Laurent Alonso

Raphaël Cerf

published or not. The documents may come

The three dimensional polyominoes of minimal area


Raphaël Cerf
email: rcerf@math.u-psud.fr

Francesco Manzo
email: manzo.fra@gmail.com

A d-dimensional nucleation and growth model

We analyze the relaxation time of a ferromagnetic d-dimensional growth model on the lattice. The model is characterized by d parameters which represent the activation energies of a site, depending on the number of occupied nearest neighbours. This model is a natural generalisation of the model studied by Dehghanpour and Schonmann [DS97a], where the activation energy of a site with more than two occupied neighbours is zero.

Introduction

Growth models have been extensively studied in many cases of physical relevance. Our model can be obtained with a particular choice of the parameters for Richardson's model on the lattice [START_REF] Richardson | Random growth in a tessellation[END_REF] and it is closely related to the models studied by Eden [START_REF] Eden | A two-dimensional growth process[END_REF], Kesten and Schonmann [START_REF] Kesten | On some growth models with a small parameter[END_REF], and specifically Dehghanpour and Schonmann [START_REF] Dehghanpour | Metropolis dynamics relaxation via nucleation and growth[END_REF], with which it shares the same physical motivation, i.e., the study of the relaxation from a metastable state to the stable phase of a thermodynamic ferromagnetic system. In many physical cases, this event is triggered by the formation, growth and coalescence of many droplets of the stable phase in the midst of the metastable one. The model we study in this paper is inspired by the metastable behavior of the kinetic Ising model in the infinite-volume regime for small magnetic field and vanishing temperature. This regime was studied by Dehghanpour and Schonmann in the two dimensional case [START_REF] Dehghanpour | A nucleationand-growth model[END_REF]. The main ideas were presented in a simplified model in [START_REF] Dehghanpour | Metropolis dynamics relaxation via nucleation and growth[END_REF]. We study here the model corresponding to the d-dimensional case. There are several problems to extend the approach of Dehghanpour and Schonmann when there are more than two activation energies. One of them is to control the speed of growth of large supercritical droplets. In the model with two activation energies, this was achieved with the technology of the "chronological paths" introduced by Kesten and Schonmann [START_REF] Kesten | On some growth models with a small parameter[END_REF]. We did not manage to adapt this technology to deal with the three dimensional Ising model. In this paper, we present an alternative new strategy to control the speed of growth. This strategy relies on coupling arguments, where we consider specific boundary conditions called sandwich boundary conditions, as was done to analyze the bootstrap percolation model [START_REF] Cerf | Finite size scaling in threedimensional bootstrap percolation[END_REF][START_REF] Cerf | The threshold regime of finite volume bootstrap percolation[END_REF]. We hope to apply this strategy to control the growth of the supercritical droplets in the context of the three dimensional Ising model in the regime of low temperatures.

The model is an irreversible gas model on the lattice Z d . Sites are occupied at exponential times with rates that depend on the number of occupied neighbors. More precisely, our model is characterized by a set of parameters Γ n , n = 0, . . . , d that represent the activation energy of a "critical droplet" in dimension n. When a site has i ≤ d occupied neighbors, its occupation rate is exp(-βΓ d-i ). When a site has d or more occupied neighbors, its occupation rate is 1. A natural choice for ferromagnetic systems is to assume

Γ 0 ≤ Γ 1 ≤ • • • ≤ Γ d .
We start from the void configuration in infinite volume or in a finite cube and look at the time τ d when a given site, for instance the origin, is occupied. The scaling behavior of τ d as β goes to ∞ can be obtained with the help of the following simplified heuristics. The rate of creation of nuclei (namely, isolated occupied sites) is exp(-βΓ d ). Once a nucleus has appeared, it starts to grow, yet its speed of growth increases with its size. Let l(τ ) be the typical diameter of a droplet grown from a nucleus after a time τ . At time 2τ , the origin is likely to have been reached by any nucleus created at distance l(τ ) before time τ . The relaxation time τ d should be such that the rate of creation of a nucleus within the space time cone l(τ d ) d × τ d is of order one. It turns out that l(τ ) behaves as follows when β goes to ∞:

l exp(βK) ∼ 1 if K < Γ d-1 exp β(K -κ d-1 ) if K ≥ Γ d-1
Suppose that τ d scales as exp(βκ d ) when β goes to ∞. The value κ d will be the smallest value K such that

l exp(βK) d exp(βK) exp(-βΓ d ) is of order 1. Since Γ d ≥ Γ d-1
, then K has to be larger than Γ d-1 , and it satisfies therefore

exp dβ(K -κ d-1 ) exp(βK) exp(-βΓ d ) = 1 .
This equation yields

K = Γ d + dκ d-1 d + 1 .
We conclude finally that

κ d = max Γ d-1 , Γ d + dκ d-1 d + 1 .

Main result

Our configuration space is {0, 1} Λ , where Λ is a subset of Z d (possibly equal to Z d itself). A configuration is thus a map σ : Λ → { 0, 1 }, and a site x ∈ Λ is empty (respectively occupied) in the configuration σ if σ(x) = 0 (respectively σ(x) = 1). Sites which are occupied remain occupied forever.

To define the dynamics, we consider a family of i.i.d. Poisson processes with rate one, associated with the sites in Z d . For x ∈ Z d , i ≥ 1, we denote by τ (x, i) the i-th arrival time of the Poisson process associated with x. With each arrival time, we associate a uniform random variable U (x, i) in [0, 1], independent of the Poisson processes and of the other uniform variables.

We build a Markov process (σ Λ,t ) t≥0 with the help of these random objects. At time 0, we start from the empty configuration:

∀x ∈ Z d σ Λ,0 (x) = 0 .
We describe now the updating procedure of our process. Let N (x, σ) be the number of occupied neighbors of the site x in the configuration σ, i.e.,

N (x, σ) = y∈Λ:|x-y|=1 σ(y) .
The rate at which a site becomes occupied depends only on the number of its occupied neighbors. These rates are given by a non-decreasing sequence

c(0) ≤ c(1) ≤ • • • ≤ c(2d) .
A site x can become occupied only at a time corresponding to an arrival of its associated Poisson process. Suppose that t = τ (x, i) for some i ≥ 1 and that x was not occupied before time t. With probability one, all the arrival times are distinct and only the state of the site x can change at time t.

If U (x, i) ≤ c N (x, σ Λ,t (x))
then x becomes occupied at time t, otherwise it stays vacant. If the set Λ is finite, the above rules define a Markov process (σ Λ,t ) t≥0 . Whenever Λ is infinite, one has to be more careful, because there is an infinite number of arrival times in any finite time interval and it is not possible to order them in an increasing sequence. However, because the rates are bounded, changes in the system propagate at a finite speed, and a Markov process can still be defined by taking the limit of finite volume processes (see [START_REF] Thomas | Interacting particle systems[END_REF] for more details). Whenever Λ = Z d , we drop it from the notation, and we write (σ t ) t≥0 for the infinite volume process in Z d . We will deal with exponentially small rates. However we need to have a sufficiently loose asymptotic condition in order to perform our inductive proof, so that we can compare the process in dimension d with a d -1 dimensional process satisfying the same condition.

Hypothesis on the rates. We suppose that the occupation rates c(n), 0 ≤ n ≤ 2d, depend on a parameter β > 0 and that the following limits exist:

∀n ∈ { 0, . . . , d } lim β→∞ 1 β ln c β (n) = -Γ d-n , ∀n ∈ { d, . . . , 2d } lim β→∞ 1 β ln c β (n) = 0 .
Moreover, we suppose that

Γ 0 ≤ Γ 1 ≤ • • • ≤ Γ d .
For 0 ≤ n ≤ d, the parameter Γ n represents the activation energy of a critical droplet in dimension n. The conditions imposed on the sequence Γ n , 0 ≤ n ≤ d, simplify substantially the analysis and they are satisfied by the growth model associated to the metastability problem for the lowtemperature Ising model. We define a sequence of critical constants κ i for 0 ≤ i ≤ d by setting κ 0 = Γ 0 and

∀i ∈ { 1, . . . , d } κ i = max Γ i-1 , Γ i + iκ i-1 i + 1 .
Thus we have

κ d = max Γ d-1 , Γ d + dΓ d-2 d + 1 , . . . , Γ d + • • • + Γ d-i + (d -i)Γ d-i-2 d + 1 , . . . , Γ d + • • • + Γ 3 + 3Γ 1 d + 1 , Γ d + • • • + Γ 2 + 2Γ 0 d + 1 , Γ d + • • • + Γ 1 + Γ 0 d + 1 .
Our main result states that, in infinite volume, the relaxation time of the system scales as exp(βκ d ).

Theorem 2.1 (Infinite volume.) Let κ > 0 and let τ β = exp(βκ).

• If κ < κ d , then lim β→∞ P σ τ β (0) = 1 = 0 . • If κ > κ d , then lim β→∞ P σ τ β (0) = 0 = 0 .
The first step of the proof consists in reducing the problem to some growth processes in a finite volume. Indeed, if κ < K and we set

τ β = exp(βκ) , Λ β = Λ(exp βK) , then lim β→∞ P σ τ β (0) = σ Λ β ,τ β (0) = 1 .
This follows from a simple large-deviation estimate based on the fact that the maximum rate in the model is 1, see lemma 1 of [START_REF] Dehghanpour | A nucleationand-growth model[END_REF] for the complete proof. Let us shift next our attention to finite volumes. We have two possible scenarios for the growth process in order to fill completely a cube. If the cube is small, the system relaxes via the formation of a single nucleus that grows until filling the entire volume. If the cube is large, a more efficient mechanism consists in creating many droplets that grow and eventually coalesce. The critical side length of the cubes separating these two mechanisms scales exponentially with β as exp(βL d ), where

L d = Γ d -κ d d .
There are three main factors controlling the relaxation time:

Nucleation. Within a box of sidelength exp(βL), the typical time when the first nucleus appears is of order exp(β(Γ d -dL)). Initial growth. The typical time to grow a nucleus into a droplet travelling at the asymptotic speed is exp(βΓ d-1 ). Asymptotic growth. A droplet travelling at the asymptotic speed covers a region of diameter exp(βL) in a time exp(β(L + κ d-1 )).

The statement concerning the nucleation time contains no mystery. Let us try to explain the statements on the growth of the droplets. Once a nucleus is born, it starts to grow at speed exp(-βΓ d-1 ). As the droplet grows, the speed of growth increases, because the number of choices for the creation of a new protuberance attached to the droplet is of order the surface of the droplet. Thus the speed of growth of a droplet of size exp(βK) is

exp(β(K(d -1) -Γ d-1 )) .
When K reaches the value L d-1 , the speed of growth is limited by the time needed for the protuberance to cover an entire face of the droplet. This time corresponds to the d -1 relaxation time and the droplet reaches its asymptotic speed, of order exp(-βκ d-1 ). The time needed to grow a nucleus into a droplet travelling at the asymptotic speed is

1≤i≤exp(βL d-1 ) exp β Γ d-1 - d -1 β ln i
and it is still of order exp(βΓ d-1 ). With the help of the above facts, we can obtain easily an upper bound on the relaxation time in a box Λ β of sidelength exp(βL). Indeed, the relaxation time is smaller than the sum time for nucleation in the box Λ β +   time to grow a nucleus into a droplet travelling at the asymptotic speed

  + time to cover the box Λ β ∼ exp(β(Γ d -dL)) + exp(βΓ d-1 ) + exp(β(L + κ d-1 ))
which is of order

exp β max Γ d -dL, Γ d-1 , L + κ d-1 .
Optimizing over the size of the box Λ β , we conclude that the relaxation time in infinite volume satisfies

τ d ≤ exp β inf L max Γ d -dL, Γ d-1 , L + κ d-1 .
Let us now try to obtain a lower bound on the relaxation time. Suppose that we examine the state of the origin at a time exp(βκ). The origin becomes occupied when it is covered by a droplet. This droplet can result either from the growth of a single nucleus or from the coalescence of several droplets. Since the speed of propagation of the effects is finite, the state of the origin at time exp(βκ) is unlikely to have been influenced by any event occurring outside the box of sidelength exp(2βκ). Thus all the subsequent computations can be restricted to this box. In particular, a droplet which covers the origin before time exp(βκ) has to be born inside this box, meaning that the oldest site of the droplet belongs to this box. Let us consider the box Λ β of sidelength exp(βL). We can envisage two scenarios. If the droplet which covers the origin is born inside the box Λ β , then nucleation has occurred inside this box. If the droplet which covers the origin is born outside the box Λ β , then it has grown into a droplet of diameter at least 1 2 exp(βL) in order to reach the origin. Thus the relaxation time is larger than min time for nucleation in the box Λ β , time to grow a nucleus into a droplet of diameter 1 2 exp(βL)

∼ min exp(β(Γ d -dL)) , exp(βΓ d-1 ) + 1 2 exp(β(L + κ d-1 ))
which is of order

exp β min Γ d -dL, max(Γ d-1 , L + κ d-1 ) .
By optimizing over the size of the box Λ β , we conclude that the relaxation time in infinite volume satisfies

τ d ≥ exp β sup L min Γ d -dL, max(Γ d-1 , L + κ d-1 ) .
Since the optimal value of L solves Γ d -dL = L + κ d-1 , the two constants appearing in the exponential in the lower and upper bounds for the relaxation time coincide, they are equal to

κ d = max Γ d-1 , Γ d + dκ d-1 d + 1 .
We state next precisely the finite volume results that we will prove.

Terminology. We say that a probability P(•) is exponentially small in β (written in short ES) if it satisfies lim sup

β→∞ 1 β ln P(•) < 0 .
We say that a probability P(•) is super-exponentially small in β (written in short SES) if it satisfies

lim β→∞ 1 β ln P(•) = -∞ .
Theorem 2.2 (Exponential volume.) Let L > 0 and let Λ β = Λ(exp(βL)) be a cubic box of sidelength exp(βL). Let κ > 0 and let τ β = exp(βκ).

• If κ < max(Γ d -dL, κ d ), then lim β→∞ P σ Λ β ;τ β (0) = 1 = 0
and this probability is exponentially small in β.

• If κ > max(Γ d -dL, κ d ), then lim β→∞ P ∃ x ∈ Λ β σ Λ β ;τ β (x) = 0 = 0
and this probability is super-exponentially small in β.

The hardest part of theorem 2.2 is the upper bound on the relaxation time, i.e., the first case where κ < max(Γ d -dL, κ d ). The first ingredient in the proof is a lower bound on the time needed to create a large droplet.

Proposition 2.3 Let L > 0 and let Λ β = Λ(exp(βL)) be a cubic box of sidelength exp(βL). Let κ < Γ d-1 and let τ β = exp(βκ). The probability that an occupied cluster in σ Λ β ;τ β has diameter larger than β is superexponentially small in β.

The key result for the inductive proof is the following control on the size of the clusters in the configuration. We set

L d = Γ d -κ d d .
Theorem 2.4 Let L > 0 and let Λ β = Λ(exp(βL)) be a cubic box of sidelength exp(βL). Let κ < κ d and let τ β = exp(βκ). The probability that an occupied cluster in σ Λ β ;τ β has diameter larger than exp(βL d ) is super-exponentially small in β.

By using theorem 2.4 inductively, we are able to show that the asymptotic speed of the droplets inside the box Λ β is of order exp(-βκ d-1 ). The proofs of proposition 2.3 and of theorem 2.4 involve both a bootstrap argument to control the coalescence of the droplets. In fact, one could make a general statement to control the maximal size of an occupied cluster at a given time. Yet it turns out that only the initial growth and the asymptotic speed of the droplets are relevant to compute the relaxation time, the intermediate stage of growth of the droplets is not a limiting factor.

Graphical construction

Throughout the paper, we use the standard graphical construction [START_REF] Dehghanpour | A nucleationand-growth model[END_REF]. All our processes are defined on the same probability space and they are built with the help of the arrival times of independent Poisson processes and the associated uniform random variables

τ (x, i) , U (x, i) , i ≥ 1 , x ∈ Z d .
This provides a natural coupling between the different growth processes. The process in a set Λ with boundary conditions ρ is denoted by

(σ ρ Λ,t ) t≥0 .
This coupling preserves the natural order on the configurations. A configuration α is included in a configuration ρ, which we denote by α ≤ ρ, if every site occupied in α is also occupied in ρ. The growth process in a box Λ starting from the configuration α will always remain smaller than the growth process in Λ starting from a larger configuration ρ. The growth processes in a box Λ associated to different boundary conditions are also coupled in the same way, and the coupling respects the order on the boundary conditions, meaning that larger boundary conditions lead to larger growth processes.

We rely repeatedly on this coupling in order to compare our model with simpler or lower-dimensional processes.

Bootstrap

Following [START_REF] Dehghanpour | A nucleationand-growth model[END_REF], we control the effect of the coalescence of the droplets with a bootstrap-percolation argument. We recall next the standard bootstrap procedure. Let A be a finite subset of Z d . We start with a configuration η ∈ {0, 1} A and we occupy iteratively all the sites which have at least two occupied neighbors, until exhaustion. Since the procedure is monotonic and the volume is finite, the algorithm will stop after a finite number of steps. We denote by bootstrap(η) the final configuration obtained by bootstraping η. This final configuration is an union of occupied parallelepipeds, which are pairwise at distance larger than or equal to two. Following [START_REF] Aizenman | Metastability effects in bootstrap percolation[END_REF], we say that a set E ⊂ Z d is internally spanned in the configuration η if it is entirely covered in the final configuration of the dynamics restricted to E. More precisely, the initial configuration is the restriction of η to E and the dynamics runs on the sites of E without taking into account sites outside E.

We will use the supremum norm, given by

∀x = (x 1 , . . . , x d ) ∈ Z d |x| ∞ = max 1≤i≤d |x i | .
We denote by d ∞ the distance associated to the supremum norm and we define the

d ∞ diameter diam ∞ C of a subset C of Z d by diam ∞ C = sup |x -y| ∞ : x, y ∈ C .
Thus diam ∞ C is the sidelength of the minimal cube surrounding C. The following lemma is a key observation of Aizenman and Lebowitz [START_REF] Aizenman | Metastability effects in bootstrap percolation[END_REF].

Lemma 4.1 If a set C is internally spanned in a configuration η then for all integer k ≥ 1 such that 2k + 1 < diam ∞ C there exists a subset D of C which is internally spanned in η and such that k ≤ diam ∞ D ≤ 2k + 1.

We give the sketch of the proof, which can be found in [START_REF] Aizenman | Metastability effects in bootstrap percolation[END_REF]. It relies on the fact that if η ≤ ξ ≤ bootstrap(η), then bootstrap(ξ) = bootstrap(η).

For this reason, we are free to change the updating order without affecting the final configuration. The idea is then to realize the bootstrap percolation by occupying a single site at each step. If the maximal diameter of the clusters present in the configuration is k before one step of the algorithm, then right after occupying one site, the new maximal diameter is between k and 2k + 1. Looking at the evolution of the maximal diameter of the occupied clusters, we get the thesis.

Proof of proposition 2.3

Let L > 0 and let Λ β = Λ(exp(βL)) be a cubic box of sidelength exp(βL). Let κ < Γ d-1 and let τ β = exp(βκ). Let α be the random configuration defined as follows. For x ∈ Λ β , we set α(x) = 1 if there exists i ≥ 1 such that τ (x, i) ≤ τ β and U (x, i) ≤ c β (1), otherwise we set α(x) = 0. The law of the configuration α is the Bernoulli product law with parameter p β given by p β = 1 -exp -c β (1)τ β .

Taking logarithm, we see that

lim β→∞ 1 β ln p β = -Γ d-1 + κ < 0.
Let bootstrap(α) be the configuration obtained by bootstraping α. The configuration σ Λ β ;τ β is smaller than or equal to bootstrap(α). Indeed, in order to grow beyond bootstrap(α), the process would have to occupy a site outside bootstrap(α) having 0 or 1 occupied neighbors, but all these events until time τ β were already recorded in the initial configuration α. Proposition 2.3 is therefore implied by the following lemma.

Lemma 5.1 The probability that there exists an occupied cluster in the configuration bootstrap(α) whose d ∞ diameter is larger than β is superexponentially small in β.

Proof.

We say that a box is crossed if, after applying the bootstrap operator restricted to the box, there is an occupied connected set joining two opposite faces of the box. By lemma 4.1, if there is an occupied cluster in bootstrap(α) whose d ∞ diameter is larger than β, then there exists an internally-spanned cluster in bootstrap(α) with diameter between β and 2β + 1. Let Q β be a cube of minimal side length containing such a cluster. The cube Q β has to be crossed in one of the d directions parallel to the axis, say for instance the vertical one. If there is an horizontal strip in Q β of height 2 which is void in the configuration α then the box Q β cannot be crossed vertically. Thus

P Q β is crossed vertically ≤ P each horizontal strip in Q β of height 2 is non void in the initial configuration α ≤ P one fixed horizontal strip in Q β of height 2 is non void in the initial configuration α β/2-1 ≤ 1 -(1 -p β ) 2(2β+1) d-1 β/2-1 .
To complete the estimate, we count the number of possible choices for the box Q β : P there is an occupied cluster in bootstrap(α) whose d ∞ diameter is larger than β

≤ |Λ β | × 3β × d P Q β is crossed vertically ≤ 3dβ exp(βdL) 1 -(1 -p β ) 2(2β+1) d-1 β/2-1
and this last bound is SES.

6 Proof of theorem 2.4

In this section we prove theorem 2.4 with the help of an induction over the dimension d. The main point here is the bound on the asymptotic speed of growth of a droplet. Our approach gives a bound on the probability of a "too fast" growth. Since this bound is super-exponential, while both the volume and the time we are considering are exponential, we end up with a deterministic computation rather than a large-deviation estimate as in [START_REF] Dehghanpour | A nucleationand-growth model[END_REF]. This fact allows to avoid all combinatorial problems like counting the number of "chronological paths" and it is the main technical difference with the method used in [START_REF] Dehghanpour | A nucleationand-growth model[END_REF]. Heuristically, the process evolves as if the droplets were growing one shell after the other, filling the sites on one face before passing to the next. Since all the sites on a face are neighbors of an occupied site in the droplet, this growth mechanism is analogous to a nucleation and growth mechanism in dimension d -1. We use the d -1 dimensional bound on the size of the clusters to show that, up to SES events, a too-fast growth has to take place into a parallelepiped with "small" base. This is a SES bound, and the result holds in any exponential volume. Throughout the section, we let

Λ β = Λ(exp(βL))
be a cubic box of sidelength exp(βL), where L > 0. Let κ < κ d and let τ β = exp(βκ). Coalescence is a nontrivial effect only if L ≥ L d , since otherwise the number of droplets formed in Λ β before time τ β is finite. Theorem 2.4 needs to be proved only for L ≥ L d .

Dilation, bootstrap and erosion

The procedure we are going to define is a modified version of standard bootstrap percolation and is specifically suited to our setting. The same results can be obtained by rescaling the lattice as in [START_REF] Dehghanpour | A nucleationand-growth model[END_REF] and using the standard bootstrap percolation arguments developed in [START_REF] Aizenman | Metastability effects in bootstrap percolation[END_REF][START_REF] Cerf | The threshold regime of finite volume bootstrap percolation[END_REF]. We denote by d ∞ the distance associated to the supremum norm, given by ∀x,

y ∈ Z d d ∞ (x, y) = |x -y| ∞ = max 1≤i≤d |x i -y i | .
Let Λ be a subset of Z d , let η be a configuration in {0, 1} Λ and let l ≥ 0. We define the dilated configuration dilate(η, l) by occupying all the sites of Λ which are at a d ∞ distance strictly less than l from a site occupied in η:

∀x ∈ Λ dilate(η, l)(x) = 1 if ∃ y ∈ Λ d ∞ (x, y) < l , η(y) = 1 0 otherwise
We define the eroded configuration erode(η, l) by emptying all the sites of Λ which are at a d ∞ distance strictly less than l from an empty site in η:

∀x ∈ Λ erode(η, l)(x) = 0 if ∃ y ∈ Λ d ∞ (x, y) < l , η(y) = 0 1 otherwise
Dilation and erosion are classical operations in mathematical morphology. Let η be the random configuration defined as follows. For x ∈ Λ β , we set η(x) = 1 if there exists i ≥ 1 such that τ (x, i) ≤ τ β and U (x, i) ≤ c β (0), otherwise we set η(x) = 0. The law of the configuration η is the Bernoulli product law with parameter p β given by

p β = 1 -exp -c β (0)τ β .
Taking logarithm, we see that

lim β→∞ 1 β ln p β = -Γ d + κ .
Let bootstrap(η) be the configuration obtained by bootstraping η.

Proposition 6.1 Let ρ be the configuration obtained by dilating η with a distance β -1 exp(βL d ) and then bootstraping it:

ρ = bootstrap(dilate(η, β -1 exp(βL d ))) .
The probability that there is an occupied cluster in ρ whose d ∞ diameter is larger than exp (βL d ) is super-exponentially small in β.

Proof. By lemma 4.1, if there is an occupied cluster in ρ whose d ∞ diameter is larger than exp (βL d ), then there exists an internally-spanned cluster in ρ with diameter between exp (βL d ) and 2 exp (βL d ) + 1. Let Q β be a cube of minimal side length containing such a cluster. The cube Q β has to be crossed in one of the d directions parallel to the axis, say for instance the vertical one. Let Q ′ β be the parallelepiped having the same center and the same height as Q β and whose sidelengths in the other directions are three times the sidelength of Q β . If there is an horizontal strip in Q ′ β of height 3β -1 exp (βL d ) which is void in the initial configuration η, then there is an horizontal strip in Q β of height 2 which is void in the intermediate configuration dilate(η, β -1 exp(βL d ))

and the box Q β cannot be crossed vertically after the bootstraping. Thus

P Q β is crossed vertically in ρ ≤ P each horizontal strip in Q ′ β of height 3β -1 exp (βL d ) is non void in the initial configuration η ≤ P one fixed horizontal strip in Q ′ β of height 3β -1 exp (βL d ) is non void in the initial configuration η β/3 ≤ 1 -exp 9 d-1 exp((d -1)βL d ) × 3β -1 exp (βL d ) × ln(1 -p β ) β/3 ≤ -9 d β -1 exp(dβL d ) × ln(1 -p β ) β/3
and this last bound is SES because dL d + Γ d = κ d . To complete the estimate, we count the number of possible choices for the box Q β :

P

there is an occupied cluster in ρ whose d ∞ diameter is larger than exp βL d

≤ |Λ β | × 3 exp(βL d ) × d P Q β is crossed vertically in ρ
and the last term is SES.

Let ξ be the erosion of ρ with a distance (2β) -1 exp(βL d ), i.e.,

ξ = erode(ρ, (2β) -1 exp(βL d )) . (6.2)
Since ρ was obtained after applying the bootstrap procedure, it is a union of occupied parallelepipeds, which are pairwise at distance larger than two.

After applying the erosion operator, we obtain again an union of occupied parallelepipeds, which are pairwise at distance larger than or equal to (2β) -1 exp(βL d ). Moreover we dilated η with a distance β -1 exp(βL d ) before the bootstrap, thus the configuration η is still included in ξ, so that all the sites where nucleation has occurred before time τ β are occupied in the configuration η. By attractivity of the process, we have

σ Λ β ,τ β ≤ σ η Λ β ,τ β ≤ σ ξ Λ β ,τ β
and because of the definition of η, no nucleation occurs in the growth process starting from η until the time τ β . We are thus able to compare σ Λ β ,τ β with a process where nucleation events are cancelled, that we define in the next section. The crucial problem is then to control the speed of growth of the droplets and to show that, up to a SES event, the nonnucleating process starting from ξ is still included in ρ at time τ β .

Control of the speed of growth

In this section, we study the growth process where the nucleation is cancelled and we prove our key estimate to control the speed of growth of the droplets. The initial speed of growth of a nucleus is exp(-βΓ d-1 ). For a droplet of size exp(βK), the speed is

exp β((d -1)K -Γ d-1 )
for K < L d-1 and exp(-βκ d-1 ) for K ≥ L d-1 . It turns out that the time needed to create a droplet travelling at the asymptotic speed is exp(βΓ d-1 ), which is of the same order as the time needed to grow the initial nucleus into a droplet of diameter β. Hence we need only to control the speed of droplets having a diameter larger than exp(βL d-1 ), which travel at the asymptotic speed. Non-nucleating processes. We define a non-nucleating process ( σ Λ β ,t ) t≥0 associated to the rates

c(0) = 0 , c(n) = c(n) , 1 ≤ n ≤ 2d .
In this process, a site cannot become occupied unless one of its neighbors is occupied. The activation energies for this process are given by

Γ(d) = ∞ , Γ(n) = Γ(n) , 0 ≤ n < d .
The process ( σ t ) t≥0 is a (d -1)-dimensional process with rates satisfying

c β (n + 1) ≤ c β (n) ≤ 2c β (n + 1) + (β -2)c β (n) , 0 ≤ n ≤ d -1 .
In terms of activation energies,

Γ(n) = Γ(n) , 0 ≤ n ≤ d -1 .
The idea is to use the (d -1)-dimensional bounds on the size of ( σ t ) t≥0 and attractivity to bound the size of the clusters of ( σ - t ) t≥0 . Let Large be the event Large =    there is an occupied cluster in σ - R β ;τ β whose projection on the floor of R β has a diameter larger than exp (βL d-1 )

   .
By theorem 2.4 in dimension d -1, since κ < κ d-1 , the probability that an occupied cluster in σ τ β has diameter larger than exp(βL d-1 ) is SES. Since the volume of R β is exponential, the probability of the event Large is SES. We write then

P( σ - R β ;τ β crosses R β ) ≤ P(Large) + P { σ - R β ;τ β crosses R β } \ { Large } ≤ SES + P there is a translate y + T β of T β included in R β such that σ - y+T β ;τ β crosses y + T β ≤ SES + |R β | P( σ - T β ;τ β crosses T β ) .
In the last step, we used the fact that the model is translation invariant. We conclude by showing that

P( σ - T β ;τ β crosses T β )
is also SES, reducing ourselves to the case where K ≤ L d-1 . We shall couple the process ( σ - T β ;t ) t≥0 with floor boundary conditions in T β with another simpler process. Sandwich boundary conditions. We call slice a parallelepiped with height 2 and basis Λ d-1 β , which is a translate of

Σ = Λ d-1 β × { 0, 1 } .
We call sandwich boundary conditions on Σ the boundary condition defined by the following configuration ρ:

∀x ∈ Z d ρ(x) = 1 if x ∈ Λ d-1 β × { -1, 2 } 0 otherwise
We denote by ( σ = Σ;t ) t≥0 the process in Σ evolving with the sandwich boundary conditions ρ. Multilayer process. Let us partition the cylinder T β into translated slices as

T β = β/2 i=0 Σ i , where Σ i = Λ d-1 β × { 2i, 2i + 1 } = Σ + (0, . . . , 0, 2i) .
We define the multilayer process ( σ ≡ T β ;t ) t≥0 using the same graphical construction as ( σ T β ;t ) t≥0 but we use sandwich boundary conditions in each slice. More precisely, we set

∀i ∈ { 1, . . . , β/2 } ∀x ∈ Σ i ∀t ≥ 0 σ ≡ T β ;t (x) = σ = Σ i ;t (x) .
A key point is that, once we put sandwich boundary conditions around each slice, the processes in the slices become independent of each other. Thanks to the coupling, the process ( σ ≡ T β ;t ) t≥0 is always above the process ( σ - T β ;t ) t≥0 . Therefore, if σ - T β ;τ β crosses T β , so does σ ≡ T β ;τ β and at least a nucleus must appear in each slice. Thus and there exists a positive constant δ such that, for β large enough,

P σ - T β ;τ β crosses T β ≤ P σ ≡ T β ;τ β crosses T β ≤ P σ = Σ i ;τ β is not void for 1 ≤ i ≤ β/2 ≤ P σ = Σ;τ β is not void β/2-1 . Yet P σ = Σ;τ β is
P σ = Σ;τ β is void ≥ exp -2 exp(-βδ) .
Reporting in the previous inequality, we get

P σ = T β ;τ β crosses T β ≤ 1 -exp -2 exp(-βδ) β/2-1 .
Hence the above probability is also SES.

Corollary 6.4 Let d ≥ 2 and let K, L > 0. Let R β be the cylinder

R β = Λ d-1 (exp (βK)) × { 0, . . . , exp(βL) } .
Let κ > 0 be such that κ < L + κ d-1 and τ β = exp(βκ). Suppose that theorem 2.4 has been proved in dimension d -1. Then the probability that σ - R β ;τ β crosses R β is SES.

Proof. For i ∈ N, let τ i be the first time when a site of the layer With these definitions, we see that if

σ = R β ;τ β crosses R β , then τ l ≤ τ β . Yet τ l = 0≤i<l τ i+1 -τ i
and moreover, by using the Markov property and the attractivity of the process, we see that, for any i ≥ 0, the time τ i+1 -τ i stochastically dominates the time τ 1 . Therefore

P(τ l ≤ τ β ) ≤ P ∃i < l τ i+1 -τ i ≤ l -1 exp(βκ) ≤ l P τ 1 ≤ l -1 exp(βκ) .
By hypothesis, we have κ -L < κ d-1 . Proposition 6.3 implies that this last bound is SES.

Conclusion of the proof of theorem 2.4

We proceed now by induction over the dimension d. The case of dimension 0 is straightforward. In this case the lattice Z 0 is reduced to the singleton {0} and κ 0 = Γ 0 , L 0 = 0. In particular, it is impossible to see an occupied cluster of diameter strictly larger than 0. Let d ≥ 1. Suppose that the result has been proved in dimension d -1. Let L > 0 and let Λ β = Λ(exp(βL)) be a d-dimensional cubic box of sidelength exp(βL). Let κ < κ d and let τ β = exp(βκ). We apply corollary 6.4 to show that, up to a SES event, σ Λ β ,τ β is included in the configuration ρ. Indeed, suppose that it is not the case. Then the configuration σ ξ Λ β ,τ β is also not included in ρ. Yet the configuration ξ is an union of occupied parallelepipeds, which are pairwise at distance larger than or equal to (2β)

-1 exp(βL d ) (see (6.2)), and the configuration ρ is obtained from ξ by dilating these parallelepipeds with a distance (2β)

-1 exp(βL d ). We consider the first time and place when the process ( σ ξ Λ β ,t ) t≥0 occupies a site not occupied in ρ. This happens close to the boundary of a face F of one of the parallelepipeds Q occupied in ρ. Let R β be the cylinder included in Q having for basis this face F and for height (2β)

-1 exp(βL d ). By corollary 6.4, the probability that σ - R β ;τ β crosses R β is SES. Since the number of choices of times and places above is exponential in β, we conclude that, up to a SES event, the configuration σ Λ β ,τ β is included in ρ. This estimate, together with proposition 6.1, implies theorem 2.4.

Proof of the upper bound of theorem 2.2

Let L > 0 and let Λ β = Λ(exp(βL)) be a cubic box of sidelength exp(βL). Let κ < max(Γ d -dL, κ d ) and let τ β = exp(βκ). We distinguish three different cases.

• First case: κ < Γ d -dL. If the origin is occupied at time τ β for the growth process in Λ β , then a nucleation must have taken place in the box Λ β before the time τ β , thus

P σ Λ β ;τ β (0) = 1 ≤ |Λ β | 1 -exp -c β (0)τ β .
Taking logarithm, we see that lim sup

β→∞ 1 β ln P σ Λ β ;τ β (0) = 1 ≤ dL -Γ d + κ .
Yet κ < Γ d -dL and the probability that the origin is occupied at time τ β for the growth process in Λ β is therefore ES in β.

• Second case: κ < Γ d-1 . Let Λ ′ β = Λ(3β) be a cubic box of sidelength 3β. Suppose that the origin is occupied at time τ β for the growth process in Λ β . The droplet which has reached the origin is either born inside the box Λ ′ β or outside of it. In the first scenario, a nucleation event must have taken place in the box Λ ′ β before the time τ β . In the second scenario there is an occupied cluster in σ Λ β ;τ β with diameter larger than β. We have thus

P σ Λ β ;τ β (0) = 1 ≤ P a nucleation event takes place in Λ ′ β before τ β + P there is an occupied cluster in σ Λ β ;τ β whose d ∞ diameter is larger than β .
Proceeding as in the first case, we bound the probability of a nucleation by

|Λ ′ β | 1 -exp -c β (0)τ β which is ES in β since κ < Γ d-1 ≤ Γ d .
By proposition 2.3, the probability that an occupied cluster in σ Λ β ;τ β has diameter larger than β is superexponentially small in β.

• Third case: κ < κ d . Let Λ ′ β = Λ 3 exp(βL d ) be a cubic box of sidelength 3 exp(βL d ). Suppose that the origin is occupied at time τ β for the growth process in Λ β . The droplet which has reached the origin is either born inside the box Λ ′ β or outside of it. In the first scenario, a nucleation event must have taken place in the box Λ ′ β before the time τ β . In the second scenario there is an occupied cluster in σ Λ β ;τ β with diameter larger than exp(βL d ). We have thus

P σ Λ β ;τ β (0) = 1 ≤ P a nucleation event takes place in Λ ′ β before τ β + P there is an occupied cluster in σ Λ β ;τ β whose d ∞ diameter is larger than exp(βL d ) .
Proceeding as in the first case, we bound the probability of a nucleation by

|Λ ′ β | 1 -exp -c β (0)τ β .
Taking logarithm, we see that lim sup

β→∞ 1 β ln P a nucleation event takes place in Λ ′ β before τ β ≤ dL d -Γ d + κ < 0 .
By proposition 2.3, the probability that an occupied cluster in σ Λ β ;τ β has diameter larger than exp(βL d ) is super-exponentially small in β.

In the three cases, the probability

P σ Λ β ;τ β (0) = 1 is ES in β.
8 Proof of the lower bound of theorem 2.2

We prove here part 2 of theorem 2.2 by induction over the dimension d.

Let us consider first the case d = 0. We have then κ 0 = Γ 0 . The box Λ β is reduced to the singleton { 0 }. Let κ > κ 0 and let τ β = exp(βκ). We have we conclude that the above probability is SES. We suppose now that d ≥ 1 and that the result has been proved in dimension d -1. Let L > 0 and let Λ β = Λ(exp(βL)) be a cubic box of sidelength exp(βL). Let κ > 0 and let τ β = exp(βκ). Let ε > 0. We define the nucleation time τ nucleation in Λ β as τ nucleation = inf t ≥ 0 : ∃ x ∈ Λ β σ Λ β ;t (x) = 1 .

P
We have ∀t > 0 P(τ N > t) = exp -|Λ β | c β (0) t .

Therefore, up to a SES event, the first nucleus in the box Λ β appeared before time exp β Γ d -dL + ε .

For i ≥ 1, we define the first time τ i when there is an occupied parallelepiped of diameter larger than or equal to i in Λ β , i.e., τ i = inf t ≥ 0 : there is an occupied parallelepiped included in Λ β whose d ∞ diameter is larger than or equal to i

The restriction of the process (σ Λ β ;t ) t≥0 to the sites which are the neighbors of a face of an occupied parallelepiped is a d -1 dimensional growth process whose rates satisfy the hypothesis of our model. From the induction hypothesis, we know that, up to a SES event, the d -1 dimensional process in a box of sidelength exp(βK) is fully occupied at a time exp β max(Γ d-1 -(d -1)K, κ d-1 ) + ε .

This implies that, up to a SES event, the box Λ β is fully occupied at time We conclude that, in both cases, for any ε > 0, up to a SES event, the box Λ β is fully occupied at a time The last probability being SES, we are done.

  void = P for any x ∈ Σ, there is no nucleation at x before τ β = P there is no nucleation at the origin before τ β |Σ| = exp -c β (1) τ β |Σ| = exp -2 |Λ d-1 β | c β (1) τ β = exp -2 exp β(d -1)K + ln c β (1) + βκ . Since K ≤ L d-1 and κ < κ d-1 , we have lim β→∞ 1 β β(d -1)K + ln c β (1) + βκ = (d -1)K -Γ d-1 + κ < 0

Λ d- 1

 1 (exp (βK)) × { iβ } becomes occupied in the process ( σ = R β ;t ) t≥0 . Let us set l = exp(βL) β .

  σ Λ β ;τ β (0) = 0 = exp -(c β (0)τ β ) = exp -(c β (0) exp(βκ)) .Since by hypothesis,lim β→∞ 1 β ln c β (0) = -Γ 0

τ

  exp(βL) ≤ τ nucleation + 1≤i<exp(βL)(τ i+1 -τ i ) ≤ exp β Γ d -dL + ε + 1≤i<exp(βL) 2d exp β max(Γ d-1 -d -1 β ln i, κ d-1 ) + εWe consider two cases.• First case: L ≤ L d-1 . Notice that L 0 = 0, hence this case can happen only whenever d ≥ 2. In this case, we have∀i < exp(βL) κ d-1 ≤ Γ d-1 -d -1 β ln i and 1≤i<exp(βL) exp β max(Γ d-1 -d -1 β ln i, κ d-1 ) ≤ exp(βΓ d-1 ) 1≤i<exp(βL) 1 i d-1 ≤ exp(βΓ d-1 ) 1≤i<exp(βL) 1 i ≤ βL exp(βΓ d-1) .• Second case: L > L d-1 . We have thenexp(βL d-1 )≤i<exp(βL) exp β max(Γ d-1 -d -1 β ln i, κ d-1 )≤ exp(βL) -exp(βL d-1 ) exp(βκ d-1 )≤ exp β(L + κ d-1 ) .

  2dβL exp(βε) exp β(Γ d -dL) + exp(βΓ d-1 ) + exp β(L + κ d-1 ) .Therefore, for any κ such thatκ > max Γ d -dL, Γ d-1 , L + κ d-1the probability that the box Λ β is not fully occupied at a time exp(βκ) is SES. If L ≤ L d then max Γ d -dL, Γ d-1 , L + κ d-1 = Γ d -dL and we have the desired estimate. Suppose next that L > L d . By the previous result, we know that, for any κ > κ d , up to a SES event, a box of sidelength exp(βL d ) is fully occupied at a time exp(βκ). We cover Λ β by boxes of sidelength exp(βL d ). Such a cover contains at most exp(βdL) boxes, thus P (Λ β is not fully occupied at time τ β ) ≤ P there exists a box included in Λ β of sidelength exp(βL d ) which is not fully occupied at time τ β ≤ exp(βdL) P the box Λ(exp(βL d )) is not fully occupied at time τ β .

Acknowledgements: Raphaël Cerf thanks Roberto Schonmann for discussions on this problem while he visited UCLA in 1995.

In the sequel, the various processes where nucleation is suppressed are denoted with a tilde above the symbol of the process. Floor boundary conditions. Let R be a cylinder with basis a d -1 dimensional cubic box Λ d-1 and height H, i.e., of the form

We call floor of R its bottom face Λ d-1 × { 0 } and ceiling of R its top face Λ d-1 × { H }. We call floor boundary conditions on R the boundary condition defined by the following configuration ρ:

The process ( σ ρ R;t ) t≥0 in R with the floor boundary conditions is denoted by ( σ - R;t ) t≥0 . We say that a configuration crosses R if it contains a cluster included in R which connects the floor and the ceiling. Proposition 6.3 Let d ≥ 2 and let K > 0. Let R β be the cylinder

Let κ < κ d-1 and τ β = exp(βκ). Suppose that theorem 2.4 has been proved in dimension d -1. Then the probability that σ - R β ;τ β crosses R β is SES.

Proof. We start with the case K > L d-1 and we set

We use theorem 2.4 to show that, most likely, the cluster that crosses R β is contained in a smaller parallelepiped of basis Λ d-1 β , i.e., a parallelepiped which is a translate of

To this end, let us consider the process obtained from ( σ - R;t ) t≥0 by occupying all the sites in each non empty column, and its projection ( σ t ) t≥0 on the floor R β defined for t ≥ 0 by ∀ x ∈ Λ d-1 β σ t ( x) = 0 if σ - R β ;t ( x, i) = 0 for all i ∈ { 0, . . . , β } 1 if σ - R β ;t ( x, i) = 1 for some i ∈ { 0, . . . , β }


Chady Kharrat
email: [chady.kharrat@cea.fr

Eric Colinet
email: eric.colinet]@cea.fr

Alina Voda
email: alina.voda@inpg.fr

A robust control method for electrostatic microbeam dynamic shaping with capacitive detection

A robust closed-loop control and observation methodology for an electrostatic dynamic shaping of a microbeam using N small separate electrodes is described. After decomposing the displacements vector on the n eigenmodes using the modal analysis, n controllers are designed to control the dynamic coefficients of each mode and thus to deliver the stresses that must be distributed throughout the beam. In previous works, we considered direct access to non noisy displacement measurements. In this paper, we investigate the capacitive measurement of the local displacements done by each small electrode, which gives a noisy readout. Robust control methodology applied on extended standard model permits to design n observers associated to n controllers and guarantees precise shape tracking, free from noise and robust against parameters incertitude.

INTRODUCTION

Electrostatically actuated MEMS are widely used for positioning, sensing and signal filtering purposes. It involves simple circuitry and has a major advantage in the fact that it offers both electrostatic actuation as well as integrated detection, without the need of an additional position sensing device [START_REF] Napoli | A Capacitive microcantilever: Modelling, validation, and estimation using current meaurements[END_REF]. Its main drawbacks are the noninearity remaining in the equation linking the voltage input to the force output and the "pull-in" instability which occurs when the voltage applied between the fixed electrode and the microbeam creates an electrostatic force higher than any other restoring force for any displacement. Capacitive sensors have many advantages such as high sensitivity, low temperature dependence, low noise, large dynamic range, and potential monolithic integration with CMOS circuits [START_REF] Chu | Measurements of material properties using differential capacitive strain sensors[END_REF]. In the other hand, these associated electronic circuits for output readout and the ADC converters generate electronic noise that reaches, in some cases, significant magnitudes in relative to the capacitive measured signal. Also, due to the uncertainties resulting from manufacturing processes, material properties, and modelling assumptions, these microsystems may exhibit significant variations in their performance compared to nominal designs [START_REF] Liu | Multiobjective design optimization of electrostatically actuated microbeam resonators with and without parameter uncertainty[END_REF]. Also, residual stresses and large deflections make the nonlinearity stretching terms significant factors governing the microbeam behaviour [START_REF] Younis | The response of clamped-clamped microbeams under mechanical shock[END_REF]. In addition, arrays of MEMS begin to play an important role in several applications. Thus mechanical and electrostatic couplings between individual microstructures appear and increase the complexity of the model. In consequence, and for all these problems, the control of such microsystems becomes a difficult task to accomplish because of the important computation capacity needed for the processor to find the appropriate nonlinear decoupling command signals and to ensure the stability and the control of the whole system. The flexible structures are generally susceptible to structural vibration and deformation, and thus vibration suppression and shape control are important. Shape control represents one of the most important applications of smart materials and structures. In lightweight adaptive optics, piezoelectric (PZT) actuators were bonded in optical mirrors to achieve designed surface shapes [START_REF] Liu | Adaptive lightweight mirrors for the correction of self-weight and thermal deformations[END_REF][START_REF] Huonker | Fast and compact adaptive mirror[END_REF]. Precise shape control for a flexible circular plate mirror was achieved by [START_REF] Philen | Active stiffener actuators for high-precision shape control of circular plate structure[END_REF] using PZT strips with a decoupled actuation direction, while high precision in static shape control of smart structures was achieved by using the orthotropic actuators in [START_REF] Luo | High precision shape control of plates using orthotropic piezoelectric actuators[END_REF]. The FEM (finite element method) was used by [START_REF] Isobe | A Parallel Control System for Continuous Architecture Using Finite Element Method[END_REF] for its capability to express the behaviour of the whole system by evaluating the stiffness equations but with reducing the number of dots. Recently, dynamic shape tracking has gained attention; this integrates structural shape control with motion control. In [START_REF] Krommer | Sensor and actuator design for displacement control of continuous systems[END_REF], dynamic displacement tracking of smart beams is studied using distributed selfstresses strain sensors and actuators or piezoelectric ones [START_REF] Irschik | A review on static and dynamic shape control of structures by piezoelectric actuation[END_REF]. In [START_REF] Luo | A segment based sequential least squares algorithm with optimum energy control for tracking the dynamic shapes of smart structures[END_REF], an efficient algorithm for dynamic shape tracking with optimum energy control is proposed. However all these works still require important computational time and algorithms as well as a complex controller network for the control of each local displacement. In this work, a novel automatic based-mode-control method is proposed to ensure the dynamic displacement reference tracking on each position along the microbeam with good regulation dynamics, robustness against parameters incertitude and rejection of the measurement noise influence. This is done by controlling only the modal dynamic coefficients and using distributed electrostatic actuators and capacitive sensors. The nonlinearities as well as the couplings between the displacements in each position are also taken into consideration. First, the system is described, and practical limitations are exposed. Then the detailed modelling of the microbeam and the modal analysis of its behaviour are explained. In the third part, the "regulator problem" [START_REF] Wonham | Linear multivariable control: A geometric approach[END_REF] and the LTR (Loop Transfer recovery) control technique are exposed for designing the mode-based-controllers and observers guarantying the control specifications. In previous work [START_REF] Kharrat | Microbeam dynamic shaping by closed-loop electrostatic actuation using modal control[END_REF], the controllers used were PIDs who were limited in performance and for whom noisy measurements had strong influence on shape outputs. Finally, simulation results of the whole controlled system on Matlab are shown, with tests of robustness, noise rejection and performances.

SYSTEM DESCRIPTION

The studied system consists of a continuous deformable microbeam, clamped on both extremities and subjected to distributed time-variant electrostatic forces generated by the application of distributed time-variant voltages on N electrodes chosen from the two sets disposed on the both sides of the microbeam. is the transverse displacement of the microbeam which is measured using the capacitive principle. The 1 st order approximative equation relating the measured capacitance to the displacement is:

) , ( ) , ( 0 t x w g S t x C e - = ε (2)
When on a fixed instant t, an electrode is used as an electrostatic actuator and is subjected to a voltage, the other one is connected to a capacitive electronic measurement circuit, while a low voltage is applied on the microbeam for the detection aim. The electric measurement circuitries as well as the ADC converters add noise to the output signals. Smaller are the electrode surfaces, smaller are the measured capacitance magnitudes with the same electric noise and thus the signal to noise ratio decreases with the increasing number of the electrodes used. In addition, the complexity of fabrication, miniaturization and computation increases and the controller becomes more difficult to integrate when using classical control methods. In the other hand, the bigger is the number of the electrodes and the more continuous is the generated local forces along the microbeam which allows more precise reference tracking. In this work, 1000 electrodes are used.

DEFORMATION MODELLING

The behaviour of the deformation of a rectangular microbeam with length l, thickness e, width h, subjected to an external distributed strength obeys to the following differential equation: 4) in (3) and projecting on each vector i w , we get n equations representing the n modes that can be written in a matrix form:

F X M X B X N KX = + + + & & & ) ( (5) where [ ] T n t a t a t a X ) ( ) ( ) ( 2 1 L = , n I S M . . ρ = ,               = 4 4 2 4 1 0 0 0 0 0 0 n EI K λ λ λ L O M M L ,             = n b b b B 0 0 0 0 0 0 2 1 L O M M L ,               = n w f w f w f F M 2 1
In this work, n is limited to 5 because the effect of the superior modes become negligible on the shape. The stress due to the elongation of the beam can be expressed by the equation

l l S E w T ∆ = . . ) (
, where

) . . ( 2 1 2 1 2 1 0 1 1 2 0 X A X dx a dx dw dx dw a dx dx dw l T l l l N k N l k k l - = =       = ∆ ∫ ∑ ∑ ∫ = =
and thus one has

"REGULATOR PROBLEM" AND "LTR" TECHNIQUE FOR MODE-BASED-CONTROL

To control the microbeam shape having the N actuators with the displacements reference vector and the displacements measures in each position along the microbeam, a rudimentary idea is to calculate each local tracking error and to use it as controller input for the computation of the required local force to be applied, at a cost of a huge network of N controllers. Also, one will be dealing with the equation ( 1) directly which has unknown coupling terms with the other displacements and no desired exact dynamics can be imposed on displacements responses. By controlling the dynamic mode coefficients , the reference mode coefficients.

Any vibrating shape reference is described by different sinusoidal dynamic coefficients whose model is added to the system state space model to obtain the so-called standard model. To take into consideration a possible disturbance action on the input, a model of a constant disturbance d is also added so that the standard model becomes: where 1

x are the system states and 2 x are the exosytstem states (disturbance and reference). A is unstable. Finding a regulator K that tends e towards 0 for any initial conditions and keeps the closed loop system stable is what was referred to as "regulator problem with internal stability" (RPIS) by [START_REF] Wonham | Linear multivariable control: A geometric approach[END_REF]. If the solutions ( ) Having the appropriate standard model (the number of reference and disturbance states included in the standard model is equal or bigger than the number of measured output) a robust asymptotic reference tracking can be achieved [START_REF] De Larminat | The sufficient duplication principle : an alternative issue to the internal model principle[END_REF]. For robust stability, one has to adjust the gains 1 F and T represents the desired response speed for the closed-loop system. Once the poles are chosen, resolving the Ackerman's formula allows to obtain the gains of 1 F .

3 1 3 2 , × × ℜ ∈ ℜ ∈
[ ] r xd K K K = for system + disturbance (
Fig. 4. The poles placement technique that assures the robust stability of the closed loop system. When we don't have access to all the "system+disturbance" states (which is our case), an observer is needed which leads to a new

) ( jw L n
. For an exact LTR, this latter is equal to the target one but this can only be obtained with a derivative regulator that amplifies the noise influence on control signals and outputs. That's why an asymptotic LTR is adopted and a parameter o T / 1 -is chosen, which defines the axis in the left side of the complex space. After projection of the unstable open-loop zeros to the left quadrant, the projections on the [START_REF] Saberi | Loop transfer recovery : Analysis and design[END_REF]. Generally, o T is chosen 3, 5 or 10 times smaller than c T . Once the poles are chosen, resolving the Ackerman's formula allows to obtain the gains of sd K . As for the reference observer's poles, they can be chosen with dynamics faster than those defined by the reference vibration frequency, which allows precise reconstruction of the reference and of its derivative. Using Ackerman's formula, the gain of r K is calculated. The two parameters c

T and o T are chosen by taking into consideration the control specifications and the reference dynamics. The smaller c T is and the faster is the response speed which can be useful in case of disturbance rejection. In addition, robustness via the parameters uncertainties is improved. But, in the other hand, the noise influence is higher on output and on control signal and big solicitation of the actuators is predicted. Once the target loop is selected with c T , o T is tuned so that we approach the specified performances and the robustness of the target loop but without having an important derivative action, to obtain a better noise rejection. Guarantying one of the two objectives is easy to obtain, but having both on the same time is much more complex and a compromise must be done. will not generate exactly the required forces for the real displacements. This limits the intended advantages just when these required forces have big magnitudes which amplify the noise in the calculated voltages.

PRACTICAL CASE OF SIMULATIONS AND TESTS

The microbeam model is implemented in Simulink/Matlab with a length m l for a sampling frequency of 1 GHz. A constant disturbance is added to the electrostatic forces. In addition to all these specifications, a modification of 20% on all the microbeam's parameters were considered to test the robustness of the control scheme even in case of noisy measurement and disturbing environment with uncertain systems.

Depending on their initial dynamics and parameters, different Tc/4 5 th mode 5.10 -9 Tc/3

The results obtained for the shape tracking without disturbance application are shown in the following figures:

CONCLUSION

This paper details the design of a fully integrable control loop for a microbeam dynamic shaping by electrostatic actuation done with two sets of 1000 small electrodes disposed on both sides of the microbeam. First of all, the detailed modelling of the structure using modal analysis is presented. Then the control problem of each of the first five modes was formulated as a "regulator problem with internal stability" for extended systems models containing the modes dynamics models as well as the "exosystems", considering a known reference trajectory after projection on the modal space of dimension 5, and a constant disturbance to be rejected. Then the controllers are designed by placing the poles of the closed-loop systems such that good robustness, stability and performance are expected and observers are designed accordingly to the "LTR" technique to approach the closed target loops. This procedure was successfully simulated on Matlab and control specifications are obtained.

Fig. 1 .

 1 Fig. 1. The deformable microbeam surrounded with two sets of N electrodes on each side, subjected to different variant distributed voltages. The two sets are necessary to make possible the generation of attractive forces on both directions depending on the calculated control signals. The 1 st order approximative relation linking the electrostatic force to the applied voltage on a determinate position x, is:

  of the beam, E the Young's modulus, I the moment of inertia, ρ the density, b the friction coefficient associated to the interaction with the surrounding fluid, transverse displacement at position x, and T(w) is the stress associated to the elongation of the beam.

Fig. 2 .

 2 Fig. 2. The clamped-clamped microbeam with dimensions l, h and e. Following Galerkin procedure of standard modal analysis, ) , ( t x w can be written as: ∑ =

  between all the modes.

Fig

  Fig. 3. The first five ) (x w k representing the first five modal shapes of the microbeam.

  vibration pulsation. This standard model with its tracking errors e and outputs y can be written as:

  the regulator of an FSF (Full State Feedback)observer type with the gains [ ] the stability of the closed-loop system and the reference tracking. Resolving these equations in our case gives

  observation. The LTR technique was used to choose these gains: Considering possible access to the all the states, and by choosing the closed-loop poles that i the circle of centre -1 and beam equal to 1 in the Nyquist diagram, which guaranties good stability margins. This can be done by choosing a parameter c T / 1 -which defines the axis in the left side of the complex space[START_REF] Larminat | Contrôle d'état standard[END_REF]. After projection of the unstable open-loop poles to the left quadrant, the projections on who are on its right + those who are on its left represent the closed-loop system poles. In other words, c

-

  axis of those who are on its left + those who are on its right + the remaining observer poles set on o observer poles are set exactly on the open-loop zeros and the remaining ones on ∞ -

  This control scheme is applied to each of the five dynamic coefficients of the first five modes, requiring five regulators instead of N ones. The calculated control signals k f allow the calculation of the distributed desired electrostatic force ∑

  parameters of the model are:

  the 5 modes go from 10 Mhz to 134 Mhz. The chosen reference coefficients define a certain shape, vibrating with a frequency f = 10 Mhz. The couplings between the displacements represented by the couplings between the different modes are taken into consideration. The electrical noise is modelled as a white noise of spectral power equal

Fig. 5 .

 5 Fig. 5. The 5 dynamic coefficients ak of the noisy measured shape used as regulators inputs (on top) compared to the real shape coefficients of the closed-loop system (on bottom). When a constant disturbance is added to the actuators forces as an instantaneous step (which can represent a mechanical shock or a curt acceleration), the designed controllers exhibit good performance (for the same values of c T , o T ) and results

Fig. 6 .

 6 Fig.6. The comparison between the reference shape (blue curve) and the output shape (red curve) of the microbeam on a fixed instant tf (on top) and the correspondent desired force distribution of the actuators (on bottom).

Fig. 7 .

 7 Fig. 7. The modal tracking errors coefficients for the 5 modes. One can notice that errors don't exceed the 10% of the reference coefficients.

Fig. 8 .

 8 Fig. 8. Microbeam shape evolution with time.

Fig. 9 .

 9 Fig. 9. The 5 dynamic error coefficients ek in case of disturbance application on force inputs, having the same Tc and To chosen for other control specifications.

Table 1 .

 1 Control and observation parameters

	Mode num 1 st mode 2 nd mode 3 rd mode 4 th mode	Tc 2.10 -8 1.10 -8 8.10 -9 7.10 -9	To Tc/4 Tc/5 Tc/4

& & & & & & &