Corentin Léna
email: corentin.lena@math.u-psud.fr

Spectral minimal partitions for a family of tori

Keywords: Spectral theory, minimal partitions, shape optimization, nodal domains, numerical simulations, finite difference method, projected gradient algorithm, tilings of the torus

We study the spectral minimal k-partitions of the two-dimensional flat torus (R/Z)×(R/bZ) , with b a parameter in b ∈ (0, 1]. We give a heuristic argument to compute a transition value when k is odd. We support this conjecture by looking for candidates to be minimal partitions using an optimization algorithm adapted from [2]. Guided by these numerical results, we construct k-partitions that are tilings of the torus by hexagons. We compute their energy and thus obtain an upper bound of the minimal energy.

which we call the energy, is minimal. Here λ 1 (D i ) is the lowest eigenvalue of the Dirichlet Laplacian on D i . This problem was studied by Bernard Helffer, Thomas Hoffmann-Ostenhof, and Susanna Terracini in [START_REF] Helffer | Nodal domains and spectral minimal partitions[END_REF], where existence and regularity were proved for a large class of domains. Similar optimization problems were studied in [START_REF] Bucur | Existence results for some optimal partition problems[END_REF] and [START_REF] Caffarelli | An optimal partition problem for eigenvalues[END_REF]. In the specific setting that we consider here, B. Helffer and T. Hoffmann-Ostenhof showed, in [START_REF] Helffer | Minimal partitions for anisotropic tori[END_REF], that when b is small enough, a minimal k-partition is obtained by dividing the torus in k equal 'vertical' strips.

One of the motivations of this work is to get information on the minimal partitions when b increases. In particular, we are interested in the value of b for which the 'vertical' partition mentioned above is no longer minimal. We present some conjectures on these transition values, inspired by the methods in [START_REF] Helffer | Minimal partitions for anisotropic tori[END_REF].

To check these conjectures, and to look for minimal partitions for larger b's (with b ∈ (0, 1]), we need a numerical method to search for minimal partitions. We have adapted the optimization algorithm used by Blaise Bourdin, Dorin Bucur and Édouard Oudet in [START_REF] Bourdin | Optimal partitions for eigenvalues[END_REF]. Our numerical computations support the conjectures concerning the transition values, improve known upper bounds of the minimal energy, and produce some quite remarkable candidates to be minimal partitions.

Guided by the numerical results, we have constructed families of k-partitions, for k ∈ {3 , 4 , 5} . These partitions are tilings of the torus by isometric polygons (in general hexagons) that satisfy a geometrical constraint called the 'equal angle property'. For some values of b, we obtain in this way partitions that closely resemble the best candidates produced by the optimization algorithm. For other values of b, such a tiling does not seem to be optimal. In any case, however, the first eigenvalue of the Dirichlet Laplacian of the fundamental domain of the tiling gives an upper bound of the minimal energy, which often improves the previously known ones.

The paper is organized in the following way. In Section 2, we recall some relevant facts concerning minimal partitions, including the results of [START_REF] Helffer | Minimal partitions for anisotropic tori[END_REF] for flat tori. In section 3, we present a heuristic method, inspired by [START_REF] Helffer | Minimal partitions for anisotropic tori[END_REF], to compute the transition values. When b is equal to the conjectured transition value, we also present some candidates to be minimal partitions which are obtained from the nodal set of eigenfunctions of the Laplacian on a covering of the torus. In Section 4, we describe the numerical method that we have used. The numerical results are presented in Section 5. For a fixed k (k ∈ {3 , 4 , 5}), we compute candidates to be a minimal k-partition of T (1, b), for a discrete set of values of b in (0, 1] . These computations suggest the existence of hexagonal tilings of some tori. In Section 6, we compute the energy of these tilings (which will be constructed in Appendix A), and compare it to the formerly obtained bounds. This allows us to improve the bounds of the minimal energy deduced from the general theorems of Section 2. In some cases, we are even able to formulate a conjecture on the value of the minimal energy. We conclude with a summary of the results and some suggestions for future research.

I would like to thank my advisors, Virginie Bonnaillie-Noël and B. Helffer, for their guidance and their encouragements. I also thank V. Bonnaillie-Noël and É. Oudet for their help with the numerical methods.

Minimal partitions 2.1 Definitions

In the following, M is a compact two-dimensional Riemannian manifold without boundary. For any open set ω ⊂ M , the sequence (λ k (ω)) k≥1 denotes the eigenvalues of the Laplace-Beltrami operator on ω with Dirichlet boundary conditions. They are arranged in increasing order and counted with multiplicity. The following definitions and results are taken from [START_REF] Helffer | Nodal domains and spectral minimal partitions[END_REF]. Compared with this reference, our statements are somewhat simplified due to the absence of a boundary. In the rest of the paper, we call the D i 's domains of the k-partition and denote by P k the set of all k-partitions for any given k. Furthermore, we simply use the term partition if we do not want to specify the number of domains.

Definition 2.2. For any integer

k ≥ 1 , we set L k (M ) = inf D∈P k Λ k (D) . A partition D ∈ P k such that Λ k (D) = L k (M ) is called a minimal k-partition.
The following existence result is proved in the paper [START_REF] Conti | On a class of optimal partition problems related to the Fučík spectrum and to the monotonicity formulae[END_REF].

Theorem 2.3. For any integer k ≥ 1 , there exists a minimal k-partition of M . This paper in fact prove a much stronger result, namely the existence of a regular minimal partition in the sense of Definition 2.5.

Its boundary is then defined as

N (D) = k i=1 ∂D i .
Definition 2.5. A partition D is called regular if it is strong and its boundary N (D) satisfies the following properties.

i. The set N (D) is locally a C 1,1 -= ∩ α<1 C 1,α curve, except in the neighborhood of a finite set {x i ; 1 ≤ i ≤ ℓ} . The elements of this set are called singular points.

ii. For each 1 ≤ i ≤ ℓ , there is an integer ν i ≥ 3 such that, in a neighborhood of x i , the set N (D) is the union of ν i half curves of class C 1,+ = ∪ α>0 C 1,α , which meet at x i .

iii. At each singular point, the half curves meet with equal angle.

The third property in Definition 2.5 is what we have called the 'equal angle property'. As proved in [START_REF] Helffer | Nodal domains and spectral minimal partitions[END_REF], we have the following regularity result: Theorem 2.6. For any k ≥ 1 , minimal k-partitions are strong and regular (up to zero capacity sets).

We give additional definitions that help us to describe the topology of a partition. Definition 2.7. Let D = {D i ; 1 ≤ i ≤ k} be a strong partition. Two domains D i and D j are said to be neighbors if Int D i ∪ D j is connected. A strong partition is said to be bipartite if one can color its domains, using only two colors, in such a way that two neighbors have different colors.

The paper [START_REF] Helffer | Nodal domains and spectral minimal partitions[END_REF] shows that there exists a connection between the minimal partitions of M and the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on M . Since we will use this connection in the rest of this work, we recall some relevant definitions and results. With an eigenfunction u of the Laplace-Beltrami operator on M we associate the nodal set

N (u) = {x ∈ M ; u(x) = 0}
and we call nodal domains the connected components of M \ N (u). We denote by µ(u) the number of nodal domains and the nodal domains themselves by D i . The following result was proved by Courant (cf. [START_REF] Courant | Methods of Mathematical Physics[END_REF]).

Theorem 2.8.

If k ≥ 1 and u is an eigenfunction associated with λ k (M ), µ(u) ≤ k .

The set {D i ; 1 ≤ i ≤ µ(u)} is therefore a partition of M in the sense of Definition 2.1. According to classical results on the nodal set, it is a regular partition, that we will call the nodal partition associated with u. Following [START_REF] Helffer | Nodal domains and spectral minimal partitions[END_REF], we introduce a new definition. Definition 2.9. Let k ≥ 1 . An eigenfunction u for the Laplace-Beltrami operator, associated with

λ k (M ), is said to be Courant-sharp if µ(u) = min{ℓ ; λ ℓ (M ) = λ k (M )} .
To give an upper-bound of L k (M ), one can use k-partitions obtained from eigenfunctions. To make this explicit, we introduce a new spectral element. Definition 2.10. For k ≥ 1 , we denote by L k (M ) the smallest eigenvalue of the Dirichlet Laplacian that has an eigenfunction with k nodal domains. If there is no such eigenvalue, we set L k (M ) = +∞ .

We can now state the results of [START_REF] Helffer | Nodal domains and spectral minimal partitions[END_REF] that link minimal and nodal partitions.

Theorem 2.11. A minimal partition is nodal if, and only if, it is bipartite. Furthermore, for any integer k ≥ 1 ,

λ k (M ) ≤ L k (M ) ≤ L k (M ) , and, if L k (M ) = L k (M ) or λ k (M ) = L k (M ) , then λ k (M ) = L k (M ) = L k (M ) .
In this latter case, any minimal k-partition is nodal.

Let us point out a few consequences of Theorem 2.11. Remark 2.12. A nodal partition associated with a Courant-sharp eigenfunction is minimal. Minimal 2-partitions are the nodal partitions of eigenfunctions associated with λ 2 (M ) .

Results on tori

In order to apply Theorem 2.11 we need the eigenvalues of the torus, which we recall here for future reference. As we will see in Section

λ m,n (a, b) = 4π 2 m 2 a 2 + n 2 b 2 ,
with m and n in N .

The following notation will also be useful.

D i = i -1 k a, i k a × ]0, b[ , for i = 1, . . . , k .
We have

Λ k (D k (a, b)) = k 2 π 2 a 2 .

Any

= 2ℓ . If b ≤ 1/ℓ , D k (1, b) is a minimal k-partition of T (1, b) . Furthermore, if b < 1/ℓ , D k (1, b) is, up to a translation, the only minimal k-partition of T (a, b) . Finally, if b > 1/ℓ , the partition D k (1, b) is not minimal. If k = 2ℓ and b = 1/ℓ , D k (1, b)
is not the only minimal partition of T (1, b) up to a translation, as we will see in Section 3.

The answer is less clear when k is odd. However, a more complicated analysis, performed by B. Helffer and T. Hoffmann-Ostenhof in [START_REF] Helffer | Minimal partitions for anisotropic tori[END_REF], gives a range of values of b for which

D k (1, b) is minimal. Proposition 2.16. Let k > 1 be an odd integer. If b < 1/k , then D k (1, b) is a minimal k-partition of T (1, b) .
Let us conclude this section by a monotonicity result.

A : T (1, b ′ ) → T (1, b) (x, y) → x, b b ′ y . It is easy to see that if ω is an open set in T (1, b ′ ) , A(ω) is an open set in T (1, b) and λ 1 (A(ω)) ≤ λ 1 (ω) . Indeed, let us consider a smooth function u supported in ω . Then v = u • A -1 is a smooth function supported in A(ω) , with T (1,b) |v| 2 dxdy = b b ′ T (1,b ′ ) |u| 2 dxdy and T (1,b) |∇v| 2 dxdy ≤ T (1,b) |∂ x v| 2 + b b ′ 2 |∂ y v| 2 dxdy = b b ′ T (1,b ′ ) |∇u| 2 dxdy .
By the variational definition of λ 1 (A(ω)) ,

T (1,b) |∇v| 2 dxdy T (1,b) |v| 2 dxdy ≥ λ 1 (A(ω)) .
From this we deduce

T (1,b ′ ) |∇u| 2 dxdy T (1,b ′ ) |u| 2 dxdy ≥ λ 1 (A(ω)) ,
and thus λ 1 (ω) ≥ λ 1 (A(ω)) .

As a consequence, a minimal partition of

T (1, b ′ ) is transformed by A into a partition of T (1, b) whose energy is lower than L k (T (1, b ′ )). This implies L k (T (1, b)) ≤ L k (T (1, b ′ )) .
b k = sup{b > 0 ; D k (1, b) is a minimal k-partition of T (1, b)} . Proof. Let us write b k = sup{b > 0 ; D k (1, b) is a minimal k-partition of T (1, b)} . We obviously have b k ≤ b k . Let us now consider b > b k . There exists b ′ ∈ (0, b) such that D k (1, b ′ ) is not a minimal partition of T (1, b ′ ) . Then L k (T (1, b ′ )) < Λ k (D k (1, b ′ ))) = k 2 π 2 .
According to Proposition 2.17,

L k (T (1, b)) ≤ L k (T (1, b ′ )) and therefore L k (T (1, b)) < k 2 π 2 , which implies that D k (1, b) is not minimal. We have shown that b k ≤ b k .
Using Proposition 2.15 and Proposition 2.16, we obtain the following result.

Proposition 3.3. If k is even, b k = 2/k . If k is odd, b k ≥ 1/k .
We are interested in finding b k for k odd, and also in understanding the transition from D k (1, b) to another type of minimal partition when b becomes greater than b k .

Heuristic analysis for b 3

In this subsection, we present a conjecture on the value of b 3 , based on a heuristic analysis of the possible deformations of the partition D 3 (1, b) . It is inspired by the methods used in [START_REF] Helffer | Minimal partitions for anisotropic tori[END_REF].

By

definition of b 3 , D 3 (1, b) is a minimal 3-partition of T (1, b) for b < b 3 . The partition D 3 (1, b)
is not bipartite and therefore not nodal. Let us however consider the torus

T (2, b) = (R/2Z) × (R/bZ) .
There is a natural map Π : (x, y) → (x, y) from T We can define the partition Π -1 (D 3 (1, b)) as the set of all the connected components of Π -1 (D i ) for all i ∈ {1 , 2 , 3} . We say that we have lifted By antisymmetric, we mean that u(x + 1, y) = -u(x, y) . We assume that this is true even when b = b 3 , for any minimal 3-partition, that is to say we make the following conjecture. At this point, we make an additional hypothesis inspired by the paper [START_REF] Bonnaillie-Noël | Aharonov-Bohm Hamiltonians, isospectrality and minimal partitions[END_REF], in which a similar study is performed on the rectangle. We assume that for b = b 3 , there exists a continuous family of minimal partitions of T (1, b 3 ) , of which D 3 (1, b 3 ) is a member. This allows the existence of a partition whose boundary contains singular points, which, for b greater than b 3 and close to b 3 , will give a minimal 3-partition of T (1, b) after a small deformation (we give more details on this point at the end of this subsection). We therefore make the following conjecture. We find that the smallest suitable b satisfies

D 3 (1, b) to Π -1 (D 3 (1, b)) . We have Π -1 (D 3 (1, b)) = D 6 (2,
λ 1,1 (2, b) = λ 3,0 (2, b) ,
that is to say

4π 2 1 4 + 1 b 2 = 9π 2 .
This leads us to conjecture that

b 3 = 1 √ 2 .
We can also make a guess on the shape of minimal 3-partitions on T (1, b) , with b = 1/ √ 2 + ε and 0 < ε ≪ 1 . Assuming some continuity of the shape of the minimal partitions with respect to b, we can conjecture a plausible mechanism, adapted from [START_REF] Bonnaillie-Noël | Aharonov-Bohm Hamiltonians, isospectrality and minimal partitions[END_REF], for the transition from D 3 (1, b 3 ) to another type of partition. We start from a 3-partition of T (1, 1/ √ 2) whose boundary contains singular points. According to Conjecture 3.4, the boundary is the projection on T (1, 1/ √ 2) of the nodal set of an eigenfunction on T (2, 1/ √ 2). As a consequence, an even number of lines meet at each singular point. The image of this 3-partition of T (1, 1/ √ 2) by the mapping

(x, y) → x, 1/ √ 2 + ε 1/ √ 2 y is a 3-partition of T (1, 1/ √ 2 + ε)
. This latest partition cannot be minimal, since it does not satisfy the equal angle property. We conjecture that, by a small deformation of this partition, we obtain a minimal 3-partition of T (1, 1/ √ 2 + ε) . We also conjecture that this deformation splits a singular point where an even number of lines meet into two singular points where an odd number of lines meet (see Figure 7, to be compared with Figure 1, for numerical simulations with b close to 1/ √ 2).

Examples of 3-partitions

Assuming b 3 = 1/ √ 2 , we obtain L 3 (T (1, 1/ √ 2)) = 9π 2 .
In this subsection, we study families of 3partitions with energy 9π 2 , and look for those that have singular points in their boundary.

To obtain these partitions, let us consider the eigenspace of the Laplace-Beltrami operator on T (2, 1/ √ 2) associated with the eigenvalue 9π 2 . It has dimension 6, and is spanned by the eigenfunctions

(x, y) → cos (3πx) , (x, y) → sin (3πx) , (x, y) → cos (πx) cos 2 √ 2 πy , (x, y) → cos (πx) sin 2 √ 2 πy , (x, y) → sin (πx) cos 2 √ 2 πy , (x, y) → sin (πx) sin 2 √ 2 πy .
Due to this high multiplicity, there is a great variety of possible nodal partitions. Let us note that all the functions in the list above are antisymmetric. This implies that every eigenfunction associated with the eigenvalue 9π 2 is antisymmetric. As a consequence, if u is such an eigenfunction on T (2, 1/ √ 2) and has 6 nodal domains, the projection of its nodal partition by Π is a 3-partition of T (1, 1/ √ 2) that has the energy 9π 2 .

We will not give all the possible nodal 6-partitions of T (2, 1/ √ 2) , but Figure 1 presents an example with singular points which will be useful to interpret the results of our numerical simulations.

Generalization

If we apply the heuristics of Subsection 3.2 to the determination of b k for any odd integer k ≥ 3 , the transition value b k should be the smallest b such that there exists an antisymmetric eigenfunction associated with λ k,0 (2, b) which is not a linear combination of the functions (x, y) → cos (kπx) and (x, y) → sin (kπx) . This leads us to the conclusion that b k should be solution of the equation (in b)

λ k,0 (2, b) = λ 1,1 (2, b) . Conjecture 3.6. For any odd integer k ≥ 3 , b k = 2 √ k 2 -1 .
This would be an improvement of Proposition 3.3.

In the case k = 5 , Conjecture 3.6 implies b 5 = 1/ √ 6 . As in Subsection 3.2, we try to build a nodal 10-partition of T (2, 1/ √ 6) , associated with the eigenvalue 25π 2 , that has singular points in its boundary. The projection on T (1, 1/ √ 6) of such a partition would give a good candidate for a minimal 5-partition of T (1, 1/ √ 6) that could be deformed into a minimal 5-partition of T (1, 1/ √ 6 + ε) , with 0 < ε ≪ 1 . An example of a nodal 10-partition of T (2, 2/ √ 6) with ten singular points is represented in Figure 2. Numerical simulations presented in Subsection 5.3 support the idea that its projection is minimal, and is a starting point to produce other minimal partitions by a small deformation (see Figure 13).

Examples of minimal 4-partitions

According to Proposition 2.15, b 4 = 1/2 . Any nodal partition of T (1, 1/2) associated with the eigenvalue 16π 2 that has four nodal domains is therefore minimal. The eigenspace of the Laplace-Beltrami operator on T (1, 1/2) associated with the eigenvalue 16π 2 is spanned by the eigenfunctions (x, y) → cos (4πx) , (x, y) → sin (4πx) , (x, y) → cos (4πy) , (x, y) → sin (4πy) .

This allows us to compute a series of nodal sets, represented on Figure 3. It can be shown that, up to translations, there is only one nodal set, represented on Figure 3(c), that contains singular points. The associated nodal partition is a 4-partition, and is therefore minimal. We conjecture that this partition is a starting point for the apparition of non-nodal 4-partitions of

T (1, b) when b = 1/2 + ε , with 0 < ε ≪ 1 .
More precisely, we conjecture that each singular point of order four splits into two singular points of order three (see Figure 10 in Subsection 5.2 for numerical simulations).

4 Numerical method

Relaxed formulation

Our numerical investigations are based on the method introduced by B. Bourdin, D. Bucur, and É. Oudet in [START_REF] Bourdin | Optimal partitions for eigenvalues[END_REF]. Let us start with relaxing the original minimization problem. In this subsection, we still denote by M a two-dimensional Riemannian manifold without boundary.

The first step of the relaxation consists in modifying the energy. We define a family of energies associated with a partition. In [START_REF] Bourdin | Optimal partitions for eigenvalues[END_REF], the authors study the minimization of the sum of the lowest eigenvalues, which is to say they look for L k,1 (M ) . We have modified their algorithm to look for L k,p (M ) for any 1 ≤ p < ∞ . To approach the minimal energy L k (M ) , corresponding to the case p = ∞ , we look for the minimal energy L k,p (M ) with 1 < p < ∞ large enough. This procedure is justified by the following result, proved in [START_REF] Helffer | Nodal domains and spectral minimal partitions[END_REF].

Definition 4.1. For any 1 ≤ p < ∞ and k ∈ N * , if D = {D i ; 1 ≤ i ≤ k} is a k-partition, we define Λ k,p (D) = 1 k k i=1 λ 1 (D i ) p
F k (M ) = F = (f 1 , . . . , f k ) ; f i : M → [0, 1] measurable, k i=1 f i (x) = 1 a.e. .
We associate with such a k-tuple an energy that is a relaxed version of Λ k,p . Definition 4.4. Let f : M → [0, 1] be a measurable function and ε a positive constant. The quantity λ 1 (f, ε) is defined as the first eigenvalue of the operator -∆ + ε -1 (1 -f ). To be more precise, λ 1 (f, ε) is defined, using a Rayleigh quotient, as

λ 1 (f, ε) = inf u∈H 1 0 (M )\{0} M (|∇u| 2 + 1 ε (1 -f )u 2 ) dxdy M |u| 2 dxdy .
Definition 4.5. Let F = (f 1 , . . . , f k ) be an admissible k-tuple, and let us choose p ≥ 1 and ε > 0 . We define the energy of F as

E k,p,ε (F) = 1 k k i=1 λ 1 (f i , ε) p 1 p . (4.1)
Definition 4.6. Let k ∈ N * , p ≥ 1 , and ε > 0 . We define the minimal energy as

E k,p,ε (M ) = inf F ∈F k (M ) E k,p,ε (F) . (4.2)
The parameter ε can be seen as a penalization of the overlapping of the supports of the functions f i . This idea is supported by the following result, proved in [START_REF] Bourdin | Optimal partitions for eigenvalues[END_REF] in the case p = 1 . Proposition 4.7. Let F = (f 1 , . . . , f k ) be a k-tuple that solves the optimization problem (4.2). Then, for all i ∈ {1 , . . . , k} , f i (x) = 0 or f i (x) = 1 for almost every x in M . This approach by penalization is justified in the case p = 1 by the following result, proved in [START_REF] Bourdin | Optimal partitions for eigenvalues[END_REF].

Proposition 4.8. Let k ∈ N * . Then lim ε→0 E k,1,ε (M ) = L k,1 (M ) .
This suggests the following procedure for determining numerically a minimal k-partition (in the sense of Definition 2.1), for a given domain M and integer k. We choose some p ≫ 1 and 0 < ε ≪ 1 and compute an approximation of a minimal admissible k-tuple

F = (f 1 , . . . , f k )
for the optimization problem (4.2). We then consider the sets

D i = {x ∈ M ; f i (x) = 1}
as domains of an approximate minimal partition.

To make this procedure effective, we need to specify how to represent the functions f i numerically and how to construct the domains D i from this representation. This question is addressed in Subsections 4.2 and 4.3.

Discretization

To perform numerical optimization, we replace Problem (4.2) by a discrete version. We approximate the torus T (1, b) by a m × n grid of points in the rectangle [0, 1] × [0, b] (with a constant step in each direction). Let us note N = mn the total number of points. An admissible k-tuple of functions is represented, in this setting, by a N × k matrix Φ, whose columns φ 1 , . . . , φ k contain the values of the functions f 1 , . . . , f k at each point of the grid. For 1 ≤ i ≤ k and 1 ≤ I ≤ N , we will sometime denote the element [Φ] I,i by [φ i ] I if we want to focus on the vector φ i . We impose the following constraints on Φ:

• ∀1 ≤ i ≤ k , ∀1 ≤ I ≤ N , 0 ≤ [Φ] I,i ; • ∀1 ≤ I ≤ N , k i=1 [Φ] I,i = 1 .
Let us give a geometric formulation of these conditions. For k ≥ 1 , we define the simplex S k in R k by

S k =    (x 1 , . . . , x k ) ; 0 ≤ x i , k j=0 x i = 1    .
We can express the constraints as

([Φ] I,1 , . . . , [Φ] I,k ) ∈ S k for 1 ≤ I ≤ N .
Let us now define a discrete analogue of the energy (4.1) that we will try to minimize in this discretized problem. With each column φ ∈ {φ j ; 1 ≤ j ≤ k} , we associate λ1 (φ, ε) the lowest eigenvalue of the matrix

L(φ) = L + 1 ε (Id N -Diag(φ)) .
In this formula:

• φ is an N × 1 vector;

• L is the finite difference approximation of -∆ on the m×n grid with periodic boundary conditions;

• Id N is the N × N identity matrix;

• Diag(φ) is the N × N diagonal matrix such that for 1 ≤ I ≤ N , [Diag(φ)] I,I = φ I .

Finally, we replace the energy E(p, k, ε, F) by its discrete version:

Ẽ(p, k, ε, Φ) = 1 k k i=1 λ1 (φ i , ε) p 1 p . ( 4.3)
We now want to minimize Ẽ(p, k, ε, Φ) with respect to the N k variables [Φ] i,I , under the constraint ([Φ] I,1 , . . . , [Φ] I,1 ) ∈ S k for 1 ≤ I ≤ N . As seen in [START_REF] Bourdin | Optimal partitions for eigenvalues[END_REF], the derivatives of Ẽ(p, k, ε, Φ) with respect to [Φ] i,I can be expressed with the help of eigenvalues and eigenvectors of the matrices L(φ i ) . We can therefore compute those derivatives and apply a projected gradient algorithm to the problem.

To make the convergence easier, we adopt the multi-step procedure described in [START_REF] Bourdin | Optimal partitions for eigenvalues[END_REF], which seems empirically very effective1 . We begin with applying the projected gradient algorithm on a very small grid, starting from a randomly generated Φ . We do not use the orthogonal projection operator but the so-called simple projection operator S k , which is defined below. Definition 4.9. Let k ∈ N * . The simple projection operator on the simplex S k is defined by

[S k x] i = |x i | k j=1 |x j | , for each x ∈ R k and 1 ≤ i ≤ k .
Once this step has converged, we double the number of points of the grid in each direction and extend the optimal Φ to the new grid by linear interpolation. We reiterate the same procedure, using the interpolated Φ as a starting point. Once we have repeated this step a certain number of times, we perform one last iteration using the orthogonal projection operator.

In practice, coefficients in the final Φ are either 0 or 1. There is, as far as we know, no proof of this fact in this discrete version of the problem, but it is consistent with Proposition 4.7. Taking it for granted, Φ gives us a partition of the points of the grid into k subsets { Di ; 1 ≤ i ≤ k} of points. From these subsets, we construct, in Subsection 4.3, a k-partition {D i ; 1 ≤ i ≤ k} of T (1, b) and compute its energy.

Construction of a k-partition

The implementation of the iterative gradient algorithm gives us a discrete k-partition { Di ; 1 ≤ i ≤ k} and a numerical relaxed energy M (p, k, ε, Φ) (see (4.3)). To approximate L k,p (T (1, b)), we have two further steps to perform:

• construct a k-partition D = {D i ; 1 ≤ i ≤ k} of T (1, b) from { Di ; 1 ≤ i ≤ k};
• compute the associated energy Λ k,p (D).

Since we want to keep the same size of discretization and since the torus is represented with a m × n grid, we construct the domains D i of the new partition as the union of elementary cells, of the form

p -1 m , p m × q -1 n b , q n b , with 1 ≤ p ≤ m and 1 ≤ q ≤ n .
Then we compute numerically the first eigenvalue of the Dirichlet Laplacian on each domain D i by a finite difference method. We can then, for any 1 ≤ p ≤ ∞ , compute an approximation of the energy Λ k,p (D) without relaxation.

In the rest of this section, we use a convenient graphical representation of the data in the matrix Φ . Let us explain it on an example. We choose m = 7 , n = 6 (therefore N = 42) and k = 3 . A matrix Φ of the type produced by the optimization algorithm is represented in the following way: for 1 ≤ I ≤ N and 1 ≤ i ≤ k , if [φ i ] I = 1 , we label the point indexed by I by the integer i (see Figure 4(a)).

3 1 1 1 3 3 2 2 1 1 1 2 2 2 2 2 1 1 2 2 2 2 2 1 1 2 2 3 3 1 1 1 3 3 3 3 1 1 1 3 3 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •
(a) Representation of a 7 × 6 matrix. We can imagine each point of the grid to be the center of a rectangular cell, which is of the form

3 3 1 1 1 3 3 2 2 1 1 1 2 2 2 2 2 1 1 2 2 2 2 2 1 1 2 2 3 3 1 1 1 3 35 3 3 1 1 1 3 3 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • (b) Dual mesh. 3 3 1 1 1 3 3 2 2 1 1 1 2 2 2 2 2 1 1 2 2 2 2 2 1 1 2 2 3 3 1 1 1 3 3 3 3 1 1 1 3 3 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • (c) Associated partition.
p -1/2 m , p + 1/2 m × q -1/2 n b , q + 1/2 n b ,
with 0 ≤ p ≤ m -1 and 0 ≤ q ≤ n -1 . We obtain a new mesh (which is in some sense dual to the first one), represented on Figure 4 We can describe the boundary of this partition. It is a union of some edges of the cells. More precisely, an edge is in the boundary if, and only if, it separates two cells that belong to different domains. To construct the boundary, we consider each cell successively. One of its edges is added to the boundary if the cell on the other side is labeled by a different integer. Actually, since all the cells are considered in turn, it is enough to look at the right and upper edges (for instance) of each cell. Let us finally note that by translating the boundary by the vector

v = - 1 2m , - b 2n ,
we can use the points of the original grid to represent the boundary. In our example, the boundary is represented on Figure 5. To find an approximation of Λ k (D) we compute

⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄
Λ k (D) = max 1≤i≤k λ 1 (D i ) ,
where λ 1 (D i ) is the first eigenvalue of the finite differences approximation of the Dirichlet Laplacian on D i .

Implementation of the method

Let us summarize the algorithm that we apply to find candidates to be minimal k-partitions. We fix J be the number of steps, ε > 0 , and p ≥ 1 .

Initialization:

• let m (1) , n (1) be two integers;

• let Φ (1) = (φ

(1) 1 , . . . , φ (1)
k ) be a random matrix of size N (1) × k with N (1) = m (1) n (1) ; • construct a uniform grid on the torus T (1, b) with N (1) = m (1) × n (1) points;

• construct the matrices L (1)

+ ε -1 Id N (1) -Diag(φ (1)
i ) , for i = 1, . . . , k ;

• apply the projected gradient algorithm, using the simple projection, with the initial data Φ (1) , to obtain a new matrix Φ (1) .

Induction For j = 2, . . . , J: 1) , n (j) = 2n (j-1) , N (j) = 4N (j-1) ;

• set m (j) = 2m (j-
• construct Φ (j) by interpolation of Φ (j-1) in the uniformly refined grid with N (j) points;

• construct the matrix

L (j) + ε -1 Id N (j) -Diag(φ (j)
i ) , for i = 1, . . . , k ; • apply the projected gradient algorithm, using the simple projection, with the initial data Φ (j) to obtain a new matrix Φ (j) .

Finalization:

• apply the projected gradient algorithm with orthogonal projection to Φ (J) ;

• construct a k-partition D by the method of Subsection 4.3;

• compute the energy Λk (D) .

To be complete, we must also specify which criterion is used to terminate the projected gradient algorithm at each step. For 1 ≤ j ≤ J , we denote by Φ (j,r) the matrix Φ , at the step j , after r iterations of the projected gradient algorithm. In this notation, the projected gradient algorithm stops when one of the two following conditions is satisfied:

• 1 kN I,i Φ (j,r+1) I,i -Φ (j,r) I,i
< δ , where δ is a fixed threshold;

• r ≥ R, where R is the maximal number of iterations, chosen beforehand.

We insist on the fact that this optimization algorithm is not necessarily successful. All the other parameters being equal, it could converges rapidly to a good candidate for some initial data, whereas for others it would terminate without reaching convergence. To overcome this problem, we have made several simulations, starting from different random initial data. We have also tested some particular initial data, for example the results of previous optimizations for different values of b . In any case, we have not accepted a result without precaution. We have checked whether the algorithm has converged, and have compared the energy of the resulting partition with the results of other runs for the same values of the parameters.

Numerical results

Let us now present the results obtained with the help of the above algorithm. We have used the following values for the parameters: p = 5, ε = 10 -4 , J = 5, n (1) = 16, m (1) = 16 .

Since J = 5 , we have refined four times the initial grid to obtain finally a 256 × 256 grid. We label the domains of the partition

D = {D i ; 1 ≤ i ≤ k} so that Λ k (D) = λ 1 (D 1 ) ≥ λ 1 (D 2 ) ≥ . . . ≥ λ 1 (D k ) .

3-partitions of the torus T (1, b)

We know from Proposition 3.3, that b 3 ≥ 1/3 . We have conjectured in Subsection 3.

2 that b 3 = 1/ √ 2 ≃ 0.707 . If this is true, D 3 (1, b) should be minimal for 1/3 < b < 1/ √ 2.
To test this, we have applied the algorithm for b ∈ {j/50, j = 17, . . . , 34}. As was expected, the lowest energy in these cases is obtained for partitions similar to the one on Figure 6. Figure 7 shows our best result for different values of b . When b is greater than 0.71 , the minimal partition seems to be a tiling of the torus by three isometric domains. These domains are roughly hexagonal, and close to the rectangles appearing in the partition of Figure 1. For brevity, we will say in the following that a partition with hexagonal domains is hexagonal. For b close to 1/ √ 2 , the final result of the optimization algorithm appears to be very sensitive to the initial data. As a consequence, the partitions of Figure 7 were not actually obtained by starting from random initial data. Rather, we ran the algorithm, starting from a random matrix, in the case b = 0.81 , where it produced an hexagonal partition similar to those of Figure 7. We then used as a starting point of the algorithm, for all the partitions of Figure 7, the matrix obtained after two steps in the case b = 0.81 . Of course, we compared our final results with those of other runs starting from random initial data, and found that they always had a lower energy.

For larger b's, up to b = 1 , the best candidates produced by the algorithm are still hexagonal partitions. Some of them are shown on Figure 8.

For each b, the energy of the best numerical candidate to be a minimal partition of T (1, b) is an upper bound for L 3 (T (1, b)) . This upper bound is plotted on Figure 9 as a function of b , and compared with 9π 2 = Λ 3 (D 3 (1, b)) . We obtain a significant improvement for large b . The third upper bound, represented by a solid line, will be discussed in Section 6.

4-partitions of the torus T (1, b)

We know from Proposition 3.3 that b 4 = 1/2 . We are interested in the nature of minimal partitions for b close to b 4 . Figure 10 shows the best candidates obtained from the algorithm.

As in the case k = 3 , we note the apparition of hexagonal partitions, shown on Figures 10(d), 10(e), and 10(f). The hexagonal domains of the partition shown on Figure 10(d

5-partitions of the torus T (1, b)

We have conjectured that b 5 = 1/ √ 6 ≃ 0.408 . Figure 13 presents the best candidates obtained numerically when b is close to this conjectured b 5 . They seem to support the conjecture. Furthermore, for b larger than 1/ √ 6 , minimal partitions seem to be hexagonal, with domains close to the rectangles appearing in the partition on Figure 2. The results of Section 4 suggest that, at least for some values of k and b, the domains of minimal kpartitions of T (1, b) are isometric polygons. In fact, except when k = 5 and b = 1, these polygons seem to be hexagons. On the other hand, according to Theorem 2.6, any minimal partition satisfies the equal angle property. This suggests the existence of partitions with a low energy, possibly minimal, that are tilings of the torus T (1, b) by k identical hexagons satisfying the equal angle property. Let us note that in this case, the equal angle property is satisfied if, and only if, the interior angle at each vertex of the hexagon is 2π/3 .

Finding these tilings is a purely geometrical problem. The task of enumerating, for given b and k, all admissible tilings seems quite complicated, and we have not been able to complete it. We can however construct families of tilings, depending on k and b , that seem close to the partitions produced by the algorithm. This is the object of Appendix A, where we prove in particular the following result. We have computed λ 1 (Hex k (b)) with high precision using the finite elements library MELINA (see [START_REF] Martin | Mélina, bibliothèque de calculs éléments finis[END_REF]). We have obtained in this way the upper bounds for

The case b = 1

Let us consider this special case, in which the tiling domains can be described simply, and let us formulate the corresponding conjectures on L k (T (1, b)) for k ∈ {3 , 4 , 5}.

3-partition

In that case, the tiling domain Hex 3 (1) is an hexagon with two axes of symmetry, described on Figure 16, with H = Figure 17: A hexagonal tiling domain of T (1, 1), for k = 4 .

4-partition

In that case, the tiling domain Hex 4 (1) is also an hexagon with two axes of symmetry, described on Figure 17, with h =

1 4 √ 3 , L = 1 2 √
3 , and

H = 1 2 -1 4 √ 3 .
Conjecture 6.4. L 4 (T (1, 1)) = λ 4 (Hex 4 (1)) ≃ 74.9467 .

5-partition

We can construct a 5-partition of T (1, 1) by identical hexagons with interior angles 2π/3 . However, numerical computations show that we obtain a lower energy when we divide T (1, 1) into five identical squares, as represented on Figure 18. Let us for instance focus on the case k = 3, for b close to 1/ √ 2 . Figure 9 shows that there exists a

b * > 1/ √ 2 such that, for b ∈ [ 1 2 , b * ) , λ 1 (Hex 3 (b)) > 9π 2 . Therefore, for b ∈ [ 1 2 , b *
) , the tiling of T (1, b) by three hexagons with straight edges that we have constructed is not minimal. This is consistent with the idea that there is some continuity of the minimal partitions with respect to b , and with the conjecture that the projection on T (1, 1/ √ 2) of the nodal partition of T (2, 1/ √ 2) shown on Figure 1 is minimal. Indeed, if we try to deform this latest partition by splitting each singular point of order four into a pair of singular points of order three, while keeping each domain close the original rectangle, the resulting partition cannot satisfy the equal angle property if all the regular parts of the boundary remain straight lines. This suggests that the boundary of the partition should be curved in the neighborhood of the singular points in such a way that the we keep the equal angle property. This seems to be the case for the partitions represented on

Conclusion

We have studied in this work the transition values b k defined in Subsection 3.1, when k is an odd integer.

We have presented a heuristic argument that leads to the conjecture

b k = 2 √ k 2 -1 .
Our numerical simulations support this conjecture when k = 3 and k = 5 .

We have also proposed a mechanism, inspired by [START_REF] Bonnaillie-Noël | Aharonov-Bohm Hamiltonians, isospectrality and minimal partitions[END_REF], that describes how a minimal 3-partition with singular points appears when b becomes larger than 1/

A Tilings of a family of tori by hexagons

The purpose of this appendix is to describe, for b ∈ (0, 1] , explicit examples of tilings of the torus T (1, b) by isometric hexagons that satisfy the equal angle property of Definition 2.5. We show that such tilings can be described by a 2 × 2 matrix with integer coefficients. We explain how to find conditions on b that allow the existence of a tiling described by a given matrix. We finally apply these results to k-partitions, with k ∈ {3 , 4 , 5}, in order to prove Proposition 2.5.

A.1 Tilings of the plane

To study tilings of the torus T (1, b) by hexagons, it will be useful to consider tilings of the plane. We recall that we consider T We will denote by (e 1 , e 2 ) the canonical basis of R 2 with e 1 = (1, 0) and e 2 = (0, 1) .

Let us consider a regular k-partition D = {D 1 , . . . , D k } of the torus T (1, b), such that all the D i 's are isometric to an hexagon that we denote by Hex. Let us note that, since D is strong, the area of Hex is b/k and, since D satisfies the equal angle property, all the interior angles of Hex are 2π/3 . Let us then consider, for any i ∈ {1 , . . . , k}, the open set Π -1 (D i ) . It has an infinite number of connected components, each one being isometric to Hex. The set of all the connected components of all the sets Π -1 (D i ), for i ∈ {1, . . . , k} , is a tiling of the plane R 2 by the hexagon Hex. This tiling is invariant under the translations associated with the vectors e 1 and be 2 .

We can see that, conversely, the image by Π of an tiling T of R 2 by an hexagon Hex is a regular k-partition of T (1, b) into domains isometric to Hex if T satisfies the following properties. i. T is invariant under the translations associated with the vectors e 1 and be 2 .

ii. The area of Hex is b k . iii. All the interior angles of Hex are 2π/3 .

We have therefore reformulated the original problem. We now look for the tilings of R 2 that satisfy properties i, ii, and iii, and, if possible, for an algorithm to construct those tilings.

A.2 Change of basis

Let T be an hexagonal tiling of R 2 , invariant under the translations associated with the vectors e 1 and be 2 , with a tiling domain Hex that satisfies properties ii and iii of Subsection A.1. We introduce the following definition, which will help us in our study.

Definition A.1. We say that a basis of R 2 is T -adapted if its vectors connect the center of a tiling domain to the centers of two neighboring domains, with these two neighboring domains having a common side (see Figure 20). Proof. Since T is invariant under the translations associated with the vectors e 1 and be 2 , there exist integers s 1,1 , s 2,1 , s 1,2 and s 2,2 such that e 1 = s 1,1 u 1 + s 2,1 u 2 ; be 2 = s 1,2 u 1 + s 2,2 u 2 .

We use the notation S = s 1,1 s 1,2 s 2,1 s 2,2 , u 1 = (u 1,1 , u 2,1 ) , u 2 = (u 1,2 , u 2,2 ) and

U = u 1,1 u 1,2 u 2,1 u 2,2 .
We One can give a geometrical interpretation of the coefficients in the matrix V in the following way. Let us go back to the torus T (1, b) . Let us assume that the matrix V = (v i,j ) has been obtained in the T -adapted basis (u 1 , u 2 ) (see Figure 20 for an example).

If we start from some hexagonal domain and translate it k times in the u 1 direction, it returns to its original position after turning v 1,1 times around the torus in the horizontal direction and v 2,1 times in the vertical direction. Similarly, if we translate the domain k times in the u 2 direction, it returns to its original position after turning v 1,2 times around the torus in the horizontal direction and v 2,2 times in the vertical direction.

We can therefore say that the matrix V describes how the hexagonal tiling T wraps around the torus T (1, b) .

Let us now choose some domain of the tiling T and denote by P 0 the vertex that connects the sides of this domain associated with u 1 and u 2 . Let us note P 1 = P 0 + u 1 and P 2 = P 0 + u 2 (see Figure 21). Since the basis (u 1 ,u 2 ) is T -adapted, both P 1 and P 2 are vertices of the tiling. Additionally, there is a vertex P of the tiling that is contained in the triangle P 1 P 2 P 3 and is connected to the points P 0 , P 1 , and P 2 . The segments P 0 P , P 1 P , and P 2 P thus meet at P with equal angles.

Conversely, let us consider (u 1 , u 2 ) a basis of R 2 such that there exists a 2 × 2 matrix V = (v i,j ) with integer coefficients that satisfies Let P 0 be some point in R 2 and let us note P 1 = P 0 + u 1 and P 2 = P 0 + u 2 . Let us now assume that there exists a point P in the triangle P 0 P 1 P 2 such that the segments P 0 P , P 1 P , and P 2 P meet at P with equal angles. After translating the three segments P 0 P , P 1 P , and P 2 P according to all the vectors in Zu 1 + Zu 2 , we obtain the boundary of a tiling T which satisfies the three properties stated in Subsection A.1. Furthermore, the basis (u 1 , u 2 ) is T -adapted.

A.3 Reconstruction of the tiling domain

Let us now recall, without proof, a very classical geometrical result (see for instance [START_REF] Coxeter | Introduction to Geometry[END_REF]).

Theorem A.3. Let A 1 , A 2 , and A 3 be three non-collinear points in R 2 (see Figure 22). One of the two following situations occurs.

1 Introduction

 1 This paper studies minimal partitions of the two-dimensional flat torusT (1, b) = (R/Z) × (R/bZ) , with b ∈ (0, 1]. To be more precise, for a given integer k, we look for k open, connected, and mutually disjoint subsets D 1 , . . . , D k of T (1, b) such that the quantity max 1≤i≤k λ 1 (D i ) ,

Definition 2 . 1 .

 21 For any integer k ≥ 1 , a k-partition is a finite set D = {D i ; 1 ≤ i ≤ k} whose k elements are open, connected, and mutually disjoint subsets of M . We define its energy by Λ k (D) = max 1≤i≤k λ 1 (D i ) .

  3, the torus T (2, b) plays a part in our study of T (1, b) . It will therefore be useful to have statements dealing with the more general torus T (a, b) = (R/aZ) × (R/bZ) , with 0 < b ≤ a . The Laplace-Beltrami operator on T (a, b) is of course unitarily equivalent to the Laplacian on the rectangle ]0, a[×]0, b[ with periodic boundary conditions. We will therefore often describe T (a, b) as the rectangle [0, a] × [0, b] with the opposite sides identified. The eigenvalues are then easy to compute using separation of variables and the periodic boundary condition. Proposition 2.13. The eigenvalues of T (a, b) are

Definition 2 .

 2 14. For 0 < b ≤ a and k ∈ N * , we denote by D k (a, b) the k-partition of T (a, b) with domains

  partition obtained from D k (a, b) by a translation has the same energy. Let us note that if k is even, D k (a, b) is the nodal partition associated with the eigenfunction (x, y) → cos kπ a x of the Laplace-Beltrami operator on T (a, b) . Let us now go back to the study of T (1, b) with b ∈ (0, 1] . We want to know for which values of b the partition D k (1, b) is minimal. We have to distinguish between an even and an odd k . If k is even, we can answer the question by applying Theorem 2.11. Proposition 2.15. Let k be an even integer, k

Proposition 2 .

 2 17. Let k ≥ 1 . The function b → L k (T (1, b)), defined on (0, 1] , is non-increasing. Proof. Let us pick b and b ′ in (0, 1] , with b ′ < b . We define a mapping A from T (1, b ′ ) to T (1, b) by

Definition 3 . 1 .

 31 Let k ≥ 1 . We define the transition value b k by b k = sup{b > 0 ; ∀b ′ ∈ (0, b) , D k (1, b ′ ) is a minimal k-partition of T (1, b ′ )} . Due to the monotonicity of b → L k (T (1, b)), we can give the following alternative definition of b k . Proposition 3.2. For k ≥ 1 ,

  (2, b) to T (1, b), where x = x mod 1 . Since every point of T (1, b) has two antecedents by Π , T (2, b) can be seen has a covering of T (1, b) with two sheets. If D i is a domain of D 3 (1, b) , the set Π -1 (D i ) has two connected components, which are open and connected subsets of T (2, b).

  b) , and D 6 (2, b) is a nodal partition of the torus T (2, b) , associated with the eigenvalue λ 3,0 (2, b) = 9π 3 . Let us also note that for 0 < b < b 3 , the nodal 6-partition D 6 (2, b) obtained by lifting the minimal partition D 3 (1, b) is associated with the antisymmetric eigenfunction u(x, y) = sin (3πx) .

Conjecture 3 . 4 .

 34 All minimal 3-partitions of T (1, b 3 ) can be lifted to nodal 6-partitions of T (2, b 3 ) associated with antisymmetric eigenfunctions.

Conjecture 3 . 5 .

 35 There exists a minimal 3-partition of T (1, b 3 ) that is not a translation of D 3 (1, b 3 ) . Let us assume that b = b 3 . If the eigenspace associated with the eigenvalue λ 3,0 (2, b 3 ) is spanned by the eigenfuctions (x, y) → cos (3πx) , (x, y) → sin (3πx) , the nodal partitions of any eigenfunction associated with λ 3,0 (2, b 3 ) is a translation of D 6 (2, b 3 ) . If Conjecture 3.4 is true, this implies that any minimal partition of T (1, b 3 ) is a translation of D 3 (1, b 3 ) , which contradict Conjecture 3.5. Therefore, if both conjectures are true, there exists (m, n) = (3, 0) such that λ m,n (2, b 3 ) = λ 3,0 (2, b 3 ) , and the eigenfunctions associated with λ m,n (2, b 3 ) are antisymmetric. This last condition simply means that m is odd. We finally conjecture that the transition from D 3 (1, b) to another type of partition occurs for the smallest b that satisfies the previous property.

Figure 1 :

 1 Figure 1: Nodal set of u(x, y) = cos(3πx) -cos(πx) cos(2 √ 2 πy) + sin(πx) sin(2 √ 2 πy) .

Figure 2 :

 2 Figure 2: Example of a nodal 10-partition of T (2, 1/ √ 6) (nodal set of u(x, y) = cos(5πx) + sin(πx) sin(2π √ 6y) -cos(πx) cos(2π √ 6y)).

  Nodal set of u(x, y) = sin (4πx) . (b) Nodal set of u(x, y) = 2 sin(4πx) + sin (4πy) .(c) Nodal set of u(x, y) = sin(4πx) + sin (4πy) . (d) Nodal set of u(x, y) = sin(4πx)+ 2 sin (4πy) . (e) Nodal set of u(x, y) = sin (4πy) .

Figure 3 :

 3 Figure 3: Examples of nodal partitions of T (1, 1/2).

Proposition 4 . 2 .

 42 Let k ∈ N * . The minimal energy L k,p (M ) is increasing with respect to p and lim p→+∞ L k,p (M ) = L k (M ) . The second step of the relaxation consists in replacing the optimal partition problem, concerning k-partitions of M , by a problem concerning k-tuples of functions. Definition 4.3. For k ∈ N * , the set of admissible k-tuples is given by

3

 3

Figure 4 :

 4 Figure 4: Partition obtained from the matrix Φ.

  (b). The torus can, in our example, be divided into three domains D 1 , D 2 , D 3 : for each i ∈ {1 , 2 , 3} , the domain D i is the union of all the cells whose center is labeled by the integer i . The partition,D = {D 1 , D 2 , D 3 } ,is strong in the sense of Definition 2.1. It is represented on Figure 4(c).

Figure 5 :

 5 Figure 5: Boundary of the partition.

Figure 6 : 3 -

 63 Figure 6: 3-partition obtained with the algorithm, b = 0.64.

Figure 7 : 3 -

 73 Figure 7: 3-partitions for b ∈ {j/100 ; j = 70 , . . . , 75} .

  (a) b = 0.8 (b) b = 0.9 (c) b = 1

Figure 8 : 3 -Figure 9 :

 839 Figure 8: 3-partitions for b ∈ {0.8 , 0.9 , 1}.

  ) seem close to the square domains of the nodal 4-partition shown on Figure 3(b). This suggests that the partition of T (1, 1/2) into four squares, shown on Figure 3(b), is the starting point for the apparition of non-nodal 4-partitions of T (1, b) , when b becomes greater than 1/2 . For larger values of b , up to b = 1 , the minimal partitions are apparently still hexagonal. Figure 11 shows for instance the best candidate for a minimal 4-partition of T (1, 1) . Its energy gives us an upper bound for L 4 (T (1, b)) , represented on Figure 12 as a function of b .

Figure 10 : 4 -

 104 Figure 10: 4-partition for b ∈ {j/100 ; j = 48 , . . . , 52} .

Figure 11: 4 -Figure 12 :

 412 Figure 11: 4-Partition of T (1, 1).

9 Figure 13 : 5 -

 9135 Figure 13: 5-partitions for some values of b.

1 Figure 14 : 5 -

 1145 Figure 14: 5-partition for b ∈ {0.98 , 0.99 , 1} .

Figure 15 :

 15 Figure 15: Upper bounds of L 5 (T (1, b)) for b ∈ {j/100 ; j = 18 , . . . , 100} .

Proposition 6 . 1 .

 61 For k ∈ {3 , 4 , 5} , there exists b c k ∈ (0, 1) such that, for any b ∈ (b c k , 1], there exists a tiling of T (1, b) by k hexagons that satisfies the equal angle property. We denote by Hex k (b) the tiling domain.

  L k (T (1, b)) represented by solid lines on Figures 9, 12, and 15. For b ∈ (b c k , 1] and k ∈ {3 , 4 , 5}, we can therefore bound L k (T (1, b)) from above either by the energy of D k (1, b) or by the energy of our hexagonal tiling, as summarized in the following statement. Proposition 6.2. For k ∈ {3 , 4 , 5} and b ∈ (b c k , 1] , L k (T (1, b)) ≤ min k 2 π 2 , λ 1 (Hex k (b)) .

λ 3 (L

 3 Figure 16: A hexagonal tiling domain of T (1, 1), for k = 3 .

Conjecture 6. 5 .Figure 18

 518 Figure 18: A 5-partition of T (1, 1) .

Figure 7 .

 7 The same phenomenon occurs for 4-partitions of T (1, b) when b is close to 1/2 , as seen on Figures 12 and 10, and for 5-partitions of T (1, b) when b is close to 1/ √ 6 or 1 , as seen on Figures 15, 13, and 14.

√ 2 .

 2 For b = 1/ √ 2 , we have constructed examples of 3-partitions of T (1, b) with energy 9π 2 that are projections of nodal 6-partitions of T (2, b) . We have found partitions whose boundary contains singular points of order four, which can split into two singular points of order three when b increases. Our numerical simulations singled out a partition of T (1, b) into three rectangles, which seems, when b increases, to change into a 3-partition with six singular points and three isometric, hexagonal domains, with slightly curved sides. We have proposed similar deformation mechanisms when k = 4 and k = 5 . They also seem to be supported by our numerical simulations. For k ∈ {3 , 4 , 5} , we have improved the formerly known upper bounds of L k (T (1, b)) by constructing explicit examples of partitions: hexagonal tilings of T (1, b) (see Figures 9, 12, and 15 for numerical computations of the new upper bounds). However, these tilings do not seem to be minimal partitions when k = 3 and b is close to 1/ √ 2 , when k = 4 and b is close to 1/2 , nor when k = 5 and b is close to 1/ √ 6 or 1 . Generally speaking, it seems interesting to investigate further for which values of k and b such tilings are minimal partitions.

  (1, b) as the quotient T (1, b) = (R/Z) × (R/bZ) , with the natural projection map Π : R 2 → T (1, b) (x, y) → (x mod 1, y mod b) .

1 =k

 1 Let us now denote by (u 1 ,u 2 ) a T -adapted basis.Proposition A.2. There exists a 2 × 2 matrix V = (v i,j ) with integer coefficients such that u (be 2 ) , and det(V ) = ±k.

  and therefore det(U ) det(S) = b . But | det(U )| is the area of the tiling domain Hex (see Figure19). Therefore det(U ) = ±b/k and det(S) = ±k . SinceS -1 = 1 det(S) s 2,2 -s 1,2 -s 2,1 s 1,1 ,we obtain the desired result by setting V = kS -1 .

u 1 u 2 Figure 19 :

 219 Figure 19: Area of the tiling domain.

u 1 u 2 Figure 20 : 3 -

 2203 Figure 20: 3-partition of a torus with an adapted basis.

1 P 2 PFigure 21 :

 1221 Figure 21: Construction of the tiling domain.

k

  (be 2 ) , and det(V ) = ±k .

We thank Édouard Oudet for giving us detailed explanations on this point.

i. If all three angles α 1 , α 2 , and α 3 of the triangle A 1 A 2 A 3 are smaller than 2π 3 , there is a unique point F belonging to the interior of the triangle A 1 A 2 A 3 such that the segments A 1 F , A 2 F and A 3 F meet with equal angles at F . The point F is called the Fermat point of the triangle A 1 A 2 A 3 . It is the point of minimum for the function

3 for some i ∈ {1 , 2 , 3}, then there is no point in the interior of A 1 A 2 A 3 at which the segments from the vertices meet with equal angles. In that case, the function P → P A 1 + P A 2 + P A 3 reaches its minimum at A i .

Using the vocabulary of Theorem A.3, we can summarize the above discussion in the following statement.

Proposition A.4. Let V = (v i,j ) 1≤i,j≤2 be an invertible matrix with integer coefficients and let b ∈ (0

There exists a regular k-partition of T (1, b) by isometric hexagons for which (u 1 , u 2 ) is an adapted basis if, and only if, the triangle spanned by u 1 and u 2 has a Fermat point.

The following result gives an easy criterion for the existence of a Fermat point.

Proposition A.5. Let u 1 and u 2 be two non-zero vectors in R 2 . Let A 3 be a point in R 2 . We set

We use the notation p = u1•u2 u1 u2 and r = u1 u2 . The triangle A 1 A 2 A 3 has a Fermat point if, and only if,

Proof. We compute cos(α i ) for i ∈ {1 , 2 , 3}, as a function of p and r. Writing down the condition

we show that it is equivalent to (A.1).

A 1

A.4 Algorithm

The results stated above give us an algorithm to build a tiling of the torus T (1, b) by k hexagons:

• choose a 2 × 2 matrix V with integer coefficients such that det(V ) = ±k ;

• check whether the triangle generated by the vectors u 1 and u 2 (defined from V as in Proposition A.4) has a Fermat point, using the conditions (A.1);

• if the triangle has a Fermat point, compute its coordinates;

• use the coordinates of the Fermat point to build the tiling domain.

Let us describe in more details how we perform the last two steps. Let us first recall a geometric construction of the Fermat point (see for instance [START_REF] Coxeter | Introduction to Geometry[END_REF]).

Theorem A.6. Let A 1 A 2 A 3 be a triangle in R 2 such that each of the angles α i , i ∈ {1 , 2 , 3}, is smaller than 2π 3 . Let us consider the three equilateral triangles lying outside of A 1 A 2 A 3 and having one side in common with it. For each of these triangles, let us consider the line passing through the outer vertex and the vertex of A 1 A 2 A 3 that does not belong to it (see Figure 23). The three lines meet at the Fermat point F .

Let us assume that we have performed the first two steps of the algorithm. We now have two vectors u 1 and u 2 such that the basis (u 1 , u 2 ) is adapted to some tiling T . We chose (arbitrarily) some point P 0 in R 2 . We can then build the points P 1 and P 2 of Figure 21. Using the construction of Theorem A.6, we can find the coordinates of the Fermat point P of the triangle P 0 P 1 P 2 . The vectors c 1 = P 0 P , c 2 = P P 2 , and c 3 = P 1 P then define three successive sides of the tiling domain, which is enough to construct the tiling domain itself.

A.5 Examples

We have shown that for any given b ∈ (0, 1] and k ∈ N * , finding all the tilings of T (1, b) by hexagons which are regular k-partitions is equivalent to finding all the matrices V with integer coefficients that satisfy the following properties. i. det(V ) = ±k .

ii. The vectors u 1 and u 2 , defined from V as in Proposition A.4, generate a triangle that has a Fermat point.

We have not been able to solve this general problem. We have nonetheless looked for examples of tilings with given matrices V , suggested by the numerical simulations. The following results can be proved using the criterion (A.1).

Proposition A.7. There exists a tiling of the torus T (1, b) with an associated matrix


Ibrahim Zidane

Dominique Guines

Lionel Leotoing

Eric Ragneau

A biaxial test for rheological and formability identification


Mariem Sahnoun

Hassan Hammouri

Nonlinear observer based on observable cascade form

Keywords: Nonlinear systems, output injection, nonlinear observer

come

I. INTRODUCTION

The implementation of linear or nonlinear observers in control systems design, fault detection and other domains is well understood by now. To design an observer for nonlinear systems, many approaches have been developed. Among them, the geometric approaches consist in characterizing nonlinear systems which can be transformed by a change of coordinates to a special class of systems for which a simple observer can be designed. The observer error linearization problem consists of transform a nonlinear system into a linear one plus a nonlinear term depending only on the known inputs and outputs. For such systems, a Luenberger observer can be designed. This problem has attracted a good deal of attention, since its formulation by [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF] (see for instance [START_REF] Boutat | New algorithm for observer error linearization with a diffeomorphism on the outputs[END_REF], [START_REF] Glumineau | New algebraic-geometric conditions for the linearization by input-output injection[END_REF], [START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF]- [START_REF] Xia | Nonlinear observer design by observer error linearization[END_REF]. Using immersion technics, an extension of this problem has been stated in [START_REF] Jouan | The time-varying linearization up to output injection[END_REF] in the single output case. In the same spirit as for the error linearization problem, the authors in [START_REF] Hammouri | Bilinearization up to output injection[END_REF]- [START_REF] Hammouri | A new procedure for time-varying linearization up to output injection[END_REF] characterized nonlinear systems which can be steered by a change of coordinates to state affine systems up to output injection. For these systems, a Kalman-like observer can be designed.

In this paper, we will characterize nonlinear systems which can be transformed by local coordinate systems into the following cascade form:

       ż = A(u)z + ψ(u, y) ˙ z = A(u) z + ψ(u, z, y) Y = y y = Cz C z (1)
For these systems, an observer structure may take the M. Sahnoun and H. Hammouri are with Université de Lyon, F-69622, Lyon, France; Université Lyon 1, Villeurbanne; CNRS, UMR 5007, LAGEP (Laboratoire d'Automatique et de Génie des Procédés). 43 bd du 11 novembre, 69100 Villeurbanne, France sahnoun@lagep.univ-lyon1.fr, hammouri@lagep.univ-lyon1.fr following form:

         ˙ z = A(u) z + ψ(u, y) -S -1 C T R(C z -y) ˙ z = A(u) z + ψ(u, z, y) -S -1 C T R( C z -y) Ṡ = -θ S -A T (u)S -SA(u) + C T RC ˙ S = -θ S -A T (u) S -S A(u) + C T R C (2)
where S(0), S(0), R and R are symmetric positive definite matrices, θ > 0, θ > 0 are parameters. The proof of the convergence of this observer has been stated in [START_REF] Besanon | Observer synthesis for a class of nonlinear systems[END_REF]. This paper is organized as follows: In section II, the problem under consideration is formalized and an existence theorem is stated. In section III, an algorithm permitting to calculate a system of coordinates in which a nonlinear system takes the desired cascade form is proposed.

II. PRELIMINARY RESULTS AND EXISTENCE THEOREM

A. Preliminary results

For the sake of simplicity, we only consider the case where the outputs y and y are scalars. The following classes of nonlinear systems will be considered:

   ẋ = f (u, x) y = h(x) y = h(x) (3)
where x ∈ R n , the input u(t) ∈ R m and the outputs y(t) and y(t) are belong to R. f , h and h are assumed to be of class C ∞ . We adopt the following definition. Definition 1: System (1) is said to be cascadeobservable, if system (1) together with its associated reduced system in z are observable.

The following geometric notions will be used in the sequel. In the system of coordinates

(x 1 , . . . , x n ), let X = ∑ n i=1 α i ∂ ∂ x i
be a vector field and let ω = ∑ n i=1 a i dx i a one-differential form, then the following operations will be considered:

• Lie derivative action: L X (ω) = ∑ n i=1 α i L X (a i )dx i + ∑ n i=1 a i dα i • The duality product: ω(X) = ∑ n i=1 α i a i
The above duality product can be extended to kdifferential forms as follow:

If ω = ∑ 1≤i 1 <...<i k ≤n a (i 1 ,...,i k ) dx i 1 ∧ . . . ∧ dx i k is a k-differential form and X = (X 1 , . . . , X k ) is a k- tuple of vector fields, with X i = ∑ n l=1 α il ∂ ∂ x l , then ω(X) = ∑ 1≤i 1 <...<i k ≤n a (i 1 ,...,i k ) α 1i 1 . . . α ki 1 . . . . . . . . . α 1i k . . . α ki k .
• Inner product: Let X = (X 1 , . . . , X l ) be a l-tuple of vector fields, with l ≤ k. Then i X (ω) is the (kl)differential form defined by:

i X (ω)(Y 1 , . . . ,Y k-l ) = ω(X 1 , . . . , X l ,Y 1 , . . . ,Y k-l ).
In particular, if k = l, then i X (ω) is a function (a 0-differential form).

Let f u be the vector field defined by f u (x) = f (u, x) and let X be a vector field on R n . We define the family of real vector spaces Ω X k of 2-differential forms as follows:

• Ω X 0 = 0 and Ω X 1 = Span{dL f u (h) ∧ dh; u ∈ R m }.
Noticing that these two spaces do not depend on X,

• for k ≥ 1, we set Ω X k+1 = Span{L f u (i X (ω)) ∧ dh; u ∈ R m ; ω ∈ Ω X k } + Ω X k . Now setting π = dϕ 1 ∧ . . . ∧ dϕ q
, where ϕ k are C ∞ functions, and let X = ( X 1 , . . . , X q+1 ) be a (q + 1)-tuple of vector fields. As above, we define the vector spaces Ω X k,π of (q + 2)differential forms as follows:

• Ω X 0,π = 0 and

Ω X 1,π = Span{dL f u ( h) ∧ d h ∧ π; u ∈ R m }, • for k ≥ 1, Ω X k+1,π = Span{L f u (i X ( ω)) ∧ d h ∧ π; u ∈ R m ; ω ∈ Ω X k,π } + Ω X k,π .

B. Existence theorem

In the single output case (see [START_REF] Hammouri | Bilinearization up to output injection[END_REF], [START_REF] Hammouri | Global time-varying linearization up to output injection[END_REF]), ( [5] for the the multi-output case) the authors gave necessary and sufficient conditions under which nonlinear systems can be transformed in a state affine system up to output injection. The following theorem states an existence theorem which extends those stated in [START_REF] Hammouri | Bilinearization up to output injection[END_REF], [START_REF] Hammouri | The time-varying linearization up to output injection[END_REF]:

Theorem 1:

Observable system (3) can be transformed by a local change of coordinates around some x 0 ∈ R n to a cascadeobservable system (1) in which C and C are of rank 1, if and only if, the following conditions hold on some neighborhood of x 0 :

1) It exists a vector field X satisfying the following conditions:

1-i) L X (h) = 1. 1-ii) The algebraic sum Ω X = ∑ k≥1 Ω X k is a real vector space of dimension q -1. 1-iii) For every ω ∈ Ω X , d(i X (ω)) = 0. 1-iv) The dimension of [ q-1 (i X (Ω X )) ∧ dh]| x 0 is equal to 1, where [ q-1 (i X (Ω X )) ∧ dh]| x 0 = {i X (ω 1 ) ∧ . . . ∧ i X (ω q-1 ) ∧ dh(x 0 ); ω i ∈ Ω X , 1 ≤ i ≤ q -1}.
2) Consider the following functions ϕ 1 , . . . , ϕ q+1 defined by: ϕ 1 = h ϕ q+1 = h (dϕ 1 , . . . , dϕ q ) forms a basis of i X (Ω X ) + Rdh

Setting π = dϕ 1 ∧ . . . ∧ dϕ q , then there exists a (q + 1)- tuple of vector fields X = ( X 1 , . . . , X q+1 ) satisfying the following conditions on some neighborhood of x 0 : 2-i) L X i (ϕ j ) = δ i j , where δ i j = 1 if i = j and 0 otherwise.

2-ii)

The algebraic sum

Ω X π = ∑ k≥1 Ω X k,π is a real vector space of dimension n -q -1. 2-iii) For every ω ∈ Ω X π , d(i X ( ω)) = 0. 2-iv) The dimension of [ n-q-1 (i X (Ω X π )) ∧ dϕ 1 ∧ . . . ∧ dϕ q+1 ]| x 0 is equal to 1.
The proof of theorem 1 can be obtained by following the same approach as the one proposed in the works [START_REF] Hammouri | Bilinearization up to output injection[END_REF], [START_REF] Hammouri | The time-varying linearization up to output injection[END_REF]. The outline of the proof is summarized as follows:

1) Sufficient condition: i X (Ω X ) and i X (Ω X π ) are vector spaces of dimension q -1 and nq -1 respectively, and (i X (ω 1 ), . . . , i X (ω q-1 )), (i X ( ω 1 ), . . . , i X ( ω n-q-1 )) are their respective bases. Setting

dz 1 = dh, dz i = i X (ω i+1 ), d z 1 = d h and d z i = i X ( ω i+1 ). It can be shown that L f u (z i ) = q ∑ j=2 a i j (u)z j + ψ i (u, z 1 ) and L f u ( z i ) = n-q ∑ j=2
a i j (u) z j + ψ i (u, z, z 1 ). Consequently, in the (z, z) sys- tem of coordinates system (3) takes the cascade form (1).

2) Necessary condition: Since conditions 1), 2) of theorem 1 are intrinsic (they do not depend on the system of coordinates), it suffices to show them for the cascade observable system (1). After a simple linear change of coordinates, we can assume that y = Cz = z 1 and y = C z = z 1 , and it can be shown that

X = ∂ ∂ z 1 and X = ( X 1 , . . . , X q+1 ) = ( ∂ ∂ z 1 , . . . , ∂ ∂ z q , ∂ ∂ z 1
) satisfy conditions 1) and 2) of theorem 1.

In the following, we focus on the development of an algorithm permitting to calculate vector fields X, X 1 , . . . , X q+1 which meet conditions 1) and 2) of theorem 1.

III. PROCEDURE OF CALCULATION OF VECTOR FIELDS

X, X 1 , . . . , X q+1

A. Preliminary results

The following notations will be used in the sequel:

• Let V be a vector space, and W a subspace of V , then for ξ , ξ ′ ∈ V , the notation ξ = ξ ′ modulo (W ) means that ξ = ξ ′ + w, for some w ∈ W . • Setting F (resp. V ) to be a set of one-differential form (resp. of vector fields). D = Span(F ) (resp. ∆ = Span(V )) will denote the co-distribution (resp. the distribution) spanned by F (resp. by V ).

• The orthogonal of a co-distribution D is the distribution ∆ = Ker(D) = Span({X; ω(X) = 0, ∀ω ∈ F }),
where ω(X) is the duality product between one-form and vector fields. In particular, if F is spanned by a family of one-exact form {dϕ;

ϕ ∈ F }, then ∆ = Ker(D)
is the distribution spanned by the set of vector fields

{X; L X (ϕ) = 0, ∀ϕ ∈ F }. • Let D, D ′ be two co-distributions, with D ′ ⊂ D, then the quotient D/D ′ will denote the set of equivalent class of differential forms [ω] = ω + D ′ = {ω + ω ′ ; ω ′ ∈ D ′ }, where ω ∈ D. Similarly, if ∆ ⊂ ∆ ′ are two distributions, elements of the quotient ∆ ′ /∆ will be denoted by [X] = X + ∆ where X ∈ ∆ ′ . If [ω] ∈ D/D ′ and χ ∈ D such that [ω] = [χ], then we set ω = χ modulo (D ′ ).
Finally, if X, Z are two vector fields, [X, Z] will denote the Lie bracket of these vector fields.

The following flag of co-distributions and distributions will be considered:

D 0 ⊂ . . . ⊂ D k ⊂ . . . ∆ 0 ⊃ . . . ⊃ ∆ k ⊃ . . . D 0 ⊂ . . . ⊂ D k ⊂ . . . ∆ 0 ⊃ . . . ⊃ ∆ k ⊃ . . . (5)
Where,

• D 0 = 0 the null co-distribution, D 1 = Span({dh}), by induction D k+1 = D k + Span({dL f u k . . . L f u 1 (h); u 1 , . . . , u k ∈ R m }), and
D ♯ = ∑ k≥1 D k . • D 0 = D ♯ , D 1 = D 0 + Span({d h}), for k ≥ 1, D k+1 = D k + Span({dL f u k . . . L f u 1 ( h); u 1 , . . . , u k ∈ R m }), and
D ♯ = ∑ k≥1 D k . • ∆ k = Ker(D k ), and ∆ ♯ = Ker(D ♯ ). • ∆ k = Ker( D k ), and ∆ ♯ = Ker( D ♯ ). • The quotient co-distribution D k /D k-1 (resp. D k / D k-1 )
is the dual of the quotient distribution

∆ k-1 /∆ k (resp. ∆ k-1 / ∆ k ). The duality product [ω]([X]) = ω(X) is well defined.
In the two following claims,

f u = q ∑ i=1 (A i (u)z + ψ i (u, y)) ∂ ∂ z i + n-q ∑ i=1 ( A i (u) z + ψ i (u, z, y)) ∂ ∂ z i
, and the outputs h, h are respec-

tively y = Cz = z 1 , y = C z = z 1 .
Considering the rings H k , H k such that:

• H 0 = C ∞ {z 1 } (resp. H 0 = C ∞ {z 1 , . . . , z q , z 1 }) is the ring of C ∞ -functions ϕ(z 1 ) (resp. ϕ(z 1 , . . . , z q , z 1 )). • C ∞ {z} (resp. C ∞ {z, z}) denotes the ring of C ∞ - functions ϕ(z 1 , . . . , z q ) (resp. ϕ(z 1 , . . . , z q , z 1 , . . . , z n-q )). Then for k ≥ 1, H k (resp. H k ) is the smallest sub- ring of C ∞ {z} (resp. of C ∞ {z, z}) containing H k-1 ∪ {CA(u 1 ) . . . A(u k )z; u 1 , . . . , u k ∈ R m } (resp. H k-1 ∪ { C A(u 1 ) . . . A(u k ) z; u 1 , . . . , u k ∈ R m }).
Then we have:

Claim 1: i) L f u k . . . L f u 1 (Cz) = CA(u 1 ) . . . A(u k )z modulo (H k-1 ). ii) L f u k . . . L f u 1 ( C z)) = C A(u 1 ) . . . A(u k ) z modulo ( H k-1 ).
The following claim can be deduced from the above one.

. . A(u l ) z; 1 ≤ l ≤ k -1, u j ∈ R m }.
• System (1) is cascade observable iff: dim D ♯ = q (q is the dimension of the z-space), and dim D ♯ /D ♯ = nq (nq is the dimension of z-space).

In the sequel, we set ν (resp ν) to be the smallest integer such that

D ν = D ♯ (resp. D ν /D ♯ = D ♯ /D ♯ ): D 0 ⊂ . . . ⊂ D ν = D ν+1 D 0 /D ♯ ⊂ . . . ⊂ D ν /D ♯ = D ν+1 /D ♯ (6)
This subsection will be ended by the two following technical results:

Lemma 1:

If dϕ ∈ D k-1 (resp. d ϕ ∈ D k-1 ) and X ∈ ∆ k-1 (resp. X ∈ ∆ k-1 ), then dϕ([ f u , X]) = -d(L f u (ϕ))(X) = -L X (L f u (ϕ)) (resp. dϕ([ f u , X]) = -d(L f u ( ϕ))( X) = -L X (L f u ( ϕ))). Proof of lemma 1. Let dϕ ∈ D k-1 and X ∈ ∆ k-1 , the equality dϕ([ f u , X]) = -d(L f u (ϕ))(X)
follows from the following facts:

• dϕ([ f u , X]) = L f u (L X (ϕ)) -L X (L f u (ϕ)) = d(L X (ϕ))( f u ) -d(L f u (ϕ))(X), • X ∈ ∆ k ⊂ ∆ k-1 = Ker(D k-1 ), • L X (ϕ) = dϕ(X) = 0 Similar argument can be used to prove dϕ([ f u , X]) = -d(L f u ( ϕ))( X).
Claim 3: Let Z = (Z 1 , . . . , Z k ) be a k-tuple of vector fields, let g, ϕ 1 , . . . , ϕ k be C ∞ -functions such that dϕ 1 ∧ . . . ∧ dϕ k is nowhere vanish and that L Z j (ϕ i ) = δ i j , then:

i Z (dg ∧ dϕ 1 ∧ . . . ∧ dϕ k ) = dg - k ∑ j=1 L Z j (g)dϕ j .
More precisely, we have:

i Z (dg ∧ dϕ 1 ∧ . . . ∧ dϕ k ) = (-1) q (dg - k ∑ j=1 L Z j (g)dϕ j ).

B. Algorithm

In this subsection, we will give an algorithm permitting to calculate the vector fields X, X 1 , . . . , X q+1 , which meet conditions of theorem 1. This algorithm will be obtained in three steps:

1) The first step consists to calculate X using only f (u, x) and h(x).

2) The knowledge of f (u, x), h, h(x) and X allows to calculate X q+1 . 3) Finally, X 1 , . . . , X q can be computed based on the knowledge of f (u, x), h, h(x), X and X q+1 .

Assuming that the flags of co-distributions:

0 = D 0 ⊂ . . . ⊂ D ν = D ν+1 0 = D 0 /D ν ⊂ . . . ⊂ D ν /D ν = D ν+1 /D ν (7)
are of constant dimensions and that dim(

D ν ) = q, dim( D ν /D ν ) = n -q.
For k ≥ 1, we define the bases B k and B k of D k /D k-1 and D k / D k-1 as follows:

B 1 = {[dh]}, B 1 = {[d h]} for k ≥ 2 : B k = {[d(L f u k-1 . . . L f u 1 (h))]; (u 1 , . . . , u k-1 ) ∈ U k-1 } B k = {[d(L f u k-1 . . . L f u 1 ( h))]; ( u 1 , . . . , u k-1 ) ∈ U k-1 } (8) for some subsets U k-1 and U k-1 of (R m ) k-1 .
The symbol [(.)] stands for the equivalent class of (.). Now, let B * ν , B * ν be the respective dual bases of B ν and B ν (B * ν , B * ν are bases of ∆ ν-1 /∆ ν and ∆ ν-1 / ∆ ν ), the following vector fields will be required in theorem 2 below :

• The vector fields [Z u 1 ...u ν-1 ], [ Z u 1 ... u ν-1 ]: Let (u 1 , . . . , u ν-1 ), (resp. ( u 1 , . . . , u ν-1 )) be fixed elements of U ν-1 (resp. of U ν-1 ), then [Y ] = [Z u 1 ...u ν-1 ] (resp. [ Y ] = [ Z u 1 ... u ν-1 ]) is the element of B * ν (resp. of B * ν ) defined by: for (v 1 , . . . , v ν-1 ) ∈ U ν-1 , d(L f v ν-1 . . . L f v 1 (h))(Y ) = 1, if (u 1 , . . . , u ν-1 ) = (v 1 , . . . , v ν-1
), and 0 otherwise for

( v 1 , . . . , v ν-1 ) ∈ U ν-1 , d(L f v ν-1 . . . L f v 1 ( h))( Y ) = 1, if ( u 1 , . . . , u ν-1 ) = ( v 1 , . . . , v ν-1
), and 0 otherwise (9)

• The vector fields [Y u 1 ...u ν-1 ], [ Y u 1 ... u ν-1 ]: Setting [Y ] = [Z u 1 ...u ν-1 ] and [ Y ] = [ Z u 1 ... u ν-1 ], then: Y u 1 ...u ν-1 = [ f u ν-1 , [. . . , [ f u 1 ,Y ] . . .]]] Y u 1 ... u ν-1 = [ f u ν-1 , [. . . , [ f u 1 ,Y ] . . .]]] (10)
In order to state lemma 2 below, the following notations will be required:

• Let (dϕ 1 , . . . , dϕ q ) be a basis of D ν and dϕ q+1 = d h.

• Setting π = dϕ 1 ∧ . . . ∧ dϕ q+1 .

• Let X = ( X 1 , . . . , X q+1 ) be a (q + 1)-tuple of vector fields satisfying

L X i (ϕ j ) = δ i j . • For u 1 ∈ U 1 , we set ω u 1 = dL f u 1 ( h) ∧ π. • For k ≥ 2 and ( u 1 , . . . , u k ) ∈ U k , we set ω u 1 ... u k = L f u k (i X ( ω u 1 ... u k-1 )) ∧ π.
Thus we have: Lemma 2: For 1 ≤ k ≤ ν -1; for every ( u 1 , . . . , u k ) ∈ U k the following properties hold:

ω u 1 ... u k = dL f u k . . . L f u 1 ( h) ∧ π + k-1 ∑ l=1 ∑ ( u 1 ,..., u l )∈ U l g u 1 ... u l (x)dL f u l . . . L f u 1 ( h) ∧ π (11) i X ( ω u 1 ... u k ) = dL f u k . . . L f u 1 ( h) - q ∑ j=1 L X j L f u k . . . L f u 1 ( h)dϕ j + Θ k Θ k = Θ k - q ∑ j=1 k-1 ∑ l=1 ∑ ( u 1 ,..., u l )∈ U l g u 1 ... u l (x)L X j L f u l . . . L f u 1 ( h)dϕ j Θ k = k-1 ∑ l=1 ∑ ( u 1 ,..., u l )∈ U l g u 1 ... u l (x)dL f u l . . . L f u 1 ( h) + g k (x)dϕ q+1 ( 12
)
with the property that g u 1 ... u l (.), g k (.) are C ∞ -functions which do not depend on ( X 1 , . . . , X q ). Proof of lemma 2. 11), ( 12) are true for k = 1.

• For k = 1: Let u 1 ∈ U 1 , by definition ω u 1 = dL f u 1 ( h) ∧ π, and from claim 3, we know that i X ( ω u 1 ) = dL f u 1 ( h) - q+1 ∑ j=1 L X j L f u 1 ( h)dϕ j = dL f u 1 ( h) - q ∑ j=1 L X j L f u 1 ( h)dϕ j + Θ 1 , here Θ 1 = L X q+1 L f u 1 ( h)dϕ q+1 . Hence (
• Assuming that ( 11), ( 12) hold for 1 ≤ l ≤ k -1, and let us show them for k. Using the definition of ω u 1 ... u k and applying [START_REF] Plestan | Linearization by generalized inputoutput injection[END_REF] for k -1, we get:

ω u 1 ... u k = dL f u k . . . L f u 1 ( h) ∧ π -L f u k [ q ∑ j=1 L X j L f u k-1 . . . L f u 1 ( h)dϕ j ] ∧ π + L f u k (Θ k-1 ) ∧ π Θ k-1 = Θ k-1 - q ∑ j=1 k-2 ∑ l=1 ∑ ( u 1 ,..., u l )∈ U l g u 1 ... u l (x)L X j L f u l . . . L f u 1 ( h)dϕ j Θ k-1 = k-2 ∑ l=1 ∑ ( u 1 ,..., u l )∈ U l g u 1 ... u l (x)dL f u l . . . L f u 1 ( h) + g k-1 (x)dϕ q+1
(13) and g u 1 ... u l , g k-1 do not depend on ( X 1 , . . . , X q ). Using the fact that dϕ i ∈ D ν , for 1 ≤ i ≤ q, and that L f u (D ν ) ⊂ D ν , then the following equality holds for every smooth functions a 1 (x), . . . , a q (x):

L f u ( q ∑ j=1 a j (x)dϕ j ) ∧ π = 0 (14)
Combining ( 14) with expressions of Θ k-1 , Θ k-1 , we get:

ω u 1 ... u k = dL f u k . . . L f u 1 ( h) ∧ π + k-2 ∑ l=1 ∑ ( u 1 ,..., u l )∈ U l L f u k [ g u 1 ... u l (x)dL f u l . . . L f u 1 ( h) + g k-1 (x)dϕ q+1 ] ∧ π (15) By construction L f u k [ g u 1 ... u l (x)dL f u l . . . L f u 1 ( h)] and L f u k ( g k-1 (x)dϕ q+1 ) ∧ π do not depend on ( X 1 , . . . , X q )
and {dϕ 1 , . . . ,

dϕ q+1 } ∪{dL f u l . . . L f u 1 ( h); ( u 1 , . . . , u l ) ∈ U l , 1 ≤ l ≤ k -1}
∑ l=1 ∑ ( u 1 ,..., u l )∈ U l g u 1 ... u l (x)dL f u l . . . L f u 1 ( h) ∧ π
, where the g u 1 ... u l (x)'s are C ∞ -functions which do not depend on ( X 1 , . . . , X q ). Consequently, expression ( 11) is satisfied.

In order to end the proof of lemma 2, it remains only to check [START_REF] Plestan | Linearization by generalized inputoutput injection[END_REF]. Applying claim 3 to expression [START_REF] Lin | Remarks on linearization of discrete time autonomous systems and nonlinear observer design[END_REF], we get:

i X ( ω u 1 ... u k ) = dL f u k . . . L f u 1 ( h) + k-1 ∑ l=1 ∑ ( u 1 ,..., u l )∈ U l g u 1 ... u l (x) dL f u l . . . L f u 1 ( h) - q+1 ∑ j=1 L X j L f u k . . . L f u 1 ( h)dϕ j - q+1 ∑ j=1 k-1 ∑ l=1 ∑ ( u 1 ,..., u l )∈ U l g u 1 ... u l (x)L X j L f u l . . . L f u 1 ( h)dϕ j
(16) Finally, expression [START_REF] Plestan | Linearization by generalized inputoutput injection[END_REF] follows from ( 16) in which we introduce:

Θ k = k-1 ∑ l=1 ∑ ( u 1 ,..., u l )∈ U l g u 1 ... u l (x)dL f u l . . . L f u 1 ( h) + g k (x)dϕ q+1 where g k (x) = -L X q+1 L f u k . . . L f u 1 ( h) - k-1 ∑ l=1 ∑ ( u 1 ,..., u l )∈ U l g u 1 ... u l (x)L X q+1 L f u l . . . L f u 1 ( h) Θ k = Θ k - q ∑ j=1 k-1 ∑ l=1 ∑ ( u 1 ,..., u l )∈ U l g u 1 ... u l (x)L X j L f u l . . . L f u 1 ( h)dϕ j (17)
Moreover, by construction g u 1 ... u l (x) and g k do not depend on ( X 1 , . . . , X q ). This ends the proof of lemma 2.

Now we can state the algorithm which allows to calculate vector fields X, X 1 , . . . , X q , X q+1 satisfying conditions 1) and 2) of theorem 1.

Theorem 2: (Algorithm) System (3) can be steered by a local change of coordinates around some x 0 to a cascade-observable system (1), if, and only if, the following conditions hold: ] ∈ B * ν , then the following properties hold:

1) The vector X = (-1) ν-1 Y u 0 1 ...u 0 ν-1 satisfies condition 1) of theorem 1.

2) Setting X q+1 = (-1) ν-1 Y u 0 1 ... u 0 ν-1 and considering C ∞ -functions ϕ 1 , . . . , ϕ q+1 such that ϕ 1 = h, ϕ q+1 = h and that (dϕ 1 , . . . , dϕ q ) forms a basis of i X (Ω X ) + Rdh. Let X 1 , . . . , X q be vector fields satisfying L X j (ϕ i ) = δ i j , 1 ≤ j ≤ q, 1 ≤ i ≤ q + 1, and such that for every ( u 1 , . . . ,

u k ) ∈ U k , 1 ≤ k ≤ ν -1, we have: q ∑ j=1 d(L X j L f u k . . . L f u 1 ( h)) ∧ dϕ j = dΘ u 1 ... u k ( 18
)
where Θ u 1 ... u k is the one-differential form stated in [START_REF] Plestan | Linearization by generalized inputoutput injection[END_REF]. Then X 1 , . . . , X q+1 satisfy condition 2) of theorem 1. Remark 1: According to expression (12) of lemma 2, expression (18) is then equivalent to

d(i X ( ω u 1 ... u k )) = 0.
Some comments on the procedure of calculation of vector fields X, X 1 , . . . , X q , X q+1 :

1) The calculation of the vector field X requires only the knowledge of expressions of f u and h. 2) X q+1 can be directly computed from the knowledge of X, f u , h and h. 3) For 1 ≤ i ≤ q + 1, the functions ϕ i can be deduced from X, f u and h and h. 4) Finally, we end these comments by giving the algorithm of computation of ( X 1 , . . . , X q ):

Computation of ( X 1 , . . . , X q ): Based on the construction of B k and the functions ϕ 1 , . . . , ϕ q+1 , the set {ϕ 1 , . . . , ϕ q+1 } ∪ {L f u k . . .

L f u 1 ( h); 1 ≤ k ≤ ν -1, ( u 1 , . . . , u k ) ∈ U k }
forms a local system of coordinates, which we denote by (ξ , ξ ), and where ξ = (ϕ 1 , . . . , ϕ q+1 ) = (ξ 1 , . . . , ξ q+1 ) ξ = ( ξ 1 , . . . , ξ ν-1 ), ξ k = ( ξ k1 , . . . , ξ k, d k )

where {d ξ k1 , . . . ,

d ξ k, d k } = {dL f u k . . . L f u 1 ( h); ( u 1 , . . . , u k ) ∈ U k }, and {[dL f u k . . . L f u 1 ( h)]; ( u 1 , . . . , u k ) ∈ U k } = B k+1
. Therefore, we adopt the following notations:

ω u 1 ... u k = ω ki = d ξ ki L X j ( ξ ki ) = L X j L f u k . . . L f u 1 ( h) = X j ki
Using the fact that L X j (ϕ j ) = L X j (ξ j ) = δ i j , we obtain

X j = ∂ ∂ ξ j + ν-1 ∑ k=1 d k ∑ i=1 X j ki ∂ ∂ ξ ki
. Thus, the expression (12) can be rewritten:

i X ( ω ki ) = d ξ ki - q ∑ j=1 X j ki dξ j + Θ ki ( 19
)
where the Θ ki 's are one-differential forms depending at most on X j li , 1 ≤ l ≤ k -1, 1 ≤ j ≤ q + 1. The calculation of X j ki 's follows from the following recursive procedure:

• We start by computing X j 1i , 1 ≤ j ≤ q, 1 ≤ i ≤ d 1 : For k = 1, expression (19) becomes: i X ( ω 1i ) = d ξ 1i - q ∑ j=1 X j 1i dξ j + Θ 1i
, where Θ 1i is a known one-differential form which does not depend on X 1 , . . . , X q . Now condition (18) of theorem 2 yields to:

q ∑ j=1 d( X j 1i ) ∧ dξ j = dΘ 1i , f or 1 ≤ i ≤ d 1
Hence the X j 1i 's follows from the simple PDE system:

for 1 ≤ j, l ≤ q, 1 ≤ i ≤ d 1 , ∂ X j 1i ∂ ξ l - ∂ X l 1i ∂ ξ j = θ jl 1i for 1 ≤ j ≤ q, 1 ≤ i ≤ d 1 , 1 ≤ t ≤ ν -1, 1 ≤ s ≤ d t , ∂ X j 1i ∂ ξ ts = θ jt 1si
(20) where θ jl 1i and θ jt 1si are known functions depending only on the known vector field X q+1 .

• Assuming that for 1

≤ j ≤ q, 1 ≤ l ≤ k -1, 1 ≤ i ≤ d l ,
the functions X j li are calculated, and let us compute

X j ki , 1 ≤ i ≤ d k .
As for the first step, using expressions (18), (19) it follows that:

q ∑ j=1 d( X j ki ) ∧ dξ j = dΘ ki which implies: for 1 ≤ j, l ≤ q, 1 ≤ i ≤ d k , ∂ X j ki ∂ ξ l - ∂ X l ki ∂ ξ j = θ jl ki for 1 ≤ j ≤ q, 1 ≤ i ≤ d k , 1 ≤ t ≤ ν -1, 1 ≤ s ≤ d t , ∂ X j ki ∂ ξ ts = θ jt ksi
(21) where θ jl ki and θ jt ksi are known functions depending on the computed functions X j li , 1 ≤ l ≤ k -1. The proof of theorem 2 is based on the following proposition.

Proposition 1:

Assuming that system (1) is cascade-observable, then the following properties hold:

1) Let [Y ] = [Z u 1 ...u ν-1 ] ∈ B * ν and X = (-1) ν-1 Y u 1 ...u ν-1 , then X = ∂ ∂ z 1 + q ∑ i=2 a i ∂ ∂ z i + n-q ∑ i=1 b i (z, z) ∂ ∂ z i (22)
where the a i 's are constants. Moreover, we have:

i X (Ω X ) = i ∂ ∂ z 1 (Ω ∂ ∂ z 1 ) modulo(Rdz 1 ) (23) 2) Let [ Y ] = [ Z u 1 ... u ν-1
] be an element of B * ν and X q+1 = (-1) ν-1 Y u 1 ... u ν-1 , and setting ϕ i = z i , for 1 ≤ i ≤ q and ϕ q+1 = z 1 . Let ( X 1 , . . . , X q ) be a sequence of vector fields such that L X j (ϕ i ) = δ i j for 1 ≤ j ≤ q, 1 ≤ i ≤ q + 1 and satisfying condition (18) of theorem 2, then:

X q+1 = ∂ ∂ z 1 + n-q ∑ i=2 a i ∂ ∂ z i (24)
where the a i 's are constants and for j = 1, . . . , q,

X j = ∂ ∂ z j + n-q ∑ i=2 β i j (z, z) ∂ ∂ z i (25) Moreover, if π = dz 1 ∧. . .∧dz q , X 0 j = ∂ ∂ z j , for 1 ≤ j ≤ q, X 0 q+1 = ∂ ∂ z 1
, and X 0 = ( X 0 1 , . . . , . . . , X 0 q+1 ), then,

i X (Ω X π ) = i X 0 (Ω X 0 π ) modulo(dA q + Rd z 1 ) (26)
where A q = C ∞ {z 1 , . . . , z q } stands for the ring of C ∞functions of (z 1 , . . . , z q ). Proof of proposition 1. Setting A i (u), A i (u) to be the respective ith rows of A(u), A(u), and ψ i (u, y), ψ i (u, z, y) are the ith components of ψ and ψ. In the (z, z)-system of coordinates f u takes the form:

f u = q ∑ i=1 (A i (u)z+ ψ i (u, z 1 )) ∂ ∂ z i + n-q ∑ i=1 ( A i (u) z+ ψ i (u, z, z 1 )) ∂ ∂ z i (27) Let B k+1 = {[d(L f u k . . . L f u 1 (z 1 ))]; (u 1 , . . . , u k ) ∈ U k }, B k+1 = {[d(L f u k . . . L f u 1 ( z 1 ))]; ( u 1 , . . . , u k ) ∈ U k } be the respective bases of D k+1 /D k and D k+1 / D k .
From claim 1, we know that :

[d(L f u k . . . L f u 1 (z 1 ))] = [dCA(u 1 ) . . . A(u l )z] (28) [d(L f u k . . . L f u 1 ( z 1 ))] = [d C A( u 1 ) . . . A( u l ) z] ( 29
)
and from claim 2, the flags of co-distributions

D 0 = 0 ⊂ . . . ⊂ D ν = D ν+1 ; 0 = D 0 /D ν ⊂ . . . ⊂ D ν /D ν = D ν+1 /D ν are of constant dimensions. Now setting n k = dim(D k ), n k = dim( D k /D ν ), then we have n 0 = 0 < n 1 = 1 < . . . < n ν = q; n 0 = 0 < n 1 = 1 < . . . < n ν = n -q.
Moreover, after a (z, z)linear change of coordinates, it can be assumed that:

B k = ([dz 1+n k-1 ], . . . , [dz n k ]) B k = ([d z 1+ n k-1 ], . . . , [d z n k ]) (30)
and that in this new system of coordinates A(u), A(u) take the following triangular structure:

for 1 ≤ k ≤ ν -1, for n k-1 + 1 ≤ i ≤ n k , A i (u)z = a i1 (u)z 1 + a i2 (u)z 2 + . . . + a i,n k+1 (u)z n k+1 (31) for 1 ≤ k ≤ ν -1, for n k-1 + 1 ≤ i ≤ n k , A i (u) z = a i1 (u) z 1 + a i2 (u) z 2 + . . . + a i, n k+1 (u) z n k+1
(32)

• Proof of property 1) of proposition 1: Proof of expression ( 22): Let Y = Z u 1 ...u ν-1 be a fixed element of B * ν (the dual basis of B ν ), thus after reordering B ν , it can be assumed that [dL f u ν-1 . . . L f u 1 (z 1 )] = dz q , and hence:

dz n ν-1 +i (Y ) = L Y (z n ν-1 +i ) = δ qi , 1 ≤ i ≤ n ν -n ν-1 (33) Recalling that B *
ν is a basis of ∆ ν-1 /∆ ν and that ∆ ν ⊂ ∆ ν-1 ⊂ . . ., and that ∆ k = Ker(D k ). Combining this last fact with (33), it follows that:

Y = ∂ ∂ z q + n-q ∑ i=1 β * j (z, z) ∂ ∂ z j (34)
Now setting X = (-1) ν-1 Y u 1 ...u ν-1 , and recalling that

Y u 1 ...u ν-1 = [ f u ν-1 , [. . . , [ f u 1 ,Y ] . . .]], then (27), (31), ( 34
))
give rise to:

Y u 1 ...u ν-1 = q ∑ i=1 a i ∂ ∂ z i + n-q ∑ i=1 b i (z, z) ∂ ∂ z i ( 35
)
where the a i 's are constants. Finally, from lemma 1, it follows that

dz 1 (Y u 1 ...u ν-1 ) = -dL f u ν-1 (z 1 )(Y u 1 ...u ν-2 ) = . . . = (-1) ν-1 d(L f u ν-1 . . . L f u 1 (z 1 ))(Y ) = (-1) ν-1 dz q (Y ) = (-1) ν-1 , hence: X = (-1) ν-1 Y u 1 ...u ν-1 = ∂ ∂ z 1 + q ∑ i=2 a i ∂ ∂ z i + n-q ∑ i=1 b i (z, z) ∂ ∂ z i (36) Proof of expression (23):
Recalling that Ω X is the real vector space spanned by the 2-differential forms ω u

1 ...u k = L f u k (i X (ω u 1 ...u k-1 )) ∧ dz 1 , where ω u = dL f u (z 1 ) ∧ dz 1 = d(CA(u))z ∧ dz 1 .
Using the expression of X given in (36) and the fact that i X (dϕ ∧ dz 1 ) = dϕ -L X (ϕ)dz 1 (see claim 3), then a simple algebraic computation yields to:

i X (Ω X ) + Rdz 1 = Span{dCA(u 1 ) . . .CA(u k )z; k ≥ 1, u i ∈ R m } + Rdz 1 (37)
In particular, expression (37) holds for X

= ∂ ∂ z 1 , hence i X (Ω X ) = i ∂ ∂ z 1 (Ω ∂ ∂ z 1 ) modulo(Rdz 1 ).
• The proof of property 2) of proposition 1 can be obtained by following the same procedure as for the property 1).

Proof of theorem 2.

The sufficient condition is stated in theorem 1. Necessary condition: Since conditions a) and b) of theorem 2 are intrinsic (they do not depend on the system of coordinates), it suffices to check them for the cascade-observable system (1). 1) Condition a) of theorem 2 is a straightforward consequence of claim 2. 2) Condition b)-1) of theorem 2:

Let X = (-1) ν-1 Y u 0 1 ...u 0 ν-1 be the vector field stated in b)-1) of theorem 2, from ( 22)-(23) of proposition 1, we know that:

X = ∂ ∂ z 1 + ∑ q i=2 a i ∂ ∂ z i + ∑ n-q i=1 b i (z, z) ∂ ∂ z i i X (Ω X ) = i ∂ ∂ z 1 (Ω ∂ ∂ z 1 ) modulo(Rdz 1 ) (38)
Combining (38) with the fact that ∂ ∂ z 1 satisfies condition 1) of theorem 1 , it can be easily checked that X meets conditions 1-i) to 1)-iv) of theorem 1. Hence condition b)-1) of theorem 2 is satisfied. 3) Condition b)-2) of theorem 2:

Let X q+1 = ± Y u 0 1 ... u 0 ν-1 and considering linear functions ϕ 1 , . . . , ϕ q+1 satisfying ϕ 1 = h = z 1 , ϕ q+1 = h = z 1 and such that (dϕ 1 , . . . , dϕ q ) forms a basis of i X (Ω X ) + Rdz 1 . Up to a linear change of coordinates, it can be assumed that:

(dϕ 1 , . . . , dϕ q ) = (dz 1 , . . . , dz q ) (39)

Now considering vector fields X 1 , . . . , X q such that L X j (ϕ i ) = δ i j , 1 ≤ j ≤ q, 1 ≤ i ≤ q + 1, and satisfy- ing condition (18) of theorem 2. Namely, for every

( u 1 , . . . , u k ) ∈ U k , 1 ≤ k ≤ ν -1, q ∑ j=1 d(L X j L f u k . . . L f u 1 ( h)) ∧ dϕ j = dΘ u 1 ... u k ( 40
)
where Θ u 1 ... u k is the one-differential form stated in [START_REF] Plestan | Linearization by generalized inputoutput injection[END_REF]. In order to check condition b)-2) of theorem 2, we will show that ( X 1 , . . . , X q+1 ) meet condition 2) of theorem 1.

From 2) of proposition 1, we know that:

X q+1 = ∂ ∂ z 1 + n-q ∑ i=2 a i ∂ ∂ z i (41)
where the a i 's are constants and for j = 1, . . . , q,

X j = ∂ ∂ z j + n-q ∑ i=2 β i j (z, z) ∂ ∂ z i (42)
Moreover, if we set π = dz 1 ∧ . . . ∧ dz q , X 0 j = ∂ ∂ z j , for

1 ≤ j ≤ q, X 0 q+1 = ∂ ∂ z 1
, and X 0 = ( X 0 1 , . . . , . . . , X 0 q+1 ), then:

i X (Ω X π ) = i X 0 (Ω X 0 π ) modulo(dA q + Rd z 1 ) (43)
where A q = C ∞ {z 1 , . . . , z q }. Thus, we have:

• By construction, L X i (ϕ j ) = δ i j , hence condition 2-i) of theorem 1 is satisfied. • From the check of the proof of the necessary condition of theorem 1 given in the subsection II-B, we know that the (q + 1)-tuple of vector fields ( X 0 1 , . . . , . . . , X 0 q+1 ) meet conditions 2-ii) of theorem 1, namely, dim(Ω X 0 π ) = nq -1. Consequently, to show that ( X 1 , . . . , X q+1 ) meet condition 2-ii) of theorem 1 (dim(Ω X π ) = nq -1), it suffices to show that:

Ω X π = Ω X 0 π (44)
By definition, Ω X π = ∑ u∈R m L f u (i X (Ω X π )) ∧ dz 1 . . . ∧ dz q ∧ d z 1 , where

f u = q ∑ i=1 (A i (u)z + ψ i (u, y)) ∂ ∂ z i + n-q ∑ i=1 ( A i (u) z + ψ i (u, z, y)) ∂ ∂ z i
. Moreover, it is easy to see that L f u (dA q ) ∧ dz 1 . . . ∧ dz q ∧ d z 1 = 0, thus L f u (dA q + Rd z 1 )∧dz 1 . . .∧dz q ∧d z 1 ⊂ Rd( C A(u) z)∧ dz 1 . . . ∧ dz q ∧ d z 1 ⊂ Ω X 0 π . Combining this last fact with (43), we deduce (44).

• Condition 2-iii) of theorem 1 consists to verify that d ω = 0, for every ω ∈ i X (Ω X π ). This property follows from the facts that d(i X 0 (Ω X 0 π )) = 0, d(dA q + Rd z 1 )) = 0 and (43).

• Condition 2-iv) of theorem 1:

It consists to show that the real vector space n-q-1 (i X (Ω X π )) ∧ dϕ 1 ∧ . . . ∧ dϕ q+1 is of dimension 1. According to (39) and (43), we have:

n-q-1 (i X (Ω X π )) ∧ dϕ 1 ∧ . . . ∧ dϕ q+1 = n-q-1 (i X 0 (Ω X 0 π )) ∧ dz 1 ∧ . . . ∧ dz q ∧ d z 1 . From the check of the proof of the necessary condition of theorem 1 given in the subsection II-B, we know that X 0 meets condition 2-iv) of theorem 1. Hence n-q-1 (i X 0 (Ω X 0 π )) ∧ dz 1 ∧ . . . ∧ dz q ∧ d z 1 is of dimension 1. This ends the proof of theorem 2.

IV. CONCLUSION

Motivated by the existence of an observer design for a class of observable cascade systems. In this paper, we have characterized the class of nonlinear systems which can be transformed by a local change of coordinates to a cascade form. First, we have stated necessary and sufficient conditions. Next, we have derived an algorithm permitting to transform a nonlinear system into such cascade observable form. Its extension to general multi-output case is a difficult task and requires solving complex PDE.

Claim 2 :

 2 • The flags of co-distributions D 0 ⊂ . . . ⊂ D k ⊂ . . .; D 0 /D ♯ ⊂ . . . ⊂ D k /D ♯ ⊂ . . . are of constant dimensions and defined as follows: a) D 1 = Span(dCz), and for k ≥ 2, D k is spanned by the set of one-forms {dCz} ∪ {dCA(u 1 ) . . . A(u l )z; 1 ≤ l ≤ k -1, u j ∈ R m }. b) Similarly, D 1 /D ♯ can be identified with the codistribution Span(d C z), and for k ≥ 2, D k /D ♯ is isomorphic to the co-distribution spanned by the set of oneforms {d C z} ∪ {d C A(u 1 ) .

  forms a basis of D k , hence the last term of the right hand expression (15) takes the form k-1

1 ]

 1 a) The flag of co-distributions D 0 ⊂ . . . ⊂ D ν = D ν+1 , D 0 /D ν ⊂ . . . ⊂ D ν /D ν = D ν+1 /D ν are of constant dimension on some neighborhood of x 0 , and dim(D ν ) = q, dim( D ν /D ♯ ) = nq b) Let B ν and B ν be any fixed bases of D ν /D ν-1 and D ν / D ν-1 (see the construction (8)). Let Y and Y be any fixed vector fields of the form [Y ] = [Z u 0 1 ...u 0 ν-∈ B * ν and [ Y ] = [ Z u 0 1 ... u 0 ν-1