Fioralba Cakoni
email: cakoni@math.udel.edu

Houssem Haddar
email: haddar@cmap.polytechnique.fr

Isaac Harris
email: iharris@udel.edu

HOMOGENIZATION OF THE TRANSMISSION EIGENVALUE PROBLEM FOR PERIODIC MEDIA AND APPLICATION TO THE INVERSE PROBLEM

Keywords: 2000 Mathematics Subject Classification. Primary: 35R30, 35Q60, 35J40, 78A25 Interior transmission problem, transmission eigenvalues, periodic inhomogeneous medium, inverse scattering problem, homogenization

Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem

Fioralba Cakoni, Houssem Haddar, Isaac Harris

1. Introduction. We consider the transmission eigenvalue problem associated with the scattering by inhomogeneuos (possibly anisotropic) highly oscillating periodic media in the frequency domain. The governing equations possess rapidly oscillating periodic coefficients which typically model the wave propagation through composite materials with fine microstructure. Such composite materials are at the foundation of many contemporary engineering designs and are used to produce materials with special properties by combining in a particular structure (usually in periodic patterns) different materials. In practice, it is desirable to understand these special properties, in particular macrostructure behavior of the composite materials which mathematically is achievable by using homogenization approach [START_REF] Allaire | Shape Optimization by the Homogenization Method[END_REF], [START_REF] Bensoussan | Asymptotic Analysis for Periodic Structures[END_REF]. Our concern here is with the study of the corresponding transmission eigenvalues, in particular their behavior as the period in the medium approaches zero. To this end, it is essential to prove strong H 1 (D)-convergence of the resolvent corresponding to the transmission eigenvalue problem, or as known as the solution of the interior transmission problem. Transmission eigenvalues associated with the scattering problem for an inhomogeneous media are closely related to the so-called non scattering frequencies [START_REF] Blåsten | Corners Always Scatter[END_REF], [START_REF] Cakoni | Qualitative Approach to Inverse Scattering Theory[END_REF], [START_REF] Cakoni | Transmission eigenvalues in inverse scattering theory Inverse Problems and Applications[END_REF]. Such eigenvalues can be determined from scattering data [START_REF] Cakoni | On the determination of Dirichlet or transmission eigenvalues from far field data[END_REF], [START_REF] Kirsch | The inside-outside duality for scattering problems by inhomogeneous media[END_REF] and provide information about material properties of the scattering media [START_REF] Cakoni | The existence of an infinite discrete set of transmission eigenvalues[END_REF], and hence can be used to estimate the refractive index of the media. In particular, in the current work we use the first transmission eigenvalue to estimate the effective material properties of the periodic media.

More precisely, let D ⊂ R d be a bounded simply connected open set with piecewise smooth boundary ∂D representing the support of the inhomogeneous periodic media. Let > 0 be the length of the period, which is assumed to be very small in comparison to the size of D and let Y = (0, 1) d be the rescaled unit periodic cell. We assume that the constitutive material properties in the media are given by a positive definite symmetric matrix valued function A := A(x/ ) ∈ L ∞ D, R d×d and a positive function n := n(x/ ) ∈ L ∞ (D). Furthermore, assume that both A(y) and n(y) are periodic in y = x/ with period Y (here x ∈ D is refer to as the slow variable where y = x/ ∈ R d is referred to as the fast variable). We remark that our convergence analysis is also valid in the absorbing case, i.e. for complex valued A and n, but since the real eigenvalues (which are the measurable ones) exist only for real valued material properties, we limit ourselves to this case. Let us introduce the following notations:

The interior transmission eigenvalue problem for the anisotropic media (d = 2 in electromagnetic scattering and d = 3 in acoustic scattering) reads: find (w ε , v ε ) satisfying:

∇ • A ∇w ε + k 2 ε n w ε = 0 in D (3)
∆v ε + k 2 ε v ε = 0 in D (4)
w ε = v ε on ∂D (5) ∂w ε ∂ν Aε = ∂v ε ∂ν on ∂D (6)
where ∂w ∂ν A = ν • A∇w. Note that the spaces for the solution (w ε , v ε ) will become precise later since they depend on whether A = I or A = I. It is known that, provided that A -I or/and n -1 do not change sign in D and are bounded away from zero, the real transmission eigenvalues exist [START_REF] Cakoni | The existence of an infinite discrete set of transmission eigenvalues[END_REF], [START_REF] Cakoni | On the interior transmission eigenvalue problem[END_REF], [START_REF] Harris | Transmission eigenvalues and non-destructive testing of anisotropic magnetic materials with voids[END_REF]. However the transmission eigenvalue problem is non-selfajoint and this causes complications in the analysis. In this study we are interested in the behavior of eigenvalues k ε and eigenfunctions (w ε , v ε ) in limiting case as ε → 0. In particular we will be interested in the limit of the real transmission eigenvalues since they have been proven to exists and can be determined from scattering data.

1.1. Formal Asymptotic Expansion. We are interested in developing the asymptotic theory of ( 3)-( 6) as the period size → 0. To this end we need to define the space

H 1 # (Y ) := {u ∈ H 1 (Y ) | u(y)
is Y -periodic} and consider the subspace of Y -periodic H 1 -functions of mean zero, i.e.

H 1 # (Y ) := u ∈ H 1 # (Y ) | Y u(y) dy = 0 .
One expects (as our convergence analysis will confirm) that the homogenized or limiting transmission eigenvalue problem will be

∇ x • A h ∇ x w 0 + k 2 n h w 0 = 0 in D (7)
∆ x v 0 + k 2 v 0 = 0 in D (8)
w 0 = v 0 on ∂D (9) ∂w 0 ∂ν A h = ∂v 0 ∂ν on ∂D. (10)
where

A h = 1 |Y | Y A(y) -A(y)∇ y ψ(y) dy and n h = 1 |Y | Y n(y) dy, (11)
The so-called cell function

ψ i (y) ∈ H 1 # (Y ) is the unique solution to ∇ y • A∇ y ψ i = ∇ y • Ae i in Y, (12)
where e i is the i-th standard basis vector in R d . We recall that it is well known that the homogenized (otherwise known as effective) anisotropic constitutive parameter of the periodic medium A h satisfies the following estimates

[2]   1 |Y | Y A -1 (y)dy   -1 ξ • ξ ≤ A h ξ • ξ ≤   1 |Y | Y A(y)dy   ξ • ξ ξ ∈ C d (13)
hence ( 1) and ( 2) are also satisfied for A h and n h . The question now is whether the eigenvalues k and corresponding eigenfunctions v , w of (3)-( 6) converge to eigenvalues and eigenfuctions of ( 7)- [START_REF] Cakoni | The linear sampling method for anisotropic media[END_REF]. For the Dirichlet and Neumann eigenvalue problem for periodic structures the question of convergence is studied in details. In particular for these problems, the convergence is proven in [START_REF] Bensoussan | Asymptotic Analysis for Periodic Structures[END_REF], [START_REF] Kesavan | Homogenization of elliptic eigenvalue problems: part 1[END_REF] and [START_REF] Kesavan | Homogenization of elliptic eigenvalue problems: part 2[END_REF] and the rate of convergence with explicit first order correction involving the boundary layer is studied in [START_REF] Kenig | Estimates of eigenvalues and eigenfunctions in periodic homogenization[END_REF], [START_REF] Kenig | Homogenization of elliptic systems with Neumann boundary conditions[END_REF], [START_REF] Moskow | First-order corrections to the homogenized eigenvalues of periodic composite material. A convergence proof[END_REF], [START_REF] Moskow | First-order corrections to the homogenized eigenvalues of periodic composite material[END_REF] and [START_REF] Santosa | First-order corrections to the homogenized eigenvalues of periodic composite medium[END_REF]. Given the peculiarities of the transmission eigenvalue problem such as non-selfadjointness and the lack of ellipticity, the above approaches cannot be applied in a straightforward manner. Furthermore the transmission eigenvalue problem exhibits different properties in the case when A = I or A = I , hence each of these cases need to be studied separately [START_REF] Cakoni | Transmission eigenvalues in inverse scattering theory Inverse Problems and Applications[END_REF]. We remark that the existence of an infinite set of transmission eigenvalues in general settings is proven in [START_REF] Lakshtanov | Ellipticity in the interior transmission problem in anisotropic media[END_REF], [START_REF] Lakshtanov | Remarks on interior transmission eigenvalues, Weyl formula and branching billiards[END_REF] and [START_REF] Robbiano | Spectral analysis of the interior transmission eigenvalue problem[END_REF], where the existence of an infinite set of real transmission eigenvalues along with monotonicity properties are proven in [START_REF] Cakoni | The existence of an infinite discrete set of transmission eigenvalues[END_REF] and [START_REF] Cakoni | On the interior transmission eigenvalue problem[END_REF]. In the next section we justify the formal asymptotic for the resolvent corresponding to the transmission eigenvalue problem using the two scale convergent approach developed in [START_REF] Allaire | Homogenization and Two-Scale Convergence[END_REF]. This is followed by the proof of convergence results for a subset of real transmission eigenvalues in Section 3. The last section is dedicated to some preliminary numerical examples where we investigate convergence properties of the first transmission eigenvalue and demonstrate the feasibility of using the first real transmission eigenvalue to determine the effective material properties A h and n h .

2. Convergence Analysis. We start with studying the convergence of the resolvent of the transmission eigenvalue problem, i.e. of the solution to the interior transmission problem with source terms. The approach to study the interior transmission problem depends on the fact whether A(y) = I for all y ∈ Y or A(y) ≡ I.

2.1.

The case of A ε = I. We assume that A min > 1 or A max < 1 in addition to (1) and [START_REF] Allaire | Shape Optimization by the Homogenization Method[END_REF]. For f and g in L 2 (D) strongly convergent to f and g, respectively, as → 0 we consider the interior transmission problem of finding

(w ε , v ε ) ∈ H 1 (D) × H 1 (D) such that ∇ • A (x/ε) ∇w ε + k 2 n (x/ε) w ε = f in D (14)
∆v ε + k 2 v ε = g in D (15)
w ε = v ε on ∂D (16) ∂w ε ∂ν Aε = ∂v ε ∂ν on ∂D. ( 17
)
The following result is known (see [START_REF] Cakoni | The linear sampling method for anisotropic media[END_REF] and [START_REF] Cakoni | Interior transmission problem for anisotropic media[END_REF] for the proof).

Lemma 2.1. Assume that A min > 1 or A max < 1. Then the problem ( 14)- [START_REF] Cakoni | On the interior transmission eigenvalue problem[END_REF] satisfies the Fredholm alternative. In particular it has a unique solution (w , v ) ∈ H 1 (D) × H 1 (D) provided k is not a transmission eigenvalue.

The following lemma is proven in [START_REF] Bonnet-Bendhia | On the use of t-coercivity to study the interior transmission eigenvalue problem[END_REF] and [START_REF] Cakoni | The existence of an infinite discrete set of transmission eigenvalues[END_REF].

Lemma 2.2. Assume that A min > 1 or A max < 1 and either n ≡ 1 or if n ≡1 then Y (n(y) -1)dy = 0. The set of transmission eigenvalues k ∈ C is at most discrete with +∞ as the only accumulation point.

Note that (13) implies that

A h -I is positive definite if A min > 1 and I -A h is positive definite if A max < 1.
To analyze ( 14)-( 17) we introduce the variational space

X(D) := {(w, v) : w, v ∈ H 1 (D) | w -v ∈ H 1 0 (D)} equipped with H 1 (D) × H 1 (D) norm
and assume that k is not a transmission eigenvalue for all > 0 small enough. Let (w ε , v ε ) ∈ X(D) be the solution of ( 14)- [START_REF] Cakoni | On the interior transmission eigenvalue problem[END_REF] for ε ≥ 0 small enough (for ε = 0 we take the interior transmission problem with the homogenized coefficients A h and n h ) and assume that (w ε , v ε ) is a bounded sequence in X(D)-norm with respect to > 0 (this assumption will be discussed later in the paper). This solution satisfies the variational problem

D A ε ∇w ε •∇ϕ 1 -∇v ε •∇ϕ 2 -k 2 (n ε w ε ϕ 1 -v ε ϕ 2 ) dx = D g ϕ 2 dx- D f ϕ 1 dx (18)
for all (ϕ 1 , ϕ 2 ) ∈ X(D). Hence we have that there is a (w, v) ∈ X(D) such that a subsequence (w ε , v ε ) (w, v) weakly in X(D) (strongly in L 2 (D) × L 2 (D)). We now show that (w, v) solves the homogenized interior transmission problem. We adopt the formal two-scale convergence framework: we say that a sequence

α ε of L 2 (D) two-scale converges to α ∈ L 2 (D × Y ) if D α ε ϕ(x)φ(x/ε)dx → 1 |Y | D Y α(x, y)ϕ(x)φ(y)dydx
for all ϕ ∈ L 2 (D) and φ ∈ C # (Y ) (the space of Y -periodic continuous functions).

From [1, Proposition 1.14] there exists w 1 and

v 1 ∈ L 2 (D, H 1 # (Y )
) such that (up to a subsequence), ∇w ε and ∇v ε respectively two-scale converge to ∇ x w(x)+∇ y w 1 (x, y) and

∇ x v(x)+∇ y v 1 (x, y). Let θ 1 and θ 2 in C ∞ 0 (D), φ 1 and φ 2 in C ∞ # (Y ) (Y -periodic C ∞ functions) and (ψ 1 , ψ 2 ) ∈ X(D). Applying (18) to (ϕ 1 , ϕ 2 ) ∈ X(D) such that ϕ i (x) = ψ i (x) + εθ i (x)φ i (x/ε), i = 1, 2 then taking the two-scale limit implies D Y A(y)(∇w(x) + ∇ y w 1 (x, y)) • (∇ψ 1 (x) + θ 1 (x)∇φ 1 (y))dydx - D Y (∇v(x) + ∇ y v 1 (x, y)) • (∇ψ 2 (x) + θ 2 (x)∇φ 2 (y))dydx -k 2 D Y n(y)w(x)ψ 1 (x) -v(x)ψ 2 (x)dydx = |Y | D g(x)ψ 2 (x) -f (x)ψ 1 (x) dx. (19)
Taking ψ 1 = ψ 2 = 0 one easily deduces

w 1 (x, y) = -ψ(y) • ∇w(x) + w 1 (x) and v 1 (x, y) = v 1 (x). (20)
Then considering again [START_REF] Haddar | Computing estimates on material properties from transmission eigenvalues[END_REF] with

θ 1 = θ 2 = 0 implies that (w, v) ∈ X(D) satisfies D A h ∇w • ∇ψ 1 -∇v ε • ∇ψ 2 -k 2 (n h wψ 1 -vψ 2 ) dx = D gψ 2 dx - D f ψ 1 dx (21)
which is the variational formulation of the homogenized problem ( 7)- [START_REF] Cakoni | The linear sampling method for anisotropic media[END_REF].

The above analysis was based on the assumption that the sequence that solves ( 14)-( 17) is bounded with respect to > 0. Now we wish to show that any sequence that solves ( 14)-( 17) is indeed bounded independently of ε.

Theorem 2.1. Assume that either A min > 1 or A max < 1 and that k is not a transmission eigenvalue for ε ≥ 0 small enough. Then for any (w ε , v ε ) solving (14)-( 17) there exists C > 0 independent of (f ε , g ε ) and ε such that

||w ε || H 1 (D) + ||v ε || H 1 (D) ≤ C ||f ε || L 2 (D) + ||g ε || L 2 (D) .
Proof. We will prove the Fredholm property following the T-coercivity approach in [START_REF] Bonnet-Bendhia | On the use of t-coercivity to study the interior transmission eigenvalue problem[END_REF]. To this end we recall the variational formulation [START_REF] Cossonnière | Valeurs propres de transmission et leur utilisation dans l'identification d'inclusions à partir de mesures électromagnètiques[END_REF] equivalent to ( 14)- [START_REF] Cakoni | On the interior transmission eigenvalue problem[END_REF]. Let us first assume that A min > 1, which means that A -I is positive definite in D uniformly with respect to > 0, and define the bounded sesquilinear forms in X

(D) × X(D) a ε (w ε , v ε ); (ϕ 1 , ϕ 2 ) := D A ε ∇w ε • ∇ϕ 1 + A min w ε ϕ 1 dx - D ∇v ε • ∇ϕ 2 + v ε ϕ 2 dx, b ε (w ε , v ε ); (ϕ 1 , ϕ 2 ) := - D (k 2 n ε + A min )w ε ϕ 1 -(k 2 + 1)v ε ϕ 2 dx.
Then [START_REF] Cossonnière | Valeurs propres de transmission et leur utilisation dans l'identification d'inclusions à partir de mesures électromagnètiques[END_REF] can be written as

a ε (w ε , v ε ); (ϕ 1 , ϕ 2 ) + b ε (w ε , v ε ); (ϕ 1 , ϕ 2 ) = F ε (ϕ 1 , ϕ 2 )
where F ε (ϕ 1 , ϕ 2 ) is the bounded linear functional on X(D) defined by the right hand side of [START_REF] Cossonnière | Valeurs propres de transmission et leur utilisation dans l'identification d'inclusions à partir de mesures électromagnètiques[END_REF]. Let us consider A : X(D) → X(D) and B : X(D) → X(D) the bounded linear operators defined from a ε (w ε , v ε ); (ϕ 1 , ϕ 2 ) and b ε (w ε , v ε ); (ϕ 1 , ϕ 2 ) by means of Riesz representation theorem. It is clear that B is compact. We next show that A is invertible with bounded inverse uniformly with respect to > 0.

To this end we consider the isomorphism T(w, v) = (w, -v + 2w) : X(D) → X(D) (it is easy to check that T = T -1 ) and show that a ε (w ε , v ε ); T(ϕ 1 , ϕ 2 ) is coercive in X(D). Note that the isomorphism T does not depend on . Hence, we have that

a ε (w ε , v ε ); T(w ε , v ε ) ≥ D A ε ∇w ε • ∇w ε + A min |w ε | 2 dx + D |∇v ε | 2 + |v ε | 2 dx -2 D ∇v ε • ∇w ε + v ε w ε dx .
But we can estimate

2 D ∇v ε • ∇w ε + v ε w ε dx ≤ 1 δ ||w ε || 2 H 1 (D) + δ||v ε || 2 H 1 (D)
for any δ > 0.

Hence we obtain

a ε (w ε , v ε ); T(w ε , v ε ) ≥ A min - 1 δ ||w ε || 2 H 1 (D) + (1 -δ)||v ε || 2 H 1 (D) .
So for any δ ∈ 1 Amin , 1 we have that there is a constant α > 0 independent of ε such that

a ε (w ε , v ε ); T(w ε , v ε ) ≥ α ||w ε || 2 H 1 (D) + ||v ε || 2 H 1 (D) .
Next we assume that A max < 1 which means that I -A is positive definite in D uniformly with respect to > 0. Similarly we define

a ε (w ε , v ε ); (ϕ 1 , ϕ 2 ) := D A ε ∇w ε • ∇ϕ 1 + A max w ε ϕ 1 dx - D ∇v ε • ∇ϕ 2 + v ε ϕ 2 dx b ε (w ε , v ε ); (ϕ 1 , ϕ 2 ) := - D (k 2 n ε + A max )w ε ϕ 1 -(k 2 + 1)v ε ϕ 2 dx
and the corresponding bounded linear operator A : X(D) → X(D) and B : X(D) → X(D). To show that A is invertible we now consider the isomorphism

T(w, v) = (w -2v, -v) : X(D) → X(D) (again it is easy to check that T = T -1 ).
We then have that

a ε (w ε , v ε ); T(w, v) ≥ D A ε ∇w ε • ∇w ε + A max |w ε | 2 dx + D |∇v ε | 2 + |v ε | 2 dx -2 D A ε ∇w ε • ∇v ε + A max w ε v ε dx .
Using that A ε is symmetric positive definite we have that for any δ > 0:

2 D A ε ∇w ε • ∇v ε dx ≤ δ D A ε ∇w ε • ∇w ε dx + A max δ D |∇v| 2 ε dx
We also use that for any µ > 0:

2 D A max w ε v ε dx ≤ A 2 max µ ||w ε || 2 L 2 (D) + µ||v ε || 2 L 2 (D)
From the above inequalities we see that:

a ε (w ε , v ε ); T(w ε , v ε ) ≥ A min (1 -δ) ||∇w ε || 2 L 2 (D) + 1 - A max δ ||∇v ε || 2 L 2 (D) + A max 1 - A max µ ||w ε || 2 L 2 (D) + (1 -µ)||v ε || 2 L 2 (D) for any µ , δ ∈ (A max , 1). Hence A -1 ε : X(D) → X(D) exists for all ε > 0 with ||A -1 ε || L(X(D))
bounded independently of ε. The above analysis also proves that the Fredholm alternative can be applied to the operator (A + B ) and equivalently to ( 14)- [START_REF] Cakoni | On the interior transmission eigenvalue problem[END_REF]. Therefore if k is not a transmission eigenvalue for ε ≥ 0 we have that there is a constant C ε that does not depend on (f ε , g ε ) but possibly on > 0 such that the unique solution (w ε , v ε ) of ( 14)-( 17)

||w ε || H 1 (D) + ||v ε || H 1 (D) ≤ C ε ||f ε || L 2 (D) + ||g ε || L 2 (D) .
The above analysis show that if (w , v ) ∈ X(D) solves ( 14)-( 17) then

(I + K )(w , v ) = (α , β ) where K is compact such that ||K ε (w ε , v ε )|| X(D) ≤ M 1 ||w ε || L 2 (D) + ||v ε || L 2 (D) (22)
and

(α ε , β ε ) ∈ X(D) is such that ||α ε || H 1 (D) + ||β ε || H 1 (D) ≤ M 2 ||f ε || L 2 (D) + ||g ε || L 2 (D) (23)
with M 1 and M 2 independent of (Note that [START_REF] Kenig | Estimates of eigenvalues and eigenfunctions in periodic homogenization[END_REF] holds for

K = A -1 B since obviously B (w ε , v ε ) X(D) is bounded by the L 2 (D) × L 2 (D) norm of (w ε , v ε ) and A -1 is uniformly bounded with respect to ).
Next we need to show that C ε is bounded independently of ε. Assume to the contrary that C ε is not bounded as → 0. If this is true we can find a subsequence such that

||w ε || L 2 (D) + ||v ε || L 2 (D) ≥ γ ε ||f ε || L 2 (D) + ||g ε || L 2 (D)
where the sequence γ ε ε→0 -→ ∞. So we define the sequence ( wε , ṽε ) ∈ X(D)

wε := w ε ||w ε || L 2 (D) + ||v ε || L 2 (D)
and ṽε :=

v ε ||w ε || L 2 (D) + ||v ε || L 2 (D)
.

Notice that || wε || L 2 (D) + ||ṽ ε || L 2 (D) = 1 and ( wε , ṽε ) solves ( 14)-( 17) with ( fε , gε ) ∈ L 2 (D) × L 2 (D) given by

fε := f ε ||w ε || L 2 (D) + ||v ε || L 2 (D)
and gε :=

g ε ||w ε || L 2 (D) + ||v ε || L 2 (D)
.

Furthermore we have that

|| fε || L 2 (D) + ||g ε || L 2 (D) ≤ 1 γε ε→0 -→ 0 and (I + K ε )( wε , ṽε ) = (α ε , βε )
, where αε , βε are defined from f and g as above. Now from ( 22) and [START_REF] Kenig | Convergence rates in L 2 for elliptic homogenization problems[END_REF] we have that for all ε sufficiently small

|| wε || H 1 (D) + ||ṽ ε || H 1 (D) ≤ ||K ε ( wε , ṽε )|| X(D) + ||(α ε , βε )|| X(D) , ≤ M 1 || wε || L 2 (D) + ||ṽ ε || L 2 (D) + M 2 || fε || L 2 (D) + ||g ε || L 2 (D) , ≤ M 1 + M 2 .
Since M 1 and M 2 are independent of ε we have that ( wε , ṽε ) is a bounded sequence in X(D) and therefore has a subsequence that converges to ( w, ṽ) weakly in X(D) (strongly in L 2 (D)×L 2 (D)). Also we have that ( w, ṽ) solves ( 21) with (f, g) = (0, 0). Since k is not a transmission eigenvalue for ε = 0 we have that ( w, ṽ) = (0, 0) which contradicts the fact that || w|| L 2 (D) + ||ṽ|| L 2 (D) = 1 which proves the claim.

Notice that Theorem 2.1 gives that any sequence (w ε , v ε ) that solves ( 14)-( 17) is bounded in X(D) since f ε and g ε are assumed to converge strongly in L 2 (D). We can now state the following convergence result given by the above analysis.

Theorem 2.2. Assume that either A min > 1 or A max < 1 and that k is not a transmission eigenvalue for ε ≥ 0 small enough. Then we have that (w ε , v ε ) solving ( 14)-( 17) converges weakly in X(D) strongly in L 2 (D) × L 2 (D) to (w, v) that is a solution of (21). If we assume in addition that w ∈ H 2 (D) then, v ε strongly converges to v in H 1 (D) and w ε (x) -w(x) -εw 1 (x, x/ε) strongly converges to 0 in H 1 (D) where w 1 (x, y) := -ψ(y) • ∇w(x).

Proof. The first part of the theorem is a direct consequence of the above analysis and the uniqueness of solutions to [START_REF] Harris | Transmission eigenvalues and non-destructive testing of anisotropic magnetic materials with voids[END_REF]. The corrector type result is obtained using the T-coercivity property as follows. We first observe that, due to the strong convergence of the right hand side of the variational formulation of interior transmission problem, we have that

(a ε + b ε ) (w ε , v ε ); T(w ε , v ε ) → F (T(w, v)) = (a + b) (w, v); T(w, v)
as ε → 0 where a and b have similar expressions as a ε and b ε with A ε and n ε respectively replaced by A h and n h and F has the same expression as F ε with f ε and g ε respectively replaced with f and g. The L 2 strong convergence implies that

b ε (w ε , v ε ); T(w ε , v ε ) → b (w, v); T(w, v) .
We therefore end up with,

a ε (w ε , v ε ); T(w ε , v ε ) → a (w, v); T(w, v) (24)
as ε → 0. Let us set w ε 1 (x) := w 1 (x, x/ε). From the expression of w 1 one has (see for instance [START_REF] Moskow | First-order corrections to the homogenized eigenvalues of periodic composite material. A convergence proof[END_REF])

ε 1/2 w ε 1 H 1/2 (∂D) ≤ C
for some constant C independent of ε. Therefore we can construct a lifting function

v ε 1 ∈ H 1 (D) such that v ε 1 = w ε 1 on ∂D and ε v ε 1 H 1 (D) → 0 as ε → 0. (25)
Now, taking as test functions ϕ 1 = wε and ϕ 2 = ṽε where wε (x) := w(x) + εw 1 (x, x/ε) and ṽε (x

) := v(x) + εv ε 1 (x), one has (a ε + b ε ) (w ε , v ε ); T( wε , ṽε ) → F (T(w, v)).
Using the two-scale convergence of the sequences w ε and v ε together with the form (and regularity) of w 1 as well as [START_REF] Kesavan | Homogenization of elliptic eigenvalue problems: part 1[END_REF], we easily see that

b ε (w ε , v ε ); T( wε , ṽε ) → b (w, v); T(w, v) while a ε (w ε , v ε ); T( wε , ṽε ) → L(w, w 1 , v) with L(w, w 1 , v) = 1 |Y | D Y A(y)(∇w(x) + ∇ y w 1 (x, y)) • (∇w(x) + ∇ y w 1 (x, y))dydx + D |∇v(x)| 2 + A min |w(x)| 2 + |v(x)| 2 -2∇w(x)∇v(x) -2w(x)v(x)dx in the case A min > 1 and L(w, w 1 , v) = 1 |Y | D Y A(y)(∇w(x) + ∇ y w 1 (x, y)) • (∇w(x) + ∇ y w 1 (x, y))dydx -2 1 |Y | D Y A(y)(∇w(x) + ∇ y w 1 (x, y)) • ∇v(x)dydx + D |∇v(x)| 2 + A min |w(x)| 2 + |v(x)| 2 -2A min v(x)w(x)dx
in the case A max < 1. Hence we can conclude that

F (T(w, v)) = L(w, w 1 , v) + b (w, v); T(w, v) and therefore a (w, v); T(w, v) = L(w, w 1 , v). (26)
Using ( 24) and ( 26) and the T-coercivity, we can apply similar arguments as in [START_REF] Allaire | Homogenization and Two-Scale Convergence[END_REF]Theorem 2.6] to obtain the result. Indeed, the T-coercivity shows that it is sufficient to prove that

a ε (w ε -wε , v ε -v); T(w ε -wε , v ε -v) → 0. (27)
Now, using the two-scale convergence of the sequences v ε and w ε , we observe that each of the quantities

a ε (w ε , v ε ); T( wε , v) , a ε ( wε , v); T(w ε , v ε ) and a ε ( wε , v); T( wε , v) converges to L(w, w 1 , v)
Finally, using [START_REF] Kenig | Homogenization of elliptic systems with Neumann boundary conditions[END_REF] we can conclude that

a ε (w ε -wε , v ε -v); T(w ε -wε , v ε -v) → a (w, v); T(w, v) -L(w, w 1 , v)
and then the result is a direct consequence of [START_REF] Kirsch | The inside-outside duality for scattering problems by inhomogeneous media[END_REF]. Here we now assume that either n min > 1 or 0 < n max < 1. For the case where A ε ≡ I the interior transmission problem becomes:

Find (w ε , v ε ) ∈ L 2 (D) × L 2 (D) such that ∆w ε + k 2 n (x/ε) w ε = 0 in D ( 28
)
∆v ε + k 2 v ε = 0 in D (29) w ε -v ε = f ε on ∂D (30) ∂w ε ∂ν - ∂v ε ∂ν = g ε on ∂D (31)
for the boundary data

(f ε , g ε ) ∈ H 3/2 (∂D) × H 1/2 (∂D) converging strongly to (f, g) ∈ H 3/2 (∂D) × H 1/2 (∂D) as → 0.
Just as in the case for anisotropic media we require that k 2 is not a transmission eigenvalue for ε ≥ 0 small enough. We formulate the interior transmission problem for the difference

U ε := w ε -v ε ∈ H 2 (D).
Using the interior transmission problem one can show that this U satisfies

0 = ∆ + k 2 n ε 1 n ε -1 ∆ + k 2 U ε in D (32)
where

v ε = - 1 k 2 (n ε -1) ∆U ε + k 2 n ε U ε in D (33)
w ε = - 1 k 2 (n ε -1) ∆U ε + k 2 U ε in D (34)
Theorem 2.3. Assume that either (n min -1) > 0 or (n max -1) < 0 and

U ε ∈ H 2 (D) is a bounded sequence, then there is a subsequence such that U ε U in H 2 (D) and (w ε , v ε ) (w, v) in L 2 (D) × L 2 (D) strongly in L 2 loc (D) × L 2 loc (D)
. Moreover we have that the limit U satisfies

∆ + k 2 n h 1 n h -1 ∆ + k 2 U = 0 in D, (35)
U = f and ∂U ∂ν = g on ∂D, (36)
U = w -v, and (w, v) satisfy

∆v + k 2 v = 0 and ∆w + k 2 n h w = 0 in D, (37)
w -v = f and ∂w ∂ν - ∂v ∂ν = g on ∂D. ( 38
)
Proof. Since U ε is a bounded sequence in H 2 (D), from ( 33) and [START_REF] Sun | Iterative methods for transmission eigenvalues[END_REF] we have that

(w ε , v ε ) is a bounded sequence in L 2 (D) × L 2 (D)
. Therefore we have that there is a subsequence still denoted by (w ε , v ε ) that is weakly convergent in L 2 (D) × L 2 (D). So we have that for all ϕ ∈ C ∞ 0 (D), there is a v ∈ L 2 (D) such that:

0 = D v ε (∆ϕ + k 2 ϕ) dx ε→0 -→ D v(∆ϕ + k 2 ϕ) dx.
This gives that ∆v + k 2 v = 0 in the distributional sense. By interior elliptic regularity (see e.g. [START_REF] Wloka | Partial Differential Equations[END_REF]) for all Ω ⊂ Ω ⊂ D and all ε > 0 we have

||v ε || H 1 (Ω) ≤ C
for some constant independent of ε which implies (using an increasing sequence of domains Ω n that converges to D and a diagonal extraction process of the subsequence) that a subsequence v ε converges to v strongly in L 2 loc (D). Next since

w ε = U ε + v ε and U is bounded in H 2 (D)
, we have that w ε converges to some w weakly in L 2 (D) and strongly in L 2 loc (D). Now using the strong convergence we have that for all ϕ ∈ C ∞ 0 (D) such that supp(ϕ) ⊂ D we obtain that

0 = D w ε (∆ϕ + k 2 n ε ϕ) dx ε→0 -→ D w(∆ϕ + k 2 n h ϕ) dx,
which gives that ∆w + k 2 n h w = 0 in the distributional sense. Now, the fact that -k 2 (n ε -1)w ε = ∆U ε + k 2 U ε , the weak convergence of U to U in H 2 (D) and the local strong convergence of w to the above w imply that the limit U satisfies

∆ + k 2 n h 1 n h -1 ∆ + k 2 U = 0 in D and U = w -v.
Finally, integration by parts formulas together with [START_REF] Moskow | First-order corrections to the homogenized eigenvalues of periodic composite material. A convergence proof[END_REF] and [START_REF] Moskow | First-order corrections to the homogenized eigenvalues of periodic composite material[END_REF] guaranty that U := w -v satisfies the boundary conditions [START_REF] Wloka | Partial Differential Equations[END_REF] and (38) which ends the proof.

The above result that connects w , v and U with the respective limits requires that U is a bounded sequence. Next we show that this is the case for every solution to the interior transmission problem. To this end, since

(f ε , g ε ) ∈ H 3/2 (∂D) × H 1/2 (∂D) there is a lifting function φ ε ∈ H 2 (D) such that φ ε ∂D = f and ∂φε ∂ν ∂D = g ε and ||φ ε || H 2 (D) ≤ C ||f ε || H 3/2 (∂D) + ||g ε || H 1/2 (∂D) (39)
where the constant C is independent of ε and φ ε → φ strongly in H 2 (D) where φ ∂D = f and ∂φ ∂ν ∂D = g. Now following [START_REF] Cakoni | The existence of an infinite discrete set of transmission eigenvalues[END_REF] and [START_REF] Cakoni | On the existence of transmission eigenvalues in an inhomogenuous medium[END_REF] we define the bounded sesquilinear forms on H 2 0 (D) × H 2 0 (D):

A ε (u, ϕ) = D 1 n ε -1 ∆u + k 2 u ∆ϕ + k 2 ϕ + k 4 uϕ dx, (40)
A ε (u, ϕ) = D n ε 1 -n ε ∆u + k 2 u ∆ϕ + k 2 ϕ + ∆u∆ϕ dx, (41)
B(u, ϕ) = D ∇u∇ϕ dx. ( 42
)
With the help of the lifting function φ , we have that u ε ∈ H 2 0 (D) where U ε = u ε +φ ε and that u ε solve the variational problems

A ε (u ε , ϕ) -k 2 B(u ε , ϕ) = L ε (ϕ) (43) A ε (u ε , ϕ) -k 2 B(u ε , ϕ) = L ε (ϕ) (44)
where the conjugate linear functionals are defined as follows

L ε (ϕ) = k 2 B(φ ε , ϕ) -A ε (φ ε , ϕ) and L ε (ϕ) = k 2 B(φ ε , ϕ) -A ε (φ ε , ϕ). Let A : H 2 0 (D) → H 2 0 (D), A : H 2 0 (D) → H 2 0 (D)
and B : H 2 0 (D) → H 2 0 (D) be bounded linear operators defined by the sesquilinear forms (40), ( 41) and ( 42) by means of Riesz representation theorem. Obviously B is a compact operator and it does not depend on , and furthermore B(u

) H 2 (D) is bounded by u H 1 (D) . In [16] it is shown that A ε (•, •) is coercive when 1 nε-1 ≥ α > 0 for all ε > 0 (which is satisfied if n min > 1) whereas A ε (•, •) is coercive when nε 1-nε ≥ α > 0 for all > 0 (which is satisfied if 0 < n max < 1)
and furthermore the coercivity constant depends only on D and α. Hence A -1 exists if n min > 1 and A -1 exists if 0 < n max < 1 and their norm is uniformly bounded with respect to .

Theorem 2.4. Assume that either n min > 1 or 0 < n max < 1, and that k is not a transmission eigenvalue for ≥ 0 small enough. If U ε ∈ H 2 (D) is a solution to [START_REF] Robbiano | Spectral analysis of the interior transmission eigenvalue problem[END_REF] such that U ε = f ε and ∂Uε ∂ν = g ε on ∂D, then there is a constant C > 0 independent of ≥ 0 and (f ε , g ε ) such that:

||U ε || H 2 (D) ≤ C ||f ε || H 3/2 (∂D) + ||g ε || H 1/2 (∂D) .
Proof. First recall that U ε = u ε + φ ε where u ε ∈ H 2 0 (D) satisfies either (43) or (44) and φ ε ∈ H 2 (D) satisfies (39). Therefore it is sufficient to prove the result for u ε . From the discussion above we know that u satisfies

(I -k 2 K )(u ) = α (45)
where

K = A -1 B and α ∈ H 2 0 (D) is the Riesz representation of L ε if n min > 1, and K = A -1 B and α ∈ H 2 0 (D) is the Riesz representation of L ε if 0 < n max < 1. In both cases K (u ) H 2 (D) ≤ M 1 u H 1 (D)
and

α H 2 (D) ≤ M 2 ||f ε || H 3/2 (∂D) + ||g ε || H 1/2 (∂D)
with M 1 and M 2 independent of > 0. Now since k 2 is not a transmission eigenvalue for ≥ 0 (small enough), the Fredholm alternative applied to (45) guaranties the existence of a constant C ε independent of f , g such that

||u ε || H 2 (D) ≤ C ε ||f ε || H 3/2 (∂D) + ||g ε || H 1/2 (∂D) .
In the same way as in Theorem 2.1, we can now show that C ε is bounded independently of ε. Indeed, to the contrary assume that C ε is not bounded as → 0. Then we can find a subsequence u such that

||u ε || H 1 (D) ≥ γ ||f ε || H 3/2 (∂D) + ||g ε || H 1/2 (∂D)
and γ → ∞ as ε → 0. Let us define the sequences ũε := . Hence we have that ( fε , gε ) → (0, 0) as ε → 0 and (I -

k 2 K )(ũ ) = α . Hence ||ũ ε || H 2 (D) ≤ k 2 ||K ε (ũ ε )|| H 2 (D) + ||α ε || H 2 (D) , ≤ M 1 ||ũ ε || H 1 (D) + M 2 || fε || H -3/2 (∂D) + ||g ε || H 1/2 (∂D) ≤ M 1 + M 2 .
Hence ũε is bounded and therefore has a weak limit in H 2 0 (D), which from Theorem 2.3 is a solution to the homogenized equation [START_REF] Sun | Computation of Maxwell's transmission eigenvalues and its applications in inverse medium problems[END_REF] with zero boundary data. This implies that ũ = 0 since k 2 is not a transmission eigenvalue for ε = 0 which contradicts the fact that ||ũ|| H 1 (D) = 1, proving the result.

We can now state the convergence result for the interior transmission problem. Theorem 2.5. Assume that either n min > 1 or 0 < n max < 1 and k is not a transmission eigenvalue for ≥ 0 small enough. Let (w , v

) ∈ L 2 (D) × L 2 (D) be such that U ε = w -v ∈ H 2 (D) is a sequence of solutions to (32) with (f ε , g ε ) ∈ H 3/2 (∂D) × H 1/2 (∂D) converging strongly to (f, g) ∈ H 3/2 (∂D) × H 1/2 (∂D) as → 0. Then U ε U in H 2 (D) and (w ε , v ε ) (w, v) in L 2 (D) × L 2 (D) strongly in L 2 loc (D) × L 2 loc (D)
, where the limit U satisfies 3. Convergence of the Transmission Eigenvalues. Using the convergence analysis for the solution of the interior transmission problem, we now prove the convergence of a sequence of real transmission eigenvalues of the periodic media, namely of those who are bounded with respect to the small parameter . The following lemmas provide conditions for the existence of real transmission eigenvalues that are bounded in .

∆ + k 2 n h 1 n h -1 ∆ + k 2 U = 0 in D (46) U = f and ∂U ∂ν = g on ∂D, (47)
Lemma 3.1. The following holds: 1. Assume that A = I for all > 0 and either n min > 1 or 0 < n max < 1.

There exists an infinite sequence of real transmission eigenvalues k j , j ∈ N of ( 3)-( 6) accumulating at +∞ such that

k j (n max , D) ≤ k j < k j (n min , D) if n min > 1 k j (n min , D) ≤ k j < k j (n max , D) if 0 < n max < 1
where k j (n, D) denotes an eigenvalue of ( 3)-( 6) with A = I and n = n. 2. Assume that n = 1 for all > 0 and either A min > 1 or 0 < A max < 1.

There exists an infinite sequence of real transmission eigenvalues k j , j ∈ N of (3)-( 6) accumulating at +∞ such that

k j (a max , D) ≤ k j ≤ k j (a min , D) if a min > 1 k j (a min , D) ≤ k j ≤ k j (a max , D) if 0 < a max < 1
where k j (a, D) denotes an eigenvalue of ( 3)-( 6) with A = aI and n = 1. Here j counts the eigenvalue in the sequence under consideration which may not necessarily be the j-th transmission eigenvalue. In particular the first transmission eigenvalue satisfies the above estimates.

Proof. The detailed proof of the above statements can be found in [START_REF] Cakoni | The existence of an infinite discrete set of transmission eigenvalues[END_REF]. We remark that the statements are not proven for all real transmission eigenvalues. For example in the case of first statement, from the proofs in [START_REF] Cakoni | The existence of an infinite discrete set of transmission eigenvalues[END_REF], real transmission eigenvalues are roots of λ j (τ, n , D) -τ = 0, where λ j , j = 1 . . . , are eigenvalues of some auxiliary selfadjoint eigenvalue problem satisfying the Rayleigh quotient. The latter implies lower and upper bounds for λ j in terms of n min and n max , and these bounds are also satisfied by the transmission eigenvalues that are the smallest root of each λ j (τ, n , D) -τ = 0. Same argument applies to the second statement also. In particular the estimates hold for the first transmission eigenvalue.

The existence results and estimates on real transmission eigenvalues are more restrictive for the case when both A = I and n = 1. The following result is proven in [START_REF] Cakoni | On the interior transmission eigenvalue problem[END_REF] (see also [START_REF] Cakoni | Qualitative Approach to Inverse Scattering Theory[END_REF]).

Lemma 3.2. The following holds:

1. Assume that either a min > 1 and 0 < n max < 1 or 0 < a max < 1 and n min > 1. There exists a infinite sequence of real transmission eigenvalues k j , j ∈ N of ( 3)-( 6) accumulating at +∞ satisfying

k j (a max , n min , D) ≤ k j < k j (a min , n max , D) if a min > 1, 0 < n max < 1 k j (a min , n max , D) ≤ k j < k j (a max , n min , D) if 0 < a max < 1, n min > 1
where k j (a, n, D) denotes an eigenvalue of ( 3)-( 6) with A = aI and n = n. 2. Assume that a min > 1 and n min > 1 or 0 < a max < 1 and 0 < n max < 1.

There exists finitely many real transmission eigenvalues k j , j = 1 • • • p of ( 3)-( 6) provided that n max is small enough. In addition they satisfy

0 < k j < k j (a min /2, D) if a min > 1, n min > 1 0 < k j < k j (a max /2, D) if 0 < a max < 1, 0 < n max < 1
where k j (a, D) denotes an eigenvalue of ( 3)-( 6) with A = aI and n = 1. Here j counts the eigenvalue in the sequence under consideration which may not necessarily be the j-th transmission eigenvalue. In particular the first transmission eigenvalue satisfies the above estimates.

Proof. The estimates follow by the same argument as in the proof of Lemma 3.1 combined with the existence proofs in [START_REF] Cakoni | On the interior transmission eigenvalue problem[END_REF]. In particular, the estimates can be obtained by modifying the proof of Theorem 2.6 and Theorem 2.10 in [START_REF] Cakoni | On the interior transmission eigenvalue problem[END_REF] in a similar way as in the proof of Corollary 2.6 in [START_REF] Cakoni | The existence of an infinite discrete set of transmission eigenvalues[END_REF].

3.1. The case of A ε = I. We assume that A min > 1 or A max < 1 in addition to (1) and ( 2) and let k be one of the transmission eigenvalues described in Lemma 3.1 and Lemma 3.2. In particular {k } is bounded and hence there is a positive number k ∈ R such that k → k as → 0. Let (w , v ) be a corresponding pair of eigenfunctions normalized such that ||w ε || L 2 (D) + ||v ε || L 2 (D) = 1. Notice from Section 2.1 that the transmission eigenfunctions satisfy

A ε ((w ε , v ε ); (ϕ 1 , ϕ 2 )) = 0 for all (ϕ 1 , ϕ 2 ) ∈ X(D)
where the sesquilinear form A (•; •) is given by

A ε (w ε , v ε ); (ϕ 1 , ϕ 2 ) := D A ε ∇w ε • ∇ϕ 1 -∇v ε • ∇ϕ 2 -k 2 ε (n ε w ε ϕ 1 -v ε ϕ 2 ) dx.
Obviously if T : X(D) → X(D) is a continuous bijection then we have that the pair of the eigenfunction (w , v ) satisfies

A ε (w ε , v ε ); T(w ε , v ε ) = 0. ( 50
)
We will use (50) to prove that the sequence (w ε , v ε ) is bounded in X(D). To do so we must control the norm of the gradients of the functions in the sequence. Indeed, assuming that A min > 1 and letting T(w, v) = (w, -v + 2w) gives that

D A ε ∇w ε • ∇w ε + |∇v ε | 2 -2∇v ε • ∇w ε dx = k 2 ε D n ε |w ε | 2 + |v ε | 2 -2v ε w ε dx, (51)
which by using Young's inequality gives that

||∇w ε || 2 L 2 (D) + ||∇v ε || 2 L 2 (D)
is bounded independently of > 0. Similarly in the case when 0 < A max < 1 we obtain the result using T(w, v) = (w -2v, -v).

Therefore, in both cases we have that (w ε , v ε ) is a bounded sequence in X(D). This implies that there is a subsequence, still denoted by (w ε , v ε ), that converges weakly (strongly in L 2 (D) × L 2 (D) to some (w, v) in X(D)). The L 2 -strong limit implies that ||w|| L 2 (D) +||v|| L 2 (D) = 1 hence (w, v) = (0, 0). Using similar argument as at the beginning of Section 2.1 we have that k is a transmission eigenvalue, with (w, v) in X(D) the corresponding transmission eigenfunctions, for the homogenized transmission eigenvalue problem

∇ • A h ∇w + k 2 n h w = 0 and ∆v + k 2 v = 0 in D, (52)
w = v and ∂w ∂ν A h = ∂v ∂ν on ∂D. (53)
Hence we have proven the following result for the transmission eigenvalue problem.

Theorem 3.1. Assume that either A min > 1 or 0 < A max < 1 and let k ε be a sequence of transmission eigenvalues for ( 3)-( 6) with corresponding eigenfunctions (w ε , v ε ). Then, if k is bounded with respect to , then there is a subsequence of

{(w ε , v ε ) , k ε } ∈ X(D)×R such that (w ε , v ε ) (w, v) in X(D) strongly in L 2 (D)× L 2 (D) and k ε → k as ε → 0, where {(w, v) , k} ∈ X(D) × R is an eigenpair for (52)-(53).
3.2. The case of A ε ≡ I. In this case we assume that either n min > 1 or 0 < n max < 1. Let k be an eigenvalue of ( 3)-( 6) with corresponding eigenfunctions

(w ε , v ε ) ∈ L 2 (D) × L 2 (D) such that u ε = w ε -v ε ∈ H 2 0 (D).
As discussed in Section 2.2, (w ε , v ε ) are distributional solutions to:

∆v ε + k 2 ε v ε = 0 and ∆w ε + k 2 ε n ε w ε = 0 in D, (54)
whereas

u ε ∈ H 2 0 (D) solves 0 = ∆ + k 2 n ε 1 n ε -1 ∆ + k 2 u ε in D, (55)
which in the variational form reads

D 1 n ε -1 ∆u ε + k 2 ε u ε ∆ϕ + k 2 ε n ε ϕ dx = 0 for all ϕ ∈ H 2 0 (D). (56)
We recall that w , v and u are related by

v ε = - 1 k 2 (n ε -1) ∆u ε + k 2 n ε u ε in D (57)
w ε = - 1 k 2 (n ε -1) ∆u ε + k 2 u ε in D. (58)
Without loss of generality we consider the first real transmission eigenvalue k := k 1 and set τ := (k ) 2 . Since the corresponding eigenfunctions are nontrivial we can take ||u ε || H 1 (D) = 1, and in addition we have the existence of a limit point τ for the set {τ ε } ε>0 . Similarly to the previous case we wish to show that the normalized sequence u ε is bounded in H 2 0 (D). We start with the case when n min > 1 and let 16 FIORALBA CAKONI, HOUSSEM HADDAR AND ISAAC HARRIS

1 nmax-1 = α > 0. Taking ϕ = u in (56) implies D 1 n ε -1 |∆u ε | 2 + 2τ ε n ε -1 (∆u ε u ε ) + τ 2 ε n ε n ε -1 |u ε | 2 dx = 0
Therefore, making use of Lemma 3.1 part 1, we have that:

D 2τ ε n ε -1 (∆u ε )u ε dx ≤ 2τ (n min , D) n min -1 D (∆u ε )u ε dx ≤ 2τ (n min , D) n min -1 D |∇u ε | 2 dx.
Which gives that:

α||∆u ε || 2 L 2 (D) ≤ τ (n min , D) 2 n max n min -1 ||u ε || 2 L 2 (D) + 2τ (n min , D) n min -1 ||∇u ε || 2 L 2 (D) . Now since ||u ε || H 1 (D) = 1 and using that ||∆ • || L 2 (D)
is an equivalent norm on H 2 0 (D) we have that u ε is a bounded sequence. By the construction of (w ε , v ε ) we have that this is a bounded sequence in L 2 (D) × L 2 (D). Note that a similar argument holds if 0 < (n max -1) < 1, by multiplying the variational form by -1. Now by similar argument as in the proof of Theorem 2.3 we can now conclude the following result. Theorem 3.2. Assume that A ≡ I for all > 0 and either n min > 1 or n max < 1, and furthermore let k ε be a transmission eigenvalue for (3)-( 6) with corresponding eigenfunctions (w ε , v ε ). Then, if k is bounded with respect to , there is a subsequence of

{(w ε , v ε ) , k ε } ∈ (L 2 (D) × L 2 (D)) × R + such that (w ε , v ε ) (w, v) in L 2 (D) × L 2 (D) and k ε → k as ε → 0, where {(w, v) , k} ∈ L 2 (D) × L 2 (D) × R + is an eigenpair corresponding to ∆v + k 2 v = 0 and ∆w + k 2 n h w = 0 in D, w -v ∈ H 2 0 (D).
The proofs of both Theorem 3.1 and Theorem 3.2 simply depend on the boundedness of the sequence of any real transmission eigenvalue in terms of , therefore the proofs hold for all the eigenvalues that satisfy bounds stated in Lemma 3.1 and Lemma 3.2.

Remark 3.1. The transmission eigenvalues of the limiting problem (52)-(53) satisfy the same type of estimates as in Lemma 3.1 and Lemma 3.2. Furthermore, from the proof of Theorem 3.1 and Theorem 3.2 one can see that the limit k of the sequence {k }, where each k is the first transmission eigenvalue of (3)-( 6), is the first transmission eigenvalue of (52)-(53).

4. Numerical Experiments. We start this section with a preliminary numerical investigation on the convergence of the first transmission eigenvalue. To this end, we fix an A ε and n ε and investigate the behavior of the first transmission eigenvalue k 1 (ε) on . More specifically, we investigate the convergence rate of k 1 (ε) to the first eigenvalue k h corresponding to the homogenized problem. The first transmission eigenvalue for the periodic media and homogenized problem is computed using a mixed finite element method with an eigenvalue-searching technique described in [START_REF] Sun | Iterative methods for transmission eigenvalues[END_REF] and [START_REF] Sun | Computation of Maxwell's transmission eigenvalues and its applications in inverse medium problems[END_REF]. In addition, we show numerical examples of determining the first few real transmission eigenvalues from the far field scattering data. This section is concluded with some examples demonstrating that the first real transmission eigenvalue provides information about the effective material properties A h and n h of the periodic media. 4.1. Numerical Tests for the Order of Convergence. We consider the case where the domain D = B R with R = 2 and for the first example assume that the periodic media is isotropic, i.e. A = I, with refractive index

n ε = sin 2 (2πx 1 /ε) + cos 2 (2πx 2 /ε) + 2.
Obviously n h = 3. If the domain is a ball of radius two in R 2 separation of variables gives that the roots of

d 0 (k) = J 0 (2k √ n h ) J 1 (2k) - √ n h J 1 (2k √ n h ) J 0 (2k)
are transmission eigenvalues. Using the secant method we see that k h ≈ 2.0820.

The values of the first transmission eigenvalue for the periodic media for different values of are shown in Table 1. To find the convergence rate we assume that the error satisfies that

ε 1/3 1/
|k 1 (ε) -k h | = Cε p which gives log |k 1 (ε) -k h | = log(C) + p log(ε)
for some constant C independent of ε. Using the polyfit command in Matlab we can find a p that approximately satisfies the above equality. The calculations give that in this case p = 2.1486 (see Figure 2). In the next example we keep the same domain D and take the periodic constitutive parameters of the media

n ε = sin 2 (2πx 1 /ε) + cos 2 (2πx 2 /ε) + 2 (59)
and

A ε = 1 3 sin 2 (2πx 2 /ε) + 1 0 0 cos 2 (2πx 1 /ε) + 1 . ( 60
)
FIORALBA CAKONI, HOUSSEM HADDAR AND ISAAC HARRIS Notice that ∇ y • Ae i = 0 which gives that A h = 1 2 I and n h = 3. In this case the first zero of

d 0 (k) = J 0 2k n h A h J 1 (2k) -n h A h J 1 2k n h A h J 0 (2k)
is the first transmission eigenvalue k 1 h for the homogenized problem which turns out to be k 1 h = 1.0582. Similarly we use polyfit in Matlab to find a p such that log |k 1 (ε) -k h | = log(C) + p log(ε). In this case we calculate that p = 1.4421. The results are shown in Table 2 and Figure 3. In these two examples the convergence rate seems to be better than of order . Notice that the boundary correction in these both cases does not appear since there is no boundary correction if A = I and in the second example we have ∇ y • A ε e i = 0 =⇒ ψ(y) = 0 which yield no boundary correction (this will become clear in the second part of this study but for the case of Dirichlet and Neumann conditions see [START_REF] Moskow | First-order corrections to the homogenized eigenvalues of periodic composite material. A convergence proof[END_REF] and [START_REF] Moskow | First-order corrections to the homogenized eigenvalues of periodic composite material[END_REF], respectively). We now wish to investigate the numerical convergence rate when ψ(y) = 0. Hence take

n ε = sin 2 (2πx 1 /ε) + 2 (61)
and Ãε = T A ε T where A is given by (60) and T is the matrix representing clockwise rotation by 1 radian. We now compute the first transmission eigenvalue with coefficients n and Ãε . Since now ψ(y) = 0, we cannot compute analytically A h (one need to solve the cell problem numerically in order to compute A h ) and hence we do not have a value for the first transmission eigenvalue of the homogenized problem. In this case, in order to obtain an idea about the convergence order of the first transmission eigenvalue we define the relative error as:

R.E. = |k 1 (ε) -k 1 (ε/2)| k 1 (ε/2)
and find the convergence rate for the relative error is a similar manner as discussed above. The Table 3 and Figure 4 show the computed first transmission eigenvalue for various epsilon in the square D := [0, 2]×[0, 2] and the circular domain The above results seem to suggest that the relative error is of order ε. In this case the boundary corrector is non-zero which explain this order of convergence.

D := B R of radius R = 1. ε 1 1/2 1/

Transmission Eigenvalues and the Determination of Effective Material Properties.

For the given inhomogeneous media, the corresponding transmission eigenvalues are closely related to the so-called non-scattering frequencies, i.e. the values of k for which there exists an incident wave doesn't scatter [START_REF] Blåsten | Corners Always Scatter[END_REF], [START_REF] Cakoni | Transmission eigenvalues in inverse scattering theory Inverse Problems and Applications[END_REF]. The scattering problem associated with our transmission eigenvalue problem in R 2 20 FIORALBA CAKONI, HOUSSEM HADDAR AND ISAAC HARRIS is given by

∇ • A(x/ε)∇w ε + k 2 n(x/ε)w ε = 0 in D ∆u s ε + k 2 u s ε = 0 in R 2 \ D w ε -u s ε = u i and ∂w ε ∂ν Aε - ∂u s ε ∂ν = ∂u i ∂ν on ∂D lim r →∞ √ r ∂u s ε ∂r -iku s ε = 0
The asymptotic behavior of u s ε (r, θ) can be shown to be [START_REF] Cakoni | Qualitative Approach to Inverse Scattering Theory[END_REF] u

s ε (r, θ) = e ikr √ r u ∞ ε (θ, φ) + O r -3/2 as r → ∞.
where the function u ∞ ε is called the far field pattern of the scattering problem with incident direction φ and observation angle θ. Recall that the far field operator

F ε : L 2 (0, 2π) → L 2 (0, 2π) is defined by (F ε g)(θ) := 2π 0 u ∞ ε (θ, φ)g(φ) dφ.
It has been shown that the transmission eigenvalues can be determine form a knowledge of the far field operator F [START_REF] Cakoni | On the determination of Dirichlet or transmission eigenvalues from far field data[END_REF] and [START_REF] Kirsch | The inside-outside duality for scattering problems by inhomogeneous media[END_REF]. Now we would like to investigate how the first transmission eigenvalue determined from the far field operator depends on the parameter . Here to find the transmission eigenvalues from the far field data, we follow the approach in [START_REF] Cakoni | On the determination of Dirichlet or transmission eigenvalues from far field data[END_REF]. To this end, let Φ ∞ (•, •) be the far field pattern for the fundamental solution to the Helmholtz equation. If g z,δ is the Tikhonov regularized solution of the far field equation, i.e. the unique minimizer of the functional:

F ε g -Φ ∞ (•, z) 2 L 2 (0, 2π) + α g 2 L 2 (0, 2π)
with the regularization parameter α := α(δ) → 0 as the noise level δ → 0, then at a transmission eigenvalue v g z,δ L 2 (D) → ∞ as δ → 0 for almost every z ∈ D, whereas otherwise bounded, where v g (x) := 2π 0 g(φ)e ik(x1 cos φ+x2 sin φ) dφ. To compute the simulated data we use a FEM method to approximate the far field pattern corresponding to the scattering problem. Using the approximated u ∞ ε we then solve: F ε g = Φ ∞ (•, z) for 25 random values of z ∈ D where the regularization parameter is chosen based on Morozov's discrepancy principle. The transmission eigenvalues will appear as spikes in the plot of ||g z || L 2 [0,2π] versus k. In our example we choose the domain D := B R to be the ball of radius R = 1 and the material properties n given by (61) and A given by (60). The effective material properties are A h = 1 2 I and n h = 3 2 and the corresponding first transmission eigenvalue is k 1 h = 2.5340. The computed transmission eigenvalue for this configuration for the choices of = 1 and = 0.1 are shown in Figure The measured first transmission eigenvalue can be used to obtain information about the effective material properties A h and n h . If A = I, it is known that k 1 h uniquely determines n h and also the transmission eigenvalue depend continuously on n h [START_REF] Cakoni | The interior transmission eigenvalue problem[END_REF][START_REF] Cossonnière | Valeurs propres de transmission et leur utilisation dans l'identification d'inclusions à partir de mesures électromagnètiques[END_REF][START_REF] Haddar | Computing estimates on material properties from transmission eigenvalues[END_REF]. From the scattering data we measure k 1 which for epsilon small enough is close to k 1 h . Hence having available k 1 we find a constant n such that the first transmission eigenvalue of the homogeneous media with refractive index n has k 1 as the first transmission eigenvalue. Then by continuity this constant n is close to n h . In Table 4 we show the calculations for the ball of radius D, A = I and Similarly, we can obtain information about the effective constant matrix A h [START_REF] Cakoni | The computation of lower bounds for the norm of the index of refraction in an anisotropic media from far field data[END_REF], [START_REF] Cakoni | The inverse electromagnetic scattering problem for anisotropic media[END_REF]. In particular, in the case when n = 1, from the first transmission eigenvalue k 1 h we can determine a constant a which is in the middle of the smallest and the largest eigenvalues (in fact earlier numerical example suggest that this constant is roughly the arithmetic average of the eigenvalues of A h ). As an example we again consider the ball D := B R of radius R = 1, n = 1 and A given by (60). Then having the measured k 1 ε , we find the constant a such that the first eigenvalue of the homogeneous media with A = aI and n = 1 is equal to k 1 ε . The calculation are shown in Table 5.

n ε = n(x/ε) = sin 2 (2πx 1 /ε) + 2. ε k ε,1 n h reconstructed n h 0.

ε k 1 ε

A h reconstructed A h 0.1 7.349 0.5I 0.4851I Table 5. Reconstruction of affective material property from FFE in unit disk

In the above both examples we see that the measured first transmission eigenvalue corresponding to the periodic highly oscillating media can accurately determine the 22 FIORALBA CAKONI, HOUSSEM HADDAR AND ISAAC HARRIS effective isotropic material properties A h = a h I or n h . Next we consider an example where A h is constant matrix. We take the ball D := B R of radius R = 1 and n = 1 and Ãε = T A ε T where A is given by (60) and T is the matrix representing clockwise rotation by 1 radian. In this case it becomes non-trivial to compute A h (one needs to solve the cell PDE problem). However the constant a found as in the above example is in between (roughly the average) of the smallest and the largest eigenvalue of A h . The results are shown in Table 6 ε k ε,1 reconstructed a 0.1 7.5499 0.4921I Table 6. Reconstruction for the unit disk and A given by (60) Furthermore, if both A ε = I and n = 1 we use a similar method as the above to obtain information about A h /n h [START_REF] Cakoni | On the interior transmission eigenvalue problem[END_REF]. Here we look for a constant α such that the first eigenvalue of coincide with k 1 ε (note that here we incorrectly drop the jump in the normal derivative), where we take n given by (61) and A given by (60) giving that the ratio n h a h = 5. The reconstruction is shown in Table 7.

ε k ε,1
reconstructed n h a h 0.1 2.5415 4.788

Table 7. Reconstruction of the ratio n h a h = 5 of effective material property for the unit disk D In all the examples so far we have considered smooth coefficients A and n . Hence, our next example concerns a checker board patterned media where the coefficients take different values in the white and black squares. Again here the scaled period for the coefficients is Y = [0, 1] 2 . The white and black squares are assumed to cover the same area in a unit cell. See Figure 6 for the definition of the coefficients. In this case we have that n h = 7/2 and A h is shown in [START_REF] Wautier | On the second-order homogenization of wave motion in periodic media and the sound of chessboard[END_REF] to be a scalar matrix, i.e. A h = a h I where a h can be computed numerically. See Table 8 for a comparison between the first transmission eigenvalue of the homogenized media and periodic media. Next we use the first transmission eigenvalue for the actual media to determine the effective material properties. The result are shown in Table 9 A(y) = I, n(y)

reconstructed n h = 3.4123 (exact n h = 3.5) A(y), n(y) = 1 reconstructed a h = 0.4472 A(y), n(y) reconstructed n h /a h = 7.4704 which gives a h = 0.4685 Table 9. Reconstructed of effective material properties for the checkerboard Lastly consider the case of a media with periodically spaced voids (subregions with n ε = 1 and A ε = I). Our analysis does not cover this type of material property (see [START_REF] Harris | Non-destructive testing of anisotropic materials[END_REF] for the case when D is a union of cells) but nevertheless we consider an example of this type (The existence of real transmission eigenvalues for media with voids is proven in [START_REF] Cakoni | The interior transmission problem for regions with cavities[END_REF][START_REF] Harris | Transmission eigenvalues and non-destructive testing of anisotropic magnetic materials with voids[END_REF]). In particular, we consider an example of isotropic media with refractive index A(y) = I and n(y) = 1 if (y 1 -0.5) 2 + (y 2 -0.5) 2 < 0.25 2 5 if (y 1 -0.5) 2 + (y 2 -0.5) 2 ≥ 0.25 2

ξ

  • A(y)ξ = A min > 0 and sup y∈Y sup |ξ|=1 ξ • A(y)ξ = A max < ∞ (1) inf y∈Y n(y) = n min > 0 and sup y∈Y n(y) = n max < ∞.

Definition 1 . 1 .

 11 The values k ε ∈ C for which (3)-(6) has a nontrivial solution are called transmission eigenvalues. The corresponding nonzero solutions (w ε , v ε ) are referred to eigenfunctions.

Figure 1 .

 1 Figure 1. A periodic domain for three different values of .

10 FIORALBA

 10 CAKONI, HOUSSEM HADDAR AND ISAAC HARRIS 2.2. The case of A ε ≡ I.

U

  = w -v, and (w, v) satisfy ∆v + k 2 v = 0 and ∆w + k 2 n h w = 0 in D The result follows from combining Theorem 2.3 and Theorem 2.4 and the uniqueness of solution for (46)-(47).

Figure 2 .

 2 Figure 2. Here is a Log-Log plot that compares the log k 1 (ε)-k h to the line with slope 2.

k 1 1 Figure 3 .

 113 Figure 3. Here is a Log-Log plot that compares the log k 1 (ε)-k h to the line with slope

Figure 4 .

 4 Figure 4. Convergence graph for relative error when ψ(y) = 0 compare to the line with slope one. On the left we have the Log-Log plot for the square and on the right for the disk.

Figure 5 .

 5 Figure 5. On the left ε = 1 and on the right is ε = 0.1. The red dot indicates k 1 h whereas the pick indicates k 1 .

  ∆w + αk 2 w = 0 and ∆v + k 2 v = 0 in D

Figure 6 .

 6 Figure 6. Definition of Checker board coefficients.

k 1 Table 8 .

 18 (n(y)) k 1 (n h ) k 1 (A(y)) k 1 (A h ) k 1 (n(y), A(y)) k 1 (n h , A h ) Media with checkerboard pattern in [-3, 3] 2

Table 1 .

 1 First TEV for various ε with A ε = I and n ε = 1

	4	1/5	1/6	1/7
	k 1 (ε) 2.0842 2.0834 2.0829 2.0828 2.0824

Table 3 .

 3 First TEV for various ε shown in the first row corresponding to Ãε and n ε . Last column shows the convergence rate.

	4	1/8	| Convergence Rate
	Circle k 1 (ε) 2.460 2.453 2.472 2.518 |	1.32
	Square k 1 (ε) 2.201 2.213 2.230 2.273 |	0.917

Table 4 .

 4 1 5.046 2.5 2.5188 Reconstruction of n h

The research of F. Cakoni is supported in part by the Air Force Office of Scientific Research Grant FA9550-13-1-0199 and NSF Grant DMS-1106972. The research of I. Harris is supported in part by the NSF Grant DMS-1106972 and the DRI of INRIA (Associate Team ISIP). I. Harris greatly acknowledge the hospitality of the DeFI Team during his visit at Ecole Polytechnique.

which gives that n h = 5 -π 4 , and an example of anisotropic case with the same n(y) and

where the period is Y = [0, 1] 2 and the is domain D = [-3, 3] 2 . See Table 10 for the comparison of the first transmission eigenvalue for the homogenized media and the actual periodic media.


R Rabah

B Bergeon

On state space representation for linear discrete-time systems in Hilbert spaces

Keywords:

For a linear continuous-time control system in Hilbert space with state x(t) is associated a discrete-time system where the state variable is z k = (x((k + 1)h) + x(kh))/2, with small h. This allows to introduce a discrete derivative ∆z k = (x((k + 1)h) -x(kh))/h. The obtained discrete-time system has structural properties with a similar formulation as continuous system. Stability is equivalent to the fact that the spectrum of the state operator of discrete-time system is in the left half plane, Lyapunov and Riccati equation are similar.

Introduction

We are concerned with systems described by equations ẋ(t) = Ax(t) + Bu(t), y(t) = Cx(t) [START_REF] Curtain | Infinite dimensional linear systems theory[END_REF] where x, u and y take values in Hilbert spaces X, U and Y respectively. A, B and C are linear operators. B is bounded and A is the infinitesimal generator of a C 0 -semi-group of bounded operators S(t), t ≥ 0. The mild solution of the system (1) is given by:

x(t) = S(t)x 0 + t 0 S(t -τ )Bu(τ )dτ
A direct discretization of the mild solution gives a discrete system with bounded operators

F = S(h), G = h 0 S(τ )Bdτ, H = C
respectively for the state, input and output:

x k+1 = F x k + Gu k , y k = Hx k , (2)
where x k = x(kh), y k = y(kh) and u k = u(kh).

Another way to obtain a discrete-time system from (1) is to use the transfer function, say T (s) = C(sI -A) -1 B. One can introduce (see for example [START_REF] Ober | Infinite dimensional continuous-time linear systems: stability and structure analysis[END_REF][START_REF] Curtain | An Introduction to infinite-dimensional linear systems theory[END_REF] for the case of infinite dimensional systems) a new transfer function T d (w). Let

T d (w) = T ( w -1 w + 1 ). (3)
Under the assumption that T (s) is exponentially stable, T d is holomorphic and stable outside the unit disc. It is well known (see for example [START_REF] Curtain | An Introduction to infinite-dimensional linear systems theory[END_REF]) that T d (w) is the transfer function of a discrete-time system (A d , B d , C d , D d ):

T d (w) = C d (wI -A d ) -1 B d + D d ,
where

A d = (I -A) -1 (I + A), B d = √ 2(I -A) -1 B and C d = √ 2C(I -A) -1 , D d = C(I -A) -1 B.
Some properties of this discrete system are given in [START_REF] Ober | Infinite dimensional continuous-time linear systems: stability and structure analysis[END_REF] (see also [START_REF] Curtain | An Introduction to infinite-dimensional linear systems theory[END_REF], pp. 212-213).

In particular, the relation of this system and the original one (1) are discussed.

In [START_REF] Ober | Bilinear transformation of infinite dimensional state space systems and balanced realizations of non-rational transfer functions[END_REF][START_REF] Ober | Infinite dimensional continuous-time linear systems: stability and structure analysis[END_REF] are investigated the problems of realization, exponential and asymptotic stability for the discrete system obtained from T d (s) which is not rational function. The corresponding state space is obtained by realization techniques. However it is not clear which connection exists between the continuous-time and the discrete-time states.

The relation (3) introduced earlier for the case of finite dimensional systems allows to obtain properties for discrete-time system from continuous-time systems. Since the early fifties, several developments of continuous-time and discrete-time systems were done in parallel ways. This is the consequence of the different formulation of generic control problems and results. However, as both forms of solutions are computable (Lyapunov and Riccati equations, linear quadratic problems, etc.), the difference between formulations does not induce difficulties. The problem induced by these difference appeared in the late 80s up to now, especially through the robust control problems. In fact, the main idea for unification of continuous and discrete time theories used in some specialized problems is the Tustin transform [START_REF] Curtain | An Introduction to infinite-dimensional linear systems theory[END_REF].

A recent contribution has been brought by Bergeon [2] in order to extend this idea and to formalize the relation between the continuous and the discrete systems. The author shaw that every problem formulation and every design available in continuous-time domain can be translated, without loss of generality and simplicity, in the discrete-time domain. Our purpose is to extend this approach to infinite dimensional systems and to study the specificity, if any, of this case. This approach amounts to putting:

z k = x k+1 + x k 2 (4)
and

∆z k = x k+1 -x k h . (5)
where x k = x(kh) as for the system (2). In this paper the following assumption is made: h > 0 is chosen such that the operator I + S(h) = I + F is bounded invertible. The following discrete-time state space system can be associated to [START_REF] Curtain | Infinite dimensional linear systems theory[END_REF]:

∆z k = F d z k + G d u k , y k = H d z k + E d u k , (6)
where the operators F d , G d , H d , E d are defined later [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]. In this system, z k is the "state" and ∆z k is the "derivative". The new term E d u k is the consequence of the discretization: strictly proper system become proper. For this system several control problems are discussed. The main results are that this system is a pseudo-continuous version of the continuous-time system [START_REF] Curtain | Infinite dimensional linear systems theory[END_REF]. It is shown also that several formulations are similar to those of continuous-time system: Lyapunov and Riccati equations, stability and stabilizability conditions.

State representation of discrete-time system

In this section, we show how the system ( 6) is obtained and how this system converge, in some sense, to the system (1).

Theorem 2.1 Let x(t) be the mild solution of the system (1) and x k = x(kh) and u k = u(kh) for h > 0 and k ∈ N and let z k and ∆z k be given by ( 4) and ( 5). Then z k is the solution of the equation

∆z k = F d z k + G d u k
and the output y k is given by

y k = H d z k + E d u k ,
where the operators F d , G d , H d and E d are bounded and expressed by the relations:

F d = 2 h (F -I)(F + I) -1 , G d = 2 h (F + I) -1 G, H d = 2H(F + I) -1 , E d = -H(F + I) -1 G. (7)
The operator F + I = S(h) + I being bounded invertible by the choice of h.

Proof.

Let us choose first h such that S(h) + I is bounded invertible. If ρ(S(h)) denote the resolvent set and σ(S(h)) the spectrum of S(h). Then it is well known [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF] that e hσ(A) ⊂ σ(S(h)), where σ(.) is the spectrum of the corresponding operator A. This means that if λ ∈ σ(A), and if hλ = 2kπi, then -1 ∈ ρ(S(h)) and S(h) + I is bounded invertible. These values of h are said admissible. It is easy to see that h may be chosen arbitrary small. From the definition of z k and the relation (2) between x k+1 and x k , we get

2z k = F x k + Gu k + x k = (F + I)x k + Gu k , which gives x k = 2(F + I) -1 z k -(F + I) -1 Gu k . (8)
In the same way, we get for ∆z k the relations:

∆z k = 1 h (x k+1 -x k ) = 1 h [(F -I)x k + Gu k ] .
From [START_REF] Przy Luski | Stability of linear infinite-dimensional systems revisited[END_REF], we obtain

∆z k = 2 h (F -I)(F + I) -1 z k - 1 h (F -I)(F + I) -1 Gu k + 1 h Gu k . As (F -I)(F + I) -1 G = (F + I -2I)(F + I) -1 G = G -2(F + I) -1 G, this gives ∆z k = 2 h (F -I)(F + I) -1 z k + 2 h (F + I) -1 Gu k .
and then

∆z k = F d z k + G d u k
with F d and G d given by [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF].

For the output relation, from (8), we have

y k = Hx k = 2H(F + I) -1 z k -H(F + I) -1 Gu k , which gives y k = H d z k + E d u k ,
with the needed operators H d and E d .

Remark 2.2

The original continuous-time system is strictly proper:

lim ℜ(s)→∞ T (s) = 0.
The discrete-time system is only proper. If the continuous-time system is proper, i.e. given by

ẋ(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t), (9)
then, in the corresponding discrete-time system, the operator E d will be given by

E d = D -H(F + I) -1 G.
This is the case when the output is not modified. One can also consider a new output y k = Hz k , but this means that we observe in fact Hx k+1 also and this system is not causal.

The above discrete-time representation may be called pseudo-continuous representation and converge, when h → 0, to the continuous-time system in the sense given by the Theorem 2.3 and related results of the following section.

The above mentioned properties show that this approach is quite different from that of the construction of pseudo-continuous system from discrete system via the Tustin transformation (see [START_REF] Curtain | An Introduction to infinite-dimensional linear systems theory[END_REF] for the infinite dimensional case).

Theorem 2.3 For all x 0 ∈ D(A), x ∈ X and u ∈ U, we have

lim h→0 F d x 0 = Ax 0 , lim h→0 G d u = Bu, lim h→0 H d x = Hx, lim h→0 E d u = 0,
the value of h being admissible. If the operator A is bounded, then the limits exist in the uniform operator topology.

Proof. Note that

F d = 2 h (F -I)(F + I) -1 = 2(F + I) -1 F -I h .
As A is the infinitesimal generator of the semigroup S(t), we have

lim h→0 F -I h x 0 = lim h→0 S(h) -I h x 0 = Ax 0 , (10)
for all x 0 ∈ D(A). On the other hand,

lim h→0 (F + I)x = lim h→0 (S(h) + I)x = 2x (11)
for all x ∈ X because of strong continuity of the semigroup. Then, for sufficiently small h, (S(h) + I)x ≥ x .

This gives

x = (S(h) + I)(S(h) + I) -1 x ≥ (S(h) + I) -1 x , which implies (S(h) + I) -1 ≤ 1. Then

2(S(h) +

I) -1 x -x ≤ (S(h) + I) -1 2x -(S(h) + I)x ≤ 2x -(S(h) + I)x .
and by (11), this gives

lim h→0 2(F + I) -1 x = lim h→0 2(S(h) + I) -1 x = x,
Then, by a simple calculation and using (10), we get, for x 0 ∈ D(A),

lim h→0 F d x 0 = lim h→0 2(S(h) + I) -1 S(h) -I h x 0 = Ax 0 .
Consider now the operator G d = 2 h (F + I) -1 G. For all u ∈ U , we have (see for example [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]):

lim h→0 1 h G = lim h→0 1 h h 0 S(τ )Budτ = Bu,

and then lim

h→0 2 h (F + I) -1 Gu = Bu.
The other limits may be calculated in the same way.

If A is bounded, then S(t) = e At is uniformly continuous, and all the limits are in the uniform operator topology.

This means that the system (F d , G d , H d , E d ) asymptotically closed to the original continuous-time system. This may be also seen by remarking that if t = kh, then for an initial condition x 0 ∈ D(A) and a control function u ∈ C 1 we have lim

h→0 z k = x(t), lim h→0 ∆z k = ẋ(t).
If the initial condition is not in D(A) and u ∈ L p , p ≥ 1, the derivative must be understood in the weak sense (see [START_REF] Curtain | An Introduction to infinite-dimensional linear systems theory[END_REF][START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]).

3 Stability and stabilizability

Stability

The first problem under investigation is that of stability. We consider here only exponential stability for continuous-time system and power stability for discretetime system. Other concepts of stability mat be considered in a similar way.

Definition 3.1 The system (1) is said exponentially stable if there exists constants M ≥ 1 and α > 0 such that

S(t) ≤ M e -αt , t ≥ 0.
The discrete-time system (2) is said power stable if there exists constants N ≥ 1 and 0 < γ < 1 such that

F n ≤ N γ n , n ∈ N.
This means that for u(t) = 0, t ≥ 0 (respectively

u k = 0, k ∈ N) the solution of both systems verify ∞ 0 S(t)x 0 2 dt < ∞, ∞ i=0 x i 2 < ∞.
As in this case z k = F +I 2 x k , power stability of systems ( 2) and ( 6) are equivalent.

Theorem 3. 2The system ( 6) is power stable if and only if the spectrum of F d , noted σ(F d ) is in the interior of the left half plane:

σ(F d ) ⊂ C -β = {s : ℜ(s) < -β}.
The corresponding Lyapunov equation is

F * d P + P F d = -Q,
for positive definite linear bounded operator Q.

Proof. It is well known (cf. for example [START_REF] Curtain | An Introduction to infinite-dimensional linear systems theory[END_REF][START_REF] Logemann | Stability and stabilizability of linear infinite-dimensional discrete-time systems[END_REF][START_REF] Przy Luski | Stability of linear infinite-dimensional systems revisited[END_REF]) that the discrete-time system (2) is power stable if and only if r(F ) < 1, where r(F ) is the spectral radius:

r(F ) = sup {|λ| : λ ∈ σ(F )} .
The transformation

F → F d = 2 h (F -I)(F + I) -1
maps in the same way the spectrum of F :

ϕ : λ → µ = 2 h (λ -1)(λ + 1) -1 , λ = -1.
The spectrum of F is in the interior of the unit ball if and only if

σ(F d ) ⊂ C -β ,

Stabilizability

The above result on stability induce similar criterion on stabilizability. Definition 3. [START_REF] Curtain | An Introduction to infinite-dimensional linear systems theory[END_REF] The system (1) is said stabilizable iff there exists a linear bounded operator K such that the semigroup generated by A + BK is exponentially stable. The system (6) is power stable iff there exists a linear bounded operator

K d such that F d + G d K d is power stable.
There are several conditions of exponential and power stabilizability. All the known conditions may be extended to the pseudo-continuous system [START_REF] Ober | Infinite dimensional continuous-time linear systems: stability and structure analysis[END_REF]

provided that F d + G d K d is power stable iff σ(F d + G d K d ) ⊂ C -β
for some positive β. In particular, the stabilizability condition may be formulated via the solution of a Riccati equation. Theorem 3. [START_REF] Ober | Bilinear transformation of infinite dimensional state space systems and balanced realizations of non-rational transfer functions[END_REF] The system (6) is stabilizable if and only if there exist a positive operator P d such that:

F * d P d + P d F d -P d G d G * d P d + Q = 0,
and the stabilizing feedback is given by K d = -G * d P d . The relation between the feedback K d and the feedback stabilizing the discrete-time system (2), say K, is given by K d = 2K(F + GK + I) -1 , and K = (2I -K d G) -1 K d (F + I), under the condition that 2 ∈ ρ(K d G), where ρ(.) is the resolvent set of the given operator.

Proof. The condition of power stability implies the condition of stabilizability using the Riccati equation (cf. [START_REF] Zabczyk | Mathematical control theory[END_REF]). Suppose that K is the stabilizing feedback for the system (2), then u k = Kx k is the stabilizing control and x k+1 = (F + GK)x k is the closed loop state. Then

2z k = (F + GK)x k + x k , which gives x k = 2(F + GK + I) -1 z k ,
the bounded invertibility of the operator F + GK + I is garanted by the stability condition of F + GK. Hence,

u k = 2(F + GK + I) -1 z k ,
is the stabilizing feedback for the pseudo-continuous system [START_REF] Ober | Infinite dimensional continuous-time linear systems: stability and structure analysis[END_REF]. In an analogous way, under the assumption that 2 ∈ ρ(K d G), one obtains

K = (2I -K d G) -1 K d (F + I),
which ends the proof.

Hence, the Riccati equation for the system (6) is of the same form as for the continuous-time system. Note that as A and A * are not defined on all the space X, the corresponding Lyapunov and Riccati equations are given on D(A) (see [START_REF] Curtain | An Introduction to infinite-dimensional linear systems theory[END_REF][START_REF] Zabczyk | Mathematical control theory[END_REF]).

Further control problems

The approach developed in Section 2 and 3 may be also extended to other control problems: linear quadratic optimal control problem, detectability, asymptotic observers, etc.

For the problem of detectability and asymptotic observers, the results can be obtained from Section 3 by duality. All the calculation are the same.

For the linear quadratic optimal control problem one can follow the example considered in [START_REF] Curtain | An Introduction to infinite-dimensional linear systems theory[END_REF]. A continuous-time system (A, B, C, D) is induced from a discrete-time system (A d , B d , C d , D d ) by the relations:

A = (A d -I)(A d + I) -1 , B = √ 2(A d + I) -1 B d , C = √ 2C d (A d + I) -1 , C = D d -C d (A d + I) -1 B d , (12)
and the linear quadratic optimal problem is considered for the continuous-time system (A, B, C, D). It is shown that the optimal solution is obtained via a continuous type Riccati equation.

The same calculation may be made for our pseudo-continuous system (6), and as in the finite dimensional case [START_REF] Bergeon | Une autre représentation d'état pour la commande à temps discret[END_REF], one can obtains similar formulation for the LQ problem.

The input-output relation

In this section we show that the transfer function may be calculated in the same way as for the continuous-time system using the state-space expression of the pseudo-continuous system.

Let L denote the discrete Laplace transform (in fact the so called z-transform). By L(x k ) we mean the Laplace transform of the sequence {x 0 , x 1 , . . .}. Assume that the initial condition x 0 = 0. Then we have L(x k+1 ) = ζL(x k ), where ζ is the discrete Laplace variable, and

L(z k ) = ζ + 1 2 L(x k ), L(∆z k ) = ζ -1 h L(x k ).
From both relations, we obtain

L(∆z k ) = 2 h ζ -1 ζ + 1 L(z k ).
Putting

ω = 2 h ζ -1 ζ + 1 , (13)
we get L(∆z k ) = ωL(z k ), which gives the Laplace transform of the pseudo-derivative. Applying this calculus to the pseudo-continuous system (6), leads to

L(y k ) = H d (ωI -F d ) -1 G d + E d L(u k ).
This means that the input-output relation is given by the transfer function

Θ(ω) = H d (ωI -F d ) -1 G d + E d .
The relation (13) between the Laplace variables of the discrete-time system (2) and the pseudo-continuous system ( 6) is also a Tustin transform like the transform (3), but is applied to the exact discrete system (2) in order to obtain a (pseudo) continuous-time system which is also exact, closed to the continuous-time system (1) and with similar properties. Hence, this approach is quite different from the classical one.

  because the application ϕ maps the open unit ball into the open left half plane. On the other hand the condition σ(F d ) ⊂ C -β is equivalent to the existence of positive solution of the Lyapunov equation [3]: F * d P + P F d = -Q, with selfadjoint, positive definite operator Q.