Tomasz Dębiec
email: t.debiec@mimuw.edu.pl

Marie Doumic
email: marie.doumic@inria.fr

AND Piotr Gwiazda
email: pgwiazda@mimuw.edu.pl

Emil Wiedemann
email: wiedemann@ifam.uni-hannover.de

Tomasz D Ębiec

Sorbonne Universités

Relative Entropy Method for Measure Solutions of the Growth-Fragmentation Equation

Keywords: measure solutions, growth-fragmentation equation, structured population, relative entropy, generalised Young measure

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INTRODUCTION

Structured population models were developed for the purpose of understanding the evolution of a population over time -and in particular to adequately describe the dynamics of a population by its distribution along some "structuring" variables representing e.g., age, size, or cell maturity. These models, often taking the form of an evolutionary partial differential equation, have been extensively studied for many years. The first age structure was considered in the early 20th century by Sharpe and Lotka [START_REF] Sharpe | A problem in age-distribution[END_REF], who already made predictions on the question of asymptotic behaviour of the population, see also [START_REF] Kermack | A contribution to the mathematical theory of epidemics[END_REF][START_REF] Kermack | Contribution to the mathematical theory of epidemics. ii. the problem of endemicity[END_REF]. In the second half of the 20th century size-structured models appeared first in [START_REF] Bell | Cell growth and division: I. a mathematical model with applications to cell volume distributions in mammalian suspension cultures[END_REF][START_REF] Sinko | A new model for age-size structure of a population[END_REF]. These studies gave rise to other physiologically structured models (agesize, saturation, cell maturity, etc.).

The object of this note is the growth-fragmentation model, which is found fitting in many different contexts: cell division, polymerisation, neurosciences, prion proliferation or even telecommunication. In its general linear form this model takes the form of the following equation.

∂ t n(t, x) + ∂ x (g(x)n(t, x)) + B(x)n(t, x) = ∞ x k(x, y)B(y)n(t, y) dy, g(0)n(t, 0) = 0, n(0, x) = n 0 (x).

(1.1)

Here n(t, x) represents the concentration of individuals of size x ≥ 0 at time t > 0, g(x) ≥ 0 is their growth rate, B(x) ≥ 0 is their division rate and k(x, y) is the proportion of individuals of size x created out of the division of individuals of size y. This equation incorporates a very important phenomenon in biology -a competition between growth and fragmentation. Clearly they have opposite dynamics: growth drives the population towards a larger size, while fragmentation makes it smaller and smaller. Depending on which factor dominates, one can observe various long-time behaviour of the population distribution.

Many authors have studied the long-time asymptotics (along with well-posedness) of variants of the growth-fragmentation equation, see e.g. [START_REF] Cáceres | Rate of convergence to the remarkable state for fragmentation and growth-fragmentation equations[END_REF][START_REF] Doumic | Eigenelements of a general aggregation-fragmentation model[END_REF][START_REF] Michel | Existence of a solution to the cell division eigenproblem[END_REF][START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF][START_REF] Perthame | Exponential decay for the fragmentation or cell-division equation[END_REF]. The studies which establish convergence, in a proper sense, of a (renormalised) solution towards a steady profile were until recently limited only to initial data in weighted L 1 spaces. The classical tools for such studies include a direct application of the Laplace transform and the semigroup theory [START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF]. These methods could also provide an exponential rate of convergence, linked to the existence of a spectral gap.

A different approach was developed by Perthame et al. in a series of papers [START_REF] Michel | General entropy equations for structured population models and scattering[END_REF][START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF][START_REF] Perthame | Exponential decay for the fragmentation or cell-division equation[END_REF]. Their Generalised Relative Entropy (GRE) method provides a way to study long-time asymptotics of linear models even when no spectral gap is guaranteed -however failing to provide a rate of convergence, unless an entropy-entropy dissipation inequality is obtained [START_REF] Cáceres | Rate of convergence to the remarkable state for fragmentation and growth-fragmentation equations[END_REF]. Recently Gwiazda and Wiedemann [START_REF] Gwiazda | Generalized entropy method for the renewal equation with measure data[END_REF] extended the GRE method to the case of the renewal equation with initial data in the space of non-negative Radon measures. Their result is motivated by the increasing interest in measure solutions to models of mathematical biology, see e.g. [START_REF] Carrillo | Structured populations, cell growth and measure valued balance laws[END_REF][START_REF] Gwiazda | A nonlinear structured population model: Lipshitz continuity of measure valued solutions with respect to model ingredients[END_REF] for some recent results concerning well-posedness and stability theory in the space of non-negative Radon measures. The clear advantage of considering measure data is that it is biologically justified -it allows for treating the situation when a population is initially concentrated with respect to the structuring variable (and is, in particular, not absolutely continuous with respect to the Lebesgue measure). This is typically the case when departing from a population formed by a unique cell. We refer also to the recent result of Gabriel [START_REF] Gabriel | Measure solutions to the conservative renewal equation[END_REF], who uses the Doeblin method to analyze the long-time behaviour of measure solutions to the renewal equatio n.

Let us remark that the method of analysis employed in the current paper is inspired by the classical relative entropy method introduced by Dafermos in [START_REF] Dafermos | The second law of thermodynamics and stability[END_REF]. In recent years this method was extended to yield results on measure-valued-strong uniqueness for equations of fluid dynamics [START_REF] Brenier | Weak-strong uniqueness for measure-valued solutions[END_REF][START_REF] Feireisl | Dissipative measurevalued solutions to the compressible Navier-Stokes system[END_REF][START_REF] Gwiazda | Weak-strong uniqueness for measurevalued solutions of some compressible fluid models[END_REF] and general conservation laws [START_REF] Christoforou | Relative entropy for hyperbolic-parabolic systems and application to the constitutive theory of thermoviscoelasticity[END_REF][START_REF] Demoulini | Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics[END_REF][START_REF] Gwiazda | Dissipative measure valued solutions for general conservation laws[END_REF]. See also [START_REF] Dębiec | Relative entropy method for measure-valued solutions in natural sciences[END_REF] and refereces therein.

The purpose of this paper is to generalise the results of [START_REF] Gwiazda | Generalized entropy method for the renewal equation with measure data[END_REF] to the case of a general growthfragmentation equation. Similarly as in that paper we make use of the concept of a recession function to make sense of compositions of nonlinear functions with a Radon measure. However, the appearance of the term H (u ε (t, x))u ε (t, y) in the entropy dissipation (see (3.8) below), which mixes dependences on the variables x and y, poses a novel problem, which is overcome by using generalised Young measures and time regularity.

The current paper is structured as follows: in Section 2 we recall some basic results on Radon measures, recession functions and Young measures as well as introduce the assumptions of our model, in Section 3 we state and prove the GRE inequality, which is then used to prove a longtime asymptotics result in Section 4.

DESCRIPTION OF THE MODEL

Preliminaries.

In what follows we denote by R + the set [0, ∞). By M (R + ) we denote the space of signed Radon measures on R + . By Lebesgue's decomposition theorem, for each µ ∈ M (R + ) we can write

µ = µ a + µ s ,
where µ a is absolutely continuous with respect to the Lebesgue measure L 1 , and µ s is singular. The space M (R + ) is endowed with the total variation norm

µ TV := R + d|µ|,
and we denote µ TV = TV (µ). By the Riesz Representation Theorem we can identify this space with the dual space to the space C 0 (R + ) of continuous functions on R + which vanish at infinity. The duality pairing is given by

ν, f := ∞ 0 f (ξ ) dµ(ξ ).
By M + (R + ) we denote the set of positive Radon measures of bounded total variation. We further define the ϕ-weighted total variation by µ TV ϕ := R + ϕd|µ| and correspondingly the space M + (R + ; ϕ) of positive Radon measures whose ϕ-weighted total variation is finite. We still denote TV (µ) = µ TV ϕ . Of course we require that the function ϕ be non-negative. In fact, for our purposes ϕ will be strictly positive and bounded on each compact subset of (0, ∞).

We say that a sequence ν n ∈ M (R + ) converges weak * to some measure ν ∈ M (R + ) if

ν n , f -→ ν, f for each f ∈ C 0 (R + ).
By a Young measure on R + × R + we mean a parameterised family ν t,x of probability measures on R + . More precisely, it is a weak * -measurable function (t, x) → ν t,x , i.e. such that the mapping

(t, x) → ν t,x , f is measurable for each f ∈ C 0 (R + ).
Young measures are often used to describe limits of weakly converging approximating sequences to a given problem. They serve as a way of describing weak limits of nonlinear functions of the approximate solution. Indeed, it is a classical result that a uniformly bounded measurable sequence u n generates a Young measure by which one represents the limit of f (u n ), where f is some non-linear function, see [?] for sequences in L ∞ and [START_REF] Ball | A version of the fundamental theorem for Young measures[END_REF] for measurable sequences.

This framework was used by DiPerna in his celebrated paper [START_REF] Diperna | Measure-valued solutions to conservation laws[END_REF], where he introduced the concept of an admissible measure-valued solution to scalar conservation laws. However, in more general contexts (e.g. for hyperbolic systems, where there is usually only one entropy-entropyflux pair) one needs to be able to describe limits of sequences which exhibit oscillatory behaviour as well as concentrate mass. Such a framework is provided by generalised Young measures, first introduced in the context of incompressible Euler equations in [START_REF] Diperna | Oscillations and concentrations in weak solutions of the incompressible fluid equations[END_REF], and later developed by many authors. We follow the exposition of Alibert, Bouchitté [START_REF] Alibert | Non-uniform integrability and generalized Young measures[END_REF] and Kristensen, Rindler [START_REF] Kristensen | Characterization of Generalized Gradient Young Measures Generated by Sequences in W 1,1 and BV[END_REF].

Suppose f : R n → R + is an even continuous function with at most linear growth, i.e.

| f (x)| ≤ C(1 + |x|)
for some constant C. We define, whenever it exists, the recession function of f as

f ∞ (x) = lim s→∞ f (sx) s = lim s→∞ f (-sx) s .
Definition 2.1. The set F (R) of continuous functions f : R → R + for which f ∞ exists and is continuous on S n-1 is called the class of admissible integrands.

By a generalised Young measure on Ω = R + × R + we mean a parameterised family (ν t,x , m) where for (t, x) ∈ Ω, ν t,x is a family of probability measures on R and m is a nonnegative Radon measure on Ω. In the following, we may omit the indices for ν t,x and denote it simply (ν, m).

The following result gives a way of representing weak * limits of sequences bounded in L 1 via a generalised Young measure. It was first proved in [START_REF] Alibert | Non-uniform integrability and generalized Young measures[END_REF]Theorem 2.5]. We state an adaptation to our simpler case. Proposition 2.2. Let (u n ) be a bounded sequence in L 1 loc (Ω; µ, R), where µ is a measure on Ω. There exists a subsequence (u n k ), a nonnegative Radon measure m on Ω and a parametrized family of probabilities (ν ζ ) such that for any even function f ∈ F (R) we have

f (u n k (ζ ))µ * ν ζ , f µ + f ∞ m (2.1)
Proof. We apply Theorem 2.5. and Remark 2.6 in [START_REF] Alibert | Non-uniform integrability and generalized Young measures[END_REF], simplified by the fact that f is even and that we only test against functions f independent of x. Note that the weak * convergence then has to be understood in the sense of compactly supported test functions ϕ ∈ C 0 (Ω, R).

The above proposition can in fact be generalised to say that every bounded sequence of generalised Young measures possesses a weak * convergent subsequence, cf. [26, Corollary 2.] Proposition 2.3. Let (ν n , m n ) be a sequence of generalised Young measures on Ω such that

• The map x → ν n x , | • | is uniformly bounded in L 1 , • The sequence (m n ( Ω)) is uniformly bounded.
Then there is a generalised Young measure (ν, m) on Ω such that (ν n , m n ) converges weak * to (ν, m).

2.2. The model. We consider the growth-fragmentation equation under a general form:

∂ t n(t, x) + ∂ x (g(x)n(t, x)) + B(x)n(t, x) = ∞ x k(x, y)B(y)n(t, y) dy, g(0)n(t, 0) = 0, n(0, x) = n 0 (x). (2.2) We assume n 0 ∈ M + (R + ).
The fundamental tool in studying the long-time asymptotics with the GRE method is the existence and uniqueness of the first eigenelements (λ , N, ϕ), i.e. solutions to the following primal and dual eigenproblems.

∂ ∂ x (g(x)N(x)) + (B(x) + λ )N(x) = ∞ x k(x, y)B(y)N(y) dy g(0)N(0) = 0, N(x) > 0, for x > 0, ∞ 0 N(x)dx = 1, (2.3) -g(x) ∂ ∂ x (ϕ(x)) + (B(x) + λ )ϕ(x) = B(x) x 0 k(y, x)ϕ(y) dy ϕ(x) > 0, ∞ 0 ϕ(x)N(x)dx = 1.
(2.4)

We make the following assumptions on the parameters of the model.

B ∈ W 1,∞ (R + , R * + ), g ∈ W 1,∞ (R + , R * + ), ∀ x ≥ 0, g ≥ g 0 > 0, (2.5) k ∈ C b (R + × R + ), y 0 k(x, y)dx = 2, y 0 xk(x, y)dx = y, (2.6) k(x, y < x) = 0, k(x, y > x) > 0. (2.7)
These guarantee in particular existence and uniqueness of a solution n ∈ C (R + ; L 1 ϕ (R + )) for L 1 initial data (see e.g. [START_REF] Perthame | Kinetic formulation of conservation laws[END_REF]), existence of a unique measure solution for data in M + (R + ) (cf. [START_REF] Carrillo | Structured populations, cell growth and measure valued balance laws[END_REF]), as well as existence and uniqueness of a dominant eigentriplet (λ > 0, N(x), ϕ(x)), cf. [START_REF] Doumic | Eigenelements of a general aggregation-fragmentation model[END_REF]. In particular the functions N and ϕ are continuous, N is bounded and ϕ has at most polynomial growth. In what follows N and ϕ will always denote the solutions to problems (2.3) and (2.4), respectively. Let us remark that in the L 1 setting we have the following conservation law

∞ 0 n ε (t, x)e -λt ϕ(x)dx = ∞ 0 n 0 (x)ϕ(x)dx.
(2.8) 2.3. Measure and measure-valued solutions. Let us observe that there are two basic ways to treat the above model in the measure setting. The first one is to consider a measure solution, i.e. a narrowly continuous map t → µ t ∈ M + (R + ), which satisfies (2.2) in the weak sense, i.e. for each

ψ ∈ C 1 c (R + × R + ) - ∞ 0 ∞ 0 (∂ t ψ(t, x) + ∂ x ψ(t, x)g(x))dµ t (x)dt + ∞ 0 ∞ 0 ψ(t, x)B(x)dµ t (x)dt = ∞ 0 ∞ 0 ψ(t, x) ∞ x k(x, y)B(y)dµ t (y)dxdt + ∞ 0 ψ(0, x)dn 0 (x).
(2.9)

Thus a measure solution is a family of time-parameterised non-negative Radon measures on the structure-physical domain R + .

The second way is to work with generalised Young measures and corresponding measurevalued solutions. To prove the generalised relative entropy inequality, which relies on considering a family of non-linear renormalisations of the equation, we choose to work in this second framework.

A measure-valued solution is a generalised Young measure (ν, m), where the oscillation measure is a family of parameterised probabilities over the state domain R + such that equation (2.2) is satisfied by its barycenters ν t,x , ξ , i.e. the following equation

∂ t ( ν t,x , ξ + m) + ∂ x (g(x)( ν t,x , ξ + m)) + B(x)( ν t,x , ξ + m) = ∞ x k(x, y)B(y) ν t,x , ξ dy + ∞ x k(x, y)B(y)dm(y) (2.10)
holds in the sense of distributions on R * + × R * + . It is proven in [START_REF] Gwiazda | A nonlinear structured population model: Lipshitz continuity of measure valued solutions with respect to model ingredients[END_REF] that equation (2.2) has a unique measure solution. To this solution one can associate a measure-valued solution -for example, given a measure solution t → µ t one can define a measure-valued solution by

δ dµ a t dL 1 , id = µ a t , m = µ s t
where dµ 1 dµ 2 denotes the Radon-Nikodym derivative of µ 1 with respect to µ 2 . However, clearly, the measure-valued solutions are not unique -since the equation is linear, there is freedom in choosing the Young measure as long as the barycenter satisfies equation (2.10). For example, a different measure-valued solution can be defined by 1 2 δ

2 dµ a t dL 1 + 1 2 δ {0} , id = µ a t .
Uniqueness of measure-valued solution can be ensured by requiring that the generalised Young measure satisfies not only the equation, but also a family of nonlinear renormalisations. This was the case in the work of DiPerna [START_REF] Diperna | Measure-valued solutions to conservation laws[END_REF], see also [START_REF] Dębiec | Relative entropy method for measure-valued solutions in natural sciences[END_REF].

To establish the GRE inequality which will then be used to prove an asymptotic convergence result, we consider the measure-valued solution generated by a sequence of regularized solutions. This allows us to use the classical GRE method established in [START_REF] Perthame | Transport equations in biology[END_REF]. Careful passage to the limit will then show that analogous inequalities hold for the measure-valued solution.

GRE INEQUALITY

In this section we formulate and prove the generalised relative entropy inequality, our main tool in the study of long-time asymptotics for equation (2.2). We take advantage of the well-known GRE inequalities in the L 1 setting. To do so we consider the growth-fragmentation equation (2.2) for a sequence of regularized data and prove that we can pass to the limit, thus obtaining the desired inequalities in the measure setting.

Let n 0 ε ∈ L 1 ϕ (R + ) be a sequence of regularizations of n 0 converging weak * to n 0 in the space of measures and such that TV(n 0 ε ) → TV(n 0 ). Let n ε denote the corresponding unique solution to (2.2) with n 0 ε as an initial condition. Then for any differentiable strictly convex admissible integrand H we define the usual relative entropy

H ε (t) := ∞ 0 ϕ(x)N(x)H n ε (t, x)e -λt N(x) dx
and entropy dissipation

D H ε (t) = ∞ 0 ∞ 0 ϕ(x)N(y)B(y)k(x, y) H n ε (t, y)e -λt N(y) -H n ε (t, x)e -λt N(x) -H n ε (t, x)e -λt N(x) n ε (t, y)e -λt N(y) - n ε (t, x)e -λt N(x) dxdy.
Then, as shown e.g. in [START_REF] Michel | General entropy equations for structured population models and scattering[END_REF], one can show that

d dt ∞ 0 ϕ(x)N(x)H n ε (t, x)e -λt N(x) dx = -D H ε (t) (3.1)
with right-hand side being non-positive due to convexity of H. Hence the relative entropy is non-increasing. It follows that

H ε (t) ≤ H ε (0) and, since H ≥ 0, ∞ 0 D H ε (t) dt ≤ H ε (0). (3.2)
In the next proposition we prove corresponding inequalities for the measure-valued solution generated by the sequence n ε . This result is an analogue of Theorem 5.1 in [START_REF] Gwiazda | Generalized entropy method for the renewal equation with measure data[END_REF]. Proposition 3.1. With notation as above, there exists a subsequence (not relabelled), generating a generalised Young measure (ν, m) with m = m t ⊗ dt for a family of positive Radon measures m t , such that

lim ε→0 ∞ 0 χ(t)H ε (t) dt = ∞ 0 χ(t) ∞ 0 ϕ(x)N(x) ν t,x (α), H(α) dx + ∞ 0 ϕ(x)N(x)H ∞ dm t (x) dt (3.3) for any χ ∈ C c ([0, ∞)), and
lim ε→0 ∞ 0 D H ε (t) dt = ∞ 0 ∞ 0 ∞ 0 ϕ(x)N(y)B(y)k(x, y) ν t,y (ξ ) ⊗ ν t,x (α), H(ξ ) -H(α) -H (α)(ξ -α) dxdydt + ∞ 0 ∞ 0 ∞ 0 ϕ(x)N(y)B(y)k(x, y) ν t,x (α), H ∞ -H (α) dm t (y)dxdt ≥ 0. (3.4)
We denote the limits on the left-hand sides of the above equations by ∞ 0 χ(t)H (t) dt and ∞ 0 D H (t) dt, respectively, thus defining the measure-valued relative entropy and entropy dissipation for almost every t. We further set

H (0) := ∞ 0 ϕ(x)N(x)H (n 0 ) a (x) N(x) dx + ∞ 0 ϕ(x)H ∞ (n 0 ) s |(n 0 ) s | (x) d|(n 0 ) s (x)|. (3.5)
We then have d dt H (t) ≤ 0 in the sense of distributions, (3.6)

and ∞ 0 D H (t)dt ≤ H (0). (3.7)
Proof. The function t → ∞ 0 n ε (t, x)e -λt ϕ(x)dx is constant and the function N is strictly positive on (0, ∞). Therefore the sequence u ε (t, x) := n ε (t,x)e -λt N(x) is uniformly bounded in L ∞ (R + ; L 1 ϕ,loc (R + )). Hence we can apply Proposition 2.2 to obtain a generalised Young measure (ν, m) on

R + × R + . Since u ε ∈ L ∞ (R + ; L 1 ϕ,loc (R + )), we have m ∈ L ∞ (R + ; M (R + ; ϕ))
. By a standard disintegration argument (see for instance [START_REF] Evans | Weak convergence methods for nonlinear partial differential equations[END_REF]Theorem 1.5.1]) we can write the slicing measure for m, m(dt, dx) = m t (dx) ⊗ dt, where the map t → m t ∈ M + (R + ; ϕ) is measurable and bounded.

By Proposition 2.2 we have the weak * convergence

H(u ε (t, x))(dt ⊗ ϕ(x)dx) * ν t,x , H (dt ⊗ ϕ(x)dx) + H ∞ m.
Testing with (t, x) → χ(t)N(x) where χ ∈ C c (R + ), we obtain (3.3). Further, the convergence [START_REF] Brenier | Weak-strong uniqueness for measure-valued solutions[END_REF], since for H ε we have the corresponding inequality (3.1).

∞ 0 χ(t)H ε (t)dt → ∞ 0 χ(t)H ε (t)dt implies (3.

We now investigate the limit as

ε → 0 of ∞ 0 D H ε (t)dt. Denoting Φ(x, y) := k(x, y)N(y)B(y) we have D H ε (t) = ∞ 0 ∞ 0 Φ(x, y)ϕ(x)[H(u ε (t, y)) -H(u ε (t, x)) -H (u ε (t, x))u ε (t, y) + H (u ε (t, x))u ε (t, x)]dxdy. (3.8)
We consider each of the four terms of the sum separately on the restricted domain [0, T ] × [η, K] 2 for fixed T > 0 and K > η > 0. Let D H ε,η,K denote the entropy dissipation with the integrals of (3.8) each taken over the subsets [η, K] of R + .

We now apply Proposition 2.2 to the sequence u ε , the measure dt ⊗ ϕ(x)dx on the set [0, T ] × [η, K]. The first two and the last integrands of D H ε,η,K (t) depend on t and only either on x or on y. Therefore we can pass to the limit as ε → 0 by Proposition 2.2 using a convenient test function. More precisely, testing with (t, x) → K η Φ(x, y)dy, we obtain the convergence

- T 0 K η K η Φ(x, y)ϕ(x)H(u ε (t, x))dydxdt -→ - T 0 K η K η Φ(x, y)ϕ(x) ν t,x , H dydxdt - T 0 K η K η Φ(x, y)ϕ(x)H ∞ dm t (x)dydt,
and, noticing that the recession function of

α → αH (α) is H ∞ , T 0 K η K η Φ(x, y)ϕ(x)H (u ε (t, x))u ε (t, x)dydxdt -→ T 0 K η K η Φ(x, y)ϕ(x) ν t,x , αH (α) dydxdt + T 0 K η K η Φ(x, y)ϕ(x)H ∞ dm t (x)dydt.
Likewise, using (t, y)

→ 1 ϕ(y) K η Φ(x, y)ϕ(x)dx, we obtain T 0 K η K η Φ(x, y)ϕ(x)H(u ε (t, y))dxdydt → T 0 K η K η Φ(x, y)ϕ(x) ν t,y , H dxdydt + T 0 K η K η Φ(x, y)ϕ(x)H ∞ dm t (y)dxdt.
There remains the term of D H ε,η,K in which the dependence on u ε combines x and y. To deal with this term we separate variables by testing against functions of the form f 1 (x) f 2 (y). We then consider

- T 0 [η,K] 2 f 1 (x) f 2 (y)H (u ε (t, x))u ε (t, y)dxdydt = - T 0 K η f 1 (x)H (u ε (t, x))dx K η f 2 (y)u ε (t, y)dy dt.
The integrands are now split, one containing the x dependence, and one the y dependence. However, extra care is required here to pass to the limit. As the Young measures depend both on time and space, it is possible for the oscillations to appear in both directions. We therefore require appropriate time regularity of at least one of the sequences to guarantee the desired behaviour of the limit of the product. Such requirement is met by noticing that since

u ε ∈ C ([0, T ]; L 1 ϕ (R + )) uniformly, we have u ε uniformly in W 1,∞ ([0, T ]; (M + (R + ), • (W 1,∞ ) * )), cf. [8, Lemma 4.1]. Assuming f 2 ∈ W 1,∞ (R + ) we therefore have t → K η f 2 (y)u ε (t, y)dy ∈ W 1,∞ ([0, T ]).
This in turn implies strong convergence of K η f 2 (y)u ε (t, y)dy in C ([0, T ]), by virtue of Arzéla-Ascoli theorem. Therefore we have (noting that (H ) ∞ ≡ 0 by sublinear growth of H)

- T 0 [η,K] 2 f 1 (x) f 2 (y)H (u ε (t, x))u ε (t, y)dxdydt = - T 0 K η f 1 (x)H (u ε (t, x))dx K η f 2 (y)u ε (t, y)dy dt -→ - T 0 K η f 1 (x) ν t,x , H dx K η f 2 (y) ν t,y , id dy dt - T 0 K η f 1 (x) ν t,x , H dx K η f 2 (y)dm t (y) dt = - T 0 [η,K] 2 f 1 (x) f 2 (y) ν t,x , H (α) ν t,y , ξ dxdy - T 0 [η,K] 2 f 1 (x) f 2 (y) ν t,x , H (α) dm t (y)dxdt.
By density of the linear space spanned by separable functions in the space of bounded continuous functions of (x, y) we obtain

- T 0 [η,K] 2 Φ(x, y)ϕ(x)H (u ε (t, x))u ε (t, y)dxdydt -→ T 0 [η,K] 2 Φ(x, y)ϕ(x) ν t,x , H (α) ν t,y , ξ dxdydt - T 0 [η,K] 2 Φ(x, y)ϕ(x) ν t,x , H (α) dm t (y)dxdt.
Gathering all the terms we thus obtain the convergence as ε → 0

T 0 D H ε,η,K (t)dt -→ T 0 D H η,K (t)dt with D H η,K (t) := [η,K] 2 Φ(x, y)ϕ(x) ν t,y (ξ ) ⊗ ν t,x (α), H(ξ ) -H(α) -H (α)(ξ -α) dxdy + [η,K] 2 Φ(x, y)ϕ(x) ν t,x (α), H ∞ -H (α) dm t (y)dx.
Observe that since Φ is non-negative and H is convex, the integrand of D H ε,η,K is non-negative. Hence so is the integrand of the limit. Therefore, by Monotone Convergence, we can pass to the limit η → 0, K → ∞, and T → ∞ to obtain

0 ≤ lim ε→0 ∞ 0 D H ε (t) dt = ∞ 0 ∞ 0 ∞ 0 ϕ(x)N(y)B(y)k(x, y) ν t,y (ξ ) ⊗ ν t,x (α), H(ξ ) -H(α) -H (α)(ξ -α) dxdydt + ∞ 0 ∞ 0 ∞ 0 ϕ(x)N(y)B(y)k(x, y) ν t,x (α), H ∞ -H (α) dm t (y)dxdt.
Finally we note that by the Reshetnyak continuity theorem, cf. [START_REF] Gwiazda | Generalized entropy method for the renewal equation with measure data[END_REF][START_REF] Kristensen | Relaxation of signed integral functionals in BV[END_REF] we have the convergence H ε (0) → H (0). Together with (3.2) this implies (3.6).

LONG-TIME ASYMPTOTICS

In this section we use the result of the previous section to prove that a measure-valued solution of (2.2) converges to the steady-state solution. More precisely we prove Theorem 4.1. Let n 0 ∈ M (R + ) and let n solve the growth-fragmentation equation (2.2). Then

lim t→∞ ∞ 0 ϕ(x)d|n(t, x) -m 0 N(x)L 1 | = 0 (4.1)
where m 0 := ∞ 0 ϕ(x)dn 0 (x) and L 1 denotes the 1-dimensional Lebesgue measure.

Proof. From inequality (3.7) we see that D H belongs to L 1 (R + ). Therefore there exists a sequence of times t n → ∞ such that

lim n→∞ D H (t n ) = 0.
Consider the corresponding sequence of generalised Young measures (ν t n ,x , m t n ). Thanks to the inequality H (t) ≤ H (0) this sequence is uniformly bounded in the sense that

sup n ∞ 0 ϕ(x)N(x) ν t,x (α), |α| dx + ∞ 0 ϕ(x)N(x)dm t n (x) < ∞. (4.2)
Therefore by the compactness property of Proposition 2.3 there is a subsequence, not relabelled, and a generalised Young measure ( νx , m) such that

(ν t n ,x , m t n ) * ( νx , m)
in the sense of measures. We now show that the corresponding "entropy dissipation"

D H ∞ := ∞ 0 ∞ 0 Φ(x, y)ϕ(x) νy (ξ ) ⊗ νx (α), H(ξ ) -H(α) -H (α)(ξ -α) dxdy + ∞ 0 ∞ 0 Φ(x, y)ϕ(x) νx (α), H ∞ -H (α) d m(y)dx (4.3)
is zero. To this end we argue that

D H ∞ = lim n→∞ D H (t n ).
Indeed this follows by the same arguments as in the proof of Proposition 3.1. In fact now the "mixed" term poses no additional difficulty as there is no time integral. It therefore follows that

D H ∞ = 0. (4.4)
As H is convex, both integrands in (4.3) are non-negative. Therefore (4.4) implies that both the integrals of D H ∞ are zero. In particular

∞ 0 ∞ 0 H(ξ ) -H(α) -H (α)(ξ -α)d νx (α)d νy (ξ ) = 0,
and since the integrand vanishes if and only if ξ = α, this implies that the Young measure ν is a Dirac measure concentrated at a constant. Then the vanishing of the second integral of D H ∞ implies that m = 0. Moreover, the constant can be identified as

m 0 := ∞ 0 ϕ(x)dn 0 (x) (4

CONCLUSION

In this article, we have proved the long-time convergence of measure-valued solutions to the growth-fragmentation equation. This result extends previously obtained results for L 1 ϕ solutions [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF]. As for the renewal equation [START_REF] Gwiazda | Generalized entropy method for the renewal equation with measure data[END_REF], it is based on extending the generalised relative entropy inequality to measure-valued solutions, thanks to recession functions. Generalised Young measures provide an adequate framework to represent the measure-valued solutions and their entropy functionals.

Under slightly stronger assumptions on the fragmentation kernel k, e.g. the ones assumed in [START_REF] Cáceres | Rate of convergence to the remarkable state for fragmentation and growth-fragmentation equations[END_REF], it has been proved that an entropy-entropy dissipation inequality could be obtained. Under such assumptions, we could obtain in a simple way a stronger result of exponential convergence, see the proof of Theorem 4.1. in [START_REF] Gwiazda | Generalized entropy method for the renewal equation with measure data[END_REF]. However the aboveseen method allows us to extend the convergence to spaces where no spectral gap exists [START_REF] Bernard | Asymptotic behavior of the growth-fragmentation equation with bounded fragmentation rate[END_REF].

A specially interesting case of application of this method would be critical cases where the dominant eigenvalue is not unique but is given by a countable set of eigenvalues. It has been proved that for L 2 initial conditions, the solution then converges to its projection on the space spanned by the dominant eigensolutions [START_REF] Bernard | Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts[END_REF]. In the case of measure-valued initial condition, due to the fact that the equation has not anymore a regularisation effect, the asymptotic limit is expected to be the periodically oscillating measure, projection of the initial condition on the space of measures spanned by the dominant eigensolutions. This is a subject for future work.

. 5 )

 5 by virtue of the conservation in time of∞ 0 ϕ(x)e -λt ν t,x , • dx + ∞ 0 ϕ(x)e -λt dm t (x).By virtue of Proposition 2.2 withH = | • -m 0 | it then follows that lim n→∞ ∞ 0 ϕ(x)d|n(t n , x)e -λt nm 0 N(x)L 1 | = 0,which is the desired result, at least for our particular sequence of times.Finally, we can argue that the last convergence holds for the entire time limit t → ∞, invoking the monotonicity of the relative entropy H . Indeed, the choiceH = | • -m 0 | in (3.5) yields the monotonicity in time of ∞ 0 ϕ(x)d|n(t, x)e -λtm 0 N(x)L 1 |, and the result follows.

Acknowledgements. T. D. would like to thank the Institute for Applied Mathematics of the Leibniz University of Hannover for its warm hospitality during his stay, when part of this work was completed. This work was partially supported by the Simons -Foundation grant 346300 and the Polish Government MNiSW 2015-2019 matching fund. The research of T. D. was supported by National Science Center (Poland) 2014/13/B/ST1/03094. M.D.'s research was supported by the Wolfgang Pauli Institute (Vienna) and the ERC Starting Grant SKIPPER AD (number 306321). P. G. received support from National Science Center (Poland) 2015/18/M/ST1/00075.


Enrico Marchetti

Laird Close

Jean-Pierre Véran

Johan Mazoyer
email: johan.mazoyer@obspm.fr

Raphaël Galicher

Pierre Baudoz

Patrick Lanzoni

Frederic Zamkotsian

Gérard Rousset

Frédéric Zamkotsian

Deformable mirror interferometric analysis for the direct imagery of exoplanets

Keywords: Instrumentation, High-contrast imaging, adaptive optics, wave-front error correction, deformable mirror

INTRODUCTION

Direct imaging of exoplanets requires the use of high-contrast imaging techniques among which coronagraphy. These instruments diffract and block the light of the star and allow us to observe the signal of a potential companion. However, these instrument are drastically limited by aberrations, introduced either by the atmosphere or by the optics themselves. The use of deformable mirrors (DM) is mandatory to reach the required performance. The THD bench (french acronym for very high-contrast bench), located in the Paris Observatory, in Meudon, France, uses coronagraphy techniques associated with a Boston Micromachines DM. [START_REF] Bifano | Adaptive imaging: MEMS deformable mirrors[END_REF] This DM is a Micro-Electro-Mechanical Systems (MEMS), composed of 1024 actuators. In March 2013, we brought this DM in Laboratoire d`Astrophysique de Marseille (LAM), France, where we studied precisely the performance and defects of this DM on the interferometric bench of this laboratory. The result of that study, conducted in collaboration with F. Zamkotsian et P. Lanzoni, from LAM are presented here.

We first describe the MEMS DM, the performance announced by Boston Micromachines and its assumed state before this analysis (Section 2). In the same section, we also present the interferometric bench at LAM. The results of this analysis are then presented in several parts. We first describe the analyzed DM overall shape and surface quality (Sections 3 and 4). We then analyze accurately the influence function of an actuator and its response to the application of different voltages (Section 5), first precisely for one actuators and then extended to all the DM. Finally, special attention will be paid to the damaged actuators that we identified (Section 6). We will present several causes of dysfunction and possible solutions.

THE MEMS DM AND THE LAM INTERFEROMETRIC BENCH 2.1 The MEMS DM: specifications and damaged actuators

Out of the 1024 actuators, only 1020 are used because corners are fixed. We number our actuators from 0 (bottom right corner) to 1023 (top left corner) as shown in Figure 1. The four fixed corner actuators are therefore numbers 31, 992 and 1023. The edges of the DM are also composed of fixed actuators, unnumbered. The inter-actuator pitch is 300µm, for a total size of the DM 9.3 mm. Boston Micromachines announces a subnanometric minimum stroke and a total stroke of 1.5 µm. All the values presented in this paper, unless stated otherwise, are in mechanical deformation of the DM surface (which are half the phase deformation introduced by a reflection on this surface). The flattened-DM surface quality is valued by Boston Micromachines at 30 nm (root mean square, RMS).

The electronics of the DM allows us to apply voltages between 0 and 300 V, coded on 14 bits. The minimum stroke is therefore 300/2 14 V or 18.3 mV. To protect the surface, the maximum voltage for this DM is 205 V.

We use percentage to express the accessible voltages: 0% corresponds to 0 V, while 100% corresponds to a voltage of 205 V. Each percent is thus a voltage of 2.05 V. The higher the voltage is the more the actuator is pulled towards the DM. A voltage of 100 % thus provides the minimum value of the stroke, a voltage of 0% its maximum. The minimum stroke of each actuators is 8.93 10 -3 %. This value was checked on the THD bench by checking the minimum voltage to produce an effect in the pupil plan after the coronagraph. Gain measurement in Section 5 will allow us to check the specifications for the maximal and minimum stroke of an actuator. The numbering starts at 0 in the bottom right corner and ends in 1023 in the top left corner. The actuators 841 and 197, in red, considered defective, were not used. Therefore, the pupil used is reduced (27 actuators along the diameter of the pupil only) and offseted on the DM.

Before March 2013, we thought that two actuators were unusable (they did not follow our voltage instructions): the 841 who could follow his neighbors if they were actuated and the 197 that seemed stuck at the 0% value. These actuators will be studied specifically in section 6. To avoid these actuators, the pupil before this analysis was reduced (27 actuators across the diameter of the pupil only) and offseted (see Figure 1).

Analysis at LAM : interferometric bench and process

The interferometric bench at LAM 2 was developed for the precise analysis of DM. Figure 2 shows the diagram of the Michelson interferometer. The source is a broadband light, which is filtered spatially by a point hole and spectrally at λ = 650 nm (this is the wavelength of the THD bench). In the interferometer, one of the mirrors is the DM to analyze (Sample in Figure 2). The other is a plane reference mirror (Reference flat in Figure 2). At the end of the other arm of the interferometer, a CCD detector (1024x1280) is placed. A lens system can be inserted in front of the camera to choose between a large field (40 mm wide, covering the whole DM) to a smaller field (a little less than 2 mm wide or 6x6 actuators). Both fields will be used in this study. The phase measurement is done using the method of Hariharan. [START_REF] Hariharan | Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm[END_REF] We introduce 5 phase differences in the reference arm:

{-2π/2, -π/2, 0, π/2, 2π/2}, (1)
and record the images with the CCD. The phase difference φ between the two arms can then be measured using:

tan(φ) = I -π/2 -I π/2 2I 0 -I -2π/2 -I 2π/2 . ( 2
)
Assuming a null phase on the reference mirror, the phase on the DM is just φ. Since the phase is only known between 0 and 2π, the overall phase is unwrapped using a path-following algorithm. This treatment can sometimes be difficult in areas with very high phase gradient. Finally, we measure the surface deformation of the DM by multiplying λ/(2π) and dividing by 2 (to measure mechanical movement of the DM from optical path difference).

The accuracy in the phase measurement is limited by the aberrations of the reference plane mirror and by the differential aberrations in the arms of the interferometer. However, the performance obtained on the measurement of the mechanical deformation of the DM are subnanometric. [START_REF] Liotard | Static and dynamic microdeformable mirror characterization by phase-shifting and time-averaged interferometry[END_REF] We can also retrieve the amplitude on the surface using:

4 M = 3 4(I -π/2 -I π/2 ) 2 + (2I 0 -I -2π/2 -I 2π/2 ) 2 2(I -2π/2 + I π/2 + I 0 + I π/2 + I 2π/2 ) . (3)
We now present the results of this analysis. Finally, the vertical lines indicate the limits of the 27x27 actuator pupil before March 2013 (red dotted line) and a pupil centered of the same size (brown dotted line). Right : Same cross sections for different voltages applied to all the actuators (from 0% to 90 %). Piston was removed and the curves were superimposed. The abscissas are measured in inter-actuator pitch and the y-axis is in nanometers.

GENERAL FORM OF THE DM

Figure 3 (left) shows, in a black solid curve, a cross section of the DM over the entire surface (in one of the main direction). We applied the same voltage of 70% to all the actuators. The x-axis is measured in inter-actuator pith and the mechanical deformation in y-axis is in nanometers. The first observation is that a uniform voltage on all the actuators does not correspond to a flat surface on the DM. The general shape is a defocus over the entire surface of approximately 500 nm (peak-to-valley, PV). The position of the 27x27 actuator pupil on the bench THD before March 2013 is drawn in red vertical lines. The brown vertical lines indicate a pupil of the same size, centered on the DM. The "natural" defocus of the DM in a pupil 27 actuators is about 350 nm (PV).

Figure 3 (right) represent the same cross section if we apply different uniform tensions to the DM (from 0% to 90%). Piston was removed and we superimposed these curves, which show that this form of defocus is present in the same proportions in all voltages. Due to slightly different gains between the actuators, there is a small variations of the actuators at the center between the various applied voltages.

The theoretical stroke of an actuator is 1.5µm and can normally compensates for this defocus by pulling the center actuators of 500 nm while letting the one on the edges at low tensions. However, this correction would be at the cost of a third of this theoretical maximum stroke on the center actuators. The chosen solution on our bench is to place the coronagraph mask outside the focal plane. Indeed, at a distance d the focal plane, the defocus introduced is:

Defoc P V = d 8(F/D) 2 , ( 4
)
in phase difference, in PV, where F/D is the opening of our bench. With the specifcations of our bench, we chose d = 7 cm, which correspond to the introduction of a defocus (in phase error) of 700 nm (PV), which exactly compensates the 350 nm (PV) of defocus (in mechanical stroke) in our 27 actuator pupil. We can therefore chose the voltages around a uniform value on the bench. Before the analysis at the LAM, we chose a voltage of 70 %, for reasons which are discussed in Section 5.2.

We also note on the black solid line on Figure 3 (left) the large variation at the edges of the DM (550 nm, PV in only one actuator pitch), when the same voltage of 70 % is applied to all actuators. This variation tends to decrease when lower voltages are applied to the actuators on the edges. However, this edge must not be included in the pupil. Note that the pupil prior to the analysis in Marseille (in red vertical lines) was very close to these edges.

On the edges, we can clearly seen the "crenelations" created by our DM actuators. To measure these deformations, I removed the lower frequencies (including all the frequencies accessible to the DM) numerically with a smoothing filter. The result is plotted in Figure 3 (left) in blue solid line. We clearly see this crenelation effect increase as we approache the edges. Once again, it is better to center the pupil on the detector to avoid the edge actuators. These effects are the main causes of the poor surface quality of MEMS DMs that we discussed in the next section. We now study the surface of our DM first through the level of aberrations, then with the study of the Power Spectral Density (PSD).

SURFACE QUALITY

In Fig 4 we show images of the surface of the DM obtained in large field (about 10 mm by 10 mm) on the left and small field (right), centered on the 4 by 4 central actuators (ie 1.2 mm by 1.2 mm). In both case, we removed all frequencies reachable by the DM (below 0.5 (inter-actuator pitch) -1 ) through a smoothing filter in post processing to observe its fine structures. For example, the defocus mentioned in the previous section has been removed. In the large field, we can observe the actuator 769 (in the lower left), which is fixed to the value 0% (see Section 6.3) and was unnoticed before this analysis but no noticeable sign of the two known faulty actuators (see Figure 1). We also note the edges and corners, very bright, due to fixed actuators.

Figure 5: On the left, cross sections on two actuators observed in small field. Each point of these cross sections is an average over a width of 0.1 inter-actuator pitch, either avoiding center and release-etch holes (curve "best case", in red) or on the contrary right in the center of an actuator (curve " worst case" in black). On the right, azimuthally averaged PSD measured on the whole DM (wide field) and on some actuators (narrow field). The frequencies on the horizontal axis are measured in µm -1 and the vertical axis is in nm 2 .µm 2 . The black dotted vertical lines indicate remarkable frequencies: the frequency of the actuators (1/300µm -1 ) and the maximum correctable frequency by the DM, of (2 inter-actuator pitch) -1 , or 1/(2 * 300)µm -1 . Finally, in red, we adjusted asymptotic curves.

Boston Micromachines announces a surface quality 30 nm (RMS) on the whole DM when it is in a "flat" position. Because of the high defocus defect that we corrected using a defocus of the coronagraphic mask, we have not tried to obtain a flat surface on the DM to verify this number. However, an estimate of the remaining aberrations in a "flat" position can be maid by removing in post processsing all the frequencies correctable by our DM. We measure the remaining aberrations without the edges and found 32 nm (RMS). This is slightly higher than the specifications of Boston Micromachines but one of the actuator at least is broken. The same measurement on our actual 27 actuators offseted pupil gives 8 nm (RMS) and 7 nm (RMS) for a same size pupil centered.

In the right image, we observe the details of the actuator. We observed three types of deformations:

• the center of the actuator, in black, with a size of about 25 µm

• the edges, which appear as two parallel lines separated by 45µm and of length one inter-actuator pitch (300µm)

• the release-etch holes of the membrane (4 in the central surface + 2 between the parallel lines of the edges).

In the principal direction of the DM, they appears very 150 µm and are only a few µm larges. A priori, these holes are a consequence of the making process by lithography.

We measured cross sections along two actuators in the small field, shown in Figure 5. The horizontal axis is in inter-actuator pitch and the vertical axis is in nanometers. Each of the points of these cross-sections is an average on a width of about 0.1 inter-actuator pitch. We placed these bands either right in the center of an actuator (curve "worst case", in black) or in a way to avoid both centers and release-etch holes (curve "best case", in red). The two bumps in 0.15 and 0.35 and in 1.15 and 1.35 inter-actuator pitch, common to both curves correspond to the parallel lines at the edges of the actuators. They produce mechanical aberrations of 12 nm (PV). The centers, in the black curve, are located in 0.65 and 1.65 inter-actuator pitch. They introduce mechanical aberrations 25 nm (PV). It is not certain that the aberrations in the release-etch holes are properly retrieved for several reasons. First, their size is a smaller than 0.1 inter-actuator pitch, so they are averaged in the cross section. We are also not sure that the phase is correctly retrieved in the unwrapping process as it encounters a strong phase gradient in these holes. They produce aberrations of 20 nm (PV). In total, on one actuator, aberrations of 30 nm (PV) and 6 nm (RMS) are obtained.

Figure 5 shows the azimuthally averaged DSP of the DM. The black curve represents the azimuthally averaged DSP for the whole DM (large field). We clearly observe the peak at the characteristic frequency of the DM (1/300 µm -1 ), indicated by a black dotted line. We can see peaks at other characteristic frequencies (1/(300 * √

2) µm -1 , 2/(300) µm -1 , ...). We took an azimuthal average to average these frequencies and observe a general trend. We repeated the same operation for DSP calculated on a small field. As shown in red in Figure 5 (right), we plotted the trends of these azimuthal DSP, which shows a asymptotic behavior in -4.4 for the big field and -3.3 for small field. Indeed, very small defects can come from differential aberrations in the interferometer, deformation of the flat reference mirror or noise in the measurement. We therefore adopt the large field value of f -4.4 for asymptotic behavior.

We now precisely study the behavior of a single actuator (Section 5), ie the influence function, the coupling with its neighbors, and the gain when we applied different voltages. For this analysis, we observe the behavior of a central actuator (number 528). We will measure its influence function and the inter-actuator coupling then study its gain, maximum and minimum strokes. These measurements were conducted by applying to the actuator 528 several voltages ranging from 10 to 90 % while the rest of the actuators are set at the value 70 %.

BEHAVIOR OF A SINGLE ACTUATOR

Influence function and coupling

We study the influence function IF of an actuator, movement of the surface when a voltage is applied. This influence function can be simulate using: 5

IF (ρ) = exp[ln(ω)( ρ d 0 ) α ], ( 5
)
where ω is the inter-actuator coupling and d 0 is the inter-actuator pitch. Figure 6 shows the influence function of a central actuator (528). At first, we apply a voltage of 40 % to the actuator (the others remaining at a voltage of 70 %) then a voltage of 70 % and made a difference, shown in the left picture. This is therefore the influence function for a voltage of -30%. We can observed that the influence function has no rotational symmetry. The main shape is a square, surrounded by a small negative halo.

We made cross section in several directions: one of the main directions of the DM, one of the diagonals. The results are presented on a logarithmic scale in Figure 6 (center). The distance to the center of the actuator is in inter-actuator pitch. We applied an offset to plot negative values in a logarithmic scale and we indicate the zero level by a dotted blue line. In the main direction, a break in the slope is observed at a distance of 1 inter-actuator pitch. The influence of the actuator in this direction is limited to 2 inter-actuator pitch in each way. In the diagonal direction, the secondary halo is about 3 nm deep, which is 0.5 % of the maximum. Due to this halo, the influence is somewhat greater (however, less than 3 inter-actuator pitch).

On Figure 6 (right), is plotted a cross section of the influence function in a principal direction of the DM. The inter-actuator coupling (height of the function at the distance of 1 inter-actuator pitch) is of 12 %. We fitted a curve using the function described in Equation 5 using this coupling and found α = 1.9. This shows that the central part of the influence function is almost a Gaussian ( alpha = 2), but do not take into account the "wings".

Gain study

In this section, we measured the maximum of the influence function for different voltages applied to the 528 actuator, the other remaining 70 %. Figure 7 (left) shows the superposition of cross sections in a principal direction of the DM for voltage values of 20 %, 30 %, 40 %, 50 %, 60 %, 70 %, 80 %, 90 %. We fitted Gaussian curves for these functions and observed that the maximum values of the peak are always located at the same place, and the width of the Gaussian is constant for the range of applied voltages. This shows that the influence function is identical for all the applied voltages. We plot the maximum of these curves as a function of the voltages in red diamonds in Figure 7 (right). The scale of these maximum can be read on the left axis, in nanometers. We then adjusted a quadratic gain (black solid curve) on this figure. This allows us to extrapolate the voltage values for 0 % and 100 %. From this figure, it can be deduced that:

• the maximum stroke is 1100 nm (1.1µ)m slightly less than the value indicated by Boston Micromachines. Temporal response to a +5% command in voltage for a normal actuator (left) and for the actuator 841 (right). Starting with a voltage of 70%, we send a +5% at 0s, wait for this command to be applied and then send a -5% command, at 5.39 s for normal actuator and at 20.14 s for the actuator 841. The vertical axis is % of the stroke, the abscissa in seconds since the update command of +5%. The dashed blue indicates the sending of the -5% command.

• the value of 70 % is the one that allows the maximum stroke in both ways (545 nm when we push and 560 nm when we pull). If the actuator is at a value of 25 %, we can only enjoy a maximum stroke of 140 nm in one direction. For this reason, we used to use the DM at values around 70 % before March 2013.

• the gain have a quadratic variation and therefore, the value of the minimum stroke in volt or in percent (8.93 10 -3 %) corresponds to different minimum strokes in nanometers depending on the location on this curve. We plotted the value of the minimum stroke in blue on the same plot (the scale of this curve, in nanometers, can be read on the right axis). We observe that a variation of 8.93 10 -3 % around 70 % produces a minimum stroke of 0.14 nm, which is twice the movement produced by the same variation around 25 % (0.07 nm).

Applying voltages around 70 % makes sense if we try to make the most of the stroke of the DM, but if we try to correct for small phase aberrations (which is our use of this DM), we should apply the lowest voltages possible.

We observed the positions of all the actuators and verified that they are evenly distributed on the surface. The gains of all actuators are very close on all the surface (variation of 20% between the minimum gain and the maximum).

We finish this study by an inventory of the different failures that we encountered and the solutions that we have fortunately been able to put in place to overcome these failures.

DAMAGED ACTUATORS

Before the analysis in March 2013, the actuators 841 and 197 were not responding correctly to our commands. A specific study on these actuators allowed us to overcome these dysfunctions and include them again in the pupil.

The slow actuator

We found that the actuator 841 responded to our voltage commands but with a very long response time. The interferometric bench in Marseille is not suited for temporal study (the successive path differences introductions limit the measurement frequency). Therefore, I used the phase measurement method developed on the THD bench: the self-coherent camera, see Mazoyer et al. (2013). [START_REF] Mazoyer | Estimation and correction of wavefront aberrations using the self-coherent camera: laboratory results[END_REF] I examined the temporal response of the 841 actuator after a command of +5% and compared it with the temporal response of a normal actuator (777). From a starting level of 70% for all of the actuator of the DM, we first sent a command to go to 75% to each of these two actuators, wait for this command to be executed and sent an order to return to the initial voltage. Figure 8 shows the results of this operation for a normal actuator (777, left) and for the slow actuator (841, right). The measurement frequency is on average 105 ms. The horizontal axis is the time (in seconds), with origin the date at which the +5% command is sent. Our phase measurement method does not give an absolute measurement of the phase and so we normalized the result (0% is the mean level before the command, 100% is the mean level after the +5%command.

For the normal actuator, the response time is inferior to the measurement period (105 ms in average). This result is consistent with the response time of an actuator announced by Boston Micromachines (< 20µs) although we cannot verify this value with this method. For the slow actuator, there is a much slower response to the rise as well as to the descent. We measured the response time to 95% of the maximum in the rise (7.5 s) and in the descent (8.1 s). However, as the static gain of the actuator is comparable with the gain of healthy actuators, we deduce that this actuator goes slowly but surely to the right position.

The coupled actuators

We realized that the actuator number 197 responded to the commands applied to the actuator 863, at the other end of the DM. It seems that the actuator 197 has a certain autonomy, but in case of large voltage differences applied on these two actuators, the 197 follows the commands applied to the 863 actuator. We carefully verified that if we apply the same voltage to them, these two actuators respond correctly to the command and have comparable gains than the other actuators. The actuator 863 is fortunately on the edge of the DM, so we can center the pupil with no influence of this actuator. Since, we have recentered the pupil to include the the 197 actuator back (see Figure 9). We systematically apply same voltages to both actuators simultaneously.

The dead actuator

Finally, we notice that the 769 actuator does not respond at all to our commands. This actuator is on the far edge of the DM. It is possible that he broke during the transportation towards LAM laboratory, but as it was far off pupil, we may have previously missed this failure. This actuator is fixed to the value 0% regardless of the applied voltage. However, we checked that it has no influence over 2 inter-actuator pitch.

CONCLUSION AND CONSEQUENCES ON THE BENCH

The identification of the faulty actuators and the solutions to overcome these dysfunctions have enabled us to recenter the pupil on our DM. Figure 9 shows the position of the pupil on the DM after the study at LAM. This centering has enabled to move away from the edges of the DM. We also saw that this centering is preferable to limit the introduction into the pupil of aberrations at high frequencies non reachable by the DM. Finally, we recently lowered the average value of the voltages on the DM from 70 % to 25 % and improve by a factor of 2 the minimum stroke reachable by each actuators. These upgrades played an important role in the improvement of our performance on the THD bench.

Figure 1 :

 1 Figure 1: Numbered actuators and position of the pupil on the DM before March 2013 (in green).The numbering starts at 0 in the bottom right corner and ends in 1023 in the top left corner. The actuators 841 and 197, in red, considered defective, were not used. Therefore, the pupil used is reduced (27 actuators along the diameter of the pupil only) and offseted on the DM.

Figure 2 :

 2 Figure 2: The interferometric bench at LAM. Figure from Liotard et al. (2005).2

Figure 3 :

 3 Figure3: DM cross sections. Left: cross sections of the whole DM surface in black when all actuators are at 70 % in voltages. Each point on this curve corresponds to an average over an actuator wide band. In blue line, we plotted the result after removing the frequencies accessible to the DM (in post-treatment, with a smoothing filter). Finally, the vertical lines indicate the limits of the 27x27 actuator pupil before March 2013 (red dotted line) and a pupil centered of the same size (brown dotted line). Right : Same cross sections for different voltages applied to all the actuators (from 0% to 90 %). Piston was removed and the curves were superimposed. The abscissas are measured in inter-actuator pitch and the y-axis is in nanometers.

Figure 4 :

 4 Figure 4: Surface of the DM in large field on the left (the whole DM is about 10 mm by 10 mm) and in small field on the right, centered on the 4 by 4 central actuators (ie 1.2 mm by 1.2 mm). In both cases, all the actuators are set to 70 % in voltages. To observe the fine structures of the DM, we removed in both cases low frequencies digitally in post-processing. On the left image, we can see actuator 769 (bottom left), which is fixed to 0% (see Section 6.3).

Figure 6 :

 6 Figure 6: Influence function. Left, measurement of the influence function of a central actuator. Center, cross section of the influence function in logarithmic scale along a principal direction of the mirror and in an diagonal direction. Right, cross section of the influence function along a principal direction, on which is superimposed cross section of a simulated influence function. The abscissas are in inter-actuator pitch and the vertical axis are in nanometers.

Figure 7 :

 7 Figure7: Study of one actuator: stroke and gain. Left: influence functions for different applied voltages. Right: maximum values of these influence functions in red and quadratic gain (black solid curve). The minimum percentage applicable (8.93 10 -3 %) can produce different minimum stroke depending on your position on this quadratic curve : we plot the minimum stroke around each voltage in blue (the scale of this curve, in nanometers, can be read on the right axis).

Figure 8 :

 8 Figure8: Study of the slow actuator. Temporal response to a +5% command in voltage for a normal actuator (left) and for the actuator 841 (right). Starting with a voltage of 70%, we send a +5% at 0s, wait for this command to be applied and then send a -5% command, at 5.39 s for normal actuator and at 20.14 s for the actuator 841. The vertical axis is % of the stroke, the abscissa in seconds since the update command of +5%. The dashed blue indicates the sending of the -5% command.

Figure 9 :

 9 Figure 9: This study allowed us to identify precisely the causes of actuator failures and recenter the pupil on the DM, including actuators 197 and 841.

1012 980 948 916 884 852 820 788 756 724 692 660 628 596 564 532 500 468 436 404 372 340 308 276 244 212 180 148 116 84 52 20 1011 979 947 915 883 851 819 787 755 723 691 659 627 595 563 531 499 467 435 403 371 339 307 275 243 211 179 147 115 83 51 19 1010 978 946 914 882 850 818 786 754 722 690 658 626 594 562 530 498 466 434 402 370 338 306 274 242 210 178 146 114 82 50 18 1009 977 945 913 881 849 817 785 753 721 689 657 625 593 561 529 497 465 433 401 369 337 305 273 241 209 177 145 113 81 49 17 1008 976 944 912 880 848 816 784 752 720 688 656 624 592 560 528 496 464 432 400 368 336 304 272 240 208 176 144 112 80 48 16 1007 975 943 911 879 847 815 783 751 719 687 655 623 591 559 527 495 463 431 399 367 335 303 271 239 207 175 143 111 79 47 15 1006 974 942 910 878 846 814 782 750 718 686 654 622 590 558 526 494 462 430 398 366 334 302 270 238 206 174 142 110 78 46 14 1005 973 941 909 877 845 813 781 749 717 685 653 621 589 557 525 493 461 429 397 365 333 301 269 237 205 173 141 109 77 45 13 1004 972 940 908 876 844 812 780 748 716 684 652 620 588 556 524 492 460 428 396 364 332 300 268 236 204 172 140 108 76 44 12 1003 971 939 907 875 843 811 779 747 715 683 651 619 587 555 523 491 459 427 395 363 331 299 267 235 203 171 139 107 75 43 11 1002 970 938 906 874 842 810 778 746 714 682 650 618 586 554 522 490 458 426 394 362 330 298 266 234 202 170 138 106 74 42 10 1001 969 937 905 873 841 809 777 745 713 681 649 617 585 553 521 489 457 425 393 361 329 297 265 233 201 169 137 105 73 41 9 1000 968 936 904 872 840 808 776 744 712 680 648 616 584 552 520 488 456 424 392 360 328 296 264 232 200 168 136 104 72 40 8 999 967 935 903 871 839 807 775 743 711 679 647 615 583 551 519 487 455 423 391 359 327 295 263 231 199 167 135 103 71 39 7 998 966 934 902 870 838 806 774 742 710 678 646 614 582 550 518 486 454 422 390 358 326 294 262 230 198 166 134 102 70 38 6 997 965 933 901 869 837 805 773 741 709 677 645 613 581 549 517 485 453 421 389 357 325 293 261 229 197 165 133 101 69 37 5 996 964 932 900 868 836 804 772 740 708 676 644 612 580 548 516 484 452 420 388 356 324 292 260 228 196 164 132 100 68 36 4 995 963 931 899 867 835 803 771 739 707 675 643 611 579 547 515 483 451 419 387 355 323 291 259 227 195 163 131 99 67 35 3 994 962 930 898 866 834 802 770 738 706 674 642 610 578 546 514 482 450 418 386 354 322 290 258 226 194 162 130 98 66 34 2 993 961 929 897 865 833 801 769 737 705 673 641 609 577 545 513 481 449 417 385 353 321 289 257 225 193 161 129 97 65 33 1 992 960 928 896 864 832 800 768 736 704 672 640 608 576 544 512 480 448 416 384 352 320 288 256 224 192 160 128 96 64 32 0 Damaged actuators Damaged actuators and pupil position before March 2013

1010 978 946 914 882 850 818 786 754 722 690 658 626 594 562 530 498 466 434 402 370 338 306 274 242 210 178 146 114 82 50 18 1009 977 945 913 881 849 817 785 753 721 689 657 625 593 561 529 497 465 433 401 369 337 305 273 241 209 177 145 113 81 49 17 1008 976 944 912 880 848 816 784 752 720 688 656 624 592 560 528 496 464 432 400 368 336 304 272 240 208 176 144 112 80 48 16 1007 975 943 911 879 847 815 783 751 719 687 655 623 591 559 527 495 463 431 399 367 335 303 271 239 207 175 143 111 79 47 15 1006 974 942 910 878 846 814 782 750 718 686 654 622 590 558 526 494 462 430 398 366 334 302 270 238 206 174 142 110 78 46 14 1005 973 941 909 877 845 813 781 749 717 685 653 621 589 557 525 493 461 429 397 365 333 301 269 237 205 173 141 109 77 45 13 1004 972 940 908 876 844 812 780 748 716 684 652 620 588 556 524 492 460 428 396 364 332 300 268 236 204 172 140 108 76 44 12 1003 971 939 907 875 843 811 779 747 715 683 651 619 587 555 523 491 459 427 395 363 331 299 267 235 203 171 139 107 75 43 11 1002 970 938 906 874 842 810 778 746 714 682 650 618 586 554 522 490 458 426 394 362 330 298 266 234 202 170 138 106 74 42 10 1001 969 937 905 873 841 809 777 745 713 681 649 617 585 553 521 489 457 425 393 361 329 297 265 233 201 169 137 105 73 41 9 1000 968 936 904 872 840 808 776 744 712 680 648 616 584 552 520 488 456 424 392 360 328 296 264 232 200 168 136 104 72 40 8 999 967 935 903 871 839 807 775 743 711 679 647 615 583 551 519 487 455 423 391 359 327 295 263 231 199 167 135 103 71 39 7 998 966 934 902 870 838 806 774 742 710 678 646 614 582 550 518 486 454 422 390 358 326 294 262 230 198 166 134 102 70 38 6 997 965 933 901 869 837 805 773 741 709 677 645 613 581 549 517 485 453 421 389 357 325 293 261 229 197 165 133 101 69 37 5 996 964 932 900 868 836 804 772 740 708 676 644 612 580 548 516 484 452 420 388 356 324 292 260 228 196 164 132 100 68 36 4 995 963 931 899 867 835 803 771 739 707 675 643 611 579 547 515 483 451 419 387 355 323 291 259 227 195 163 131 99 67 35 3 994 962 930 898 866 834 802 770 738 706 674 642 610 578 546 514 482 450 418 386 354 322 290 258 226 194 162 130 98 66 34 2 993 961 929 897 865 833 801 769 737 705 673 641 609 577 545 513 481 449 417 385 353 321 289 257 225 193 161 129 97 65 33 1 992 960 928 896 864 832 800 768 736 704 672 640 608 576 544 512 480 448 416 384 352 320 288 256 224 192 160 128 96 64 32 0 Fixed actuator (dead) Coupled actuators Slow actuator Damaged actuators and new pupil position

  To see the latest results on this high contrast bench, see Galicher et al. (2014) 7 and Delorme et al. (2014). 8

ACKNOWLEDGMENTS

J. Mazoyer is grateful to the CNES and Astrium (Toulouse, France) for supporting his PhD fellowship. The DM study at LAM was funded by CNES (Toulouse, France).