Muriel Boulakia

Elisa Schenone

Stability estimates for some parameters of a reaction-diffusion equation coupled with an ODE

Keywords: parameter identification, inverse problem, reaction-diffusion equation

In this work, we are interested by the monodomain equation which describes the evolution of the cardiac electrical potential and which corresponds to a coupled system involving a reaction-diffusion equation and an ordinary differential equation. We show Lipschitz stability inequalities for the identification of some parameters of the model from measurements on the cardiac potential and the ionic variable.

Introduction

In this paper, we consider the monodomain model which is a coupled system involving a reactiondiffusion equation and an ordinary differential equation. The monodomain model is a classical model to describe the evolution of the electrical potential u in the heart and can be directly derived from the bidomain model by assuming that the intracellular and extracellular conductivities have similar anisotropic ratio (for an introduction to the modeling in cardiac electrophysiology, we refer to [START_REF] Sundnes | Computing the electrical activity in the heart[END_REF]). In this work, we are concerned with the estimation of some parameters involved in the model from measurements on u and on the ionic variable w

Let Ω be a bounded and regular domain in dimension 3 which corresponds to the domain occupied by the heart and let T > 0 be a given final time. We assume that the conductivity of the cardiac medium is isotropic and homogeneous. The monodomain equations are given in Q := (0, T ) × Ω by:

∂ t u -∆u + I ion (u, w) = 0 in Q ∂ t w + f (u, w) = 0 in Q. (1.1)
We complete this system by the initial conditions u(0, •) = u 0 in Ω and w(0,

•) = w 0 in Ω (1.2)
and by Neumann boundary conditions on u ∇u • n = 0 on (0, T ) × ∂Ω.

(1.3)

These homogeneous Neumann conditions correspond to the assumption that the heart is electrically insulated. Monodomain equations are complemented by the choice of a ionic model which will fix the expressions of the functions I ion and f . In this work, we will assume that I ion and f are given by:

I ion (u, w) = -kg(u) + r 1 (u)w, f (u, w) = -γr 2 (u) + βw (1.4)
where g(u) = u(u -a)(1 -u) and r 1 and r 2 are polynomial functions given by

r 1 (u) = χu + 1 -χ, r 2 (u) = ςu(u -a) -u, where χ, ς ∈ [0, 1].
We could also take arbitrary polynomial functions of degree 1 for r 1 and of degree 2 for r 2 but, by this way, our writing includes the classical phenomenological models in cardiac electrophysiology like FitzHugh-Nagumo (χ = 0, ς = 0), Alief-Panfilov (χ = 1, ς = 1) and Roger-McCulloch (χ = 1, ς = 0). We want to prove for this system stability estimates for the parameters k and γ which appear in the ionic model. To prove these estimates, we follow the Bukhgeim-Klibanov approach [START_REF] Bukhgeim | Global uniqueness of a class of inverse problems[END_REF] which relies on the use of global Carleman estimates on the time derivative of the solution of the system.

We will assume that the parameters a, k, γ, β only depend on space, belong to L ∞ (Ω) and satisfy 0

≤ a(x) ≤ 1, k(x) ≥ k 0 > 0, γ(x) ≥ 0, β(x) ≥ 0, ∀x ∈ Ω, (1.5)
and that there exists M > 0 such that

k L ∞ (Ω) + γ L ∞ (Ω) + β L ∞ (Ω) ≤ M. (1.6)
Let us first give regularity results on the solution of system (1.1) completed by (1.4). These results can be proved without any major difficulty thanks to successive energy estimates on the solution and its derivatives with respect to time and thanks to elliptic energy estimates.

Proposition 1. Let a, k, γ and β satisfy (1.5)-(1.6).

• We assume that (u 0 , w 0 ) ∈ H 2 (Ω) × L ∞ (Ω) and that

∇u 0 • n = 0 on ∂Ω. (1.7)
Then the solution of (1.1)-(1.3) completed by (1.4)

satisfies u ∈ W 1,∞ (0, T ; L 2 (Ω)) ∩ H 1 (0, T ; H 1 (Ω)) ∩ L ∞ (0, T ; H 2 (Ω)) w ∈ W 1,∞ (0, T ; L ∞ (Ω)) and u W 1,∞ (0,T ;L 2 (Ω)) + u H 1 (0,T ;H 1 (Ω)) + u L ∞ (0,T ;H 2 (Ω)) + w W 1,∞ (0,T ;L ∞ (Ω)) ≤ C
where C depends on u 0 H 2 (Ω) , w 0 L ∞ (Ω) , k 0 and M .

• We assume that (u 0 , w 0 ) ∈ H 4 (Ω) × H 2 (Ω) and that

∇u 0 • n = 0 on ∂Ω and ∇(∆u 0 + kg(u 0 ) -r 1 (u 0 )w 0 ) • n = 0 on ∂Ω. (1.8)
Moreover, we take k and a in H 2 (Ω) and assume that there exists M > 0 such that

k H 2 (Ω) + a H 2 (Ω) ≤ M . (1.9)
Then the solution of (1.1)-(1.3) completed by (1.4) satisfies

u ∈ W 2,∞ (0, T ; L 2 (Ω)) ∩ H 2 (0, T ; H 1 (Ω)) ∩ W 1,∞ (0, T ; H 2 (Ω)) w ∈ W 2,∞ (0, T ; L ∞ (Ω))
and

u W 2.∞ (0,T ;L 2 (Ω)) + u H 2 (0,T ;H 1 (Ω)) + u W 1,∞ (0,T ;H 2 (Ω)) + w W 2,∞ (0,T ;L ∞ (Ω)) ≤ C
where C depends on u 0 H 4 (Ω) , w 0 H 2 (Ω) , k 0 , M and M .

Stability estimate for a reaction parameter

Due to the writing of f in (1.4), we see that we can easily get an explicit formula for the solution w of the second equation in (1.1) and that the first equation of (1.1) may be viewed as a reactiondiffusion equation with a memory term. Even in the linear case, having a memory term may lead to difficulties in the theory of control and inverse problems. In [START_REF] Guerrero | Remarks on non controllability of the heat equation with memory[END_REF], it is proved that the null controllability property fails for heat equations with memory. This is due to the fact that the observability inequality which is a key point to get controllability does not hold for this system.

A related problem is addressed in [START_REF] Chaves-Silva | Null controllability of a system of viscoelasticity with a moving control[END_REF] where the null controllability of a model coupling a linear parabolic equation and an ODE is studied. If the control domain is fixed, then the controllability result does not hold and, to overcome this problem, the authors consider a control with a time-dependent support which covers the whole domain Ω and get an observability inequality for the coupled system.

Due to these remarks, it would not be possible to derive a global observability inequality for our coupled system (1.1). However, we will see that we can get stability estimates for the identification of some parameters of the model. Getting a global inequality for the monodomain equation coupled with an ODE would have allowed to prove stability estimates for more general ionic systems. In our case, we will use the fact that w can be expressed explicitly with respect to u and this will allow to get stability estimates for the identification of some parameters of the model.

Let us mention that paper [START_REF] Cristofol | Stable estimation of two coefficients in a nonlinear Fisher-KPP equation[END_REF] gives a simultaneous stability estimate for two parameters in the Fisher-KPP equation. For the coupling between two reaction-diffusion equations, we refer to [START_REF] Cristofol | Identification of two coefficients with data of one component for a nonlinear parabolic system[END_REF] for a simultaneous stability estimate of two parameters involved in the reaction terms.

Stability estimate for a reaction parameter in the nonlinear parabolic equation

We are first interested by a stability estimate on the reaction parameter k which appears in the first equation of (1.1). Let us consider (ū, w) the solution of (1.1)-(1.4) with k replaced by k in the expression of I ion . The following theorem gives a Lipschitz stability estimate on k -k with respect to measurements on u -ū and w -w. In a classical way for parabolic equations, we have two types of measurements: one local measurement on the whole time interval and one global measurement at a positive time.

Theorem 1. Let us consider a nonempty open set ω included in Ω. Let (u, w) be the solution of (1.1), (1.3) and (1.4) with initial conditions (u 0 , w 0 ) and (ū, w) be the solution of (1.1),(1.3) and (1.4) with k replaced by k in the expression of I ion and with initial conditions (ū 0 , w0 ). We assume that (u 0 , w

0 ) ∈ H 2 (Ω) × L ∞ (Ω) satisfies (1.7) and that (ū 0 , w0 ) ∈ H 4 (Ω) × H 2 (Ω) satisfies (1.8).
We suppose that a, k, γ, β satisfy (1.5)-(1.6), a, k ∈ H 2 (Ω) and that

k H 2 (Ω) + a H 2 (Ω) ≤ M
for some M > 0. Let us assume that there exists t 0 ∈ (0, T ) and c 1 > 0 such that

|g(ū)(t 0 , x)| ≥ c 1 , ∀x ∈ Ω. (2.1)
Then, there exists C > 0 which depends on the initial conditions, c 1 , k 0 , M and M such that

π L 2 (Ω) ≤ C p(t 0 ) H 2 (Ω) + q(t 0 ) L 2 (Ω) + p H 1 (0,T ;L 2 (ω))
where π = k -k, p = u -ū and q = w -w.

In [START_REF] Boulakia | Some inverse stability results for the bistable reaction-diffusion equation using carleman inequalities[END_REF], similar parameter estimates as the one in Theorem 1 were proved for the bistable equation (the first equation of (1.1) with I ion (u, w) = -kg(u)) thanks to a new Carleman inequality for the bistable equation. In our work, we will use the classical Carleman estimates for the heat equation [START_REF] Fursikov | Controllability of evolution equations[END_REF].

Proof of Theorem 1. The difference between the systems satisfied by (u, w) and (ū, w) leads to

∂ t p -∆p + kp 3 = kh(p, ū) + πg(ū) -µ(p, q, ū, w) in Q ∂ t q = γςp 2 + γ(2ς ū -ςa -1)p -βq in Q (2.2)
where

h(p, ū) := (1 + a -3ū)p 2 + (-a + 2(1 + a)ū -3ū 2 )p, (2.3)
µ(p, q, ū, w) := r 1 (u)w -r 1 (ū) w = χūq + χpw + (1 -χ)q.

(2.4)

We see that we have an explicit formula for the solution q of the second equation: for all (t, x) ∈ Q q(t, x) = q(t 0 , x)e -(t-t0)β + t t0 γ ςp 2 + (2ς ū -ςa -1)p e -(τ -t0)β dτ.

(2.5) Second, we consider the time derivative of the first equation of system (2.2). Then, ρ := ∂ t p satisfies

∂ t ρ -∆ρ = -3kp 2 ρ + k∂ t h(p, ū) + π∂ t g(ū) -∂ t µ(p, q, ū, w) in Q ∇ρ • n = 0 on (0, T ) × ∂Ω. (2.6)
To simplify, we assume that 0 < t 0 ≤ T /2 but the result still holds if t 0 is arbitrary taken in (0, T ) (it is sufficient to modify slightly the definition of the weight θ in what follows) and we define T 0 = 2t 0 ≤ T and Q 0 = (0, T 0 ) × Ω. Following [START_REF] Fursikov | Controllability of evolution equations[END_REF], we can define a function ψ in C 2 (Ω) such that

ψ > 0 in Ω, ψ = 0 on ∂Ω, |∇ψ| > 0 in Ω \ ω. (2.7)
Next, we define, for all λ > 0, the following weights for all (t, x) ∈ Q 0 ϕ(t, x) = θ(t) e 2λ ψ L ∞ (Ω) -e λψ(x) and η(t, x) = θ(t)e λψ(x) (2.8)

where θ(t) = 1 t(T 0 -t) .

According to Proposition 1, u, ū and w belong to L ∞ (Q 0 ). Thus, if we consider the first equation of (2.2) at time t 0 , we obtain that

Ω e -2sϕ(t0,x) |π(x)| 2 |g(ū)(t 0 , x)| 2 dx ≤ Ω e -2sϕ(t0,x) |ρ(t 0 , x)| 2 dx +C Ω e -2sϕ(t0,x) (|∆p(t 0 , x)| 2 + |p(t 0 , x)| 2 + |q(t 0 , x)| 2 )dx. (2.9)
Let us consider the first term in the right-hand side. We first notice that

Ω e -2sϕ(t0,x) |ρ(t 0 , x)| 2 dx = t0 0 Ω ∂ t (e -2sϕ |ρ| 2 )dxdt ≤ C Q0 e -2sϕ (s 2 η 2 |ρ| 2 + (s 2 η) -1 |∂ t ρ| 2 )dxdt since |∂ t ϕ| ≤ C|η| 2 . According to Proposition 1, u ∈ L ∞ (Q), ū ∈ W 1,∞ (0, T ; L ∞ (Ω)) and w ∈ W 1,∞ (0, T ; L ∞ (Ω))
. Thus, if we apply the Carleman inequality for the heat equation with Neumann boundary conditions introduced in [START_REF] Fursikov | Controllability of evolution equations[END_REF] and [START_REF] Fernández-Cara | Null controllability of the heat equation with boundary Fourier conditions: the linear case[END_REF] to the solution ρ of (2.6), we get, for s and λ large enough

Q0 e -2sϕ (s 2 η 2 |ρ| 2 + (s 2 η) -1 |∂ t ρ| 2 )dxdt ≤ C s Q0 e -2sϕ |ρ| 2 dxdt + C s Q0 e -2sϕ (|p| 2 + |q| 2 + |∂ t q| 2 + |π| 2 )dxdt + C T0 0 ω e -2sϕ s 2 η 3 |ρ| 2 dxdt.
Taking s large enough, we can absorb the first term in the right-hand side. Next, replacing ∂ t q by the second equation of (2.2), we get that, for s and λ large enough

Q0 e -2sϕ (s 2 η 2 |ρ| 2 + (s 2 η) -1 |∂ t ρ| 2 )dxdt ≤ C s Q0 e -2sϕ |p| 2 dxdt + C s Q0 e -2sϕ |q| 2 dxdt + C s Q0 e -2sϕ |π| 2 dxdt + C T0 0 ω e -2sϕ s 2 η 3 |ρ| 2 dxdt. (2.10)
To estimate the weighted norm of q in the right-hand side of this inequality, we use (2.5) and obtain

Q0 e -2sϕ |q| 2 dxdt ≤ C Ω e -2sϕ(t0) |q(t 0 , x)| 2 dx + C Q0 e -2sϕ |p| 2 dxdt (2.11) since u, ū ∈ L ∞ ((0, T ) × Ω)
. By definition of ρ = ∂ t p, we have for this last term:

Q0 e -2sϕ |p| 2 dxdt ≤ C Q0 e -2sϕ |p(t 0 )| 2 + t t0 ρ(τ, x)dτ 2 dxdt ≤ C Ω e -2sϕ(t0) |p(t 0 , x)| 2 dx + C Q0 e -2sϕ |ρ| 2 dxdt.
(2.12)

Then, combining inequalities (2.10), (2.11) and (2.12), estimate (2.9) becomes: for s and λ large enough

Ω e -2sϕ(t0,x) |π(x)| 2 |g(ū)(t 0 , x)| 2 dx ≤ C s Ω e -2sϕ(t0) (|p(t 0 , x)| 2 + |∆p(t 0 , x)| 2 )dx + C s Q0 e -2sϕ |π| 2 dxdt + C s Ω e -2sϕ(t0) |q(t 0 , x)| 2 dx + C T0 0 ω e -2sϕ s 2 η 3 |ρ| 2 dxdt.
At last, using hypothesis (2.1), we see that, for s large enough, the first term in the second line can be absorbed by the left-hand side and we get the desired result.

Stability estimate for a parameter of the ODE

We are now interested in giving an estimation of the parameter γ which appears in the expression of f in the second equation of (1.1).

Let us call û, ŵ the solution of (1.1)-(1.4) with γ replaced by γ in the second equation of (1.1). The following theorem gives a Lipschitz stability estimate of γ -γ with respect to measurements on u -û and w -ŵ.

Theorem 2. Let us consider a nonempty open set ω included in Ω. Let (u, w) be the solution of (1.1), (1.3) and (1.4) with initial conditions (u 0 , w 0 ) and (û, ŵ) be the solution of (1.1), (1.3) and (1.4) with γ replaced by γ and with initial conditions (û 0 , ŵ0 ). We assume that (u 0 , w 0 ) ∈ H 2 (Ω) × L ∞ (Ω) satisfies (1.7) and that (û 0 , ŵ0 ) ∈ H 4 (Ω) × H 2 (Ω) satisfies (1.8). We suppose that a, k, γ, β satisfy (1.5)-(1.6) and a, k ∈ H 2 (Ω), γ H 2 (Ω) ≤ M for some M > 0. Let us assume that there exist 0 < t 0 < t 1 < T and c 2 > 0 such that

|r 2 (û)(t, x)| ≥ c 2 , ∀(t, x) ∈ (t 0 , t 1 ) × Ω and |r 1 (û)(t 1 , x)| ≥ c 2 , ∀x ∈ Ω.
(2.13)

Then, there exists C > 0 which depends on the initial conditions, c 2 , k 0 , M and M such that

α L 2 (Ω) ≤ C z(t 1 ) H 2 (Ω) + v(t 0 ) L 2 (Ω) + z H 1 (0,T ;L 2 (ω))
where α = γ -γ, z = u -û and v = w -ŵ.

Proof of Theorem 2. The difference between the systems satisfied by (u, w) and (û, ŵ) leads to

∂ t z -∆z + kz 3 = kh(z, û) -µ(z, v, û, w) ∂ t v = γ(ςz 2 + 2ς ûz -(ςa + 1)z) -α(ς û2 -(ςa + 1)û) -βv (2.14)
where h(z, û) and µ(z, v, û, w) are defined in (2.3)-(2.4). As in the proof of Theorem 1, the solution v of the second equation can be expressed with an explicit formula: for all (t,

x) ∈ Q v(t, x) = v(t 0 , x)e -(t-t0)β + t t0 γ(ςz 2 + 2ς ûz -(ςa + 1)z) -αr 2 (û) e -(τ -t0)β dτ.
(2.15) First, we consider the time derivative of the first equation of system (2.14)

∂ t ζ -∆ζ = -3kz 2 ζ + k∂ t h(z, û) -∂ t µ(z, v, û, w) in Q ∇ζ • n = 0 on (0, T ) × ∂Ω (2.16) where ζ = ∂ t z.
As in the proof of Theorem 1, we assume, without loss of generality that 0 < t 0 + t 1 ≤ T and we define T 0 = t 0 + t 1 ≤ T and Q 0 = (0, T 0 ) × Ω.

Let us now introduce a function ψ in C 2 (Ω) which satisfies (2.7) and consider the functions ϕ and η defined in (2.8), with a new function θ ∈ C ∞ (0, T 0 ) which satisfies

θ(t) =              1 t(T 0 -t) , 0 < t ≤ 2t 0 3 strictly decreasing, 2t 0 3 < t ≤ t 0 constant, t 0 < t ≤ t 1 θ(T 0 -t), t 1 < t ≤ T 0 .
Let us consider the first equation of (2.14) and replace µ(z, v, û, w) by its definition (2.4). We get that r 1 (û)v = -χzw -ζ + ∆z -kz 3 + kh(z, û).

(2.17)

Next, we take this equation at time t 1 and replace v(t 1 ) thanks to formula (2.15). Thanks to Proposition 1, w, u and û belong to L ∞ ((0, T ) × Ω) and we deduce that, for all x ∈ Ω α

(x) r 1 (û)(t 1 , x) t1 t0 r 2 (û)(t, x)e -(t-t0)β dt ≤ C |z(t 1 , x)| + |ζ(t 1 , x)| + |∆z(t 1 , x)| + |v(t 0 , x)| + t1 t0 |z(t, x)|dt
We take the square of this expression, multiply by e -2sϕ(t1) and integrate over Ω, we get According to Proposition 1, u ∈ L ∞ (Q), û ∈ W 1,∞ (0, T ; L ∞ (Ω)) and w ∈ W 1,∞ (0, T ; L ∞ (Ω)). Thanks to the Carleman inequality for parabolic equations with Neumann boundary conditions ( [START_REF] Fursikov | Controllability of evolution equations[END_REF] and [START_REF] Fernández-Cara | Null controllability of the heat equation with boundary Fourier conditions: the linear case[END_REF]) applied to ζ solution of (2.16), we get, for s and λ large enough We then argue as in estimates (2.12) for the integral in v in the right-hand side and replace ∂ t v by the second equation of (2.14) and thus, the last two terms of ( Using hypothesis (2.13), we conclude the proof.

Ω e -2sϕ(t1,x) |α(x)| 2 |r 1 (û)(t 1 , x)| 2 t1 t0 r 2 (û)(t, x)e -(t-t0)β dt 2 dx ≤ C z(t 1 ) 2 H 2 (Ω) + v(t 0 ) 2 L 2 (Ω) + Ω e -2sϕ (
Bibliography

  t1) |ζ(t 1 )| 2 dx + Q0 e -2sϕ |z| 2 dx dt .(2.18) For the third term in the right-hand side, we haveΩ e -2sϕ(t1) |ζ(t 1 )| 2 dx = t1 0 Ω ∂ t (e -2sϕ |ζ| 2 )dxdt ≤ C Q0 e -2sϕ (s 2 η 2 |ζ| 2 + (s 2 η) -1 |∂ t ζ| 2 )dxdt.

Q0 e -2sϕ (s 2 η 2

 2 |ζ| 2 + (s 2 η) -1 |∂ t ζ| 2 )dxdt ≤ C s Q0 e -2sϕ (|z| 2 + |v| 2 + |∂ t v| 2 )dxdt + C T0 0 ω e -2sϕ s 2 η 3 |ζ| 2 dxdt.

  2.18) become, for s and λ large enoughΩ e -2sϕ(t1) |ζ(t 1 )| 2 dx + -2sϕ s 2 η 3 |ζ| 2 dxdt.Putting this inequality in (2.18), we get, for s and λ large enoughΩ e -2sϕ(t1,x) |α(x)| 2 |r 1 (û)(t 1 , x)| 2 -2sϕ s 2 η 3 |∂ t z| 2 dxdt .

					e -2sϕ |z| 2 dx dt
				Q0
		≤ C v(t 0 ) 2 L 2 (Ω) +	C s Ω	T0 e t1 e -2sϕ(t0,x) |α(x)| 2 dx + C 0 ω 2
					r 2 (û)(t, x)e -(t-t0)β dt	dx
				t0
	≤ C	z(t 1 ) 2 H 2 (Ω) + v(t 0 ) 2 L 2 (Ω) + +	C s Ω	e -2sϕ(t0,x) |α(x)| 2 dx + C	0	T0	ω	e


Joachim Schöpfel Bionote

Joachim Schöpfel
email: joachim.schopfel@univ-lille3.fr

Part of the grey literature, electronic theses and dissertations (ETDs) represent a growing segment of open, available content in institutional repositories (IR) where they contribute to the impact and ranking of their institution.

More than half of all IRs listed in the Directory of Open Access Repositories contain ETDs. Most of these open access projects have similarities and common features, such as access to full text and compliance with the OAI metadata harvesting protocol. But more important are the differences, with regard to metadata, policy, access restrictions, representativeness, file format, status, quality and related services.

In this paper we investigate what can be done to improve the quality of content and service provision in an open environment, in order to increase impact, traffic and usage. Based on a review of 54 recent communications and articles on PhD theses in institutional repositories, this paper shows five ways in which institutions can add value to the deposit and dissemination of electronic theses and dissertations and describes two developments that are challenging institutional repositories.

Introduction

A steadily growing corpus of electronic theses and dissertations (ETDs) is available through institutional repositories (IRs), i.e. open archives "serving the interests of facultyresearchers and teachers -by collecting their intellectual outputs for long-term access, preservation and management" (Carr et al., 2008).

At the end of 2012, the international directory of open archives OpenDOAR2 listed more than 1,100 institutional repositories with ETDs, representing roughly half of all registered open archives. The Registry of Open Access Repositories (ROAR)3 lists more than 200 services with 100% ETDs. Some people consider electronic PhD theses to be a Trojan horse that pushes institutions to launch open repositories, even without full compliance or mandate from the scientific community, mainly because they need a platform for preservation and diffusion of their theses. Following published journal articles in pre-or post-print formats, ETDs are the most important document type in open archives, significantly more important for instance than working papers, reports, book chapters or conference papers.

It is difficult to estimate how many ETDs are actually available through IRs. The Union Catalogue of the Networked Digital Library of Theses and Dissertations (NDLTD) contains more than one million records of ETDs. One segment of themin fact, probably the greatestis in IRs. This is good. This is clearly a success of the green road access to scientific information [START_REF] Harnad | The Access/Impact Problem and the Green and Gold Roads to Open Access: An Update[END_REF], and in our case, to scientific grey literature (Schöpfel 2010). But open is not enough. In order to increase acceptance, access and impact, it is not enough to simply upload content on servers. There must be at least a minimum of added value (Schöpfel et al., 2011).

So our questions are: How can an institution add value to ETDs in its IR? What can be done to improve quality of content and service provision in an open environment?

Methodology

This paper is based on the state-of-the-art of scientific and professional literature on PhD theses in institutional repositories. In particular, it reviews recent communications of thirteen conferences on electronic theses and grey literature: Based on this literature review, we came to understand which features make a difference. The reviewed papersmostly case studiespresent good practices, success stories, challenges and problems. But what are best practices? Which aspects are critical to distinguishing excellent IRs from good ones, and that add value to the dissemination of ETDs?

Results: Five ways to add value to PhD theses

The reviewed sample provides very different solutions, from various countries, institutions and environments. National open access networks like the Peruvian Network of Digital Theses [START_REF] Huaroto | New trends and future applications/directions of institutional repositories in academic institutions[END_REF]) cohabit with a central 'hub' or single point access interface to all kinds of theses, such as the British Library EThOS service [START_REF] Gould | EThOS: the UK national E-Theses Online Service[END_REF]; local ETD programs, as on the Emory campus at Atlanta [START_REF] Halbert | Integrating ETD Services into Campus Institutional Repository Infrastructures Using Fedora[END_REF]; large document servers such as e-doc at the Humboldt university of Berlin (Schirmbacher 2009); and international solutions for repositories, like the European ETD portal DART [START_REF] Moyle | The DART-Europe E-theses Portal: helping the discovery of Europe's open access research theses[END_REF].

All of these solutions in our sample have two features in common: they are (or give access to) open institutional repositories, and they contain electronic theses. Yet, when they are compared to each other, one can identify five specific characteristics that make a noticeable difference in their value, and provide scientific excellence.

Quality

The first feature is quality of content. "The content (…) is the most important factor that has been cited by researchers to show the success of a repository" [START_REF] Macha | A comparative overview of the development of the institutional repositories at the University of Cape Town and at the University of Pretoria[END_REF]. A good IR not only defines a set of standards and criteria for the selection and validation of deposits but also communicates and promotes this editorial policy.

For ETDs, this means only validated versions, with minor or major revisions if required by the PhD jury, but without preprints or non-controlled self-deposits. This also means that the IR is either part of the local ETD workflow or that it includes a point of formal and institutional validation during the deposit process.

Independent of the institutional validation, and in addition, the IR standards could hold some minimal requirements for acceptance, such as unrestricted access to full text, copyright clearance or limitation to research theses. A specific review system can be helpful for quality assurance.

The same selective quality criteria should apply to retrospective digitization of print theses, when the IR contains both born-digital and older repurposed digital materials.

In networked systems such as the European DART, minimal quality standards are vital for the global service.

Metadata

A second feature that makes a difference is the description of the content and context of the ETD files, e.g. metadata. "Considerable variations in practice exist (…). Many of these variations involve local qualifiers and/or the use of metadata elements to record information that does not correspond to the prescribed meaning of the elements" (Park & Richard 2011).

A good and rich bibliographic description increases the searchability of deposited items and allows for scientometric analysis, book publishing (print on demand) and other functionalities and services.

Generally said, metadata of ETDs in IRs should be rich and standard, and the use of standards should be promoted by recommendations (good practice). Metadata standards are especially important for small and medium-sized institutions and in networks such as DART.

The DART network applies the simplified version of the OAI Dublin Core as the basic format and as a kind of backbone for all members, while the French national ETD infrastructure chose the more sophisticated TEF [START_REF] Ducloy | Metadata towards an e-research cyberinfrastructure: the case of French PhD theses[END_REF] format.

Other metadata standards are the NDLTD standards for DC/MARC (ETD-MS), the ProQuest schema UMI XML DTD [START_REF] Marsh | Ensuring Discovery of ETDs: The Hong Kong University of Science & Technology & ProQuest/UMI Case Study[END_REF] and the Czech EVSKP-MS.

Whichever metadata schema is implemented, it should remain flexible, facilitate easy conversion to other formats and must change with new policies [START_REF] Heyse | Supporting change and diversity for theses in an institutional repository[END_REF].

Format

Good institutional repositories contain full text, not just metadata. They offer different file formats for different usage and purpose. Deposit formats should be searchable, open, and appropriate for long-term preservation and intelligent exploitation of the content (review, bibliography….). "In the sciences theses should be regarded not simply as objects destined for preservation and archival status, but rather as unique resources containing potentially valuable data that must be made extractable and re-usable" [START_REF] Morgan | Extracting and re-using research data from chemistry etheses: the SPECTRa-T Project[END_REF].

Accepted standard formats for deposit and dissemination are MS Office Word and Adobe PDF.

But some repositories accept and process different formats, such as XML, LaTeX, Postscript or RDF, more suitable for some disciplines, semantic querying and reuse. For retro-digitized theses, this means OCR and production of text files before submission.

Interoperability

To increase discoverability and availability of ETDs, repositories should network and interconnect. This interconnection can be on different levels, e.g. regional or state-wide, national or international.

The Networked Digital Library of Theses and Dissertations is the best-known example for an international network. The UK EThOS hosted by the British Library is a national network, like the French STAR system.

These and other initiatives exhibit the main characteristics and conditions for networking and interconnection: a shared standard format for deposit, exchange and harvesting (OAI-PMH), an explicit policy, i.e. commitment to the network and collective will for sharing and interconnection, a state-wide (regional, national…) architecture, and a single hub for research published in PhD theses.

Describing a US project, [START_REF] Dowling | Managing Consortial ETD Repositories[END_REF] point out that the principal objective is to "demonstrate research efforts of the state (through a) central finding tool for research".

Another approach to increase and guarantee interoperability of local repositories is the certification of the local system and/or the networked architecture [START_REF] Dobratz | DINI institutional repository certification and beyond[END_REF].

Services

Certainly, any institutional repository offers some basic functionality, e.g. basic and advanced search facilities, different browsing options, visualisation, downloading, etc.

Yet, if the hosting institution or service provider wants to add value, if they want to increase usage and the impact of their production and content, they should develop innovative complementary and peripheral services and practices, what [START_REF] Halbert | Integrating ETD Services into Campus Institutional Repository Infrastructures Using Fedora[END_REF] calls a "user-centred process of service development prioritization".

In her study on open repositories, Bester (2010) distinguishes nine browsing options, nine search options and ten options for customisation and reference management, such as export of bibliographic listings in different formats, including XML. Our own survey reveals some other interesting initiatives, listed below with some examples:  Social media tools: interactivity, discussion forum and comments [START_REF] Cocciolo | Can Web 2.0 Enhance Community Participation in an Institutional Repository? The Case of PocketKnowledge at Teachers College, Columbia University[END_REF][START_REF] Millard | MePrints: Building User Centred Repositories[END_REF][START_REF] Waddington | CLIF: Moving repositories upstream in the content lifecycle[END_REF]  Federated search and more sophisticated discovery tools [START_REF] Dowling | Managing Consortial ETD Repositories[END_REF].

 Usage statistics and metrics, citations [START_REF] Huaroto | New trends and future applications/directions of institutional repositories in academic institutions[END_REF][START_REF] Walker | What We Can Learn from ETDs: Using ProQuest Dissertations & Theses as a Dataset[END_REF]).

 Video with presentation of thesis [START_REF] Huaroto | New trends and future applications/directions of institutional repositories in academic institutions[END_REF].

 Print on demand in book format [START_REF] Rajendiran | Preservation of dissertations and theses in electronic form: a case study of SRM Institute of Science and Technology in Tamilnadu (India)[END_REF].

 Options for © protection or Creative Commons licensing [START_REF] Hagen | Building Effective Discovery Tools for Academic Promotion and Tenure Evidence: The Added Value of ETD and Institutional Repository Metadata, Citations and Access[END_REF][START_REF] Harper | ETDs, Open Access and Intellectual Property Issues[END_REF].

 Preservation in multiple copies [START_REF] Mikeal | ETD Management in DSpace: A Report from the Texas ETD Repository Project[END_REF]). Some of these developments are possible even with a "very modest programming staff available to deploy on ad hoc projects" [START_REF] Halbert | Integrating ETD Services into Campus Institutional Repository Infrastructures Using Fedora[END_REF] while others ask for more investment and resources.

On a more general level, the IR services for the dissemination of ETDs should be flexible, with a capacity for rapid adaptation; the software should be user-friendly and reliable, perhaps also open to other service providers and/or integrated in another service environment, such as extended (distance) learning.

Discussion: Two ways to do better

Institutional repositories are "tools (…) for collecting, storing and disseminating scholarly outputs within and without the institution" (Jain, 2011), in other words "a set of services (…) for the management and dissemination of digital materials created by the institution and its community members" [START_REF] Lynch | Institutional Repositories: Essential Infrastructure for Scholarship in the Digital Age[END_REF].

One of their main characteristics is their great diversity. Institutional repositories have different policies, procedures, functionalities, services and metadata; they have different business models and funding strategies [START_REF] Swan | Linking UK Repositories: Technical & Organisational Models to Support User-Oriented Services Across Institutional & Other Digital Repositories[END_REF], and their content may include more than current output from faculty. [START_REF] Smith | Institutional Repositories and E-Journal Archiving: What Are We Learning?[END_REF] details a "wide variety of materials in digital form, such as research journal articles, preprints and post-prints, digital versions of theses and dissertations, and administrative documents, course notes, or learning objects." Other repositories include datasets, multimedia or cultural and scientific heritage. So far we have described five features that add value to the dissemination of electronic theses via institutional repositories. All of these features have already been tested and implemented but are not generalized and depend on the political, commercial and technical choices of the hosting institution and the service provider; on their commitment to open access; and on their willingness and capacity to invest.

Beyond these features, we distinguish two other challenges for ETDs in institutional repositories: being future-oriented and anticipating the transformation of scientific communication to come. These are global challenges for the whole academic publishing landscape, not only for ETDs, but we can already find some studies and examples in our field.

Data

We stated previously that the text format should allow for extraction and reuse of content. More generally, PhD theses are regularly deposited accompanied by supplemental material, e.g. video and multimedia, images, spreadsheets, music etc.

In the digital environment of open repositories and added value services, this material becomes a rich resource of research results and datasets.

"It may be possible to enhance both the research options and the richness of access. "Deep access" to the research could include data, visual and aural representations, links to supporting material such as notebooks or collaborative activity" [START_REF] White | Opening access and closing risk: delivering the mandate for etheses deposit[END_REF].

The impact of the dissemination of this material on institutional repositories is at least threefold:

 The IR must support other and non-textual formats, such as AV for video or mp3 for music files.

 The IR host must develop an innovative service environment, extending the basic and added value services for text files to datasets.

 The institution needs to review the legal condition of the deposit and dissemination of datasets and other material. Some content may be protected by privacy or confidentiality concerns. On the other hand, the institution and/or author must apply an open licence (CC-BY or similar) that allows the maximum reuse and exploitation of these files.

Applying the logics of e-science, supplemental material should not only be available as an appendix to, or illustration in the related PhD thesis, but also extractable and reusable when not linked to the thesis, as an independent dataset and interconnected to other data [START_REF] Morgan | Extracting and re-using research data from chemistry etheses: the SPECTRa-T Project[END_REF][START_REF] Ross | The changing landscape of dissertation supplementary and supporting content: ETDs in transformation[END_REF][START_REF] Ubogu | Management of Research Data in ETD Systems[END_REF].

Current Research Information Systems (CRIS)

Last but not least, electronic theses and dissertations in institutional repositories should, and will, contribute to the evaluation of scientific production in the emerging environment of current research information systems (CRIS).

"A well managed repository can simplify the task for the researchers and their respective departments, their university administrators and (…) their assessors" [START_REF] Hey | Leveraging the Institutional Research Repository: harnessing the drive for quality assessment[END_REF].

Publications are essential elements of a CRIS. Institutional repositories contain the metadata for their institutions" outputs, and can make it available to the CRIS for monitoring and evaluating scientific production and research trends [START_REF] Lambert | Grey literature, institutional repositories and the organisational context[END_REF].

For the connection of the IR data silo to the CRIS and the data exchange, standard formats (such as the European CERIF7 ) and rich and valid metadata are crucial. Also, the assignment of unique identifiers for authors (PersID, DAI…), structures (OrgID) and publications (RefID, DOI…) is necessary, especially when local repositories are integrated in a national infrastructure. In such an eScience and evaluation environment, PhD theses are considered part of relevant material [START_REF] Hey | Leveraging the Institutional Research Repository: harnessing the drive for quality assessment[END_REF], see also [START_REF] Sugita | Dissertations and Theses in Institutional Repositories: Case Study in Japan[END_REF] and would become a significant sources of scientometric studies and research evaluations, for scientists and research managers.

Conclusion

When compared to articles published in academic journals, PhD theses are sometimes considered to be second-level information, for at least two reasons. PhD theses do not undergo peer review, and they are produced early in an academic career. Yet, not only do these documents often produce results that reflect years of intensive research combined with a rich literature review, they also represent a growing part of available open content in institutional repositories (IR) where they contribute to the overall impact and ranking of their institution. We showed five ways in which institutions can add value to the deposit and dissemination of electronic theses and dissertations via their open repositories. We also described two developments that are challenging institutional repositories.

Institutional repositories can have very different characteristics, and any one solution may not apply to all institutions. Our research findings do not promote a must-do list, but instead proffer some perspectives and options that should be adapted to the specific context of a given institution.

It is crucial to the success of a repository that the institution clearly defines its objectives in line with its scientific strategy and environment. "Each of the reasons for setting up a repository carries implications for the content, design and funding of a repository, and the institution needs to be clear about the implications of different roles for a repository, while being prepared to change or add roles as the scholarly communication environment develops" [START_REF] Friend | Open Access Business Models for Research Funders and Universities[END_REF].

We began our paper with the question: How can an institution add value to ETDs in its IR? Our answer is that there are at least five different ways, but that an institution, beyond its commitment to the open access principle, must make informed and conscious choices on the technical level, but also and above all, on the political level.

  ETD: Six international symposia on electronic theses and dissertations from 2007 to 2012, organized by the Networked Digital Library of Theses and Dissertations (NDLTD) 4 . USETDA: Two national conferences 2010 and 2011 of the US ETD Association (part of NDLTD). GL: Five international conferences on grey literature 2007 to 2011, organized by the Grey Literature Network Service (GreyNet) 5 . The review includes other published studies on open access and grey literature and case studies on ETDs in IRs, identified via Google Scholar, Scirus and the EmeraldInsight platform. The corpus of reviewed papers contains 54 references. 6

http://www.opendoar.org

http://roar.eprints.org

http://www.ndltd.org/

http://www.greynet.org

The whole sample is available on CiteULike http://www.citeulike.org/user/Schopfel/tag/gl14theses

Common European Research Information Format see http://www.eurocris.org