ingham-type inequality for the reunion of the two families, for any time T > 0 (with a constant depending only on T ).

Hence, for α ≥ 2 the uniform controllability property holds automatically in the context of internal controls.

Main results and structure of the article

Let us consider some increasing function f : N * :→ N * (that will play the role of our filtration). Our main Theorem is the following.

Theorem 1.1. Let (u 0 , u 1 ) = k∈Z * a k 1 k α i , -1 sin(kπx) ∈ H. Assume that lim N →∞ f (N ) = ∞ and Γ := lim sup N →∞ f (N ) N < 1. (1.5)
For any (u 0 j , u 1 j ) 1≤j≤N = 1≤|n|≤f (N ) a n Φ n h there exists a uniformly bounded sequence of controls (v h ) h>0 for the discrete control problem (1.3) verifying that for any h > 0, the solution (u 1 , . . . u N ) of (1.3) verifies (u 1 (T ), . . . , u N (T )) = (0, . . . , 0). Remark that our theorem is optimal in the sense that we can work on the natural energy space H and that uniform controllability may fail if we do not filtrate the initial conditions. This last phenomenon has notably been highlighted in [START_REF] Micu | Uniform boundary controllability of a semi-discrete 1-D wave equation[END_REF] in the case of the wave equation α = 1, and is stated in the following Theorem.

Theorem 1.2. Let (u 0 , u 1 ) = k∈Z * a k 1 k α i , -1 sin(kπx) ∈ H. We introduce (u 0 j , u 1 j ) 1≤j≤N = 1≤|n|≤N a n Φ n h .
There exists initial conditions (u 0 , u 1 ) for which no sequence of controls (v h ) h>0 for the discrete control problem (1.3) (i.e., verifying that for any h > 0, the solution (u 1 , . . . u N ) of (1.3) verifies (u 1 (T ), . . . , u N (T )) = (0, . . . , 0)) can be uniformly bounded in h. Theorem 1.2 is standard and relies on the non-uniform gap property at high frequency stated in Remark 1.1. For the sake of completeness, Theorem 1.2 is proved in Appendix A.

As a corollary of Theorem 1.1, we obtain the following convergence result.

Corollary 1. Let (u 0 , u 1 ) = 1≤|n|≤f (N ) a n Φ n be the initial datum for the continuous problem (1.1) and let us consider the initial datum of the semi-discrete problem (1.3)

(u 0 j , u 1 j ) 1≤j≤N = 1≤|n|≤f (N )
a n Φn (h), (1.6) where Φn (h) is the discretization of the eigenfunction Φ n of the form

Φn (h) = 1 n α i sin(jnπh) 1≤j≤N , (-sin(jnπh)) 1≤j≤N T .
Let v ∈ L 2 (0, T ) a weak limit of a subsequence of the bounded controls (v h ) h>0 given by Theorem 1.1, with initial datum given in (1.6). Then v is a control for the continuous problem (1.1).

The structure of the article is as follows.

In Section 2.1, we will give some spectral properties of the semi-discretized problem and we will set our equivalent moment problem, that we will solve by applying the Paley-Wiener Theorem. This means that one has first to study some appropriate product (the product R m introduced in (2.26)) involving the discrete eigenfunctions and then construct an adequate multiplier. As already mentioned, contrarily to many work on the controllability of distributed systems, we do not impose a fixed profile in space and we do not work with a scalar control depending only on time. This enables to get rid of technical assumptions on the initial condition (see e.g. [START_REF] Micu | On the controllability of a fractional order parabolic equation[END_REF]Section 6]) and to work in the natural energy space, which is also a novelty of this article.

In Section 2.2, we will give crucial estimates on an auxiliary product P m , defined in (2.8) and related to R m by (2.27). The main novelty is that we are able to prove precise (and difficult) estimates on this product, that is bounded independently on the parameter h in some appropriate range (see (2.9)). This leads to a new and very precise estimate on the product R m .

It is worth to mention that, in contrast to the case of wave equation α = 1, where the Weierstrass product R m is bounded in the range less than 2 h , for α ∈ (1, 2) we have a smaller domain on boundedness. This fact is really crucial for the wave case and we need to solve this problem with a precise and smart multiplier.

Concerning the multiplier, in Section 2.3, we will use a new construction in the context of semidiscrete problems, taking into account both regimes described above. We will mix a part similar to the construction of [START_REF] Lissy | Optimal filtration for the approximation of boundary controls for the one-dimensional wave equation using finite-difference method[END_REF] for the semi-discrete wave equations in order to compensate the "bad" dependance in h of R m for large x, and another part, which is similar to the construction made in [START_REF] Lissy | Construction of Gevrey functions with compact support using the Bray-Mandelbrojt iterative process and applications to the moment method in control theory[END_REF] in the case of (continuous) fractional heat or Schrödinger equations, and will be useful to compensate the estimation of R m for small x coming from inequality (2.9). In Section 2.4, we construct our biorthogonal family using the Paley-Wiener Theorem, and we give some estimations on this biorthogonal family.

Finally, Sections 3.1 and 3.2 are devoted to the proofs of Theorem 1.1 and Corollary 1, i.e. the construction of the discrete controls, the proof of uniform estimates with respect to h for filtered initial conditions, and the proof of convergence result for the discretized control.

2 The moment problem. Construction of a biorthogonal family

Reduction to a moment problem

The aim of this subsection is to explain how the control result given in Theorem 1.1 can be reduced to the study of an adequate moment problem.

We start the study of the controllability properties for (1.3) by considering the following adjoint system:

       w j (t) + (A α h W ) j = 0 1 ≤ j ≤ N, t > 0, w 0 (t) = 0 t ∈ (0, T ), w N +1 (t) = 0 t ∈ (0, T ), w j (0) = w 0 j , w j (0) = w 1 j 1 ≤ j ≤ N, (2.1)
where

W (t) = (w 1 (t), . . . , w N (t)) T .
We can rewrite the adjoint system (2.1) in an abstract Cauchy form as

W (t) + A α h W = 0, t > 0, W (0) = W 0 , W (0) = W 1 , where W (t) = (w 1 (t), . . . , w N (t)) T , W 0 = (w 0 j ) 1≤j≤N and W 1 = (w 1 j ) 1≤j≤N .
Moreover, if we consider Z = (W, W ) T , we have the following equivalent form

Z (t) + L α h Z(t) = 0, t > 0, Z(0) = Z 0 = (W 0 , W 1 ) T , (2.2)
where L h ∈ M 2N ×2N (R) is given by

L α h :=   0 -I A α h 0   .
Let us define the corresponding discrete energy as

E(t) = h 2 N j=0 (A α h W ) j w j + |w j (t)| 2 . (2.3)
In fact, the above energy is generated by the following inner product

  W 1 U 1   ,   W 2 U 2   α = A α h W 1 , W 2 + U 1 , U 2 .
We can easily deduce that, if Z 0 = (W 0 , W 1 ) verifies

Z 0 = 1≤|n|≤N a 0 n Φ n h ,
then the solution of the system (2.2) is given by

Z(t) = 1≤|n|≤N a 0 n e iλnt Φ n h . (2.4)
Now, we have all the ingredients to derive our moment problem. Let us go back to equation (1.3).

The following lemma gives a variational characterization of the controllability problem (1.3).

Lemma 2.1. The problem (1.3) is controllable if and only if, for any initial datum

Z 0 = (u 0 j , u 1 j ) 1≤j≤N ∈ C 2N there exist a function v h ∈ L 2 (0, T ) N such that 1≤j≤N (u 0 j w 1 j -u 1 j w 0 j ) = 1≤j≤N T 0 χ (a,b) (jh)w j (t)v h (t, jh) dt, (2.5)
for any (w 0 j , w 1 j ) 1≤j≤N ∈ C 2N and w = (w 1 (t), . . . , w N (t)) solution of the adjoint system (2.1).

Proof of Lemma 2.1 Let (u 0 j , u 1 j ) 1≤j≤N ∈ C 2N and let U = (u 1 (t), . . . , u N (t)) be the corresponding solution of the primal system (1.3). We consider (w 0 j , w 1 j ) 1≤j≤N ∈ C 2N and let w = (w 1 (t), . . . , w N (t)) be the corresponding solution of the adjoint system (2.1).

For each, 1 ≤ j ≤ N , multiplying the (j + N )-th equation in (1.3) by w j (t), integrating by parts and summing up we obtain

1≤j≤N T 0 (u j (t)w j (t) + (A α h U ) j w j (t)) dt = 1≤j≤N T 0 χ (a,b) (jh)w j (t)v h (t, jh) dt,
from where it follows 1≤j≤N (u j (t)w j (t) -u j (t)w j (t))

T 0 + 1≤j≤N T 0 u j (t)(w j (t) + (A α h W ) j ) dt = 1≤j≤N T 0 χ (a,b) (jh)w j (t)v h (t, jh) dt.
Finally, since the controllability of problem (1.3) is equivalent with

u j (T ) = u j (T ) = 0 (j = 1, 2, ..., N ),
the proof of Lemma 2.1 is complete.

Remark 2.1. Note that, using similar estimates as in [START_REF] Micu | Uniform boundary controllability of a semi-discrete 1-D wave equation[END_REF]Remark 6], the left hand term in

(2.5) for (u 0 j , u 1 j ) 1≤j≤N = Φ k h and (w 0 j , w 1 j ) 1≤j≤N = Φ n h is given by 1≤j≤N (u 0 j w 1 j -u 1 j w 0 j ) = i hλ n δ nk . ( 2
= 1≤|k|≤N α k Φ k h ∈ C 2N there exist a function v h ∈ L 2 (0, T ) N such that h 1≤j≤N χ (a,b) (jh) sin(jπnh) T 0 e -iλnt v h (t, jh) dt = α n (1 ≤ |n| ≤ N ). (2.7)
Proof of Proposition 2.1 Using Lemma 2.1, we deduce that system (1.3) is controllable if and only if, for any initial datum (u 0 j , u 1 j ) 1≤j≤N , there exists a function v h ∈ L 2 (0, T ) such that (2.5) holds for (w 0 j , w

1 j ) 1≤j≤N = Φ n h (1 ≤ |n| ≤ N ),
which gives the solution from (2.4) as

Z(t) = e iλnt Φ n h .
Hence, for (u

0 j , u 1 j ) 1≤j≤N = 1≤|k|≤N α k Φ k h ∈ C 2N from (2.6) we have 1≤j≤N (u 0 j w 1 j -u 1 j w 0 j ) = iα n hλ n .
Since w j (t) = -sin(jπnh) iλn e -iλnt we get

h 1≤j≤N χ (a,b) (jh) sin(jπnh) T 0 e -iλnt v h (t, jh) dt = α n ,
hence the proof of (2.7) is finished.

From now on, our aim is to construct and evaluate an explicit biorthogonal sequence to the family (e iλnt ) 1≤|n|≤N in L 2 -T 2 , T 2 .

New product estimates

In this subsection we introduce a product R m which has the property that R m (λ n ) = δ mn . We are able to get very precise estimates of the product R m on the real axis.

For every 1 ≤ |m| ≤ f (N ), we define first the following auxiliary product, that will be of interest for what follows (see notably (2.27)):

P m (x) := 1≤n≤N n =m 1 - x λ 2 n -λ 2 m .
(2.8)

We have the following result.

Proposition 2.2. Let A > 0. There exists a constant D A > 0 (depending on α > 1 and A) such that for every h > 0 small enough, |m| f (N ) and every x such that |x| A h 2α , we have

|P m (x)| D A exp D A |x| 1 2α
.

(2.9)

Moreover, we also have the existence of C > 0 (depending only on α and Γ defined in (1.5)) such that for any x ∈ R,

|P m (x)| C exp C h|x| 1 α + |x| ln(m) m 2α-1 . (2.10) Proof of Proposition 2.2.
In all what follows, C > 0 is a constant depending only on α and Γ (and possibly A if we restrict to the range |x| A h 2α ) that may vary from inequality to inequality. We assume without loss of generality that λ m 0 (the case λ m 0 can be treated in a similar way). First of all, we remark that

|P m (x)| = 1≤n≤N n =m 1 - x λ 2 n -λ 2 m 1≤n≤N n =m 1 + |x| |λ 2 n -λ 2 m | = Q 1 m (x)Q 2 m (x), (2.11)
where

Q 1 m (x) = 1≤n≤N n =m, n+m 1 h 1 + |x| |λ 2 n -λ 2 m | , Q 2 m (x) = 1≤n≤N n =m, n+m< 1 h 1 + |x| |λ 2 n -λ 2 m | . (2.12)
We have that

λ 2 n -λ 2 m = 2 h 2α sin nπh 2 2α -sin mπh 2 2α . • First regime: n + m 1 h . Since |x α -y α | ≥ |x -y| α , for all x, y ∈ R and α > 1, it follows that |λ 2 n -λ 2 m | 2 h 2α sin nπh 2 2 -sin mπh 2 2 α
.

Following the computations made in [START_REF] Lissy | Optimal filtration for the approximation of boundary controls for the one-dimensional wave equation using finite-difference method[END_REF] and using the inequality sin( πx 2 ) x, true for any x ∈ [0, 1], we deduce that

|λ 2 n -λ 2 m | 2 h 2α sin (n -m)
N N + 1 1 + f (N ) N π 2 .
Using Hypothesis (1.5), we obtain that lim sup h→0

N N +1 1 + f (N ) N π 2 = (1 + Γ) π 2 .
Hence, there exists C < 1 such that for h small enough, we have

sin N N + 1 1 + f (N ) N π 2 sin (1 + C) π 2 .
Since C < 1, we deduce that sin (1 + C) π 2 > C > 0, so that finally, we obtain

sin (n + m)πh 2 C.
Hence, we deduce

|λ 2 n -λ 2 m | C |n -m| h α . (2.13)
We deduce by plugging this inequality into the first term of (2.12) that

Q 1 m (x) 1≤n≤N n =m 1 + Ch α |x| |n -m| α . (2.14) Then, ln Q 1 m (x) 1≤n≤N n =m ln 1 + Ch α |x| |n -m| α . 1≤n≤N n =m C|x|h α 0 dt t + |n -m| α C|x|h α 0 1≤n≤N n =m dt t + |n -m| α C|x|h α 0 1≤n≤N n =m ∞ |n-m| α ds (t + s) 2 dt C|x|h α 0 |N -m| α infn |n-m| α L m (s)ds (t + s) 2 dt, where L m (s) := card{1 ≤ n N s.t. |n -m| α s}.
It is easy to estimate this counting function. We have

|n -m| α s ⇔ m -s 1 α + n m + s 1 α .
Hence, we deduce that

ln Q 1 m (x) C|x|h α 0 (N -m) α 1 (m + s 1 α ) -m -s 1 α + (t + s) 2 dsdt C|x|h α (N -m) α 1 (m + s 1 α ) -m -s 1 α + s(s + C|x|h α ) ds.
We introduce the change of variables s = m α t. We deduce that ln

Q 1 m (x) C|x|h α m α-1 |N -m| α |m α | 1 m α 1 + t 1 α -1 -t 1 α + t t + C|x|h α m α dt C|x|h α m α-1 ∞ 0 1 + t 1 α -1 -t 1 α + t t + C|x|h α m α dt.
(2.15)

Let us introduce, for y 0,

U (y) = 1 0 2t 1 α t (t + y) dt and V (y) = ∞ 1 1 + t 1 α t(t + y) dt.
Using the change of variables t = yv α , we obtain

U (y) = y -1 α 0 2αyv α-1 y 1 α v y 2 v α (1 + v α ) dv y 1 α -1 ∞ 0 2α (1 + v α ) dv Cy 1 α -1 .
Hence, we have obtained

U (y) C y 1-1 α . (2.16)
On the other hand,

V (y) = ∞ 1 t 1 α t(t + y) dt + ∞ 1 1 t(t + y) dt.
Concerning the first integral, using as before the change of variables t = yv α , we obtain that

∞ 1 t 1 α t(t + y) dt y 1-1 α ∞ 0 α (1 + v α ) dt C y 1-1 α .
Concerning the second integral, its value is exactly log(1+y)/y C y 1-1 α , ∀y 0, by compared growth.

Hence, we can deduce that for any y 0 we have

V (y) C y 1-1 α .
(2.17)

Combining (2.16) and (2.17), we deduce that for any y 0, , so we have that

U (y) + V (y) C y 1-1 α . ( 2
Q 1 m (x) D A exp D A |x| 1 2α
.

(2.20)

• Second regime: n + m 1/h.

We use the following basic inequality, true for any x ∈ (0, π/2), any y ∈ (0, π/2 -x), and any α 1:

| sin(x) 2α -sin(y) 2α | ≥ 2 π 2α |x 2α -y 2α | .
We deduce that

|λ 2 n -λ 2 m | 2 α n 2α -m 2α . (2.21)
Plugging this inequality into the second term from (2.12), we obtain that

Q 2 m (x) 1≤n≤N n =m 1 + |x| |n 2α -m 2α | . Then, ln Q 2 m (x) 1≤n≤N n =m ln 1 + |x| |n 2α -m 2α | n≥1 n =m |x| 0 dt t + |n 2α -m 2α | |x| 0 1≤n≤N n =m dt t + |n 2α -m 2α | |x| 0 1≤n≤N n =m ∞ |n 2 -m 2 | α ds (t + s) 2 dt |x| 0 sup n |n 2α -m 2α | infn |n 2α -m 2α | L m (s)ds (t + s) 2 dt |x| 0 1 h 2α m 2α-1 L m (
|n 2α -m 2α | s ⇔ m 2α -s + 1 2α n m 2α + s 1 2α
.

Hence, we deduce that

ln Q 2 m (x) |x| 0 1 h 2α m 2α-1 m 2α + s 1 2α -m 2α -s + 1 2α (t + s) 2 dsdt |x| 1 h 2α m 2α-1 m 2α + s 1 2α -m 2α -s + 1 2α s(s + |x|) ds.
We introduce the change of variables s = m 2α t. We deduce that

ln Q 2 m (x) |x| m 2α-1 1 (mh) 2α 1 m (1 + t) 1 2α -(1 -t) + 1 2α t t + |x| m 2α
dt.

(2.22)

Let us introduce, for y > 0,

U (y) = 1 1 m (1 + t) 1 2α -(1 -t) 1 2α t (t + y) dt and V (y) = 1 (hm) 2α 1 (1 + t) 1 2α t(t + y) dt.
Using the inequality (1 + t)

1 2α -(1 -t) 1 2α
αt, true for any x ∈ [0, 1], we obtain U (y)

1 1 m α t + y dt ln(m). ( 2

.23)

On the other hand, concerning V , we remark that We see that contrary to the computations made for the wave equation in [START_REF] Lissy | Optimal filtration for the approximation of boundary controls for the one-dimensional wave equation using finite-difference method[END_REF], the product P m is only bounded in the small range |x| C. As soon as x C h s for some s > 0, we have an upper bound that may explode as h → 0.

V (y) ∞ 1 (1 + t)

Now, we consider

R m (z) = 1≤|n|≤N n =m z λ n -1 1≤|n|≤N n =m λ n λ m -λ n := R 1 m (z)S m (z ∈ C), (2.26)
where

R 1 m (z) = 1≤|n|≤N n =m z λ n -1 ,
and

S m = 1≤|n|≤N n =m λ n λ m -λ n . We remark that R m (λ n ) = δ nm .
Moreover, R m is related to the product P m defined in (2.8) by the following relation:

1 - x -λ m λ -m -λ m P m (x 2 -λ 2 m ) = R m (x), (2.27)
so that our estimates on P m will give estimates on R m . We start our estimates with the following technical result concerning the product S m .

Lemma 2.2. For every 1 ≤ |m| ≤ N , we have that

|S m | cos 2α mπh 2 .
(2.28)

Proof of Lemma 2.2. From the symmetry of the sequence (λ n ) 1≤|n|≤N (i.e. λ -n = -λ n ), it is sufficient to consider only the case 1 ≤ m ≤ N .

Firstly, we obtain that

|S m | = 1≤|n|≤N n =m λ n λ n -λ m = 1 2 1≤n≤N n =m λ 2 n λ 2 n -λ 2 m = 1 2 1≤n≤N n =m sin 2α nπh 2 sin 2α nπh 2 -sin 2α mπh 2 . Since |x α -y α | ≥ |x -y| α , for all x, y ∈ R and α > 1, it follows that |S m | 1 2 1≤n≤N n =m sin 2 nπh 2 sin 2 nπh 2 -sin 2 mπh 2 α = 1 2 1≤n≤N n =m sin 2 nπh 2 sin (m-n)πh 2 sin (m+n)πh 2 α . (2.29)
Now we proceed similar to the wave equation (see [START_REF] Lissy | Optimal filtration for the approximation of boundary controls for the one-dimensional wave equation using finite-difference method[END_REF]). We study the above product by splitting it into two parts. We have that First of all, let us rewrite R 1 m as

R 1 m (z) = z + λ m λ m 1≤n≤N n =m z 2 λ 2 n -1 .
|R 1 m (x)| = 1≤n≤N n =m x 2 λ 2 n -1 = 1≤n≤N n =m r sin 2α nπh 2 -1 = 1≤n≤N n =N +1-m r cos 2α nπh 2 -1 = cos 2α (N +1-m)πh 2 r -cos 2α (N +1-m)πh 2 1≤n≤N r cos 2α nπh 2 -1 cos 2α (N +1-m)πh 2 r -cos 2α (N +1-m)πh 2 exp N 1 ln r cos 2α tπh 2 -1 dt .
In what follows we compute the last integral by using a change of variable u = tan tπh 2 :

N 1 ln r cos 2α tπh 2 -1 dt = 2 πh ∞ Ch ln r(1 + u 2 ) α -1 1 + u 2 du.
Note that the following inequality holds ln r(1

+ u 2 ) α -1 C ln(r + 1)(1 + u 2 ) 1 4α
(r 1, u 0), so that we obtain

∞ Ch ln r(1 + u 2 ) α -1 1 + u 2 du C ln(r + 1) ∞ Ch 1 (1 + u 2 ) 1-1/(4α) du C ln(r + 1).
Using (2.26) together with (2.28) and the previous estimate, we deduce that for ant |x|

2 h α , we have |R m (x)| exp C h ln xh α 2 α + 1 exp C h ln 2xh α 2 α .
Hence the proof of the estimate of R m given in (2.30) is finished, taking R = C .

Remark 2.2. Let A > 0. By using Proposition 2.2, (2.27) and Proposition 2.3, we infer the existence of two constants R ∈ N * and C > 0 such that for N large enough we have

|R m (x)| ϕ(x),
where

ϕ(x) =        D A exp D A |x 2 -λ 2 m | 1 2α |x| ≤ A h 2α (|x|h α ) R h |x| ≥ 2 h α C exp C h|x 2 -λ 2 m | 1 α + |x 2 -λ 2 m | ln(m) m 2α-1 x ∈ R.

Construction of the multiplier

To get our multiplier, taking into account the different bad behaviour of the product R M that requires to be compensated on the real line (see Remark 2.2), we need to consider a construction in two parts. Let us remind that 2/h = 2(N + 1) ∈ N. Hence, we use a new construction, mixing a part similar to the construction of [START_REF] Lissy | Optimal filtration for the approximation of boundary controls for the one-dimensional wave equation using finite-difference method[END_REF] and another part, which is similar to the construction made in [START_REF] Lissy | Construction of Gevrey functions with compact support using the Bray-Mandelbrojt iterative process and applications to the moment method in control theory[END_REF].

For every b > 0 we set

H b := 1 [-b,b] 2b . We remark that R H b = b -b H b = 1.
We denote by

a := h 2R , so that R/h n=1 a = 1 2 .
Remind that R was introduced in Proposition 2.3. We consider the product

u a := H a * . . . * H a ,
where * represents the convolution product. The product is made of R/h terms.

We introduce

M 1 m (z) := 1/2 -1/2 u a (t)e -i T 2 (z-λm)t dt. (2.31)
One has the following properties on M 1 m (see [START_REF] Lissy | Optimal filtration for the approximation of boundary controls for the one-dimensional wave equation using finite-difference method[END_REF]Lemma 2.6]):

Lemma 2.3. M 1 m is of exponential type T /4. Moreover, one has M 1 m (λ m ) = 1 (2.32)
and for every

x ∈ R, |M 1 m (x)| 1. (2.33)
Moreover,

|M 1 m (x)| 4R |x -λ m |T h R h .
(2.34)

Now, we consider the second part of the multiplier. Let ν > 0 be some large parameter to be chosen later. Let also γ ∈ (1, α) some parameter to be chosen later close enough to 1. We consider

σ ν := lim n→∞ H a0 * H a1 * • • • * H an ,
where the sequence (a k ) k∈N is given by

a k = 1 ν if k ν , a k = ν γ-1 k γ if k ν + 1.
We consider the corresponding function σ ν constructed as before from the sequence (a k ) k∈N . We call a(ν) its support, that is to say

a(ν) := ∞ k=0 a k = ν + 1 ν + ∞ k= ν +1 ν γ-1 k γ .
Taking into account the definition of a(ν), we may observe that there exists some constant S(γ) > 0 (not depending on ν) such that a(ν) S(γ).

(

Now, we introduce

H β (z) := a(ν) -a(ν) σ ν (t)e -iβtz dt,
We have the following properties ([29, Lemma 3.4]).

Lemma 2.4. The following estimates hold:

H β (0) = 1, (2.36)
|H β (z)| e S(γ)β|Im(z)| , (2.37)
|H β (x)| Cνe γν e -γ 2 ((βν γ-1 )|x|) 1/γ , x ∈ R. (2.38)
Firstly, we choose β = S(γ)T 4 , so that by (2.35) and (2.37), H β is of exponential type T /4. We also consider any K > 0 (to be chosen later) and we choose ν such that

α 2 βν γ-1 1 γ = K.
Remark that ν is only depending on β, T and K, but not on h. Hence, from (2.38) we can write the final useful estimate

|H β (x)| Ce -K|x| 1 γ , x ∈ R, (2.39)
where C depend on T , γ and K but not on h. Our second part of the multiplier will then be

M 2 m (z) = H β (x -λ n ).

Construction and estimation of the biorthogonal family

We can now introduce the family

ψ m (z) = R m (z)M 1 m (z)M 2 m (z). (2.40)
Let us study this product in different regimes, using the previous estimates. We assume without loss of generality that λ m 0.

If |x|

2 h α , then there exists some A > 0 such that for any such x and any λ m , we have

|x 2 -λ 2 m | A h 2α .
(one may for example take A = 4).

Let us consider different subranges of x.

• Assume that |x -λ m | λ δ m for some δ ∈ (0, 1) to be chosen. We can use now (2.27), (2.9), (2.33) and (2.39) to obtain

|ψ m (x)| C(1 + |x|)D A e D A |x 2 -λ 2 m | 1 2α -K|x-λm| 1 γ .
We want to choose δ such that there exists some K > 0 (depending on δ, K, α and γ) such that for any x such that |x -λ m | λ δ m , we have .41) this is equivalent to

D A |x 2 -λ 2 m | 1 2α -K|x -λ m | 1 γ -K |x 2 -λ 2 m | 1 2α . ( 2
(D A + K )|x 2 -λ 2 m | 1 2α K|x -λ m | 1 γ -1 2α , i.e. (D A + K )|x + λ m | 1 2α K|x -λ m | 1 γ -1 2α . (2.42)
Assume in addition for the moment that |x -λ m | λ m . Then, one readily verifies that there exists some constant C > 0 (independent on m, x or h) such that for any x such that |x -λ m | λ m , we have

|x + λ m | 1 2α |x -λ m | 1 γ -1 2α C,
for some constant C > 0. Hence, we deduce that (2.42) is automatically verified for some K > 0 as soon as γ < α (which is true) and K is sufficiently large, which is assumed from now on. Assume now that |x -λ m | λ m , which implies notably that |x + λ m | 3λ m . In this case, taking into account that |x -λ m | λ δ m , we see that (2.42) is verified for some K > 0 as soon as δ 1 γ -1 2α > 1 2α and K is chosen large enough. Taking into account that we also want δ < 1, we deduce that we require

1 > δ > γ 2α -γ , (2.43)
which is possible since γ < α.

To summarize, under condition (2.43), we have the existence of some K > 0 (depending on γ, δ and α) such that (2.41) holds as soon as |x -λ m | λ δ m and K is chosen large enough, which is assumed from now on.

• Assume now that |x -λ m | λ δ m for δ verifying (2.43). In this case, we use (2.27), (2.10), (2.33) and (2.39) to obtain

|ψ m (x)| C(1 + |x|)e C h|x 2 -λ 2 m | 1 α + |x 2 -λ 2 m | ln(m) m 2α-1 -K|x-λm| 1 γ .
Hence, we notably have

|ψ m (x)| Ce C h|x 2 -λ 2 m | 1 α + |x 2 -λ 2 m | ln(λm ) λ 2-1 α m -K|x-λm| 1 γ .
We remark that since we chose δ < 1, we have |x + λ m | Cλ m , so that

h|x 2 -λ 2 m | 1 α Chλ 1 α m |x -λ m | 1 α .
Hence, taking into account that λ m C h α , we have

h|x 2 -λ 2 m | 1 α C|x -λ m | 1 α .
We deduce the existence of some K > 0 such that

h|x 2 -λ 2 m | 1 α -K|x -λ m | 1 γ -K |x -λ m | 1 α (2.44)
as soon as K is chosen large enough, which is assumed from now on. Concerning the other term, since we chose δ < 1, we have |x

+ λ m | Cλ m , hence |x 2 -λ 2 m | ln(λ m ) λ 2-1 α m -K|x -λ m | 1 γ C |x -λ m | ln(λ m ) λ 1-1 α m -K|x -λ m | 1 γ |x -λ m | 1 γ C |x -λ m | 1-1 γ ln(λ m ) λ 1-1 α m -K |x -λ m | 1 γ   C λ δ(1-1 γ ) m ln(λ m ) λ 1-1 α m -K   .
We deduce the existence of some K > 0 such that

|x 2 -λ 2 m | ln(λ m ) λ 2-1 α m -K|x -λ m | 1 γ -K |x -λ m | 1 γ (2.45)
as soon as K is chosen large enough (which is assumed from now on) and δ is chosen such that

δ 1 - 1 γ < 1 - 1 α , i.e. δ < γ(α -1) α(γ -1) . ( 2
γ 2α -γ < γ(α -1) α(γ -1) , which is equivalent to α(γ -1) < (2α -γ)(α -1).
This last inequality is true since γ ∈ (1, α).

If |x| 2 h

α , then we use (2.30), (2.34) and (2.39), with K > 0 and γ > 0 as in the previous point. Taking into account that γ < α and |x2 -λ 2 m | C for some C > 0 (independent of m but depending on Γ), we obtain

|ψ m (x)| Ce -2K|x 2 -λ 2 m | 1 2γ (|x|h α ) R h 4R |x -λ m |T h R h CD A e -2D A |x 2 -λ 2 m | 1 2α 4R|x|h α |x -λ m |T h R h .
Using that |m| f (N ), we easily deduce as in [26, Proof of Prop. 2.3] that there exists some

C 1 > 0 such that |x| |x -λ m | C 1 .
Hence, using now that α > 1, we deduce that for some K > 0,

|ψ m (x)| Ce -C|x 2 -λ 2 m | 1 2α 4RC 1 h α-1 T R h Ce -K |x 2 -λ 2 m | 1 2α . ( 2
|ψ m (x)| C max e -C|x 2 -λ 2 m | 1 2α , e -C|x-λm| 1 γ , e -C|x-λm| 1 α . (2.48)
Hence, ψ m ∈ L 2 (R) and is of exponential type at most T /2 thanks to (2.37), (2.31) and the fact that P m is obviously of exponential type 0. We can use the Paley-Wiener Theorem (see e. According to Lemma 2.1 an more precisely identity (2.7), we consider the control function v h ∈ L 2 (0, T ) N given by

v h (t) = 1≤|n|≤f (N ) a n B n h ||B n h || 2 e -iλn T
where B n h is given by

B n h = h(χ (a,b) (jh) sin(jπnh)) 1≤j≤N . (3.2)
Let us prove that here exists a constant C > 0 (independent of n or N ) such that for h small enough (depending on a and b), we have

||B n h || 2 Ch. (3.3)
First of all, we remark that

h N j=1 χ (a,b) (jh) sin(jπnh)) 2 = h b h j= a h sin(jπnh) 2 = h 4 2 + 2b h - 2a h + sin 2h a h nπ -hnπ -sin 2h b h nπ + hnπ sin(nπh) .
Let us consider two different cases:

• Assume that n is such that lim inf h→0 nh > 0. In this case, since n f (N ), using Hypothesis (1.5), we have that there exists C > 0 such that for any h > 0, sin(nπh) C. We deduce that h 4

sin 2h a h nπ -hnπ -sin 2h b h nπ + hnπ sin(nπh) C h 4 sin 2h a h nπ -hnπ -sin 2h b h nπ + hnπ 2C h 4 ,
which goes to 0 as h → 0. Hence, we deduce that in this case

h N j=1 χ (a,b) (jh) sin(jπnh) 2 → b -a 2 as h → 0.
• Assume now that n is such that lim inf h→0 nh = 0. In this case, we have

lim inf h→0 h N j=1 χ (a,b) (jh) sin(jπnh)) 2 = b -a 2 + sin (2aπ) -sin (2bπ) 4π .
Moreover, it is very easy to prove that b -a 2 + sin (2aπ) -sin (2bπ) 4π > 0 as soon as a = b. Indeed, if b-a 2 = sin(2bπ)-sin (2aπ) 4π

, then we have 2πb -sin(2bπ) = 2πasin(2aπ), which is impossible since x → x -sin(x) is strictly increasing on [0, 2π] (remind that 0 < a < b < 1).

Hence, we have the existence of C > 0 such that whatever h ∈ (0, 1) and n ∈ [|0, f (N )|] are, we have

h N j=1 χ (a,b) (jh) sin(jπnh)) 2 C.
This implies that (3.3) is verified.

Finally, we can prove the uniformly boundedness of the controls sequence (v h ) h>0 . Indeed, using the definition of v h given in (3.1) together with (3.3), (2.48) and the fact that the Fourier transform is an isometry in L 2 (R), we deduce that there exists C 1 > 0 such that

v h 2 L 2 ≤ C 1 1≤|n|≤f (N ) 1≤|m|≤f (N ) h 2 |a n ||a m χ (a,b) (jh) sin(jπnh) sin(jπmh) h 2 R |ψ n (x)ψ m (x)|dx C 1 1≤|n|≤f (N ) 1≤|m|≤f (N ) |a n ||a m | max R e -C|x 2 -λ 2 n | 1 2α -C|x 2 -λ 2 m | 1 2α dx , R e -C|x-λn| 1 γ -C|x-λm| 1 γ , dx, R e -C|x-λn| 1 α -C|x-λm| 1 α , (3.4)
where C is the constant from (2.48).

Consider some |m| |n| (otherwise, the same reasoning applies by exchanging n and m). Then, we make the change of variables

x 2 = y 2 + λ 2 m . Since λ 2 m -λ 2 n 0, we obtain that R e -C|x 2 -λ 2 n | 1 2α -C|x 2 -λ 2 m | 1 2α = R |y| y 2 + λ 2 m e -C|y 2 +λ 2 m -λ 2 n | 1 2α -C|y| 1 α dy C R e -C|λ 2 m -λ 2 n | 1 2α -C|y| 1 α dy Ce -C|λ 2 m -λ 2 n | 1 2α . (3.5)
It is very easy to prove that there exists some C > 0 (that may depend on α) such we have

e -C|λ 2 m -λ 2 n | 1 2α C 1 + |λ 2 m -λ 2 n | 2 α
.

From (2.13) and (2.21), we deduce that there exists some C > 0 such that

|λ 2 n -λ 2 m | 2 α C|n -m| 2 , from which we obtain e -C|λ 2 m -λ 2 n | 1 2α C 1 + C|n -m| 2 . (3.6)
By the same type of arguments, we can also easily obtain the estimates R e -C|x-λn|

1 γ -C|x-λm| 1 γ Ce -C|λm-λn| 1 γ , R e -C|x-λn| 1 α -C|x-λm| 1 α Ce -C|λm-λn| 1 α , (3.7
) and e -C|λm-λn|

1 γ C 1 + C|n -m| 2 , e -C|λm-λn| 1 α C 1 + C|n -m| 2 . ( 3
0 |v h (t)| 2 dt C 1≤|n|≤f (N ) 1≤|m|≤f (N ) |a n ||a m | 1 + C|n -m| 2 C 1≤|n|≤f (N ) 1≤|m|≤f (N ) |a n | 2 + |a m | 2 (1 + C|n -m| 2 ) 2 C 1≤|n|≤f (N ) |a n | 2   1≤|m|≤f (N ) 1 (1 + C|n -m| 2 ) 2   C 1≤|n|≤f (N ) |a n | 2 ,
and the proof of Theorem 1.1 is finished.

Proof of Corollary 1

The aim of this subsection is to prove the convergence of the controls sequence (v h ) h>0 to a control of the continuous problem (1.1).

We study the existence of a control function v ∈ L 2 ((0, T ) × (0, 1)) such that the solution of equation (1.1) verifies u(T ) = u (T ) = 0.

We transform the controllability of the continuous problem into a moment problem, using an additional variational lemma. Lemma 3.1. Problem (1.1) is controllable if and only if, for any initial datum (u 0 , u 1 ) ∈ H decomposed as in (1.2), there exists v ∈ L 2 ((0, T ) × (0, 1)) such that

w 1 , u 0 D((-∆ D ) α/2 ) ,D((-∆ D ) α/2 ) - 1 0 u 1 (x)w 0 (x) dx = b a T 0 v(t, x)w(t, x) dt dx, (3.9)
for all (w 0 , w 1 ) ∈ H = L 2 (0, 1)×D((-∆ D ) α/2 ) , where w is the the solution of the adjoint problem

       w (t, x) + (-∆ D ) α w xx (t, x) = 0 t ∈ (0, T ), x ∈ (0, 1), w(t, 0) = 0 t ∈ (0, T ), w(t, 1) = 0
t ∈ (0, T ), w(0, x) = w 0 (x), w (0, x) = w 1 (x) x ∈ (0, 1).

(3.10) Proof of Lemma 3.1. Assume that we have (u 0 , u 1 ) ∈ D((-∆ D ) α/2+1/2 ) × H 1 (0, 1) and (w 0 , w 1 ) ∈ D((-∆ D ) α/2+1/2 ) × H 1 (0, 1). Multiplying Finally, if we integrate in space interval [0, 1] and taking into account the hypotheses and the fact that (1.1) is controllable if and only if u(T ) = u (T ) = 0, we obtain (3.9).

The result for (w 0 , w 1 ) ∈ H easily follows by a standard density argument.

Hence, we get the following moment problem. where for the last identity we have used the fact that w(t, x) = ie -iνn t n α . The proof is now finished combining the last two identities in Lemma 3.1.

Proof of Corollary 1. First of all, we remark that initial datum from (1.6) can be written as

(u 0 j , u 1 j ) 1≤j≤N = 1≤|n|≤f (N ) a n (h)Φ n h ,
where a n (h) = Let us consider a weak limit (up to a subsequence) v of the discretized controls (v h ) h>0 .

Note that for any n ∈ Z * , we have Finally, if we combine the above estimates with the continuous moment problem (3.11), it follows that any weak limit v of a subsequence of controls (v h ) h>0 verifies A Appendix: Proof of Theorem 1.2

Assume by contradiction that for any (u 0 j , u 1 j ) 1≤j≤N = 1≤|n|≤N a n Φ n h there exists a uniformly bounded sequence of controls (v h ) h>0 for the discrete control problem (1.3) verifying that for any h > 0, the solution (u 1 , . . . u N ) of (1.3) verifies (u 1 (T ), . . . , u N (T )) = (0, . . . , 0). By duality, this is totally equivalent to proving the following observability inequality: there exists C > 0 (independent on h > 0) such that for any

Z 0 = 1≤|n|≤N a 0 n Φ n h ,
then the solution of system (2.2) given by Moreover, it is very easy to prove that λ N -λ N -1 → 0 as h → 0 (see Remark 1.1), so that by the dominated convergence Theorem, T 0 cos ((λ N -λ N -1 )t)) dt → T as h → 0. Hence, dividing by h and making h → 0 in (A.2) leads to a contradiction since the left-hand side is equal to 1 whereas the right-hand side tends to 0 as h → 0.

Z(t) =

- 1 0

 1 . . . . . . . . . 0 -1 2 -1 0 . . . . . . 0 0 -1 2 -1 0 . . . 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . . . 0 0 -1 2 -1 0 . . . . . . . . . 0 -1 2

kπh 2 , 2 = 2 .kπh 2 α= cos 2α mπh 2 ,

 22222 where for the last estimate we have used the fact that sin kπh cos (N +1-k)πhFrom the last two relations together with (2.29) we obtain that|S m | cos 2α mπh 2and the proof is complete.Now, let us study the product R m for high values of |x|.

Proposition 2 . 3 .

 23 There exist a constant R ∈ N * such that for N large enough and for |x| 2 h α , we have |R m (x)| (|x|h α )

3. 1 1 Proof of Theorem 1. 1 . 2 -T 2 Θ

 11122 g. [40, Theorem 19.3, Page 370] and we introduce the Fourier transform θ m ∈ L 2 (0, T ) to ψ m , which verifies, thanks to the definition of R m given in (2.26), (2.36) and (2.32), that θ m (λ n ) = δ nm and is compactly supported in [-T /2, T /2]. 3 Proofs of Theorem 1.1 and Corollary 1 Proof of Theorem 1.Let (Θ m ) 1≤|m|≤N be the biorthogonal sequence to the family (e iλnt ) 1≤|n|≤N in L 2 -T 2 , T 2 constructed and evaluated in Section 2.4 as the Fourier transform of ψ m introduced in (2.40), verifying by definition T m (t)e iλnt dt = δ mn (|m|, |n| = 1, . . . , N ).

. 8 )

 8 Combining (3.4), (3.5), (3.6), (3.7), (3.8) and Young's inequality, we deduce that

T

  (1.1) with w and integrating by parts from 0 to T we have(u w -uw ) T 0 + T 0 u(w + (-∆ D ) α w) dt = T 0 v(t, x)χ (a,b) (x)w(x, t) dt.

Theorem 3 . 1 . 1 0u 0

 3110 The initial datum (u 0 , u 1 ) ∈ H is controllable if and only if there exists v ∈ L 2 (0, T ) such that , x)e -iνnt dt dx = a n (n ∈ Z * ),(3.11) where ν n = sgn(n)|n| α verifies that (iν n ) n∈Z * are the eigenvalues of the elliptic operator associated to the continuous problem (3.10).Proof of Theorem 3.1. Not that it is sufficient to show (3.9) is verified only for (w 0 , w 1 ) = Φ n , n ∈ Z * , where the eigenfunction of the fractional Laplacian operator is given byΦ n = 1 n α i sin(nπx), -sin(nπx) T .Hence, by considering (w 0 , w 1 ) = Φ n , n ∈ Z * , in Lemma 3.1 we get (x)w 1 (x) -u 1 (x)w 0 (x) dx = ia n n α , v(t, x)e -iνnt dt dx,

α + 1 a n + 1 2 λn( 0 e

 20 nπ) α -1 a -n |n| ≤ f (N ), 0 |n| > f (N ).The sequence of uniformly bounded controls (v h ) h>0 given by Theorem 1.1 satisfies the discrete moment problem (2.7), which givesh 1≤j≤N (sin(jπnh)) χ a,b (jh) T -iλnt v h (t, jh) dt = a n (h) (1 ≤ |n| ≤ N ).

h 1≤j≤N ( 0 e

 1≤j≤N0 sin(jπnh)) χ a,b (jh) T -iλnt v h (t, jh) dt -→ )e -iλnt v(t, x)dt dx, a n (h) -→ h→0 a n .

  )e -iλnt v(t, x)dt dx = a n (n ∈ Z * ),hence, v is a control for continuous problem (1.1).

e| 2 - 2

 22 n e iλnt || 2 dt, (A.[START_REF] Amblard | Subdiffusion and Anomalous Local Viscoelasticity in Actin Networks[END_REF] where the definition of the discrete energy E is given in (2.3) and the definition of the discrete observation B n h is given in (3.2). Let us consider the following initial condition:Z 0 = Φ N h -Φ N -1 iλ N t sin(jπN h) -e iλ N -1 t sin(jπ(N -1)h) sin(jπN h)| 2 + | sin(jπ(N -1)h)| cos ((λ N -λ N -1 )t)) sin(jπN h) sin(jπ(N -1)h))) dt.

  -1)h) sin(jπN h) → b -a 2 + sin (2aπ) -sin (2bπ) 4π as h → 0.

  .6) Now, we are able to transform our controllability problem into a moment problem. Proposition 2.1. The problem (1.3) is controllable if and only if, for any initial datum Z 0

  Now, let us give another estimate on Q 2 m . It is easy to compute this counting function L m . We have

		s)ds (t + s) 2 dt,
	where
	L m (s) := card{n ∈ N Following the computations made in [27, Proof of Lemma 2.1], we can already infer that
	ln Q 2 m (x)	C|x|

* s.t. |n 2α -m 2α | s}.

  .[START_REF] Yao | Solving fractional Laplacian viscoacoustic wave equation using Hermite distributed approximating functional method[END_REF] In any cases, combining (2.41), (2.44), (2.45) and (2.47), we have, for any x ∈ R, the following estimate:

2α , which gives (2.9) when combined with (2.11) and (2.20).

Θ n t -T 2 . (3.1)