Matthieu Alfaro

Hiroshi Matano

ON THE VALIDITY OF FORMAL ASYMPTOTIC EXPANSIONS IN ALLEN-CAHN EQUATION AND FITZHUGH-NAGUMO SYSTEM WITH GENERIC INITIAL DATA

Keywords: 1991 Mathematics Subject Classification. 35B25, 35C20, 35R35 Singular perturbation, asymptotic expansion, front profile, Allen-Cahn equation, FitzHugh-Nagumo system, reaction-diffusion system

Formal asymptotic expansions have long been used to study the singularly perturbed Allen-Cahn type equations and reaction-diffusion systems, including in particular the FitzHugh-Nagumo system. Despite their successful role, it has been largely unclear whether or not such expansions really represent the actual profile of solutions with rather general initial data. By combining our earlier result and known properties of eternal solutions of the Allen-Cahn equation, we prove validity of the principal term of the formal expansions for a large class of solutions.

Introduction

In this paper, we study the behavior of solution u ε of an Allen-Cahn type equation of the form

(P ε )            u t = ∆u + 1 ε 2 (f (u) -εg ε (x, t, u)) in Ω × (0, ∞) ∂u ∂ν = 0 on ∂Ω × (0, ∞) u(x, 0) = u 0 (x) in Ω ,
and also that of a reaction-diffusion system of the form

(RD ε )                      u t = ∆u + 1 ε 2 (f (u) + εf 1 (u, v) + ε 2 f ε 2 (u, v))
in Ω × (0, ∞)

v t = D∆v + h(u, v) in Ω × (0, ∞) ∂u ∂ν = ∂v ∂ν = 0 on ∂Ω × (0, ∞) u(x, 0) = u 0 (x) in Ω v(x, 0) = v 0 (x) in Ω ,
where f (u), g ε (x, t, u), f 1 (u, v) and f ε 2 (u, v) satisfy the conditions to be specified later, and ε is a positive parameter. A typical example of (RD ε ) is the FitzHugh-Nagumo system:

(FHN ε )                      u t = ∆u + 1 ε 2 (f (u) -εf 1 (u) -εv)
in Ω × (0, ∞)

v t = D∆v + αu -βv in Ω × (0, ∞) ∂u ∂ν = ∂v ∂ν = 0 on ∂Ω × (0, ∞) u(x, 0) = u 0 (x) in Ω v(x, 0) = v 0 (x)
in Ω .

It is well-known that the solution u ε (x, t) of the above systems develops a steep transition layer, which converges to a "sharp interface" as ε → 0. To study such a sharp interface limit, formal asymptotic expansions of u ε are commonly used to discover, formally, the law of motion of the limit interface. Then, based on these expansions, one can construct sub-and super-solutions or use some approximation argument to prove the convergence of the transition layer -or the front -to the sharp interface, thereby establishing rigorously that the limit motion law agrees with what is anticipated from the formal asymptotics.

However, this standard approach only tells us that the transition layer of u ε is confined within a relatively narrow zone -of thickness o(1) or sometimes even O(ε)around the limit interface, but it does not say much about whether or not the actual transition layer really possesses a robust profile that matches the formal asymptotics. Known answers to this question are mainly concerned with solutions whose initial data already has a well-developed transition layer. The case of more general solutions has largely been unexplored. Our goal is to provide an affirmative answer in this direction: we shall prove that, for ε sufficiently small, the solution u ε -with rather general initial data -of both (P ε ) and (RD ε ) possesses a profile that agrees with the principal term of the formal expansion.

1.1. Notation and assumptions. The notation and assumptions stated below strictly follow those in [START_REF] Alfaro | The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system[END_REF].

In problems (P ε ) and (RD ε ) above, Ω is a smooth bounded domain in R N (N ≥ 2) and ν denotes the outward unit normal vector to ∂Ω. The nonlinearity is given by f (u) := -W ′ (u), where W (u) is a double-well potential with equal well-depth, taking its global minimum value at u = α ± . More precisely we assume that f is C 2 and has exactly three zeros α -< a < α + such that

(1.1) f ′ (α ± ) < 0 , f ′ (a) > 0 (bistable nonlinearity),
and that

(1.2) α+ α- f (u) du = 0 (balanced case).
In the Allen-Cahn equation (P ε ) we allow the balance of the two stable zeros α - and α + to be slightly broken by the function -εg ε (x, t, u) defined on Ω × [0, ∞) × R. We assume that g ε is C 2 in x and C 1 in t, u, and that, for any T > 0 there exist ϑ ∈ (0, 1) and C > 0 such that, for all (x, t, u)

∈ Ω × [0, T ] × R, ε|∆ x g ε (x, t, u)| + ε|g ε t (x, t, u)| + |g ε u (x, t, u)| ≤ C , g ε (•, •, u) C 1+ϑ, 1+ϑ 2 ( Ω×[0,T ]) ≤ C .
Moreover, we assume that there exists a function g(x, t, u) and a constant, which we denote again by C, such that

(1.3) |g ε (x, t, u) -g(x, t, u)| ≤ Cε ,
for all small ε > 0. In [START_REF] Alfaro | The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system[END_REF], we also assumed ∂g ε ∂ν = 0 on ∂Ω × [0, ∞) × R, though this last condition is only for technical simplicity. Note that these conditions, except the last one, are automatically satisfied if g ε is smooth and independent of ε.

In the reaction-diffusion system (RD ε ) we assume that f 1 (u, v), f ε 2 (u, v) are C 2 functions and that f ε 2 , along with its derivatives, remain bounded as ε → 0. We also assume that D > 0 and that h(u, v) is a C 2 function such that, for any constants L, M > 0, there exists a constant M 1 ≥ M such that h(u, -M 1 ) ≥ 0 ≥ h(u, M 1 ) for |u| ≤ L. These conditions enable us to construct a family of invariant rectangles mentioned in Remark 1.

In the FitzHugh-Nagumo system, we assume that f 1 ∈ C 2 (R) and that α and β are given positive constants so that (FHN ε ) becomes a special case of the reactiondiffusion system (RD ε ).

To complete the picture we need to specify conditions on the initial data. We assume that u 0 and v 0 belong to C 2 ( Ω). We define the "initial interface" Γ 0 by (1.4) Γ 0 := {x ∈ Ω : u 0 (x) = a} , and assume that Γ 0 is a C 3+ϑ hypersurface (0 < ϑ < 1) without boundary such that

Γ 0 ⊂⊂ Ω and ∇u 0 (x) • n(x, 0) = 0 if x ∈ Γ 0 , u 0 > a in Ω + 0 , u 0 < a in Ω - 0 ,
where Ω - 0 denotes the region enclosed by Γ 0 and Ω + 0 the region enclosed between ∂Ω and Γ 0 , and n(x, 0) denotes the outward unit normal vector at x ∈ Γ 0 = ∂Ω - 0 . Let us emphasize that we do not assume that the initial data u 0 of u ε already has well-developed transition layers depending on ε, in which case the validity of the formal expansions is more or less known (see subsection 1.2 for more details).

Remark 1 (Time-global smooth solutions). Under the above assumptions, it is classical that (P ε ) has a uniformly bounded smooth solution u ε that exists for all t ≥ 0. As for (RD ε ), the same can be shown for ε > 0 small enough, by using the method of invariant rectangles (see e.g. [START_REF] Alfaro | The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system[END_REF] for details).

1.2. Known results for the singular limit. We present here a brief overview of known results. Heuristically, in the very early stage, the diffusion term is negligible compared with the reaction term. Hence, in view of the profile of f , the value of u ε quickly becomes close to either α + or α -in most part of Ω, creating a steep interface (transition layers) between the regions {u ε ≈ α -} and {u ε ≈ α + } (Generation of interface). Once the balance between diffusion and reaction near the transition layers is established, the interface starts to propagate in a much slower time scale (Motion of interface). The interface obeys a certain law of motion, which is to be investigated.

A first step to understand this motion is to use (inner and outer) formal asymptotic expansions of u ε . This was performed in the pioneering work of Allen and Cahn [START_REF] Allen | A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening[END_REF] and, slightly later, in Kawasaki and Ohta [START_REF] Kawasaki | Kinetic drumhead model of interface I[END_REF], who revealed that the interface motion involves curvature effects. Using such arguments, one discovers that the sharp interface limit of (P ε ) obeys the following law of motion:

(P 0 )    V n = -(N -1)κ + c 0 α+ α- g(x, t, r) dr on Γ t Γ t t=0 = Γ 0 ,
where Γ t denotes the limit sharp interface at time t ≥ 0, V n is the normal velocity of Γ t in the exterior direction, κ the mean curvature at each point of Γ t , c 0 a constant determined straightforwardly from f via

c 0 := √ 2 α+ α- (W (s) -W (α -)) 1/2 ds -1
, where W (s) := -s a f (r)dr. As long as the solution Γ t of (P 0 ) exists, we denote by Ω - t the region enclosed by Γ t , and by Ω + t the region enclosed between ∂Ω and Γ t . Also we define a step function ũ(x, t) by

(1.5) ũ(x, t) := α -in Ω - t α + in Ω + t ,
to which u ε is formally supposed to converge as ε → 0. As regards (RD ε ), the limit problem is found to be

(RD 0 )                        V n = -(N -1)κ -c 0 α+ α- f 1 (r, v) dr on Γ t ṽt = D∆ṽ + h(ũ, ṽ) in Ω × (0, ∞) Γ t t=0 = Γ 0 ∂ṽ ∂ν = 0 on ∂Ω × (0, ∞) ṽ(x, 0) = v 0 (x) in Ω ,
where ũ is the step function defined in (1.5). This is a system consisting of an equation of surface motion and a parabolic partial differential equation. Since ũ is determined straightforwardly from Γ t , in what follows, by a solution of (RD 0 ) we mean a pair (Γ, ṽ).

Remark 2 (Local smooth solutions for the limit problems). Under our assumptions, there exists T max > 0 such that (P 0 ), respectively (RD 0 ), possesses a unique smooth solution Γ = ∪ 0≤t<T max (Γ t ×{t}), resp. (Γ, ṽ) = (∪ 0≤t<T max (Γ t ×{t}), ṽ). For more details we refer to [START_REF] Alfaro | The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system[END_REF] and the references therein, in particular [START_REF] Chen | Local existence and uniqueness of solutions of the Stefan problem with surface tension and kinetic undercooling[END_REF], [START_REF] Chen | Dynamics of interfaces in reaction diffusion systems[END_REF], [START_REF] Chen | Generation and propagation of interfaces for reaction-diffusion systems[END_REF]. In the sequel we select any 0 < T < T max and work on [0, T ].

Numerous efforts have been made to rigorously prove the convergence of (P ε ) and (RD ε ) to (P 0 ) and (RD 0 ), respectively. Concerning the Allen-Cahn equation, let us mention the work of de Mottoni and Schatzman [START_REF] De Mottoni | Development of interfaces in R n[END_REF] (generation of interface via sub-and super-solutions) and [START_REF] De Mottoni | Geometrical evolution of developed interfaces[END_REF] (motion of interface via construction of and linearization around an ansatz) or that of Bronsard and Kohn [START_REF] Bronsard | Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics[END_REF] (motion of interface via Γ-convergence). Chen [START_REF] Chen | Generation and propagation of interfaces for reaction-diffusion equations[END_REF][START_REF] Chen | Generation and propagation of interfaces for reaction-diffusion systems[END_REF] has established an O(ε| ln ε|) error estimate between the location of the actual transition later and the limit interface, both for scalar equations and systems for rather general initial data. More recently, in [START_REF] Alfaro | The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system[END_REF], the present authors improved this estimate to O(ε). More precisely, they show that the solution u ε develops a steep transition layer within the time scale of O(ε 2 | ln ε|), and that the layer obeys the law of motion that coincides with the formal asymptotic limit (P 0 ) or (RD 0 ) within an error margin of O(ε). Let us also mention that there are results of much finer error estimates in the literature (see for instance [START_REF] Bellettini | Quasi-optimal error estimates for the mean curvature flow with a forcing term[END_REF]), but those results are concerned with very specific initial data which already have nice transition layers consistent with formal asymptotics (hence dependent on ε).

As mentioned before, in most of the aforementioned works, approximate solutions or sub-and super-solutions are constructed by roughly following the formal expansions. Our goal is to investigate the actual validity of such expansions for solutions u ε with rather general initial data.

Remark 3 (Viscosity framework). Since the limit problem may develop singularities in finite time, the classical framework does not always allow to study the singular limit procedure for all t ≥ 0. Nevertheless -as far as the Allen-Cahn equation is concerned -following [START_REF] Evans | Motion of level sets by mean curvature I[END_REF], [START_REF] Chen | Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations[END_REF] one can define a limit problem for all t ≥ 0 that generalizes (P 0 ) in the framework of viscosity solutions. In this setting we refer to [START_REF] Evans | Phase transitions and generalized motion by mean curvature[END_REF] (convergence of Allen-Cahn equation with prepared initial data to generalized motion by mean curvature), [START_REF] Barles | Front propagation for reaction-diffusion equations of bistable type[END_REF], [START_REF] Barles | Front propagation and phase field theory[END_REF] (generalizations), [START_REF] Soner | Ginzburg-Landau equation and motion by mean vurvature, I: convergence[END_REF][START_REF] Soner | Ginzburg-Landau equation and motion by mean vurvature, II: development of the initial interface[END_REF], [START_REF] Barles | A new approach to front propagation problems : theory and applications[END_REF], [START_REF] Barles | A geometrical approach to front propagation problems in bounded domains with Neumann-type boundary conditions[END_REF] (not well-prepared initial data), [START_REF] Alfaro | Convergence rate of the Allen-Cahn equation to generalized motion by mean curvature[END_REF] (fine convergence rate).

Main results

We start by giving an outline of the formal asymptotic expansions mentioned before. See [2, Section 2] for more details. Let u ε be the solution of (P ε ), and Γ = ∪ 0≤t≤T (Γ t × {t}) be the solution of the limit geometric motion problem (P 0 ). We define the signed distance function to Γ by

(2.1) d(x, t) := -dist(x, Γ t ) for x ∈ Ω - t dist(x, Γ t ) for x ∈ Ω + t .
Then, near Γ, we make a formal inner expansion of the form (2.2)

u ε (x, t) = U 0 x, t, d(x, t) ε + εU 1 x, t, d(x, t) ε + • • • .
Some normalization conditions and matching conditions (with the outer expansion) are also imposed. By plugging the expansion (2.2) into (P ε ), we discover that U 0 (x, t, z) = U 0 (z), where U 0 (z) is the unique solution (whose existence is guaranteed by the integral condition (1.2)) of the stationary problem

(2.3) U 0 ′′ + f (U 0 ) = 0 U 0 (-∞) = α -, U 0 (0) = a , U 0 (∞) = α + .
This solution represents the first approximation of the profile of a transition layer around the interface observed in the stretched coordinates. Next the solvability condition for the equation involving U 1 provides the law of motion (P 0 ) for the limit interface Γ, which, in turn, determines the term U 1 .

It is then natural to wonder if the ansatz

u ε (x, t) = U 0 d(x, t) ε + εU 1 x, t, d(x, t) ε + • • •
is really a good approximation of the profile of the solution u ε . Note that the convergence results mentioned in subsection 1.2 do not answer this question; indeed those results simply show that the level surface of the solution u ε

(2.4) Γ ε t := {x ∈ Ω : u ε (x, t) = a} converges to the sharp interface (Γ t ) 0≤t≤T , which is a solution of (P 0 ), and that

(2.5) lim ε→0 u ε (x, t) = α -for x ∈ Ω - t α + for x ∈ Ω + t ,
without clarifying the validity of (2.2). Our main result Theorem 2.1 below provides a first answer in this direction. In the sequel we define (2.6)

t ε := f ′ (a) -1 ε 2 | ln ε| ,
which is the time needed for the transition layer of u ε to become fully well-developed (see Lemma 3.1). We define the signed distance function associated with Γ ε by (2.7)

d ε (x, t) := -dist(x, Γ ε t ) if u ε (x, t) < a dist(x, Γ ε t ) if u ε (x, t) > a .
Note that this definition of d ε is consistent with that of d in (2.1) in view of (2.5).

Theorem 2.1 (Validity for Allen-Cahn). Let the assumptions of subsection 1.1 hold (in particular the initial condition u 0 is rather generic). Let u ε be the smooth solution of Allen-Cahn equation (P ε ). Fix µ > 1. Then the following hold.

(i) If ε > 0 is small enough then, for any t ∈ [µt ε , T ], the level set Γ ε t is a smooth hypersurface and can be expressed as a graph over Γ t .

(ii) (2.8) lim ε→0 sup µt ε ≤t≤T, x∈ Ω u ε (x, t) -U 0 d ε (x, t) ε = 0 ,
where d ε denotes the signed distance function associated with Γ ε . (iii) There exists a family of functions (2.9)

θ ε : ∪ 0≤t≤T (Γ t × {t}) → R (0 < ε << 1)
whose L ∞ -norms remain bounded as ε → 0, such that

(2.10) lim ε→0 sup µt ε ≤t≤T, x∈ Ω u ε (x, t) -U 0 d(x, t) -εθ ε (p(x, t), t) ε = 0 ,
where d denotes the signed distance function associated with Γ and p(x, t) denotes a point on Γ t such that dist(x, Γ t ) = xp(x, t) .

Note that p(x, t) is an orthogonal projection of the point x onto Γ t , which is uniquely defined in a small tubular neighborhood of Γ t since Γ t is a smooth solution of (P 0 ). Note also that the presence of the perturbations -εθ ε (p(x, t), t) cannot be since it reflects the small difference between d(x, t) and d ε (x, t). Let us mention that the validity of higher order terms of the formal expansions (for generic solutions) is still unknown.

Our next theorem provides similar estimates for the reaction-diffusion systems.

Theorem 2.2 (Validity for the reaction-diffusion system). Let the assumptions of subsection 1.1 hold, and let (u ε , v ε ) be the smooth solution of the reaction-diffusion system (RD ε ). Fix µ > 1. Then the same conclusions as in Theorem 2.1 hold, with d being the signed distance function associated with (Γ, ṽ), which is the smooth solution of (RD 0 ) on [0, T ].

Proof of the main results

The proof of Theorem 2.1 relies on the following two results:

(a) the level set Γ ε t is approximated by the interface Γ t by order O(ε) ( [START_REF] Alfaro | The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system[END_REF], see subsection 3.1 of the present paper), (b) any eternal solution that lies between two planar waves is actually a planar wave ( [START_REF] Berestycki | Generalized travelling waves for reaction-diffusion equations in Perspectives in Nonlinear Partial Differential Equations[END_REF], see subsection 3.2 of the present paper), combined with a rescaling argument.

3.1. Thickness of the layers: the refined O(ε) estimate. We quote a result which is valid for both (P ε ) and (RD ε ). Lemma 3.1 ([2, Theorem 1.3 and Theorem 1.11]). Let η be an arbitrary constant satisfying 0 < η < min(aα -, α +a). Then there exist positive constants ε 0 and C 0 such that, for all ε ∈ (0, ε 0 ) and for all f ′ (a) -1 ε 2 | ln ε| = t ε ≤ t ≤ T , we have

(3.1) |u ε (x, t) -α ± | ≤ η if x ∈ Ω ± t \ N C0ε (Γ t ) , where N r (Γ t ) : = {x ∈ Ω : dist(x, Γ t ) < r} denotes the r-neighborhood of Γ t . This implies in particular that Γ ε t ⊂ N C0ε (Γ t ) for all t ε ≤ t ≤ T , hence (3.2) |d ε (x, t) -d(x, t)| ≤ C 0 ε for all (x, t) ∈ Ω × [t ε , T ] , 0 < ε << 1 .

3.2.

Eternal solutions and planar waves. We recall that a solution of an evolution equation is called eternal (or an entire solution) if it is defined for all positive and negative time. We follow this terminology to refer to a solution w(

w τ = ∆ z w + f (w) , z ∈ R N , τ ∈ R . z, τ ) of (3.3)
Stationary solutions and travelling waves are examples of eternal solutions. Crucial to our analysis is a recent result of Berestycki and Hamel [START_REF] Berestycki | Generalized travelling waves for reaction-diffusion equations in Perspectives in Nonlinear Partial Differential Equations[END_REF] asserting that "any planar-like eternal solution is actually a planar wave". More precisely, the following holds (for z ∈ R N we write z = (z (1)

, • • • , z (N ) )).
z (N ) →∞ inf z ′ ∈R N -1 w(z, τ ) > a , lim sup z (N ) →-∞ sup z ′ ∈R N -1 w(z, τ ) < a ,
where z ′ := (z (1) , • • • , z (N -1) ). Then there exists a constant z * ∈ R such that

w(z, τ ) = U 0 (z (N ) -z * ) , z ∈ R N , τ ∈ R . 3.3. Proof of (ii) in Theorem 2.1.
In what follows we fix µ > 1 and an arbitrary constant T 1 with T < T 1 < T max (see Remark 2). Obviously the conclusion of Lemma 3.1 remains valid if T is replaced by T 1 . Assume by contradiction that (2.8) does not hold. Then there is η > 0 and sequences

ε k ↓ 0, t k ∈ [µt ε k , T ], x k ∈ Ω (k = 1, 2, ...) such that (3.5) u ε k (x k , t k ) -U 0 d ε k (x k , t k ) ε k ≥ 2η .
In view of (3.1)-(3.2) and U 0 (±∞) = α ± , for (3.5) to hold it is necessary to have

(3.6) d(x k , t k ) = O(ε k ) , as k → ∞ . If u ε k (x k , t k ) = a, then this would mean that x k ∈ Γ ε k t k ,
in which case the left-hand side of (3.5) would be 0 (since U 0 (0) = a), which is impossible. Hence u ε k (x k , t k ) = a. By extracting a subsequence if necessary, we may assume without loss of generality that u ε k (x k , t k )-a has a constant sign for k = 0, 1, 2, . . .. Since the sign of this quantity is irrelevant in the later argument, in what follows we assume that (3.7)

u ε k (x k , t k ) > a (k = 0, 1, 2, . . .),
which then implies that

d ε k (x k , t k ) > 0 (k = 0,
u ε k (y k , t k ) = a (k = 0, 1, 2, . . .), (3.9
)

d ε k (x k , t k ) = x k -y k , (3.10) u ε k (x, t k ) > a if x -x k < y k -x k , x k -p k ⊥ Γ t k at p k ∈ Γ t k
, where p k := p(x k , t k ). Furthermore, (3.6) and (3.2) imply (3.11) x k -

p k = O(ε k ) , y k -p k = O(ε k ) (k = 0, 1, 2, . . .).
We now rescale the solution u ε around (p k , t k ) and define (3.12)

w k (z, τ ) := u ε k (p k + ε k R k z, t k + ε 2 k τ )
, where R k is a matrix in SO(N, R) that rotates the z (N ) axis onto the normal at p k ∈ Γ t k , that is R k : (0, . . . , 0, 1) T → n(p k , t k ) , where ( ) T denotes a transposed vector and n(p, t) the outward normal unit vector at p ∈ Γ t . Since Γ t (hence the points p k ) is uniformly separated from ∂Ω by some positive distance, there exists c > 0 such that w k is defined (at least) on the box

B k := (z, τ ) ∈ R N × R : z ≤ c ε k , -(µ -1)f ′ (a) -1 | ln ε k | ≤ τ ≤ T 1 -T ε 2 k .
Since u ε satisfies (P ε ), we see that w k satisfies (3.13)

w k τ = ∆ z w k + f (w k ) -ε k g ε k (p k + ε k R k z, t k + ε 2 k τ, w k ) in B k . Moreover, if (z, τ ) ∈ B k then t ε k ≤ t k + ε 2 k τ ≤ T 1 . Therefore (3.1) implies (3.14) d(p k + ε k R k z, t k + ε 2 k τ ) ≤ -C 0 ε k ⇒ w k (z, τ ) ≤ α -+ η , d(p k + ε k R k z, t k + ε 2 k τ ) ≥ C 0 ε k ⇒ w k (z, τ ) ≥ α + -η ,
so long as (z, τ ) ∈ B k . Since the rotation by R k of the z (N ) axis is normal to Γ t k at p k := p(x k , t k ), and since the curvature of Γ t is uniformly bounded for 0 ≤ t ≤ T , we see from (3.14) that there exists a constant C > 0 such that

(3.15) z (N ) ≤ -C ⇒ w k (z, τ ) ≤ α -+ η , z (N ) ≥ C ⇒ w k (z, τ ) ≥ α + -η ,
so long as (z, τ ) ∈ B k and z ≤ ε -1 k . Now, since w k solves (3.13), the uniform (w.r.t. k ≥ 0) boundedness of w k and standard parabolic estimates, along with the derivative bounds on g ε , imply that w k is uniformly bounded in C 2+γ,1+ γ 2 loc (B 1 ). We can therefore extract from (w k ) a subsequence that converges to some w in C 2,1 loc (B 1 ). By repeating this on all B k , we can find a subsequence of (w k ) that converges to some w in C 2,1 loc (R N × R) (note that ∪ k≥0 B k = R N × R). Passing to the limit in (3.13) yields

w τ = ∆ z w + f (w) on R N × R .
Hence we have constructed an eternal solution w(z, τ ) which -in view of (3.15)satisfies (3.4). Lemma 3.2 then implies that (3.16) w(z, τ ) = U 0 (z (N )z * )

for some z * ∈ R. Now we define sequences of points (z k ), (z k ) by

z k := 1 ε k R -1 k (x k -p k ) , zk := 1 ε k R -1 k (y k -p k ) .
By (3.11), these sequences are bounded, so we may assume without loss of generality that they converge:

z k → z ∞ , zk → z∞ , as k → ∞ .
By the definition of the z coordinates, z ∞ must lie on the z (N ) axis, that is,

z ∞ = (0, . . . , 0, z (N ) ∞ ) T . It follows from (3.8) and (3.10) that (3.17) w(z ∞ , 0) = a , w(z, 0) ≥ a if z -z ∞ ≤ z∞ -z ∞ .
Note that by (3.16), the level set w(z, 0) = a coincides with the hyperplane z (N ) = z * , and recall that U 0 ′ > 0. Therefore, in view of (3.16) and (3.17), we have either z∞ = z ∞ , or that the ball of radius z∞z ∞ centered at z ∞ is tangential to the hyperplane z (N ) = z * at z∞ . This implies that z∞ , as well as z ∞ , must also lie on the z (N ) axis. Therefore z∞ = (0, . . . , 0, z * ) T , and the inequality w(z ∞ , 0) ≥ a implies that z

= w(z ∞ , 0) -U 0 (z (N ) ∞ -z * ) = lim k→∞ u ε k (x k , t k ) -U 0 lim k→∞ d ε k (x k , t k ) ε k ≥ 2η .
This contradiction proves statement (ii) of Theorem 2.10.

3.4.

Proof of (i) and (iii) in Theorem 2.1. The proof of (i) below uses an argument similar to the proof of Corollary 4.8 in [START_REF] Matano | Large time behavior of disturbed planar fronts in the Allen-Cahn equation[END_REF]. Fix µ > 1. For a given η ∈ (0, min(aα -, α +a)) define ε 0 > 0 and C 0 > 0 as in Lemma 3.1. Then we claim that (3.18) lim inf ε→0 inf x∈NC 0 ε(Γt ), µt ε ≤t≤T ∇u ε (x, t) • n(p(x, t), t) > 0 ,

where n(p, t) denotes the outward unit normal vector at p ∈ Γ t . Indeed, assume by contradiction that there exist sequences

ε k ↓ 0, t k ∈ [µt ε k , T ], x k ∈ N C0ε k (Γ t k ) (k = 1, 2, ...) such that ∇u ε k (x k , t k ) • n(p k , t k ) ≤ 0 ,
where p k = p(x k , t k ). By rescaling around (p k , t k ) and using arguments similar to those in the proof of (ii), one can find a point z ∞ with |z (N )

∞ | ≤ C 0 such that U 0 ′ (z (N ) ∞ ) ≤ 0 , which contradicts the fact that U 0 ′ > 0 and establishes (3.18). Since, in view of Lemma 3.1, Γ ε t ⊂ N C0ε (Γ t ), the estimate (3.18) implies that ∇u ε (x, t) = 0 for all x ∈ Γ ε t ; hence by the implicit function theorem, Γ ε t is a smooth hypersurface in a neighborhood of any point on it. The fact that Γ ε t can be expressed as a graph over Γ t also follows from (3.18). This proves statement (i) of Theorem 2.1.

Finally, statement (iii) follows immediately from statements (i), (ii) and (3.2). This completes the proof of Theorem 2.1.

3.5. Proof of Theorem 2.2. As shown in [2, Section 7], the behavior of u ε in the system (RD ε ) can be treated as a special case of (P ε ), by regarding v ε as a given function and using a contraction mapping theorem. Thus the conclusion of Theorem 2.2 follows directly from Theorem 2.1.

Lemma 3 . 2 ([ 9 ,

 329 Theorem 3.1]). Let w(z, τ ) be an eternal solution of (3.3) satisfying(3.4) lim inf

∞

  ≥ z * . On the other hand (3.9)implies d ε k (x k , t k )/ε k = x ky k /ε k = z kzk → z ∞ -z∞ = z (N ) ∞z * .The assumption (3.5) then yields 0

  1, 2, . . .).Since the sequence (x k ) remains close to Γ t k by (3.6), and since Γ t k is uniformly smooth for k = 0, 1, 2, . . ., each x k has a unique orthogonal projection p(x k , t k ) ∈ Γ t k . Let y k be a point on Γ ε k t k that has the smallest distance from x k . If such a point is not unique, we choose one such point arbitrarily. Then we have(3.8)

The first author is supported by the French Agence Nationale de la Recherche within the project IDEE (ANR-2010-0112-01), and the second author by KAKENHI (23244017).


Jana Dlouhá

Alméras

Tancrède

Clair Bruno

Representativeness of wood biomechanical properties measured after storage in different conditions

Keywords:

Obtaining representative values of green wood properties is essential for studies investigating the biomechanical aspects of tree development and ecology. Here we compare the biomechanical properties of wood stored in various conditions between their collection in the field and their measurement. The study was performed on a large sample of wood specimens from different tropical species and different location in the trees, representing a wide diversity in wood structures. Elastic and viscoelastic properties are measured on green wood, and measured again after storage in different conditions: immersion in cold water during various durations, storage in an ethanol solution with or without washing in water, and air-drying with or without re-hydration. The systematic and random errors induced by these storage methods are quantified. Storage in cold water is the best way to preserve wood native properties. Soaking in ethanol is a fair alternative regarding elastic properties, but induces a significant change in viscoelastic properties. Air-drying causes important and partly irreversible changes in mechanical properties. However, regarding elastic properties, this change is a systematic bias so that the airdried elastic modulus provides a good basis for comparative studies of green wood stiffness.

INTRODUCTION

The biomechanics of woody plants is increasingly taken into account in different fields of plant biology and forest ecology [START_REF] Baraloto | Decoupled leaf and stem economics in rain forest trees[END_REF][START_REF] Chave | Towards a worldwide wood economics spectrum[END_REF][START_REF] Moulia | Posture control and skeletal mechanical acclimation in terrestrial plants: implications for mechanical modeling of plant architecture[END_REF][START_REF] Niklas | Worldwide correlations of mechanical properties and green wood density[END_REF]. In the context of a decreasing biodiversity and increasing atmospheric carbon dioxide, the rationale for this growing interest can be briefly summarised as follows: mechanical support is the primary function of wood, wood contains the largest part of forest biomass, and forest ecosystems host the largest part of terrestrial biodiversity and biomass. Evaluating functional mechanical properties of green wood is a prerequisite to many studies involving plant biomechanics. This evaluation is the basis of either quantitative estimation of composite functional traits at the tree level, such as its critical height [START_REF] Greenhill | Determination of the greatest height consistent with stability that a vertical pole or mast can be made, and the greatest height to which a tree of given proportions can grow[END_REF][START_REF] Niklas | Growth and hydraulic (not mechanical) constraints govern the scaling of tree height and mass[END_REF], gravitropic performance (Alméras et al., 2009;Alméras and Fournier, 2009;[START_REF] Alméras | Effect of circumferential heterogeneity of wood maturation strain, modulus of elasticity and radial growth on the regulation of stem orientation in trees[END_REF][START_REF] Coutand | The gravitropic response of poplar trunks: key roles of prestressed regulation and the kinetics of cambial growth versus wood maturation[END_REF][START_REF] Fournier | Tree biomechanics and growth strategies in the context of forest functional ecology[END_REF], safety factors against buckling [START_REF] Jaouen | How to determine sapling buckling risk with only a few measurements[END_REF][START_REF] Niklas | The allometry of safety-factors for plant height[END_REF] or breaking (Sterck et al., 2006;[START_REF] Van Gelder | Wood mechanics, allometry, and life-history variation in a tropical rain forest tree community[END_REF], or comparative studies assessing e.g. trade-offs between the mechanical function and other functions of wood [START_REF] Alvarez-Clare | Physical defence traits enhance seedling survival of neotropical tree species[END_REF][START_REF] Russo | Interspecific relationships among growth, mortality and xylem traits of woody species from New Zealand[END_REF][START_REF] Sterck | Mechanical branch constraints contribute to life history variations across tree species in a Bolivian rainforest[END_REF][START_REF] Van Gelder | Wood mechanics, allometry, and life-history variation in a tropical rain forest tree community[END_REF], variations in mechanical design between lifeforms [START_REF] Richardson Ba ; Spon | Diversity of mechanical architectures in climbing plants: an ecological perspective[END_REF], during ontogeny [START_REF] Isnard | Biomechanics and development of the climbing habit in two species of the siyth american palm genus Desmoncus (Arecaceae)[END_REF] or during evolution [START_REF] Ménard | Developmental plasticity and biomechanics of treelets and lianas in Manihot aff. quinquepartita (Euphorbiaceae): a branch-angle climber of French Guiana[END_REF].

A huge amount of data regarding mechanical properties of different wood species and types has been collected to characterize wood as a building material [START_REF] Bergman | Wood Handbook, Wood as an Engineering Material USDA[END_REF]. Wood in this context is mostly used and studied in air-dried conditions yielding a moisture content of approximately 12%. It is therefore generally submitted to kiln drying, and mechanical properties are measured after a given stabilisation period in room conditions. However the drying process is known to induce irreversible changes in wood structure and properties [START_REF] Kifetew | The effect of drying on wood fracture surfaces from specimens loaded in wet condition[END_REF][START_REF] Muller | Strength of dried and re-moistened spruce wood compared to native wood[END_REF]Sakagami et al., 2009a;[START_REF] Thuvander | Modeling of cell wall drying stresses in wood[END_REF]. Moreover, the amplitude of the change in mechanical properties of wood induced by drying can be affected by occurrence of some particular microstructures such as G-layer [START_REF] Yamamoto | Origin of the characteristic hygro-mechanical properties of the gelatinous layer in tension wood from Kunugi oak (Quercus acutissima)[END_REF] resulting from high stresses generated during the tree growth [START_REF] Archer | Growth stresses and strains in trees[END_REF]. Therefore, it is questionable to extrapolate mechanical properties measured in dried or rewetted state to the properties of wood in native conditions in the living tree i.e. with fully saturated cell walls. The straightforward way for obtaining such data is to perform mechanical tests in the forest immediately after harvesting, but the accuracy of experimental equipment usable in field work conditions is generally limited. Moreover, some properties such as viscoelastic parameters require longer times of measurement or finer experimental equipment and cannot be obtained out of a laboratory. Therefore the wood generally needs to be stored between harvesting and laboratory testing. However, one may wonder if the parameters measured after a given storage period are still representative of the native state and what is the relation between the native properties and those measured after storage.

Change in mechanical properties of wood during its storage may result from the attack of fungi and/or from the removal of extractives [START_REF] Brémaud | Diversité des bois utilisés ou utilisables en facture d'instruments de musique. Etude expérimentale des propriétés vibratoires en direction axiale de types de bois contrastés en majorité tropicaux[END_REF][START_REF] Matsunaga | Vibrational property changes of spruce wood by impregnation with water-soluble extractives of pernambuco (Guilandina echinata Spreng.)[END_REF][START_REF] Matsunaga | Physical and mechanical properties required for violin bow materials[END_REF][START_REF] Yano | The Changes in the Acoustic Properties of Western Red Cedar due to Methanol Extraction[END_REF]. Fungi need an optimal temperature together with appropriate moisture and oxygen content for their development. Drying removes from the material the moisture necessary for fungi development. Another way to preserve wood is to keep it in water-saturated conditions thus depriving the fungi of the oxygen [START_REF] Richardson Ba ; Spon | Diversity of mechanical architectures in climbing plants: an ecological perspective[END_REF]. Low temperature is generally preferred during storage, but the possibility of storage in an anoxic aquarium with water at room temperature was also reported for archaeological wood [START_REF] Björdal | Preservation, storage and display of waterlogged wood and wrecks in an aquarium: "Project Aquarius[END_REF]. Alternatively, samples can be immersed in alcohol, method frequently used for preserving organic materials in historic and medicine collections and occasionally used for fresh wood samples [START_REF] Coutand | Comparison of mechanical properties of tension and opposite wood in Populus[END_REF].

The aim of present study is to assess the effect of storage procedures on the mechanical properties of green wood, in order to suggest the most appropriate storage method to preserve the green wood properties. Three storage methods were studied: air-drying, storage in water at low temperature and storage in ethanol solution. In order to get rather general results, two dynamic mechanical properties (elastic modulus and damping coefficient) were measured on various wood samples taken from six tropical tree species having diverse microstructure and quality.

MATERIALS AND METHODS

Plant material

Wood material of six angiosperm tree species was collected in the tropical rainforest of French Guyana. The sampling was designed to be representative of a large diversity of wood qualities in order to draw conclusions valid for most angiosperm woods. The species were chosen to cover a wide range of wood specific gravities and mechanical properties and exhibited different extractive contents as illustrated in Table 1. Positioning of the selected species among the more than 1,000 species recorded in database of CIRAD (International Centre of Agronomic Research for Development) from the viewpoint of mechanical properties is displayed in Fig. 1.

Straight trees as well as tilted trees were sampled in order to account for a wider range of wood microstructures, including normal wood, tension wood and opposite wood. Tree diameter at the breast height ranged between 4 and 7 cm.

Wood specimens preparation

After tree felling, 50 cm-long logs were sawn in the field and its ends were covered with wet paper and plastic bags to limit the drying. In each log, 12 specimens were prepared at different location along a diameter and at two different vertical positions, in order to include specimens of varying age and quality. A total of 18 trees and 216 wood specimens were used for the study. Specimens were cut along the fibre direction (150 x 12 x 2mm, L x R x T) within a week after felling the trees, wrapped in a wet paper, sealed in plastic bags and stored in the fridge. Specimens were then transported to the laboratory in Montpellier (France) for measurements and kept in cold water until further testing.

Measurement of mechanical properties

Mechanical properties were first measured 1 month after felling the trees accounting for the time necessary for processing and transport of the specimens. This first measurement is taken as a reference and designated by index 0 in the following text and figures. The specific modulus E/ρ was measured by forced flexural vibrations of free-free beams at the first resonant frequency ranging from 200 to 650Hz. Used experimental device is pictured in Fig. 2, details about the method are given elsewhere [START_REF] Brémaud | Diversité des bois utilisés ou utilisables en facture d'instruments de musique. Etude expérimentale des propriétés vibratoires en direction axiale de types de bois contrastés en majorité tropicaux[END_REF][START_REF] Brémaud | Diversity in wood dynamic mechanical properties characterised by forced released vibration on an hundred wood types[END_REF]. The specific gravity ρ was estimated from measurements of the sample volume and mass. The dynamic elastic modulus E was calculated as a product of specific modulus and specific gravity. Volume of the specimen was estimated by double weighting to obtain maximal accuracy (especially in the case of dried specimens that can be more or less curved). The damping coefficient tanδ was calculated by the half-power bandwidth method defined as the ratio of the frequency range between the two half power points to the natural frequency at this mode.

Storage treatments

The different samples and storage treatments are summarised in Fig. 3. For investigating the effect of long-term storage in cold water (4±0.5°C), a sample of 144 specimens (6 species*2 trees*12 specimens) was measured repeatedly after 3, 8 and 12 months storage in cold water. Afterwards, half of the sample was soaked during 40 days in 40% ethanol, measured, and then washed in water during 80 days at 20°C before being measured again. In parallel, a group of 72 specimens was let drying in room conditions (20°, 65% relative humidity) during 14 days and measured in air-dry state. Afterwards, they were re-saturated in water (at 20°) during 14 days and measured in rewetted state.

RESULTS

Effect of air-drying and rewetting

Sets of measurements were performed on specimens in different conditions: in never dried state (taken as a reference), after air-drying and in rewetted condition. As we can see from Fig. 4 and Fig. 5, air-drying has significantly increased the elastic modulus (+12.7±7.8%) and strongly decreased the damping coefficient (-16.9±10%). The relationship between the elastic modulus in air-dried and never-dried state is linear with some dispersion (R²=0.97). The change in damping coefficient due to air-drying also induced some dispersion (R²=90), and is clearly non-linear: the reduction in damping due to drying is disproportionately larger for wood specimens with higher damping coefficients. These changes in mechanical properties between green and air-dry states are expected and well-documented [START_REF] Bergman | Wood Handbook, Wood as an Engineering Material USDA[END_REF]. They are due to the stiffening of the matrix components of wood (lignin and hemicelluloses) associated to water desorption. Average values of measured parameters for each species together with the density and moisture content in air dry conditions are summarized in Table 2.

The comparison between the rewetted state and the never-dried state shows that dried wood does not recover its initial properties after re-saturation. The elastic modulus measured in rewetted state was slightly lowered (-5.8±2.1%) but a very good linear correlation was observed with the never-dried state (R²>0.99). The damping coefficient was higher in rewetted state (15.9±2.1%) than in the never-dried state and the relation between these parameters was very scattered (R²=0.82). Observation of Fig. 4 shows that the drying-rewetting sequence actually induced more scattering than the drying event alone. The scatter of this relation is probably due to variations in the response to the drying-rewetting treatment between specimens having different micro-structural and chemical characteristics, for example due to differences in wood type, e.g. tension wood vs. normal wood [START_REF] Clair | Relationship between growth stresses, mechano-physical properties and proportion of fibres with gelatinous layer in chestnut (Castanea Sativa Mill[END_REF][START_REF] Coutand | Comparison of mechanical properties of tension and opposite wood in Populus[END_REF][START_REF] Yamamoto | Origin of the characteristic hygro-mechanical properties of the gelatinous layer in tension wood from Kunugi oak (Quercus acutissima)[END_REF]. The decrease in elastic modulus and increase in damping coefficient can be explained by irreversible damages occurring during drying process such as formation of micro-cracks (Sakagami et al., 2009a), that may be accentuated by the swelling of wood during re-saturation. Drying/rewetting treatment is known to reduce mechanical properties in particular wood strength of oven-dried [START_REF] Kifetew | The effect of drying on wood fracture surfaces from specimens loaded in wet condition[END_REF] as well as air-dried samples [START_REF] Muller | Strength of dried and re-moistened spruce wood compared to native wood[END_REF]. Microcraks are in generally initiated in the middle lamella and follow the ray parenchyma (Sakagami et al., 2009b). Location of microcracks in the middle lamella can explain why damping coefficient is highly affected by drying/rewetting treatment while the effect on elastic modulus is limited. Sensitivity to cracks formation is also species dependant which can explain great scatter of values (Sakagami et al., 2009b).

Effect of long-term storage in cold water

The effect of storage of never-dried wood specimens in water at low temperature (4±0.5°C) was investigated. The elastic modulus and damping coefficient were measured on the sample after 3, 8 and 12 months of storage, and compared to the initial measurement, taken as a reference state. The relation between the reference elastic modulus and that measured after different storage durations was linear with low dispersion (R²>99, Fig. 6). Mean relative changes in elastic modulus are displayed in Fig. 7. A slight decrease (-2.1±0.3%) was observed after 3 months storage. A further decrease (-4.2±0.3%) was observed after 8 months, but no further significant change was noticed after 12 months (-3.4±0.3%). No conclusion can be drawn on this change compare the initial state just after tree felling, but because the changes are very low after one year, one can suppose the difference to be low.

The relation between the damping coefficient in reference state and after different storage durations was linear with some dispersion (R²=0.98, 0.97, 0.96 for 3, 8 and 12 months; Fig. 6). Slight but significant variations in damping coefficient were noticed (-1.4±0.6%, -2.7±0.9%, -0.7±0.9% after 3, 8 and 12 month, see Fig. 7). We hypothesised that the observed increase of the damping coefficient after 8 months storage compared to 3 months storage may be an artefact due to the experimental conditions because this measurement was done during the summer period. Thus, slightly higher ambient temperature during this measurement might be responsible for the observed increase of the damping coefficient.

Moisture content during the first measurement ranged from 43 to 160% depending on the species ensuring that all sample was kept above the saturation point. Therefore, in the computation of the elastic modulus dimensions of the specimen were assumed to be stable while the variations in specimen weight were taken into account. The weight slightly increased between the first and the second measurements and remained stable afterwards. This was ascribed to the completion of the saturation process of the specimens that was not homogeneous between species as we can see from the Table 3. Weight variations did not affect the damping coefficient and were not correlated with the elastic modulus variations indicating that slight changes in the free water content do not affect mechanical properties.

The removal of water-soluble extractives possibly occurred during storage and is known to induce some modifications of wood mechanical properties for some species [START_REF] Brémaud | Diversité des bois utilisés ou utilisables en facture d'instruments de musique. Etude expérimentale des propriétés vibratoires en direction axiale de types de bois contrastés en majorité tropicaux[END_REF][START_REF] Matsunaga | Physical and mechanical properties required for violin bow materials[END_REF][START_REF] Obataya | The effects of water-soluble extractives on the acoustic properties of reed (Arundo donax L.)[END_REF]. Its effect on the elastic modulus could not be detected because of overlapping with the uptake of water. However, as the damping coefficient is known to be sensitive to the extractive content, and no significant change in damping coefficient was observed during the storage, it is likely that the removal of water soluble extractives was inhibited by the low temperature (extraction procedures are in general using hot water).

Effect of soaking in ethanol and washing in water

Ethanol was used as an alternative way to protect specimens from the biological attack. Measurements of the specimens were performed after soaking in 40% ethanol during 40 days, and after washing in water at ambient temperature during 80 days (Fig. 8). The elastic modulus after soaking in ethanol and after washing were linearly related to the reference values, with very low dispersion (R²>99, not shown). Compared to the reference value, the elastic modulus was slightly higher after soaking in ethanol (+1.8±0.9%) and slightly decreased after washing (-0.9±0.8%). The damping coefficient was more affected by the ethanol treatment. Values after soaking and after washing were linearly related to reference values, but with some dispersion (R²=0.98, not shown). The damping coefficient was significantly increased with ethanol (+18.4±1.4%), and this increase was not completely reversed after washing in water (+6.2±1.0%). Some weight variations were observed after the soaking in ethanol (-1.5±0.1%), likely due to the lower density of ethanol compared to water. Part of this weight variations remained after the washing in water (-0.2±0.1%) indicating residual ethanol content and/or removal of some extractives.

DISCUSSION

Comparison of storage methods for comparative and quantitative biomechanical studies

We studied different methods of wood storage to determine which is more suitable to obtain representative values of the mechanical properties of green wood when measurements cannot be performed in the field immediately after sampling. To assess the representativeness of measurements after storage, it is important to distinguish between two possible applications of these measurements: quantitative estimations of the mechanical parameter or comparative studies between species.

Quantitative estimations can be useful for example to provide input for biomechanical models and evaluate composite biomechanical traits at the tree level, e.g. a safety factor against buckling [START_REF] Jaouen | How to determine sapling buckling risk with only a few measurements[END_REF], or the efficiency of gravitropic reaction (Alméras and Fournier, 2009). For this kind of application, it is important to have minimal bias on the estimation of the mechanical parameter. The relevant parameter to quantify this effect is the mean relative difference between the value after storage and the reference value. Our results show that drying is prohibited in this context, regarding both the elastic modulus and the damping coefficient, since it modifies mechanical properties in a non-negligible and partly irreversible way (Fig. 5). Storage in cold water is advisable for both parameters, since their value is not much affected by this storage method (Fig. 7). For the modulus of elasticity, soaking in ethanol can be used as an alternative method e.g. if no fridge is available. This storage method does however induce a significant change in viscoelastic properties (Fig. 9).

For comparative purpose, for example in studies of diversity [START_REF] Alvarez-Clare | Physical defence traits enhance seedling survival of neotropical tree species[END_REF][START_REF] Russo | Interspecific relationships among growth, mortality and xylem traits of woody species from New Zealand[END_REF][START_REF] Sterck | Mechanical branch constraints contribute to life history variations across tree species in a Bolivian rainforest[END_REF][START_REF] Van Gelder | Wood mechanics, allometry, and life-history variation in a tropical rain forest tree community[END_REF] or comparative biology [START_REF] Richardson Ba ; Spon | Diversity of mechanical architectures in climbing plants: an ecological perspective[END_REF], the precise estimation of the value of the parameter is not that much important and some bias can be acceptable provided its effect is systematic and does not change the ranking of individuals or species, i.e. it does not induce additional dispersion. The relevant parameter in this context is the coefficient of determination (R²) between the values measured after storage and the reference values. Our results show that storage in cold water is appropriate regarding that for both elastic and viscoelastic properties. Storage in ethanol (with or without washing in water) is a fairly good alternative, since it does not lead to a substantially larger dispersion. Drying is prohibited for the study of viscoelastic properties, because it induces a large dispersion (Fig. 5). However, it can to some extent be used for comparative studies on the elastic modulus: although it induces an important bias, this bias is almost systematic and linear so that it does not change the comparison between species. For example, analysis of variance and post-hoc analysis applied to our dataset (not shown) showed that the ranking and discrimination between species elastic modulus was statistically unchanged by drying. This means that databases about elastic properties of dry wood produced by wood technologists can be used to compare biomechanical properties of different species.

Conclusions

Storage of fresh wood specimens in water at low temperature has revealed to be the most appropriate way to preserve its elastic and viscoelastic properties. Besides a slight decrease of elastic modulus (approx. -4%), no further change in mechanical properties was detected during one year. Soaking in ethanol is a satisfying alternative method regarding elastic properties, but it induces a significant change in viscoelastic properties. Drying causes important and partly irreversible changes in mechanical properties. However, in the case of elastic properties, this change is a systematic bias so that the air-dried elastic modulus provides a good basis for comparative studies on the elastic properties of green wood.

ACKNOWLEDGMENTS

This work was carried out in the framework of the Woodiversity project, funded by the French National Research Agency [ANR05-BDIV-012-04]. Authors would like to thank Iris Brémaud for useful discussions about the manuscript and Shanshan Chang for help with experimental work.


P.-C Héam

On the Complexity of Computing the Profinite Closure of a Rational Language

The profinite topology is used in rational languages classification. In particular, several important decidability problems, related to the Malcev product, reduce to the computation of the closure of a rational language in the profinite topology. It is known that given a rational language by a deterministic automaton, computing a deterministic automaton accepting its profinite closure can be done with an exponential upper bound. This paper is dedicated the study of a lower bound for this problem: we prove that in some cases, if the alphabet contains at least three letters, it requires an exponential time.

Preliminaries

.

Introduction

The profinite topology is used to characterise certain classes of rational languages: the languages of level 1/2 in the group hierarchy and the languages recognisable by reversible automata [START_REF] Pin | On the language accepted by finite reversible automata[END_REF][START_REF] Pin | Polynomial closure of group languages and open sets of the Hall topology[END_REF]. Moreover profinite topologies on the free group or on the free monoid play a crucial role in the theory of finite semigroups [START_REF] Delgado | Abelian pointlikes of a monoid[END_REF][START_REF] Henckell | Ash's type II theorem, profinite topology and Malcev products. part I[END_REF][START_REF] Pin | A topological approach to a conjecture of Rhodes[END_REF][START_REF] Delgado | A polynomial time algorithm to compute the abelian kernel of a finite monoid[END_REF]. In particular, several important decidability problems, related to the Malcev product, reduce to the computation of the closure of a rational language in the profinite topology.

It is known that the profinite closure of a rational language is rational too [START_REF] Ribes | On the profinite topology on a free group[END_REF][START_REF] Herwig | Extending partial automorphism and the profinite topology on the free groups[END_REF]. The first algorithm was given in [START_REF] Pin | A conjecture on the Hall topology on a free group[END_REF] for languages given by rational expressions, while [START_REF] Steinberg | Finite state automata: A geometric approach[END_REF][START_REF] Héam | Automata for pro-v topologies[END_REF] provide algorithms on finite automata. In this paper we are interesting in the following problem: Profinite Closure Input: A finite deterministic n-states automaton B on the alphabet A. Output: A finite deterministic automaton accepting the profinite closure of L(B). A solution to this problem is known to be computable in time O(2 n ) [START_REF] Héam | Automata for pro-v topologies[END_REF]. We prove in this paper that it can not be done, in some cases, faster than in exponential time (if the alphabet contains at least three letters).

In the first part of this paper, we introduce useful notations and definitions. In the second one, we recall an algorithm [START_REF] Steinberg | Finite state automata: A geometric approach[END_REF][START_REF] Héam | Automata for pro-v topologies[END_REF] to answer the above problem. The last section of this paper is dedicated to the main result of the paper: we will prove there exists a family or rational languages K n such that:

(1) The minimal automaton of K n has 3n states, [START_REF] Delgado | Abelian pointlikes of a monoid[END_REF] The minimal automaton of the profinite closure of K n has Ω(4 n / √ n)

states.

Notice that topological notions related to this paper are technical and required a wide mathematical background. However, the proved result can be easily understood using only automata theoretic arguments. In order to not overload the reader, we do not expose the mathematical background in this article. The interested reader is referred to [START_REF] Hall | A topology for free groups and related groups[END_REF][START_REF] Pin | Topologies for the free monoid[END_REF] for more information on profinite topologies. Particular topological properties of rational languages are studied in [START_REF] Héam | Some topological properties of rational sets[END_REF]. We just provide short definitions in the next section.

Background and notations

Let A be a finite alphabet and let A = {a | a ∈ A} be a copy of A. Finally, let à be the disjoint union of A and A. The map a → a from A onto A can be extended to a one-to-one function from à into itself by setting a = a. A word of A * is said to be reduced if it does not contain any factor of the form aa with a ∈ Ã. We denote by ≡ the monoid congruence generated by the relations aa ≡ 1 for all a ∈ Ã. The set Ã/ ≡ is a group for the quotient law, called the free group over A. Let π be the projection from à into this group, which is a monoid morphism. We denote by D( Ã) the set π -1 (ε), i.e. the set of words of à * that can be rewritten into ε using the rewriting rules aa → ε, with a ∈ Ã.

The family of normal subgroups of the free group with finite index forms a basis of open sets for the profinite topology on the free group. Similarly, the class of group languages (regular languages whose syntactic monoids are finite groups) forms a basis of open sets for the profinite topology on A * . Note that the profinite closure in A * of a language L is the intersection of A * with its profinite closure in the free group. Throughout the end of this paper, the considered profinite topology is the one on A * .

Recall that a finite automaton is a 5-tuple

A = (Q, B, E, I, F ) where Q is a finite set of states, B is the alphabet, E ⊆ Q × B × Q is the set of edges (or transitions), I ⊆ Q is the set of initial states and F ⊆ Q is the set of final states. A path in A is a finite sequence of consecutive edges: p = (q 0 , a 0 , q 1 ), (q 1 , a 1 , q 2 ), • • • , (q n-1 , a n , q n )
The label of the path p is the word a 1 a 2 • • • a n , its origin is q 0 and its end is q n . A word is accepted by A if it is the label of a path in A having its origin in I and its end in F . Such a path is said to be successful. The set of words accepted by A is denoted by L(A).

For every state q and language K we denote by q • A K (or q • K if there is no ambiguity on A), the subset of Q of all the states which are the end of a path having its origin in q and its label in K. An automaton is said to be trim if for each state q there exists a path from an initial state to q and a path from q to a final state. An automaton is deterministic if it has a unique initial state and does not contain any pair of edges of the form (q, a, q 1 ) and (q, a, q 2 ) with q 1 = q 2 . An important result of automata theory states that for any automaton A there exists exactly one deterministic automaton (up to isomorphism) with a minimal number of states which accepts the same language. It is called the minimal automaton of L(A). Two states p and q of an automaton are Nerode-equivalent if for every word u, p • u is final if and only if q • u is final too. It is well known that a trim deterministic automaton is minimal if and only if all classes of the Nerode equivalence are singletons.

Let A be an automaton with set of states Q and set of transitions E. A subset P of Q is said to be strongly connected if, for each pair p and q of states in P , there exist a path from p to q and a path from q to p. A strongly connected component of A is a maximal (for the inclusion) set of states which is strongly connected. The strongly connected components of A form a partition of Q. A transition (p, a, q) of A is internal to a strongly connected component if p and q belongs to the same strongly connected component. It is said internal if it is internal to some strongly connected component and external otherwise.

The class of rational languages of A * is the smallest class of languages closed under product, finite union and star operation. It is well known that a language of A * is rational if and only if it can be accepted by a finite automaton.

Profinite Closure of a Rational Language

It is known that the profinite closure of a rational language is rational too [START_REF] Ribes | On the profinite topology on a free group[END_REF].

In this direction, we use the following algorithm [START_REF] Steinberg | Finite state automata: A geometric approach[END_REF][START_REF] Héam | Automata for pro-v topologies[END_REF], called Profinite-Closure, working on a finite trim automaton A = (Q, A, E, I, F ) in order to compute the profinite closure of L(A).

Compute

E 1 = {(q, a, p) | (p, a, q) ∈ E \ T }. Let A 1 = (Q, Ã, E ∪ E 1 , I, F ). 4. Compute E 2 = {(p, ε, q) | p = q, q ∈ p • A 1 u, u ∈ D( Ã)}. Let A 2 = (Q, A, E ∪ E 2 , I, F ).
5. Return A 3 , the automaton obtained for (Q, A, E ∪ E 2 , I, F ) by a classical ε-transitions elimination.

In order to obtain a resulting deterministic automaton, one can use the standard determinization algorithm, which is known to be exponential in the worst case [START_REF] Hopcroft | Introduction to automata theory, languages, and computation[END_REF]. We illustrate how this algorithm is working on a graphical example on Fig. 1.

Main result

We will prove in this section there exists a family of rational languages K n such that:

(1) The minimal automaton of K n has 3n states,

(2) The minimal automaton of the profinite closure of K n has Ω(4 n / √ n)

states.

We first need the following technical lemma.

Lemma 1 Let a and b be the two following permutations of

Q n = {1, . . . , 2n} (with n ≥ 2): a = (2, 3, • • • , 2n, 1) b = (1, 2)
.

Let E n = {(i, a, a(i)) | i ∈ Q n } ∪ {(i, b, b(i)) | i ∈ Q n } and A n = (Q n , {a, b}, E n , {1, . . . , n}, 2n).
The minimal automaton of L(A n ) has (2n)!/(n!) 2 states.

Proof. Notice first that a and b generate the symmetric group of {1, . . . , 2n}.

For every word u ∈ {a, b} * , we define σ u by σ ua = a • σ u and σ ub = b • σ u . Therefore, for every permutation σ of A there exists a word u such that σ = σ u . Consider the following automata A ′ n obtained by the classical determinization algorithm:

-The set of states of A ′ n is composed by all the images of {1, . . . , n} by all permutations of Q n . We obtain all the subsets of A with n elements. Therefore, since Q n has 2n elements, A ′ n has (2n)!/(n!) 2 states .

- -Initial state of A ′ n is {1, . . . , n}.

-Final states of A ′ n are all states containing 2n. [START_REF] Hopcroft | Introduction to automata theory, languages, and computation[END_REF]).

By construction L(A

′ n ) = L(A n ) (see

Now we claim that

A ′ n is minimal. Consider the two distinct states R = {a 1 , • • • , a n } and S = {b 1 , • • • , b n } of A ′
n . We will prove that R and S are not Nerode-equivalent. Following cases arise:

-If R is final and S is not final, then R • ε is final but not S • ε. Thus R and S are not Nerode-equivalent. The same argument holds if S is final and R is not final.

-If R and S are either both final or both non final, then there exists a k ∈ R such that a k = 2n and a k / ∈ S. Consider a word u such that σ u = (a k , 2n). The unique state of R•u contains 2n; it is final. However the unique state of S • u doesn't contain 2n since a k / ∈ S. Since A ′ n is trim, we proved that the minimal automaton of L(A n ) has (2n)!/(n!) 2 states.

2

With previous lemma notations, we consider now the following family of automata denoted (B n ) n∈N and illustrated on Fig. 2. -Alphabet of B n is {a, b, c}.

-Transitions of B n are the following tuples:

-(2n + k, b, 2n + k + 1) for 1 ≤ k ≤ n -1, -(2n + k, c, k) for 1 ≤ k ≤ n -1, -(k, a, a(k)) and (k, b, b(k)) for 1 ≤ k ≤ 2n, - (3n
, b, 3n), (2n + 1, a, 2n + 1) and (3n, a, 2n + 1), -The unique initial state of B n is 2n + 1, -The unique final state of B n is 2n.

Theorem 2 With above notations, B n is minimal, has 3n states and the minimal automaton of the profinite closure of L(B n ) has Ω(4 n / √ n) states.

Proof. In this proof A = {a, b, c}. We first prove that B n is minimal. Indeed consider two distinct states k 1 and k 2 of B n . The following cases arise:

-If 1 ≤ k 1 ≤ 2n and 1 ≤ k 2 ≤ 2n, then k 1 • a 2n-k 1 is final. Without loss of generality, we may assume that k 2 < k 1 . Therefore k 2 • a 2n-k 1 is 2n -(k1 -k2) which is not final. Thus the two states are not Nerode-equivalent.

-If 1 ≤ k 1 ≤ 2n and 2n + 1 ≤ k 2 ≤ 3n, then one can reach a final states from k 2 using a word u containing the letter c. By construction, k 1 • u is not final. Thus k 1 and k 2 are not Nerode-equivalent.

-If 1 ≤ k 2 ≤ 2n and 2n + 1 ≤ k 1 ≤ 3n, we conclude similarly.

-

If 2n + 1 ≤ k 2 ≤ 3n and 2n + 1 ≤ k 1 ≤ 3n, then k 1 • ca 4n-k 1 is final and k 2 • ca 4n-k 1 is not final.
Thus the two states are not Nerode-equivalent.

Since B n is trim, it follows that B n is a 3n states minimal automaton.

We will now apply the algorithm ProfiniteClosure on B n .

1. The strongly connected components of B n are {1, . . . , 2n} on one hand and {2n + 1, . . . , 3n} on the other hand.

2. The set of external transitions is reduced to all transitions labelled by c.

3. We add the reverse transitions labelled in A of internal transitions. Notice we only add transitions labelled by a and b. We have the automaton represented on Fig. 3.

4. In the strongly connected component {2n + 1, . . . , 3n}, one has a path from k 1 to k 2 labelled by b 3n-k 1 (b) n-k 1 +k 2 -1 aab k 2 -2n-1 . Therefore there is an ε-transition between all pairs of distinct states of {2n + 1, . . . , 3n}.

Since every word labelling a path from a state of {2n + 1, . . . , 3n} to a state of {1, . . . , 2n} contains an occurrence of c and no occurrence of c, there is no ε-transition between the two strongly connected components.

Now for every state p of {1, . . . , 2n}, p•aa = p•aa = p•bb = p•bb = {p}, thus we add no ε-transition in {1, . . . , 2n};

5. It follows that the profinite closure of L(B n ) is the language {a, b} * cL(A n ) whose minimal automaton is given on Fig. 4.

Observe that, using lemma 1, this automaton is minimal. Consequently, the profinite closure of L(B n ) has 1 + (2n)! (n!) 2 states. Thus, using Stirling Formula, the minimal automaton of the profinite closure of L(B n ) has Ω(4 n / √ n) states. We proved that the Profinite Closure problem can not be solved, in some cases, faster than in exponential time (if the alphabet contains at least three letters).

Conclusion

In this paper we studied the complexity of computing the profinite closure of a rational language when the chosen representation for rational languages is finite deterministic automata. We proved this computation requires an exponential time in the worst case (if the alphabet contains at least three letters). One can easily verify that the problem is polynomial for unary alphabet. However, as far as we know, the problem is still open for two letters alphabets.

The author would like to thank J.-E. Pin and P. Weil for helpful discussions. This work was partially granted by the french project ANR-2010-BLAN-0202-02-FREC.

Figure 1 :

 1 Figure 1: Algorithm ProfiniteClosure on an example

  We have a transition between two parts R and S labelled by a [resp. par b], if and only if a(R) = S [resp. b(R) = S].

Figure 2 :

 2 Figure 2: Automaton B n -Set of states of B n is {1, . . . , 3n}.

Figure 3 :Figure 4 :

 34 Figure 3: Automaton B n


Corentin Briat
email: cbriat@math.kth.se.url

Olivier Sename

J-F Lafay

Design of LPV observers for LPV time-delay systems: an algebraic approach

Keywords: Time-delay systems, Linear parameter varying systems, Observers, LMIs

The design of reduced-order observer for Linear Parameter Varying (LPV) time-delay systems is addressed. Necessary conditions guaranteeing critical structural properties for the observation error dynamics are first provided through nonlinear algebraic matrix equalities. An explicit parameterization of the family of observers fulfilling these necessary conditions is then derived. Finally, an approach based on Linear Matrix Inequalities (LMIs) is provided and used to select a suitable observer within this family, according to some criterion; e.g. maximization of the delay-margin or guaranteed suboptimal L 2 -gain. Examples from the literature illustrate the efficiency of the approach.

Introduction

Time-delay systems are omnipresent: from engineering to economics passing through biology, ecology and social sciences. This ubiquity has made these systems more and more attractive over the last century [START_REF] Kolmanovskii | Introduction to the Theory and Applications of Functional Differential Equations[END_REF][START_REF] Niculescu | Delay effects on stability. A robust control approach[END_REF][START_REF] Gu | Stability of Time-Delay Systems[END_REF][START_REF] Fridman | New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems[END_REF][START_REF] Michiels | Stability and stabilization of time-delay systems. An eigenvalue based approach[END_REF]. However, most of the breakthroughs in that topic have been made within the past three decades and many problems remain open. Amongst other, the observation is a relatively few studied topic compared to stability, filtering and control. The observation of time-delay systems has been studied for instance in [START_REF] Fiagbedzi | A state observer for systems described by functional differential equations[END_REF][START_REF] Fattouh | H ∞ observer design for time-delay systems[END_REF], 1999b[START_REF] Aggoune | Observers design for a class of non linear systems with time-varying delay[END_REF][START_REF] Fattouh | Robust observer design for linear uncertain time-delay REFERENCES 13 systems: A factorization approach[END_REF],b,a,d, Sename 2001[START_REF] Darouach | Linear functional observers for systems with delays in state variables[END_REF], Sename et al. 2001[START_REF] Koenig | Unknown inputs proportional integral observers for descriptors systems with multiple delays and unknown inputs[END_REF][START_REF] Darouach | Reduced-order observers for linear neutral delay systems[END_REF][START_REF] Koenig | Delay dependent H ∞ observer for linear delay descriptor systems[END_REF][START_REF] Sename | Observer-based H ∞ control for time-delay systems: a new LMI solution[END_REF][START_REF] Sename | Is a mixed design of observer-controllers for time-delay systems interesting[END_REF][START_REF] Sename | Is a mixed design of observer-controllers for time-delay systems interesting[END_REF].

More recently, many works have been devoted to the analysis of LPV systems due to the generic framework they offer for the control of complex systems, e.g. nonlinear and Linear Time-Varying (LTV) systems [START_REF] Shamma | Analysis of nonlinear gain scheduled control systems[END_REF][START_REF] Shamma | Guaranteed properties of gain scheduled control of linear parametervarying plants[END_REF][START_REF] Shamma | Gain scheduling: potential hazards and possible remedies[END_REF][START_REF] Packard | Gain scheduling via Linear Fractional Transformations[END_REF][START_REF] Gahinet | A linear matrix inequality approach to H ∞ control[END_REF][START_REF] Apkarian | A convex characterization of gain-scheduled H ∞ controllers[END_REF][START_REF] Apkarian | Advanced gain-scheduling techniques for uncertain systems[END_REF], Wu 2001a[START_REF] Scherer | LPV control and full-block multipliers[END_REF][START_REF] Iwasaki | LPV system analysis via quadratic separator for uncertain implicit systems[END_REF][START_REF] Mehendale | A new approach to LPV gain-scheduling design and implementation[END_REF]. The emergence of the 'LPV trend' has taken the advantage of the rise of LMI-based techniques [START_REF] Boyd | Linear Matrix Inequalities in Systems and Control Theory[END_REF]) which provide a convenient framework for solving LPV-related problems.

The stability analysis and control of LPV time-delay systems have been explored in [START_REF] Wu | LPV systems with parameter-varying time delays: analysis and control[END_REF], Wu 2001b[START_REF] Zhang | Stability analysis of LPV time-delayed systems[END_REF][START_REF] Zhang | Delay-dependent stability analysis and H ∞ control for statedelayed LPV system[END_REF], Briat et al. 2007a, 2008b[START_REF] Briat | Memory resilient gain-scheduled state-feedback control of uncertain LTI/LPV systems with time-varying delays[END_REF]. However, the problem is far from being easy since this class of problems inherits of the difficulty of each subclass but, additionally, new troubles occur. As an example, many analy-sis tools developed for LPV systems fail when applied to LPV time-delay systems (e.g. projection lemma, dualization lemma). Conversely, many results obtained for Linear Time Invariant (LTI) time-delay systems cannot be applied to LPV ones due to their time-varying nature, e.g. frequency domain methods or eigenvalue based analysis techniques. Finally, amongst the remaining possible approaches, Lyapunov-Krasovskii-Functionals-based techniques are certainly the most useful.

The problem of observation of LPV time-delay systems has been relatively few studied in the literature, several works are devoted to the filtering of LPV time-delay systems (Mohammadpour and Grigoriadis 2006a,b, 2007, Briat et al. 2009a). In [START_REF] Briat | Full order LPV/H ∞ observers for LPV time-delay systems[END_REF]) is developed a simple method for observers design based on a 'free-weighting matrices' approach, extended to the LPV case. Compared to the authors previous work, the provided approach is much more efficient since it involves fewer decision variables and convex constraints (LMIs).

We propose in this paper to attack the problem using an algebraic approach, extended from [START_REF] Darouach | Linear functional observers for systems with delays in state variables[END_REF] to the LPV case. Since the system is not time-invariant anymore, the problem is slightly more difficult than in the LTI case due to the parameter independence of some of the observer matrices. Structural necessary conditions reflecting the constraints on the desired structure for the dynamic model of the observation error are obtained. From these conditions, implicitly defining a family of observers, an explicit parameterization of all these observers is obtained. Finally, LMI conditions (Briat et al. 2008b[START_REF] Briat | Memory resilient gain-scheduled state-feedback control of uncertain LTI/LPV systems with time-varying delays[END_REF] are then derived in order to choose a suboptimal observer which guarantees given objectives, such as a maximal delay-margin or a minimal suboptimal L 2 -gain of the transfer mapping the exogenous inputs to the observation error. Similarly, as in (Briat et al. 2009b[START_REF] Briat | Memory resilient gain-scheduled state-feedback control of uncertain LTI/LPV systems with time-varying delays[END_REF], a 'slackvariable' approach is considered in order to make the design problem easier than the original LMI obtained directly from the Lyapunov-Krasovskii Theorem. The obtained conditions are both more computationally attractive and less conservative than the ones in [START_REF] Briat | Full order LPV/H ∞ observers for LPV time-delay systems[END_REF]).

The paper is structured as follows: in Section 2 definitions and preliminary results are stated. Section 3 is devoted to the derivation of the main results of the paper. Finally, examples are considered in Section 4.

The notations are quite standard, for symmetric matrices A, B: A ≺ B stands for A -B negative definite, S n ++ is the cone of symmetric positive definite matrices of dimension n. For a general matrix A, A + stands for its Moore-Penrose pseudoinverse, and for a square matrix A we define A S = A + A T .

Definitions and Preliminary Results

The objective of the paper is to determine whether there exists an r-order LPV observer of the form:

ξ(t) = M 0 (ρ)ξ(t) + M h (ρ)ξ(t -h(t)) + S(ρ)u(t) + N 0 (ρ)y(t) + N h (ρ)y(t -h(t)) ẑ(t) = ξ(t) + Hy(t) (1)
for the class of LPV time-delay systems defined by

ẋ(t) = A(ρ)x(t) + A h (ρ)x(t -h(t)) + B(ρ)u(t) + E(ρ)w(t) y(t) = Cx(t), C full row rank z(t) = T x(t), T full row rank x(θ) = φ(θ), θ ∈ [-h m , 0] (2) where x ∈ R n , w ∈ R m , ξ ∈ R r , u ∈ R q , z ∈ R r , y ∈ R s and φ ∈ C([-h m , 0], R n ) are
the system state, the exogenous input, the observer state, the known system control input, the signal to estimate, the measured output and the functional initial condition. In order to obtain an observer which depends on the parameter values ρ(t) only (and not on their derivative ρ(t)), the matrices C, H and T are set as parameter independent. Note that C can always be made parameter independent through an appropriate filtering of the measured output y(t).

The parameter vector ρ(•) is assumed to belong to the set:

P := {ρ : R + → U ρ ⊂ R p , ρ ∈ co{U ν }} (3)
where U ρ is a compact and connected set of R p , U ν is the set of vertices of the parameter derivative space and co{•} is the convex-hull operator. The time-varying delay h(t) is assumed to belong to the set

H := h : R + → [0, h m ], ḣ(t) ≤ µ < 1 (4)
where 0 < h m < +∞.

With the above ingredients in mind, the main problem of the paper can be stated as follows:

Problem 2.1 Find an LPV time-delay observer of the form (1) for system (2) which 1) asymptotically stabilizes the observation error e(t) = z(t) -ẑ(t), i.e. ||e(t)|| → 0 as t → +∞ when w ≡ 0 and ||φ|| = 0; and 2) provides a guaranteed (suboptimal) L 2 gain from the exogenous input w to the estimation error e, i.e. ||e|| L2 ≤ γ||w|| L2 , ∀w ∈ L 2 and for some γ > 0 when ||φ|| = 0.

The observation error defined above is governed by the following dynamical equation:

ė(t) = M 0 (ρ)e(t) + M h (ρ)e(t -h(t)) + (T A(ρ) -M 0 (ρ)F -N 0 (ρ)C -HCA(ρ))x(t) +(T A h (ρ) -M h (ρ)F -N h (ρ)C -HCA h (ρ))x(t -h(t)) + (F B(ρ) -S(ρ))u(t) +F E(ρ)w(t) (5)
where F = T -HC. In order to make the observer error independent of the system state and the control input, the following nonlinear algebraic conditions must be fulfilled:

T A(ρ) -M 0 (ρ)F -N 0 (ρ)C -HCA(ρ) = 0, (6)
T A h (ρ) -M h (ρ)F -N h (ρ)C -HCA h (ρ) = 0, (7)
S(ρ) -F B(ρ) = 0. (8)
If one the above conditions is not satisfied, say the first one, then the observation error cannot converge to 0 when the current state x(t) is different from 0. Indeed, in such a case, the state will act as an exogenous input on the observation error dynamical model ( 5). This suggests that when the matrices are uncertain, it is unlikely possible to observe exactly the state of the system due to the practical impossibility of fulfilling the above equalities. This also suggests that when the system matrices are exactly known, it is theoretically possible to observe unstable systems1 . This is basically not true when the system matrices are partially unknown. Note that the condition for the cancelation of the control input can always be satisfied since S(ρ) is only involved in (8). So, assuming all the above conditions hold, the resulting dynamical model for the observation error writes

ė(t) = M 0 (ρ)e(t) + M h (ρ)e(t -h(t)) + (T -HC)E(ρ)w(t). (9)
This leads to the following proposition:

Proposition 2.2: There exists an r-order LPV observer of the form (1) for system (2) ensuring an L 2 gain for the transfer w → e lower than γ if and only if 1) the observation error e(t) is asymptotically stable. i.e. e(t) → 0 as t → +∞ when w ≡ 0 and ||φ|| = 0; 2) The nonlinear algebraic equalities 2a

) T A(ρ) -M 0 (ρ)(T -HC) -N 0 (ρ)C -HCA(ρ) = 0 2b) T A h (ρ) -M h (ρ)(T -HC) -N h (ρ)C -HCA h (ρ) = 0 2c) S(ρ) = (T -HC)B(ρ) hold for all ρ ∈ U ρ ; 3) ||e|| L2 ≤ γ||w|| L2 for all w ∈ L 2 and ||φ|| = 0.
Hence, the above result incorporates all the conditions an observer must fulfill to solve Problem 2.1. Indeed, the stability and performance constraints are given by 1) and 3) respectively while the constraints 2) are structural constraints imposed on the observer to decouple the observation error from the known inputs and the system state.

Observer design

We state in this section the conditions for the existence of an observer (1) for system (2).

Theorem 3.1 : The conditions 2) of Proposition 2.2 can be satisfied if and only if one of the following equivalent statements hold:

1) The constant matrix H ∈ R r×s is such that the equality

[ϕ(ρ) -Hψ(ρ)][I -φ + φ] = 0 (10)
holds for all ρ ∈ U ρ where

φ =     T 0 0 T C 0 0 C     , ψ(ρ) = CA(ρ) CA h (ρ) and ϕ(ρ) = T A(ρ) T A h (ρ) .
2) The constant matrix H ∈ R r×s is such that the equality

rank φ ϕ(ρ) -Hψ(ρ) = rank[φ] (11)
holds for all ρ ∈ U ρ with the matrices defined above.

Proof : Since 2c) in Proposition 2.2 can always be satisfied, the remaining problem is the existence of solutions to 2a) and 2b). First, consider the linear algebra problem

O(ρ)φ = ϕ(ρ) -Hψ(ρ) (12)
where the parameter-independent matrix H is placed on the right-hand side and the remaining parameter-dependent unknown matrices

O(ρ) = M 0 (ρ) M h (ρ) K 0 (ρ) K h (ρ) , K 0 (ρ) = N 0 (ρ) - M 0 (ρ)H and K h (ρ) = N h (ρ) -M h (ρ)
H are parameterized in terms of this matrix H. According to [START_REF] Skelton | A Unified Algebraic Approach to Linear Control Design[END_REF], there exist solutions to such an equation if and only if (10) holds for some H ∈ R r×s and for all ρ ∈ U ρ . The rank condition (11) is obtained using standard linear algebra by considering the equality:

O(ρ) H φ ψ(ρ) = ϕ(ρ). (13)
An immediate conclusion of the above result concerns the case when the matrix φ is full column rank. In such a case, the equality Iφ + φ = 0 holds and hence statement 1) of Theorem 3.1 holds for any H ∈ R r×s . Since φ ∈ R 2(r+s)×2n , then the matrix can be full column rank only when r + s ≥ n (i.e. dim(z) + dim(y) ≥ dim(x)). For instance, if a full-order observer is sought, then r = n, T = I n and φ is automatically full-column rank. When the matrix φ is not full-column rank, i.e. Iφ + φ = 0, then the matrix H ∈ R r×s must be chosen such that one of the statements of Theorem 3.1 is satisfied.

Conditions 2) of Proposition 2.2 implicitly define a matching family of observers for system (2). Assuming one of the conditions of Theorem 3.1 is satisfied, it is possible to explicitly parameterize this family, as stated in the following lemma: Lemma 3.2: When the conditions of Theorem 3.1 are fulfilled, the observer matrices are given by the expressions M 0 (ρ) = Θ(ρ) -L(ρ)Ξ, M h (ρ) = Υ -L(ρ)Ω where L(ρ) is an arbitrary matrix with appropriate dimensions and

Θ(ρ) = T A(ρ)U -HCA(ρ)U -[ϕ(ρ) -Hψ(ρ)]φ + ∆ 0 CU Ξ = -(I -φφ + )∆ 0 CU Υ(ρ) = T A h (ρ)U -HCA h (ρ)U -[ϕ(ρ) -Hψ(ρ)]φ + ∆ h CU Ω = -(I -φφ + )∆ h CU S(ρ) = F B(ρ) N 0 (ρ) = O s (ρ)∆ 0 + M 0 (ρ)H N h (ρ) = O s (ρ)∆ h + M h (ρ)H F = T -HC O s (ρ) = [ϕ(ρ) -Hψ(ρ)]φ + -L(ρ)(I -φφ + ) (14) with ∆ 0 =     0 0 I s 0     and ∆ h =     0 0 0 I s     .
Proof : Assuming a solution to the equation ( 13) exists, then the general solution is expressed as [START_REF] Skelton | A Unified Algebraic Approach to Linear Control Design[END_REF]:

O s (ρ) = [ϕ(ρ) -Hψ(ρ)]φ + -L(ρ)(I -φφ + ) ( 15
)
where L(ρ) is an arbitrary matrix function. Then, rewrite (13) as

M 0 (ρ)T = T A(ρ) -K 0 (ρ) H C CA(ρ) , (16)
M h (ρ)T = T A h (ρ) -K h (ρ) H C CA h (ρ) . ( 17
)
Using the fact that T is a full row rank matrix, it is thus possible to build a matrix U such that T U = I r . Right multiplying ( 16) and ( 16) by U yields

M 0 (ρ) = T A(ρ)U -K 0 (ρ) H C CA(ρ) U, M h (ρ) = T A h (ρ)U -K h (ρ) H C CA h (ρ) U. (18)
We the above defined ∆ 0 and ∆ h , it comes K 0 (ρ) = O s (ρ)∆ 0 and K h (ρ) = O s (ρ)∆ h . Finally, the algebraic relations of Lemma 3.2 are obtained after substitution of K 0 (ρ) and K h (ρ) into the expressions (18).

After substitution of the explicit expressions of the matrices M 0 (ρ), M h (ρ) and H into the dynamical model ( 9), we get

ė(t) = (Θ(ρ) -L(ρ)Ξ)e(t) + (Υ(ρ) -L(ρ)Ω)e(t -h(t)) + (T -HC)E(ρ)w(t). (19)
Obviously, the obtained dynamical model for the observation error is parameterized by the free matrix function L(ρ) and, possibly, the constant matrix H if not already been chosen to have Theorem 3.1 satisfied. So, the remaining problem consists of suitable choice for L(ρ) and H which leads to stability and performance for (19).

Remark 1 : It seems important to note that, under some circumstances, it can be possible to remove completely the influence of the exogenous inputs independently of L(ρ) by choosing appropriately the matrix H ∈ R r×s . Indeed, if the matrix H can be chosen such that both (T -HC)E(ρ) = 0 and Theorem 3.1 are satisfied, then the observation error evolution is totally decoupled from the exogenous input. In such a case, it is clear that the L 2 gain from w to e is 0 and L(ρ) is the only remaining degree of freedom and must be chosen according to some stability constraints only, e.g. pole placement, maximal delay margin.

The following theorem provides a way of choosing the matrices L(ρ) and H such that both the state observation error is asymptotically stable for all (ρ, h) ∈ P × H and the L 2 gain from the exogenous input to the observation error is lower than γ > 0.

Theorem 3.3 : There exists a parameter dependent observer (1) such that Problem 2.1 is solved if there exist a continuously differentiable matrix function P :

U ρ → S r ++ , a matrix function L : U ρ → R r×(2r+2s) , constant matrices Q, R ∈ S r
++ , X ∈ R r×r , H ∈ R r×r and a positive scalar γ such that the LMI:

          -X S Σ 12 (ρ) Σ 13 (ρ) Σ 14 (ρ) 0 X T h m R Σ 22 R 0 I r 0 0 Σ 33 0 0 0 0 -γI m 0 0 0 -γI r 0 0 -P (ρ) -h m R -R           ≺ 0 (20)
holds for all ρ ∈ U ρ , ν ∈ U ν where

Σ 12 (ρ) = X T Θ 0 (ρ) -HΘ H (ρ) -L(ρ)Ξ Σ 13 (ρ) = X T Υ 0 (ρ) -HΥ H (ρ) -L(ρ)Ω Σ 14 (ρ) = (X T T -HC)E(ρ) Σ 22 (ρ) = ∂P (ρ) ∂ρ ν -P (ρ) + Q -R Σ 33 (ρ) = -(1 -µ)Q -R Θ 0 (ρ) = T A(ρ)U -ϕ(ρ)φ + ∆ 0 CU Θ H (ρ) = CA(ρ)U -ψ(ρ)φ + ∆ 0 CU Υ 0 (ρ) = T A h (ρ)U -ϕ(ρ)φ + ∆ H CU Υ H (ρ) = CA h (ρ)U -ψ(ρ)φ + ∆ H CU (21)
In such a case, the observer ensures ||e|| L2 ≤ γ||w|| L2 and is obtained using Lemma 3.2 with L(ρ) = X -T L(ρ) and H = X -T H.

Proof : The proof is given in Appendix A.

Remark 2 : If H has already been chosen to satisfy Theorem 3.1, then it is sufficient to change H by X T H in the above result. In such a case, the matrix X can now be made parameter dependent, i.e. X : U ρ → R r×r and the LMI is solved for P (ρ), X(ρ), Q, R, L(ρ) and γ.

Such a parameter dependent LMI can be solved by different techniques including for instance gridding [START_REF] Apkarian | Advanced gain-scheduling techniques for uncertain systems[END_REF] or any other technique [START_REF] Scherer | LMI relaxations in robust control[END_REF].

y(t) = 1 0 x(t) z(t) = x(t) ρ(t) ∈ [-1, 1] ρ(t) ∈ [-1, 1] (22
) slightly modified from [START_REF] Mohammadpour | Less conservative results of delay-depdendent H ∞ filtering for a class of time-delayed LPV systems[END_REF].

The matrix φ in Theorem 3.1 is full-column rank, hence H does not need to be chosen a priori. The structure of L(ρ) is arbitrarily chosen to be L(ρ) = L 2 ρ 2 + L 1 ρ + L 0 and we also pick P (ρ) = P 0 + P 1 ρ + P 2 ρ 2 /2. Using Theorem 3.3 and a gridding approach, we seek the maximal delay value for which there exists an observer and we find h m = 213.238 seconds using a bisection approach. Now, letting h m = 0.8, we find the minimal γ = 1.39 • 10 -4 with the observer For simulation, the constant delay is set to h(t) = 0.5, t ∈ R + , the parameter is chosen as ρ(t) = sin(t), a known input signal of the form u(t) = sin(10t) is applied at t = 6 seconds while a step disturbance w(t) of amplitude 10 is applied at t = 12 seconds. The state trajectories and their respective estimated values are depicted in Figure 2 where we can observe that, using the knowledge of the known input u, the state estimation is accurate even in presence of a disturbance of large amplitude, due to the high L 2 attenuation (note that ||(T -HC)E(ρ)|| 2 10 -6 ).

M 0 (ρ) = -0.0006ρ 2 + 0.00001ρ -5.1879 0 0 -0.1ρ -4 , H = 1 1 (23) M h (ρ) = 0 0 0 -0.4 , N h (ρ) = 0 -0.1ρ -0.6 , N 0 (ρ) = 0 -0.1ρ -6 , S(ρ) = 0 0.000014ρ + 1 (24)

Example 2: Reduced order observer for stable systems

Let us consider now the same system as in Section 4.1 with the difference that y(t) = 0 1 x(t) and z(t) = 1 0 x(t).

(25)

Hence, we would like to obtain a reduced-order controller estimating the first state from the measurement of the second state only. Since the matrix φ is full-column rank, then H is not restricted. Using Theorem 3.3 with h m = 0.8 and a griding of the conditions, we find a minimal γ = 0.2565, showing that total decoupling is not possible in this case. The observer matrices are given by

M 0 (ρ) = -2.3306, M h (ρ) = 0.3165ρ -0.23306, N 0 (ρ) = 0.3165ρ + 0.2199, N h (ρ) = 0.36885ρ + 0.02199, H = -1.1653, S(ρ) = 2.16528ρ + 3.3306. ( 26
)
The simulation result is depicted in Fig. 3 where we have used the same scenario as in Example 1. In this case, however, the disturbance is responsible of an observation error, as seen in Fig. 3. Let us consider a milling process [START_REF] Zhang | Stability analysis of LPV time-delayed systems[END_REF] described by the model

ẋ(t) = (A 0 + ρ(t)A 1 )x(t) + (A h0 + ρ(t)A h1 )x(t -h) (27)
where the delay h = π/ω is constant, ρ(t) = cos(2ωt + β), β > 0 and ω > 0 is the angular velocity. The system matrices are given by

A 0 =     0 0 1 0 0 0 0 1 -(10 + 0.1710k) 10 0 0 5 -15 0 -0.25     , A 0 =     0 0 0 0 0 0 0 0 0.5k 0 0 0 0 0 0 0     , A h0 =     0 0 0 0 0 0 0 0 0.1710k 10 0 0 0 0 0 0     , A h1 =     0 0 0 0 0 0 0 0 -0.5k 0 0 0 0 0 0 0     (28)
where k > 0 is the cutting stiffness constant. We assume that the displacement of the blade x 1 (t) and the tool x 2 (t) are measured while the whole state is desired to be estimated, that is

y(t) = 1 0 0 0 0 1 0 0 x(t) and z(t) = x(t). ( 29
)
According to the discussion in [START_REF] Zhang | Stability analysis of LPV time-delayed systems[END_REF], the parameter has unbounded derivative, hence we pick P (ρ) = P 0 and L(ρ) = L 0 + L 1 ρ + L 2 ρ 2 . For the observer design, we pick h m = 0.8 (hence h ≤ h m , ω ≥ π/0.8), k = 0.2 and use Theorem 3.3 where H is unconstrained since φ is full-column rank. For simulation, we pick ω = 10 rad/s and β = 7π/18. The results are depicted in Fig. 4 where we can see that the observer is able to track the system state accurately.

Remarks on Examples

It is important to explain how the above matrices have been obtained. Actually, since many matrix manipulations (pseudoinverse, inverse and arithmetical operations) have been performed numerically, many errors also accumulated. Initially, the obtained matrices are rational functions of high order (approx. 10) but after a closer analysis, we can see that many poles/zeros cancelations are prevented by small numerical errors on the coefficients of the rational functions. Hence, using a least mean square approximation on each coefficient of the matrices, it is possible to simplify the overall expression of the observer. In the current example, the L 2 norm of the error between the initial rational function and the approximant is of order 10 -6 .

Conclusion

The design of a class of observers for LPV time-delay systems has been addressed. The approaches are based on the use of algebraic techniques to parameterize a family of observers which satisfy a set of structural constraints. Then, an observer which guarantees stability and performance criteria is finally determined using LMIs. The approach has been finally illustrated through examples demonstrating the relevance of the approach.

The current limitations of the approach lie in the difficulty of knowing the time-varying delay exactly. This knowledge is necessary to the implementation of the observers, extensions to the uncertain delay knowledge cases might be possible using techniques inspired from [START_REF] Briat | Memory resilient gain-scheduled state-feedback control of uncertain LTI/LPV systems with time-varying delays[END_REF]). Other possible extensions are possible, e.g. widening the class of systems by allowing the measured output to depend on the parameters, the exogenous inputs and the delayed state. Finally, the class of observers could also be improved by making H parameter-dependent. and will be used throughout the proof. Let us also consider the Lyapunov-Krasovskii functional (Briat et al. 2008a(Briat et al. ,b, 2009a,b) ,b) extended from (Han 2005, Gouaisbaut and[START_REF] Gouaisbaut | Delay dependent robust stability of time delay-systems[END_REF]:

V (x t ) := x(t) T P (ρ)x(t) + t t-h(t) x(θ) T Qx(θ)dθ + h m 0 -hm t t+θ ẋT (s)R ẋ(s)dsdθ (A2)
playing the role of the storage function in the dissipativity criterion. Define also the supply-rate s(w, z) := γw T wγ -1 z T z which characterizes the L 2 -gain from the input w to the output z of the system (A1). Then, considering the function H d (t) := V (x t ) - Using the Jensen's inequality [START_REF] Gu | Stability of Time-Delay Systems[END_REF] on the term I, we get I ≤ -(x(t) -x(th(t))) T R(x(t) -x(th(t))).

(A3)

Then expanding the above relations, adding them up and using Schur complements, the following LMI is obtained

     
(ρ, ρ) -R P (ρ)A h (ρ) + R P (ρ)E(ρ) C(ρ) T h m A(ρ

) T R -(1 -µ)Q -R 0 0 h m A h (ρ) T R -γI 0 h m E(ρ) T R -γI 0 -R       ≺ 0 (A4)
and ensures the stability of system (A1) by virtue of the Lyapunov-Krasovskii Theorem. Due to the multiple products P (ρ)A(ρ) and RA(ρ) between system and Lyapunov matrices, the linearization after substitution of the matrices of ( 19) is not possible. Following (Briat et al. 2009a(Briat et al. ,b, 2010)), the following dilated version of the above inequality is used instead:

           -X(ρ) S X(ρ) T A(ρ) + P (ρ) X(ρ) T A h (ρ) X(ρ) T E(ρ) 0 X T h m R ∂P ∂ρ ρ -P (ρ) + Q -R R 0 C(ρ) T 0 0 -Q µ -R 0 0 0 0 -γI m 0 0 0 -γI r 0 0 -P (ρ) -hR -R            ≺ 0 (A5)
where X(ρ) ∈ R r×r is an additional free matrix. As explained in the above references, the LMI (A5) is not equivalent to (A4) but is only sufficient.

Substitute the observation error dynamic model in (A5) with C(ρ) = I r , expand the expression and perform the change of variables H = X T H, L(ρ) = X T L(ρ) (where X is set constant since H is constant) to obtain the LMI (20). Finally, since the LMI is affine in ρ, then using a convexity argument, it is enough to check the feasibility of the LMIs at the vertices of co{U ν }, namely on the set U ν . This completes the proof.

Figure 1 .

 1 Figure 1. Time evolution of the parameter ρ (top), the known input u (center) and the disturbance w (bottom) for Examples 1 and 2.

Figure 2 .

 2 Figure 2. Actual (plain) estimated (dashed) outputs for Example 1 -First approach (Theorem 3.1)

Figure 3 .

 3 Figure 3. Actual (plain) and estimated (dashed) outputs for Example 2 4.3 Example 3: Observation of a milling process

Figure 4 .

 4 Figure 4. Actual (plain) and estimated (dashed) outputs for the milling process.

t 0 s

 0 (w(θ), z(θ))dθ, and differentiating it along the trajectories solutions of the system (A1) yieldsḢd ≤ ξ(t) ρ) P (ρ)A h (ρ) P (ρ)E(ρ) ρ) = A(ρ) T P (ρ) + P (ρ)A(ρ) + ∂P (ρ) ∂ρ ρ + Q, s(w, z) = γw T wγ -1 z T z, I = -h m t t-h(t) ẋ(s) T R ẋ(s)ds, ξ(t) = col[x(t), x(th(t)), w(t)].

This is however hardly possible in practice due to approximation/numerical errors.

Appendix A: Proof of theorem 3.3

For convenience, the following LPV time-delay system is introduced: