Fazia Bedouhene
email: fbedouhene@yahoo.fr

Youcef Ibaouene
email: youcef.ibaouene@etu.univ-rouen.fr

Omar Mellah
email: omellah@yahoo.fr

Paul Raynaud De Fitte

semilinear equations with Weyl almost periodic coefficients

Keywords: Weyl almost periodic, linear and semilinear differential equation, mild solution

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

In 1927, Hermann Weyl [START_REF] Weyl | Integralgleichungen und fastperiodische Funktionen[END_REF] introduced a generalization of Bohr and Stepanov almost periodic functions. Since then, generalized almost periodic functions bear his name and numerous contributions to theory of almost periodic functions have been brought, see for instance [START_REF] Andres | Hierarchy of almost-periodic function spaces[END_REF][START_REF] Andres | On some almost-periodicity problems in various metrics[END_REF][START_REF] Besicovitch | Almost periodicity and general trigonometric series[END_REF][START_REF] Besicovitoh | Almost periodic functions[END_REF][START_REF] Danilov | On Weyl almost periodic selections of multivalued maps[END_REF][START_REF] Guter | Elements of the theory of functions[END_REF][START_REF] Ursell | Parseval's theorem for almost-periodic functions[END_REF]. Weyl-almost periodic functions are very important and have several applications in dynamical systems, in particular in symbolic dynamics [START_REF] Iwanik | Weyl almost periodic points in topological dynamics[END_REF], where an important class of Weyl almost periodic trajectories are the regular Toeplitz sequences of Jacobs and Keane [START_REF] Jacobs | 0-1-sequences of toeplitz type[END_REF].

In recent years, many authors have been interested in the study of existence and uniqueness of almost periodic solutions to different types of differential equations with Stepanov almost periodic coefficients in both deterministic and stochastic cases. We can mention, in the deterministic case, the work by Andres and Pennequin [START_REF] Andres | On Stepanov almost-periodic oscillations and their discretizations[END_REF][START_REF] Andres | On the nonexistence of purely Stepanov almost-periodic solutions of ordinary differential equations[END_REF], Long and Ding [START_REF] Long | Composition theorems of stepanov almost periodic functions and stepanov-like pseudo-almost periodic functions[END_REF], Ding et al. [START_REF] Ding | Almost periodic solutions to abstract semilinear evolution equations with Stepanov almost periodic coefficients[END_REF], Hu and Mingarelli [START_REF] Hu | Bochner theorem and stepanov almost periodic functions[END_REF], Henriquez [START_REF] Henríquez | On stepanov-almost periodic semigroups and cosine functions of operators[END_REF], Rao [START_REF] Rao | On the Stepanov-almost periodic solution of a second-order operator differential equation[END_REF], Zaidman [START_REF] Zaidman | An existence result for Stepanoff almost-periodic differential equations[END_REF]. Particularly, in [START_REF] Ding | Almost periodic solutions to abstract semilinear evolution equations with Stepanov almost periodic coefficients[END_REF], [START_REF] Long | Composition theorems of stepanov almost periodic functions and stepanov-like pseudo-almost periodic functions[END_REF], and [START_REF] Ait Dads | Behavior of bounded solutions for some almost periodic neutral partial functional differential equations[END_REF], the authors show the existence and uniqueness of almost periodic solution for an abstract semilinear evolution equation with Stepanov almost periodic coefficients. They prove the existence and uniqueness of a Bohr almost 1 periodic mild solution. Nevertheless, in the work [START_REF] Andres | On the nonexistence of purely Stepanov almost-periodic solutions of ordinary differential equations[END_REF], Andres and Pennequin proved the nonexistence of purely Stepanov almost periodic solutions of ordinary differential equations in uniformly convex Banach spaces. Until now, to our knowledge, there are only few works dedicated to the study of existence and uniqueness of Weyl almost periodic solutions to some differential equations. The first investigation in this direction is due to Lenka Radova's [START_REF] Radová | Theorems of Bohr-Neugebauer-type for almost-periodic differential equations[END_REF], but her study was limited to the scalar equation x ′ = ax + f (t), where a ∈ R and f : R → R is an essentially bounded Weyl almost periodic function. The second one is due to Andres et al. [START_REF] Andres | Almost-periodic solutions in various metrics of higher-order differential equations with a nonlinear restoring term[END_REF], and can be seen as an extension of Radova's work. Precisely, the authors investigate almost periodic solutions in various senses (Stepanov, Weyl, Besicovitch) of higher-order scalar differential equations with a nonlinear Lipschitz restoring term and a bounded Stepanov, Weyl or Besicovitch almost periodic forcing term. One can also mention the recent paper by Kostic [START_REF] Kostic | Generalized almost periodic solutions and generalized asymptotically almost periodic solutions of inhomogenous evolution equations[END_REF], where Radova's study was addressed in the context of linear evolution equations with bounded and (asymptotically) Weylalmost periodic coefficients.

Motivated by the previous papers, the aim of this paper is to generalize the previous investigations to the case of semilinear evolution equations, without any boundedness restriction on the forcing term. More precisely, we consider the following abstract semilinear evolution equation:

u ′ (t) = Au(t) + f (t, u(t)) ∀t ∈ R, (1)
where A : D (A) ⊂ X → X is a linear operator (possibly unbounded) which generates an exponentially stable C 0 -semigroup on a Banach space X, and (t, x) → f (t, x) is a parametric Weyl almost periodic function of degree p ≥ 1. We show that, under some conditions, (1) has a unique solution which is bounded and Weyl almost periodic. Note that, contrarily to the Stepanov case, the solution can be purely Weyl almost periodic.

The major problem encountered in our study is that the space of Weyl almost periodic functions endowed with the Weyl semi-norm is not complete [START_REF] Andres | On some almost-periodicity problems in various metrics[END_REF], which makes the approach and classical tools of functional analysis inapplicable, especially the fixed-point theorem in Banach spaces. To overcome this difficulty, we have opted for the method used by Kamenskii et al [START_REF] Kamenskii | Weak averaging of semilinear stochastic differential equations with almost periodic coefficients[END_REF], which consists in showing "manually" that the unique bounded mild solution to (1), obtained using the fixed-point theorem, is Weyl almost periodic.

This work is organized as follows: In the second section, we recall some definitions and results related to Stepanov almost periodic and Weyl almost periodic functions. In section 3, we present three essential parts. In the first part, based on a compactness property established by Danilov [START_REF] Danilov | On Weyl almost periodic selections of multivalued maps[END_REF], we give a new superposition result in the space of Weyl almost periodic functions. In the second part, after providing examples of bounded and purely Weyl almost periodic solutions, we tackle the problem of existence and uniqueness of a bounded Weyl almost periodic solution to an abstract linear differential equation. Finally, in the third part, inspired from [START_REF] Kamenskii | Weak averaging of semilinear stochastic differential equations with almost periodic coefficients[END_REF], we prove that equation (1) has a unique bounded mild solution which is Weyl almost periodic.

Notations and Preliminaries

First, we give some basic definitions and results on Stepanov almost periodic functions and Weyl almost periodic functions (for more details, see [START_REF] Andres | Hierarchy of almost-periodic function spaces[END_REF][START_REF] Besicovitch | Almost periodicity and general trigonometric series[END_REF][START_REF] Besicovitoh | Almost periodic functions[END_REF]).

We denote by R the set of real numbers and by X a Banach space endowed with the norm . . We denote also by CB(R, X) the Banach space of continuous and bounded functions from R to X, endowed with the norm

u ∞ = sup t∈R u(t) .

Stepanov and Weyl norm

Given l > 0, the Stepanov norm associated with l of a locally integrable function f : R → X, i.e f ∈ L p loc (R, X) (p ≥ 1), is defined by

f S p l = sup ξ∈R 1 l ξ+l ξ f (t) p dt 1 p
.

The Weyl norm is defined by

f W p = lim l→+∞ f S p l .
The limit always exists for l → +∞ [

Almost periodicity in Stepanov and Weyl senses

Let us recall some definitions of Stepanov and Weyl almost periodic functions. Recall that a set T of real numbers is relatively dense if there exists a real number k > 0, such that T ∩ [a, a + k] = ∅, for all a in R. Definition 2.1 ([2, p.76-77], [START_REF] Andres | Hierarchy of almost-periodic function spaces[END_REF][START_REF] Andres | On some almost-periodicity problems in various metrics[END_REF], [10, p.77], [11, p.173], [14, p.188]) A function f ∈ L p loc (R, X) is said to be almost periodic in the sense of Stepanov (or p-Stepanov almost periodic if we want to highlight the degree p) if, for every ε > 0, the set

T S p (ε, f ) =    τ ∈ R; sup ξ∈R ξ+1 ξ f (t + τ ) -f (t) p dt 1 p < ε  
 is relatively dense. The number τ is called an ε-Stepanov almost-period (or Stepanov ε-translation number of f ). We denote the set of all such functions by S p AP(R, X).

Definition 2.2 ([4, 5], [10, p.78], [14, p.190]) A function f ∈ L p loc (R, X) is called Weyl almost periodic if, for every ε > 0, there exist l = l(ε) > 0 such that, the set

T S p l (ε, f ) =    τ ∈ R; sup ξ∈R 1 l ξ+l ξ f (t + τ ) -f (t) p dt 1 p < ε    (2)
is relatively dense. This means that for every ε > 0, there exists l = l(ε) > 0 such that f ∈ S p l AP(R, X). The number τ is called a Weyl ε-translation number of f . We denote the set of all such functions by W p AP(R, X).

There is, however, another interesting definition given by Ursell [START_REF] Ursell | Parseval's theorem for almost-periodic functions[END_REF]:

Definition 2.3 ([27]) A function f ∈ L p loc (R, X
) is said to be Weyl *-almost periodic (or W.a.p following Ursell's notations) if, for every ε > 0, there exists l = l(ε) such that the set

T * S p l (ε, f ) =    τ ∈ R; 1 l l 0 f (t + τ ) -f (t) p dt 1 p < ε  
 is relatively dense. We denote the set of all such functions by W p * AP(R, X).

Ursell [START_REF] Ursell | Parseval's theorem for almost-periodic functions[END_REF] has shown that Theorem 2.4 The classes W p AP(R, X) and W p * AP(R, X) coincide.

This identification is very important when establishing our main result.

Remark 2.5 1. Some authors name the Weyl almost periodicity by equi-Weyl almost periodicity (e.g [START_REF] Andres | Hierarchy of almost-periodic function spaces[END_REF][START_REF] Andres | On some almost-periodicity problems in various metrics[END_REF][START_REF] Radová | Theorems of Bohr-Neugebauer-type for almost-periodic differential equations[END_REF]), distinguishing between the above definition and the following one introduced by Kovanko [START_REF] Kovanko | Sur la compacité des systèmes de fonctions presque périodiques généralisées de H. Weyl[END_REF]: A function f ∈ L p loc (R, X) is said to be almost periodic in the sense of Weyl according to [START_REF] Andres | Hierarchy of almost-periodic function spaces[END_REF][START_REF] Andres | On some almost-periodicity problems in various metrics[END_REF][START_REF] Radová | Theorems of Bohr-Neugebauer-type for almost-periodic differential equations[END_REF] if, for every ε > 0, the set

T W p (ε, f ) =    τ ∈ R; lim l→+∞ sup ξ∈R 1 l ξ+l ξ f (t + τ ) -f (t) p dt 1 p < ε  
 is relatively dense. We denote the set of all such functions by W p (R, X).

Of course the space W p AP(R, X) is an intermediate space between S p AP(R, X) and W p (R, X), i.e. we have

S p AP(R, X) ⊂ W p AP(R, X) ⊂ W p (R, X).
According to [START_REF] Andres | Hierarchy of almost-periodic function spaces[END_REF][START_REF] Andres | On some almost-periodicity problems in various metrics[END_REF], the previous inclusions are strict.

2. Every Weyl almost periodic function of degree p is S p -bounded and also W p -bounded and also (see [START_REF] Andres | Hierarchy of almost-periodic function spaces[END_REF]). Actually, the spaces

BS p (R, X) :=    f ∈ L p loc (R, X); sup ξ∈R 1 L ξ+L ξ f (t) p dt 1 p < ∞    ∀L > 0,
and

BW p (R, X) :=    f ∈ L p loc (R, X); lim l→∞ sup ξ∈R 1 l ξ+l ξ f (t) p dt 1 p < ∞    , coincide, see [4].

Weyl almost periodic functions depending on a parameter

1. We say that a parametric function f : R×X → X is Weyl almost periodic if, for every x ∈ X, the function f (., x) is Weyl almost periodic, we denote by W p AP(R × X, X) the space of such functions.

2. A function f : R × X → X, (t, u) → f (t, u) with f (., u) ∈ L p loc (R, X) for each u ∈ X is said to be Weyl almost periodic in t ∈ R uniformly with respect compact subsets of X if, for each ε > 0, there exists l = l(ε) > 0 and for all compacts K in X, the set

T S p l (ε, f, K) =    τ ∈ R; sup u∈K sup ξ∈R 1 l ξ+l ξ f (t + τ, u) -f (t, u) p dt 1 p < ε  
 is relatively dense. We denote by W p AP K (R × X, X) the set of such functions.

It is clear that

W p AP K (R × X, X) ⊂ W p AP(R × X, X).
Let us introduce some notations due to Danilov [START_REF] Danilov | On Weyl almost periodic selections of multivalued maps[END_REF]. We denote by

M * p (R, X) the set of functions f ∈ BS p (R, X) such that lim δ→0 + lim l 0 →∞ sup l>l 0 sup ξ∈R 1 l sup T ⊆[ξ,ξ+l]: κ(T )≤δl T f (t) p dt 1 p = 0
holds, where κ is the Lebesgue measure on R.

On the space X we also consider the truncated norm . * = min{1, . }. Let W 0 AP(R, X) := W 1 AP(R, (X, . * )) denote the space of Weyl almost periodic functions f : R → X (of degree 1), when X is endowed with . * . From Danilov [START_REF] Danilov | On Weyl almost periodic selections of multivalued maps[END_REF], we have Theorem 2.6 (Danilov [START_REF] Danilov | On Weyl almost periodic selections of multivalued maps[END_REF])

1. The following characterization holds true:

W p AP(R, X) = W 0 AP(R, X) ∩ M * p (R, X). (3)
2. If f ∈ W p AP(R, X), then for every ε, δ > 0 there exist a number l > 0 and a compact set K ε,δ ⊂ X such that

sup ξ∈R 1 l κ {t ∈ [ξ, ξ + l] : d (f (t), K ε,δ ) ≥ δ} < ε.
Note that, as for almost periodicity in Stepanov sense, which is a metric property [8, Remark 2.4], it can be easily shown that almost periodicity in Weyl sense is a metric property too. In fact, using the arguments in [8, Remark 2.4], let g = exp +∞ n=2 g n , where g n is a 4n-periodic function given by

g n (t) = β n 1 - 2 α n |t -n| 1l [n-2 αn ,n+ 2 αn ] (t), t ∈ [-2n, 2n],
where α n ∈]0, 1 2 [, and β n > 0. As shown in [START_REF] Andres | On the nonexistence of purely Stepanov almost-periodic solutions of ordinary differential equations[END_REF], for α n = 1/n 5 and β n = n 3 , we have

g S 1 ≥ αnβ 2 n 6 → ∞.
Which means that g is not S 1 -bounded. Hence g cannot be in W 1 AP(R), . However, the function g is almost periodic in Lebesgue measure, that is, it belongs to S 0 AP(R) := S 1 AP R, (R, . * ) . In view of the inclusion S 0 AP(R) ⊂ W 0 AP(R), we deduce that g belongs to W 0 AP(R). This shows also that the inclusion W p AP(R) ⊂ W 0 AP(R) is strict, and that M * p (R, X) depends on the norm of the space X.

3 Main results

A superposition theorem in W p AP(R, X)

In this section, we establish a composition theorem for Weyl almost periodic functions.

Our result is based on Theorem 2.6. In the following, we assume that 1 p = 1 q + 1 r with p, q and r ≥ 1.

Theorem 3.1 Let f ∈ W p AP K (R × X, X). Assume that there exists a positive function L(.) ∈ BW r (R), such that f (t, u) -f (t, v) ≤ L(t) u -v ∀t ∈ R, u, v ∈ X. (4)
Then, for every x(.) ∈ W q AP(R, X), we have

f (., x(.)) ∈ W p AP(R, X).
Proof For a measurable set A ⊂ R and l > 0, let us introduce the notation

κ l (A) = sup ξ∈R 1 l κ {t ∈ [ξ, ξ + l] ∩ A} ,
and let us denote by A c the complementary set of A. Fix ε > 0. Let x(.) ∈ W q AP(R, X).

In view of (3), we can find δ ε > 0 and l 0 = l 0 (ε) > 0 such that, for every measurable set

A, if κ l 0 (A) ≤ δ ε , then sup x∈R 1 l 0 A∩[x,x+l 0 ] x(s) p ds 1 p ≤ ε. (5)
Using the second item of Theorem 2.6, for the above ε > 0 and the corresponding δ ε > 0, there exist a compact subset K ε ⊂ X and a positive number

l 1 = l 1 (ε) > 0 such that κ l 1 (T ε ) < δ ε , (6)
where

T ε = t ∈ R : d(x(t), K ε ) ≥ ε 24 L W r .
Hence, we obtain by taking l 2 = max(l 0 , l 1 ),

sup x∈R 1 l 2 Tε∩[x,x+l 2 ] x(s) p ds 1 p ≤ ε 24 L W r . ( 7
)
The compactness of K ε yields that for every t ∈ R, there exists

x * t ∈ K ε such that d(x(t), K ε ) = x(t) -x * t .
Now, since x(.) ∈ M * p (R, X) and {x * t ; t ∈ R} ⊂ K ε , we obtain by ( 7) that,

sup ξ∈R 1 l 2 Tε∩[ξ,ξ+l 2 ] x(t) -x * t q dt 1 q < ε 24 L W r . (8)
We also have

sup ξ∈R 1 l 2 T c ε ∩[ξ,ξ+l 2 ] x(t) -x * t q dt 1 q < ε 24 L W r . (9)
Using again the compactness of K ε , we can find a finite sequence

y 1 , y 2 , • • • , y n ∈ K ε such that K ε ⊂ n i=1 B y i , ε 24 L W r . ( 10
) Since x * t ∈ K ε , it follows that, for every t ∈ R, there exists i(t) ∈ {1, 2, • • • , n} such that sup ξ∈R 1 l 1 ξ+l ξ x * t -y i(t) q dt 1 q < ε 24 L W r . ( 11
)
On the other hand, as f ∈ W p AP K (R×X, X) and x(.) ∈ W q AP(R, X), for the above ε > 0, we can chose l 3 = l 3 (ε) > 0 and a common relatively dense set

T S p l (ε, x, f ) ⊂ R such that sup ξ∈R 1 l 3 ξ+l 3 ξ x(t + τ ) -x(t) q dt 1 q < ε 2 L W r , (12)
and

sup ξ∈R 1 l 3 ξ+l 3 ξ f (t + τ, v) -f (t, v) p dt 1 p < ε 12n , (13)
for all τ ∈ T S p l (ε, x, f ) and v ∈ K ε . Furthermore, by triangular inequality, we have, for l = max(l 2 , l 3 ) and for every τ ∈ T S p l (ε, x, f ),

sup ξ∈R 1 l ξ+l ξ f (t + τ, x(t + τ )) -f (t, x(t)) p dt 1 p ≤ sup ξ∈R 1 l ξ+l ξ f (t + τ, x(t + τ )) -f (t + τ, x(t)) p dt 1 p + sup ξ∈R 1 l ξ+l ξ f (t + τ, x(t)) -f (t, x(t)) p dt 1 p =:I 1 + I 2 .
Let us estimate I 1 and I 2 . For I 1 , by ( 4), ( 12) and Hölder's inequality ( r p , q p ), we have, for every τ ∈ T S p l (ε, x, f ),

I 1 = sup ξ∈R 1 l ξ+l ξ f (t + τ, x(t + τ )) -f (t + τ, x(t)) p dt 1 p ≤ L W r sup ξ∈R 1 l ξ+l ξ x(t + τ ) -x(t) q dt 1 q < ε 2 .
For I 2 , for all τ ∈ T S p l (ε, x, f ), we have

I 2 ≤ sup ξ∈R 1 l Tε∩[ξ,ξ+l] f (t + τ, x(t)) -f (t, x(t)) p dt + sup ξ∈R 1 l T c ε ∩[ξ,ξ+l] f (t + τ, x(t)) -f (t, x(t)) p dt 1 p := I 1 2 + I 2 2 .
First, we consider I 1 2 . Using triangular inequality, the condition (4), ( 8) and Hölder's inequality ( p r , p q ), we get, for all τ ∈ T S p l (ε, x, f ),

I 1 2 = sup ξ∈R 1 l Tε∩[ξ,ξ+l] f (t + τ, x(t)) -f (t, x(t)) p dt 1 p ≤ ε 24 + sup ξ∈R 1 l ξ+l ξ f (t + τ, x * t ) -f (t, x * t ) p dt 1 p + ε 24 := ε 12 + I * . By (13), for each τ ∈ T S p l (ε, x, f ) and i = 1, 2, • • • , n, we obtain that sup ξ∈R 1 l ξ+l ξ f (t + τ, y i ) -f (t, y i ) p dt 1 p < ε 12n . (14)
Using ( 4) and ( 11), it follows that, for every τ ∈ T S p l (ε, x, f ),

I * ≤2 L W r sup ξ∈R 1 l ξ+l ξ y i(t) -x * t q dt 1 q + sup ξ∈R 1 l ξ+l ξ f (t + τ, y i(t) ) -f (t, y i(t) ) p dt 1 p ≤ ε 12 + n l j=1 sup ξ∈R 1 l ξ+l ξ f (t + τ, y j ) -f (t, y j ) p dt 1 p < ε 6 .
We thus have, for all τ ∈ T S p l (ε, x, f ),

I 1 2 = sup ξ∈R 1 l Tε∩[ξ,ξ+l] f (t + τ, x(t)) -f (t, x(t)) p dt 1 p < ε 4 .
For I 2 2 , by ( 4), ( 9), ( 11), ( 14) and Hölder's inequality, we follow the same steps than for I 1 2 , we obtain, for every τ ∈ T S p l (ε, x, f ),

I 2 2 = sup ξ∈R 1 l T c ε ∩[ξ,ξ+l] f (t + τ, x(t)) -f (t, x(t)) p dt 1 p < ε 4 .
Then, for all τ ∈ T S p l (ε, x, f ), we get that

I 2 = sup ξ∈R 1 l ξ+l ξ f (t + τ, x(t)) -f (t, x(t)) p dt 1 p < ε 2 .
Finally, combining I 1 and I 2 , we obtain, for every τ ∈ T S p l (ε, x, f ),

sup ξ∈R 1 l ξ+l ξ f (t + τ, x(t + τ )) -f (t, x(t)) p dt 1 p < ε, that is, f (., x(.)) ∈ W p AP(R, X).
Now, we discuss the Weyl almost periodic solutions to linear differential equations.

Existence and uniqueness of bounded Weyl almost periodic solution to abstract linear differential equation

We investigate now the Weyl almost periodic solutions to

u ′ (t) = Au(t) + f (t), (15)
where A : D (A) ⊂ X → X is a linear operator which generates an exponentially stable C 0 -semigroup (T (t)) t≥0 on a Banach space X, and f : R → X is Weyl almost periodic. We recall that a function u : R → X is a mild solution to [START_REF] Henríquez | On stepanov-almost periodic semigroups and cosine functions of operators[END_REF] if

u(t) = T (t -a)u(a) + b a T (t -s)f (s)ds, t ≥ s.
Let us also recall that a semigroup (T (t)) t∈R of linear operators is exponentially stable if there exist δ > 0 and M ≥ 1 such that

T (t) ≤ M e -δt . (16)
The main motivation of this study is that, contrarily the Stepanov case (see [START_REF] Andres | On the nonexistence of purely Stepanov almost-periodic solutions of ordinary differential equations[END_REF] and [START_REF] Bedouhene | Almost periodic solution in distribution for stochastic differential equations with Stepanov almost periodic coefficients[END_REF]), where it is shown the nonexistence of purely Stepanov almost periodic solutions for ordinary differential equations with Stepanov almost periodic coefficients, we can find, in the Weyl case, a bounded and purely Weyl almost periodic mild solution to Equation [START_REF] Henríquez | On stepanov-almost periodic semigroups and cosine functions of operators[END_REF]. The problem is closely related to Bohl-Bohr theorem which fails in the Weyl case. Indeed, the following example shows that there exist Weyl almost periodic functions such that their integrals are bounded but not Bohr almost periodic, nor Stepanov almost periodic. However, they are (purely) Weyl almost periodic.

Example 3.2 We consider the following function f : R → R defined by

f (t) = 1 for 0 < t < 1 2 ; 0 elsewhere, ( 17
)
which is purely Weyl almost periodic (see [4, p.145]). Let F : R → R with

F (t) = t -∞ f (s)ds =    0 for t ≤ 0; t for 0 < t < 1 2 ;
1 2 elsewhere, be a primitive of f . Clearly F is bounded on R. Moreover, for every t 1 , t 2 ∈ R, we have

|F (t 1 ) -F (t 2 )| ≤ |t 1 -t 2 | .
Thus F is uniformly continuous. So by [START_REF] Radová | Theorems of Bohr-Neugebauer-type for almost-periodic differential equations[END_REF]Proposition 4], F is Weyl almost periodic. On the other hand, for ε < 1 4 , there exists t ∈ [0, 1 4 ] such that, for every τ > ε, we have

|F (t + τ ) -F (t)| ≥ 1 4 > ε.
So, the function F is not Bohr almost periodic. We can show more, that is [START_REF] Iwanik | Weyl almost periodic points in topological dynamics[END_REF]. Its unique bounded solution is given by

F
x(t) =    0 for t ≤ 0; 1 -exp(-t)
for 0 < t < 1 2 ; ( exp(1) -1) exp(-t) elsewhere. . Then, every τ

such that τ + 1 2 > -ln ε √ exp(1)-1
, satisfies

x 1 2 -x τ + 1 2 = 2ε -( exp(1) -1) exp(-τ - 1 2 ) ≥ 2ε -ε = ε.
Which means that x is not almost periodic, nor Stepanov almost periodic.

Theorem 3.4 There exists a unique bounded mild solution to [START_REF] Henríquez | On stepanov-almost periodic semigroups and cosine functions of operators[END_REF]. This solution is Weyl almost periodic of degree p and is given by

u(t) = t -∞ T (t -s)f (s)ds. ( 19
)
Proof First, we show that the function u(t) = t -∞ T (ts)f (s)ds is well defined, i.e, lim a→-∞ t a T (ts)f (s)ds exists. We set

G(a) = t a T (t -s)f (s)ds.
Let us show that (G(a)) a∈R is Cauchy at -∞. If a < b, then by ( 16), we have

G(b) -G(a) ≤ b a T (t -s) f (s) ds ≤ M b a e -δ(t-s) f (s) ds.
Since f is Weyl almost periodic, we have f ∈ BS p L (R, X), for every L > 0. Thus, there exists N ∈ N, such that, by Hölder's inequality ( 1p , 1 q ), we have

G(b) -G(a) ≤ M N k=0 b-kL b-(k+1)L e -δ(t-s) f (s) ds ≤ M N k=0 b-kL b-(k+1)L
e -δq(t-s) ds

1 q b-kL b-(k+1)L f (s) p ds 1 p ≤ M L f S p L e -δq(t-b) 1 -e -δq(N +1)L 1 -e -δqL .
Consequently,

lim b→-∞ G(b) -G(a) ≤ lim b→-∞ M L f S p L e -δq(t-b) 1 -e -δq(N +1)L
1e -δqL = 0.

Thus the limit lim a→-∞ t a T (ts)f (s)ds exists. This means that u is well-defined. Moreover, it is easy to see that u given by ( 19) is a mild solution to [START_REF] Henríquez | On stepanov-almost periodic semigroups and cosine functions of operators[END_REF].

In the following, we show that u is bounded. By Hölder's inequality and ( 16), we obtain

u(t) ≤ t -∞ T (t -s) f (s) ds ≤ M ∞ k=0 e -δk t-k t-k-1 f (s) p ds 1 p . Then, we have sup t∈R u(t) ≤ M ∞ k=0 e -δk sup t∈R t-k t-k-1 f (s) p ds 1 p .
Since Weyl almost periodic functions of degree p are S p -bounded, we have

sup t∈R u(t) ∞ ≤ M f S p ∞ k=0 e -δk < ∞.
Thus u is bounded. Now, we prove that the solution to (15) is Weyl almost periodic. Using a change of variable and Hölder's inequality ( 1 p + 1 q = 1), we have any τ ∈ R,

u(t + τ ) -u(t) ≤ M 0 -∞ e δs f (t + s + τ ) -f (t + s) ds ≤ M 2 δq 1 q 0 -∞ e δps 2 f (t + s + τ ) -f (t + s) p ds 1 p
.

Let l > 0. Using Fubini's theorem, we get, for every τ ∈ R,

sup ξ∈R 1 l ξ+l ξ u(t + τ ) -u(t) p dt 1 p ≤ M 2 δq 1 q sup ξ∈R   1 l ξ+l ξ 0 -∞ e δps 2 f (t + s + τ ) -f (t + s) p ds 1 p p dt   1 p ≤ M 2 δq 1 q 0 -∞ e δps 2 sup ξ∈R 1 l ξ+l ξ f (t + s + τ ) -f (t + s) p dt ds 1 p
.

Since f is Weyl almost periodic, for any ε > 0, we can choose positive number l = l(ε) and there exists a relatively dense set

T S p l (ε, f ) as in (2), such that 0 -∞ e δps 2 sup ξ∈R 1 l ξ+l ξ f (t + s + τ ) -f (t + s) p dt ds 1 p ≤ 2 δp 1 p ε ∀τ ∈ T S p l (ε, f ). Hence, sup ξ∈R 1 l ξ+l ξ u(t + τ ) -u(t) p dt 1 p ≤ Cε,
where

C = M 2 δq 1 q 2 δp 1 p .
Summing up, we have shown that, for every ε > 0, there corresponds a positive number l = l(ε) and there exists a relatively dense set

T S p l (ε, u) = T S p l ( ε C , f ) such that sup ξ∈R 1 l ξ+l ξ u(t + τ ) -u(t) p dt 1 p < ε; ∀τ ∈ T S p l (ε, u).
Which implies the Weyl-almost-periodicity of the solution u. Uniqueness of the solution follows from the same arguments as those in [START_REF] Ding | Almost periodic solutions to abstract semilinear evolution equations with Stepanov almost periodic coefficients[END_REF][START_REF] Long | Composition theorems of stepanov almost periodic functions and stepanov-like pseudo-almost periodic functions[END_REF]. We omit the details.

Existence and uniqueness of bounded Weyl almost periodic solution to abstract semilinear differential equation

In this section we study the existence and uniqueness of the Weyl almost periodic mild solution to equation

u ′ (t) = Au(t) + f (t, u(t)), (20)
where A : D (A) ⊂ X → X is unbounded linear, of domain D(A) dense in X, and f : R × X → X measurable function.

We consider the following hypothesis:

(H1) The operator A generates a C 0 -semigroup (T (t)) t≥0 , exponentially stable.

(H2) f ∈ W p AP K (R × X, X).
(H3) The function f is L(.)-Lipschitz i.e., there exists a nonnegative function L(.) ∈ BW p (R) such that

f (t, u) -f (t, v) ≤ L(t) u -v ; ∀t ∈ R, ∀ u, v ∈ X.
Theorem 3.5 Let the assumptions (H1)-(H3) be fulfilled. For p ≥ 2, the equation (1) has a unique mild solution in CB(R, X) given by

u(t) = t -∞ T (t -s)f (s, u(s))ds, (21)
provided that

L S p < δq M q 1 q e δ -1 e δ , ( 22
)
where

1 p + 1 q = 1. If, furthermore L S p < pδ M p 2 p+1 e pδ 4 -1 e pδ 4 pδ 2p -4 p-2 for p > 2, ( 23
)
or

L S 2 < pδ 8M 2 e 2δ 4 -1 e 2δ 4 for p = 2, (24)
then, u is Weyl almost periodic.

Before giving the proof of Theorem 3.5, we need the following technical results.

Lemma 3.6 ([19]

) Let g : R → R be a continuous function such that, for every t ∈ R,

0 ≤ g(t) ≤ α(t) + β 1 t -∞ e -δ 1 (t-s) g(s)ds + ... + β n t -∞ e -δ 1 (t-s) g(s)ds, (25)
for some locally integrable function α : R → R, and for some constants β 1 , ..., β n ≥ 0, and some constants δ 1 , ..., δ n > β, where β = i=n i=1 β i . We assume that the integrals in the right hand side of (25) are convergent. Let δ = min 1≤i≤n δ i . Then, for every γ ∈]0, δβ] such that 0 -∞ e γs α(s)ds converges, we have, for every t ∈ R,

g(t) ≤ α(t) + β t -∞ e -γ(t-s) α(s)ds.
In particular, if α is constant, we have

g(t) ≤ α δ δ -β . Proposition 3.7 Let f be in W p AP K (R × X, X) (2 ≤ p < ∞)
and satisfying (H1). Let u ∈ CB(R, X). Then, for every ε > 0 there exists l = l(ε) > 0 such that for every

τ ∈ T S p l (ε, f, K), we have 0 -∞ e δ 1 s 1 l s+l s f (t + τ, u(t)) -f (t, u(t)) p dt ds < ε p (26)
and

0 -∞ e γr r -∞ e -δ 1 (r-s) 1 l s+l s f (t + τ, u(t)) -f (t, u(t)) p dt ds dr < ε p , (27)
for all δ 1 > 0 and γ > 0.

Proof Let fix δ 1 > 0 and γ > 0. For s ∈ R, ξ ∈ R, l > 0 and τ ∈ R, let

h τ,l (s) = 1 l s+l s f (t + τ, u(t)) -f (t, u(t)) p dt, α τ,l (ξ) = ξ -∞ e -δ 1 (ξ-s) h τ,l (s)ds.
Inequalities ( 26) and ( 27) become α τ,l (0) < ε p and 0 -∞ e γr α τ,l (r)dr < ε p , respectively. First, we deal with the estimation [START_REF] Rao | On the Stepanov-almost periodic solution of a second-order operator differential equation[END_REF]. Using the boundedness of both u and f in the Weyl sense (f is W p -bounded), we have for every τ ∈ R, l > 0 and for every s ∈] -∞, 0],

h τ,l (s) = 1 l s+l s f (t + τ, u(t)) -f (t, u(t)) p dt ≤2 2p-2 1 l s+l s f (t + τ, u(t)) -f (t + τ, 0) p dt + 1 l s+l s f (t + τ, 0) p dt + 2 2p-2 1 l s+l s f (t, 0) p dt + 1 l s+l s f (t, 0) -f (t, u(t)) p dt ≤2 2p-1 L(.) p W p u(.) p ∞ + f (., 0) p W p < +∞.
The function s → e δ 1 s h τ,l (s) is bounded and continuous on ] -∞, 0], hence

α τ,l (0) = 0 -∞ e δ 1 s h τ,l (s)ds = ∞ n=0 -n -n-1 e δ 1 s h τ,l (s)ds ≤ ∞ n=0 e -δ 1 n max s∈[-n-1,-n]
h τ,l (s) .

(28) Due to the uniform boundedness of the sequence max s∈[-n-1,-n] h τ,l (s) n , the series in ( 28) is uniformly convergent. Thus, for every ε > 0, we can find an integer N 1 = N 1 (ε) ≥ 1 such that for every τ ∈ R, l > 0, and δ 1 > 0, we have

n>N 1 e -δ 1 n max s∈[-n-1,-n] h τ,l (s) < ε p 2 . (29)
Moreover, we claim that there exists l = l(ε) > 0 such that every τ ∈ T S p l (ε, f, K) satisfies

N 1 n=0 e -δ 1 n max s∈[-n-1,-n] h τ,l (s) < ε p 2 . ( 30
)
In fact, for the above ε > 0, let l = l(ε) > 0 provided by the condition that T S p l (ε, f, K) is relatively dense. In view of the continuity of the mapping s → h τ,l (s) on the compact set [-n -1, -n], there exists s * n ∈ [-n -1, -n] such that, for every τ ∈ R and

δ 1 > 0, max s∈[-n-1,-n] h τ,l (s) = h τ,l (s * n ). Put s * = s * ε := arg max n∈{0,••• ,N 1 } h τ,l (s * n ). We have then N 1 n=0 e -δ 1 n max s∈[-n-1,-n] h τ,l (s) ≤ N 1 n=0 e -δ 1 n h τ,l (s * ).
Now, since u is continuous, we have that K * = u ([s * , s * + l]) is a compact set that depends only on ε. Hence, there exist

u 1 , • • • , u k ∈ K 1 such that K * ⊂ i=k i=1 B u i , ε 2 p √ 6 L W p . It follows that, for each t ∈ [s * , s * + l], there exists i(t) ∈ {1, 2, • • • , k} such that u(t) -u i(t) < ε 2 p √ 6 L W p . (31)
On the other hand, since

f ∈ W p AP K (R × X, X), every τ ∈ T S p l (ε, f, K) is such that 1 l s * +l s * f (t + τ, u i(t) ) -f (t, u i(t) ) p dt < ε p 2 p+1 3k . ( 32
)
To combine the previous arguments, we use the triangle inequality:

h τ,l (s * ) ≤2 p 1 l s * +l s * f (t + τ, u(t)) -f (t + τ, u i(t) ) p dt + 2 p 1 l s * +l s * f (t + τ, u i(t) ) -f (t, u i(t) ) p dt + 2 p 1 l s * +l s * f (t, u i(t) ) -f (t, u(t)) p dt =:2 p (A 1 + A 2 + A 3 ) . ( 33
)
Let us estimate A 1 . By (H2) and (31), we have

A 1 = 1 l s * +l s * f (t + τ, u(t)) -f (t + τ, u i(t) ) p dt ≤ 1 l s * +l s * L(t + τ ) p u(t) -u i(t) p dt ≤ ε 2 p √ 6 L W p p L p W p < ε p 2 p+1 3 . (34)
We follow the same steps as for A 1 , we obtain

A 3 ≤ ε 2 p √ 6 L W p p L p W p < ε p 2 p+1 3 . ( 35
)
Now, we estimate A 2 . By (32), we have

A 2 ≤ k j=1 1 l s * +l s * f (t + τ, u j ) -f (t, u j ) p dt ≤ ε p 2 p+1 3 . ( 36
)
Taking into account (33)-(36), we obtain that

h τ,l (s * ) ≤ ε p 2 , for every τ ∈ T S p l (ε, f, K). (37)
This implies that

N 1 n=0 e -δ 1 n max s∈[-n-1,-n] h τ,l (s) ≤ (1 -e -δ 1 ) ε p 2 ≤ ε p 2 .
Having disposed the previous preliminaries steps, we now return to estimation [START_REF] Weyl | Integralgleichungen und fastperiodische Funktionen[END_REF], we get in view of ( 29) and (30) that for every ε > 0, there is l = l(ε) > 0 (provided by the condition f ∈ S p l AP K (R × X, X)) such that any τ ∈ T S p l (ε, f, K) verifies

α τ,l (0) = 0 -∞ e pδ 4 s 1 l s+l s f (t + τ, u(t)) -f (t, u(t)) p dt ds ≤ ε p . (38)
Next, let us consider 0 -∞ e γr α τ,l (r)dr. The arguments we use are in part the same as those mentioned above. The function r → α τ,l (r) is continuous and uniformly bounded with respect to both τ and l, thus the series n≥0 e -γn α τ,l (r * n ) is uniformly convergent, where r * n = arg max

r∈[-n-1,-n] α τ,l (r). Therefore, 0 -∞ e γr α τ,l (r)dr ≤ n≥0 e -γn α τ,l (r * n ). (39)
Now, let ε > 0. We can find an integer

N 2 = N 2 (ε) > 1 such that n>N 2 e -γn α τ,l (r * n ) < ε p 2 . (40)
Furthermore, by putting r * = arg max n∈{0,...,N 2 } α τ,l (r * n ), we have

N 2 n=0 e -γn α τ,l (r * n ) ≤ (1 -e -γ ) -1 α τ,l (r * ). (41)
Repeating the same argument as for α τ,l (0) enables us to write that for the positive number l = l(ε) provided by the condition f ∈ S p l AP K (R × X, X) and for every τ belonging to the relatively dense set ∈ T S p l (ε, f, K) we have

α τ,l (r * ) ≤ (1 -e -γ ) ε p 2 . ( 42
)
Consequently, combining (39), ( 40) and (42), we obtain for the above l and τ the desired estimation:

0 -∞ e γr α τ (r)dr ≤ ε p .
Which achieves the proof of Proposition 3.7.

Proof of Theorem 3.5 We can break down the demonstration into two part. In the first part, we that, assuming [START_REF] Kufner | Function spaces[END_REF], equation ( 1) has a unique mild solution belonging to CB(R, X), and given by [START_REF] Kovanko | Sur la compacité des systèmes de fonctions presque périodiques généralisées de H. Weyl[END_REF]. In the second, using the technique by Kaminski et al. [START_REF] Kamenskii | Weak averaging of semilinear stochastic differential equations with almost periodic coefficients[END_REF], we show that solution [START_REF] Kovanko | Sur la compacité des systèmes de fonctions presque périodiques généralisées de H. Weyl[END_REF] is Weyl almost periodic.

First part

Clearly, following the same steps as in the proof of Theorem 3.4, u is solution to

u(t) = t -∞ T (t -s)f (s, u(s))ds
if, and only if,

u(t) = T (t -a)u(a) + t a T (t -s)f (s, u(s))ds,
for all t ≥ a for each a ∈ R, which means that u is a mild solution to [START_REF] Ait Dads | Behavior of bounded solutions for some almost periodic neutral partial functional differential equations[END_REF].

We introduce an operator Γ by

Γu(t) = t -∞ T (t -s)f (s, u(s))ds.
Let us show that, assuming [START_REF] Kufner | Function spaces[END_REF], the operator Γ maps CB(R, X) into CB(R, X) and has a unique fixed point. First step. Let us show that Γu is bounded, for any u ∈ CB(R, X). From the hypotheses (H1), Hölder's inequality ( 1p , 1 q ) and the triangular inequality, we have

sup t∈R Γu(t) ≤M sup t∈R t -∞ e -δ(t-s) f (s, u(s)) ds ≤M sup t∈R ∞ k=1 t-k+1 t-k e -qδ(t-s) ds 1 q t-k+1 t-k f (s, u(s)) -f (s, 0) p ds 1 p + M sup t∈R ∞ k=1 t-k+1 t-k e -qδ(t-s) ds 1 q t-k+1 t-k f (s, 0) p ds 1 p .
Since the singleton {0} is compact, f (., 0) is bounded with respect to Stepanov's norm. By (H2) and Hölder's inequality, we have

sup t∈R Γu(t) ≤M sup t∈R ∞ k=1 t-k+1 t-k e -qδ(t-s) ds 1 q   t-k+1 t-k L(s) p u(s) p ds 1 p + f (., 0) S p   ≤M e δ e δ -1 sup t∈R   t-k+1 t-k L(s) p ds 1 p u(.) ∞ + f (., 0) S p   ≤M e δ e δ -1 L(.) S p u(.) ∞ + f (., 0) S p < ∞.
Thus Γu is bounded.

Second step Let us show that the mapping t → Γu is continuous on R, for every u ∈ CB(R, X). For, let fix u ∈ CB(R, X). We have

Γu(t) = t -∞ T (t -s)f (s, u(s))ds = ∞ n=0 t-n t-n-1 T (t -s)f (s, u(s))ds. Let us set, for n ∈ N, Γ n u(t) = t-n t-n-1 T (t -s)f (s, u(s))ds, t ∈ R.
Let us check that Γ n (u) ∈ CB(R, X). To this end, let t 0 ∈ R. Put g(s) =: f (s, u(s)). Clearly g ∈ L p loc (R, X). Hence, in view of [

Γ n u(t 0 + h) -Γ n u(t 0 ) ≤ lim h→0 M t 0 -n t 0 -n-1 e -δ(t 0 -s) g(s + h) -g(s) ds ≤M t 0 -n t 0 -n-1 e -qδ(t 0 -s) ds 1 q lim h→0 t 0 -n t 0 -n-1 g(s + h) -g(s) p ds 1 p
.

The continuity of Γ n u is then a straightforward consequence of (43). The boundedness of Γ n u is a simple consequence of the following estimation

Γ n u ∞ ≤ sup t∈R t-n t-n-1 e -qδ(t-s) ds 1 q L(.) S p u(.) ∞ + f (., 0) S p ≤e -nδ L(.) S p u(.) ∞ + f (., 0) S p ,
from which we deduce in addition that the series ∞ n=0 Γ n u(t) is uniformly convergent. The continuity of Γu is provided by the continuity of Γ n u for each n and the uniform convergence of the series ∞ n≥0 Γ n u(t) . Thus Γ maps CB(R, X) into CB(R, X). Third step Let us show that, assuming [START_REF] Kufner | Function spaces[END_REF], Γ has a fixed point in CB(R, X). For every t ∈ R, we have from (H1)

Γu(t) -Γv(t) ≤ M t -∞ e -δ(t-s) f (s, u(s)) -f (s, v(s))ds ≤ M +∞ k=1 t-k+1 t-k e -δ(t-s) f (s, u(s)) -f (s, v(s)) ds.
Using Hölder's inequality and (H2), we get for every t ∈ R,

Γu(t) -Γv(t) ≤M +∞ k=1 t-k+1 t-k e -δq(t-s) ds 1 q t-k+1 t-k L(s) p u(s) -v(s) p ds 1 p ≤ M +∞ k=1 t-k+1 t-k e -δq(t-s) ds 1 q t-k+1 t-k L(s) p ds 1 p u -v ∞ ≤ M q δq 1 q e δ e δ -1 L S p u -v ∞ .
Hence, it follows that, for each t ∈ R,

Γu -Γv ∞ ≤ k u -v ∞ .
Since k < 1, we conclude that Γ is a contraction operator. We conclude that there exists a unique mild solution to (1) in CB(R, X).

Second part

In this part, we show that, under ( 22) and ( 23), the solution u ∈ CB(R, X) to ( 1 (44) For the second term, using again a change of variables and Fubini's theorem, we have, for all τ ∈ R, ξ ∈ R and l > 0, 1 l

( 18 )

 18 The solution x is clearly uniformly continuous. Moreover, by[START_REF] Radová | Theorems of Bohr-Neugebauer-type for almost-periodic differential equations[END_REF] Proposition 4], x is Weyl almost periodic. Now, we can get more, namely that the solution x is purely Weyl almost periodic. Indeed, for every ε > 0, the following inequality ( exp(1) -1) exp(-t) < ε holds for all t ≥ln

I 1 I 2 I 1

 121 4 (t-s) f (s + τ, u(s))f (s, u(s)) p ds 1 p . Thus, for all τ ∈ R, ξ ∈ R and l > 0, we have1 l ξ+l ξ u(t + τ )u(t) p dt ≤M p 2p -(t) + I 2 (t) p dt ≤M p 2 p-1 2p -(t) p dt .Let us estimate the first term of the last inequality. We have, for every τ ∈ R, ξ ∈ R and l 4 (t-s) u(s + τ )u(s) p ds 4 (t-s) u(s + τ )u(s) p ds dt.By a change of variables and Fubini's theorem, it follows that 1 l ξ+l ξ (t) p dt ≤ L p + τ )u(t) p dt ds.

f

  4 (t-s) f (s + τ, u(s))f (s, u(s)) p ds (t + τ, u(t))f (t, u(t)) p dt ds 4 (ξ-s) h τ,l (s)ds, whereh τ,l (s) := 1 l s+l s f (t + τ, u(t))f (t, u(t)) p dt.Summing up, for all ξ, τ ∈ R and l > 0, we obtain0 ≤ g τ,l (ξ) := 1 l ξ+l ξ u(t + τ )u(t) p dt ≤ Cα τ,l (ξ) + β ξ -∞ e -δ 1 (ξ-s) g τ,l(s)ds, with 4 (ξ-s) h τ,l (s)ds, δ 1 := pδ 4 , and β := M p 2 p-1 L p S p

  is notStepanov almost periodic. Indeed, if F is Stepanov almost periodic, thanks to its uniform continuity, we get would have that F was Bohr almost periodic (Bochner's Theorem, see. Let the scalar differential equation x ′ = -x + f (t), t ∈ R, with f given by

	e.g [4], [10, p.82], [23, Lemme.4 p.34] and [11, Th 6.16]). A contradiction. Hence F is not
	Stepanov almost periodic.
	Example 3.3

  ) is Weyl almost periodic.By (H1), we obtain that, for any τ ∈ R and any t ∈ R,u(t + τ )u(t) ≤

									t
									-∞	T (t -s) f (s + τ, u(s + τ )) -f (s, u(s)) ds
										t
								≤ M	-∞
	By Hölder's inequality ( 2 p , p-2 p ) and triangular inequality, we obtain
	u(t + τ ) -u(t) ≤M	2p -4 pδ	p-2 p	-∞ t	2 ds p	2 p
					+ M	2p -4 pδ	p-2 p	-∞ t	2 ds p	2 p
				=:M	pδ 2p -4	p-2 p
	For I 1 (t), by (H2) and Hölder's inequality ( 1 2 , 1 2 ), we have
										2
	I 1 (t) =									2 ds p	p
	≤	k=0 +∞	e -pδ 4 k		t-k-1 t-k	L(s + τ ) p ds	1 p	-∞ t	1 p
	≤	e	e pδ 4 -1 pδ 4	1 p	L S p		t -∞	e -pδ 4 (t-s) u(s + τ ) -u(s) p ds	1 p	.
	Next, let us consider I 2 (t). By Hölder's inequality ( 1 2 , 1 2 ), we have
	I 2 (t) =		t -∞	e -pδ 4 (t-s) f (s + τ, u(s)) -f (s, u(s))	2 ds p	2 p
		≤		t -∞	e -pδ 4 (t-s) ds	1 p	t -∞

e -δ(t-s) f (s + τ, u(s

+ τ ))f (s, u(s)) ds. e -pδ 4 (t-s) f (s + τ, u(s + τ ))f (s + τ, u(s)) e -pδ 4 (t-s) f (s + τ, u(s))f (s, u(t)) I 1 (t) + I 2 (t) . t -∞ e -pδ 4 (t-s) f (s + τ, u(s + τ ))f (s + τ, u

(s))

e -pδ 4 (t-s) u(s + τ )u(s) p ds e -pδ 4 (t-s) f (s + τ, u(s))f (s, u(s)) p ds 1 p

p

It should be noted that the hypothesis [START_REF] Levitan | Almost periodic functions and differential equations[END_REF] is equivalent δ 1 > β. We conclude by Lemma 3.6 that, for all γ ∈]0, δ 1β] such that 0 -∞ e γr α τ,l (r)dr converge, we have, for every ξ ∈ R,

e -γ(ξ-r) α τ,l (r)dr.

We use Theorem 2.4 to get the Weyl almost periodicity of u. We take ξ = 0, then for all τ ∈ R and l > 0, we have

e γr α τ,l (r)dr.

Therefore, in view of Proposition 3.7, for all ε > 0, there exist l = l(ε) > 0 and a relatively dense set

Weyl almost periodicity of u is then proved thanks to the inclusion

For p = 2, the proof is similar to that for p > 2. The details are omitted.

Remark 3.8

The results obtained here can also be applied to the more general case p ≥ 1 provided that L(.) in (H1) is independent of t.


INTRODUCTION

Steel structures in sea or estuary area are subjected to corrosion process. The phenomenon is very complex due to the nature of the environment and material, and the type of the structure. This phenomenon needs to be characterized and modeled for structural analysis which accounts for the loss of thickness. Moreover, due to the randomness of the corrosion process, probabilistic models are needed for structural reliability. Ultrasonic residual measurements allow determining profiles of loss of thickness, identifying the areas which are the most affected by corrosion and stating the assumptions for modeling the random corrosion process. Several inspection campaigns have been performed during the three last decades in some French harbours. Thus a great number of ultrasonic residual measurements are now available for structures in various environments. The protocol is recommended by the CETMEF (French Center for Maritime and Fluvial Technical Studies :Engineering centre of the French Ministry of Public Works): at a given level, this protocol suggest to perform three geometrical measurements at a given location to assess the loss of thickness as the average of these three readings. Generally, the amount of data is more than 1000 per structure and allows performing a statistical analysis of the spatial distribution of the loss of thickness according to the different areas (mainly tidal and immersion areas).

Moreover environmental parameters (temperature, PH, oxygen, salinity, conductivity, nutrients) have been measured since a decade and are available.

This study began within the MEDACHS framework (Marine Environment Damage to Atlantic Coast Historical and transport works or Structures-Interreg IIIB Project funded by EC 2005EC -2007) ) and is performed now in the GEROM framework (Risk management of French harbor structures: stakes, current practices and needs -Experience feedback of owners). The steel structures are sheet-piles, onpile and on-sheet-pile wharves. This paper focuses on sheet-piles seawalls and coffer-dams. More than 30 000 data are available: they correspond to measurements in 4 harbours and concern more than 20 quays. This paper focuses on the spatial dependence of the corrosion process and we present only results of three of them where the distance between data is small (0.20 m horizontally and 0.10 m vertically).

In the first part, the paper presents the data available to study the spatio(-temporal) fields of corrosion for steel sheet-piles.

The second part presents the geo-statistical modeling of the spatial variability of the steel sheet piles corrosion; and more particularly the study of the stationarity of the process of corrosion according to the length and the depth along three marine structures. Finally, a first level of stochastic modeling is suggested.

Statistical analysis of corrosion process along French coasts

R. Melchers

Center of Infrastructure Performance and Reliability, The University of Newcastle, Australia Modeling of corrosion for steel structures in sea or estuary area is still a great challenge in view to analyze their safety and optimize their maintenance. Several inspection campaigns have been performed during the three last decades in some French harbours. Thus a great number of ultrasonic residual measurements are now available for structures in various environments. This information allows to perform a statistical analysis in view to best understand and model the distribution of loss of matter on those structures according to different areas (mainly tidal and immersion areas). Results confirm that the loss of thickness is higher at the interface between the tidal area and immersion area. An analysis of the spatial process is performed: stationarity is analyzed on three marine structures where the amount of data is significant. Finally, a first level of spatial dependent corrosion modeling of steel in marine environment is suggested.

Corrosion measurements

Corrosion of steel structures is classically assessed by ultrasonic non destructive testing. In France, a protocol is given by the CETMEF and classically used by several companies to perform inspections of coastal steel structures. The CETMEF is part of the French ministry of building. It is devoted to the diffusion of knowledge, to provide technical and research studies as well as engineering and expert's reports. This protocol consists in performing residual thickness measurements for several heights of the structure using ultrasonic testing. The corrosion products are removed by grinding. By using this technique and considering the harsh conditions for marine inspections, the error on the measurements at a given point on the structure cannot be neglected. Error comes both from the physical measurement (around 0.1 mm) and from the protocol (grinding, link diver-operator, etc.). Thus three measurements are performed on a given location: the three measurements are distributed on a circle, with diameter of about 5 cm, every 120° as shown on figure 1. The average of these three readings is considered as the "true" loss of thickness. For on-pile wharves, in view to analyze the corrosion around the pile, measurements are performed at several cardinal positions of the pile's section (see figure 1). For on-sheet-pilewall wharves, they are performed along the length of the wharf around every 10 meters, which is classically the mean distance between two piles for onpile wharves. Starting from these residual thickness values, and knowing the initial thickness of piles or sheet-piles, we deduce the thickness loss mainly due to corrosion. This protocol involves recommendations only and in some cases, this protocol is not followed because of the experience of owners or the results of the risk based inspection analysis. That is the case for structures considered in the lat part of the structures where the distance between measurements is closer. Over a total amount of 23 structures, 11 are located in docks at constant seawater level, 9 in tidal docks and 3 in wet docks. In this last case the water level changes with the same period as the tide but with lower amplitude.

The variation of the seawater level in the various docks is detailed on figure 4.

The tide level corresponds to the range between the HAT (Highest Astronomical Tide) and the (LAT (Lowest Astronomical Tide). The variation of the height of water is important in tidal docks of BO and HA harbours. The variation is of course weaker in harbours located along the Mediterranean coast (i.e. PL and SE harbours).

Most of the measurements are made in the immersed zone, including the zone of low seawater level and the mud zone. Only the HA harbour began some measurements in the tidal/spray and in the atmospheric zones. The amount of data according to the studied harbours and for the various exposure zones is detailed on the figure 5. The structure of the BO harbour was alone the subject of large campaign of measurements: it represents 99 % of all the available measurements.

Selected structures for spatial variability assessment: inspection campaign

The study of the spatial variability of corrosion is performed from the data of three structures extracted from the database. This selection is based on the fact that the distance between horizontal or vertical measurements are much weaker than everywhere else. Consequently, the quantity of information is consequent to perform a spatial analysis of the corrosion process. In the three cases, it is important to remind that these specific campaigns were devoted to the discovery of perforations on steel sheet piles. In the case of the structure HA7d, the control was also started after the observation of accidental acids discharges.The main technical characteristics of these structures are detailed in tables 1. Table 2 gives more details concerning the type of sheet piles and the auscultation strategy.

Environment of selected structures

The physico-chemical parameters of seawater near the studied structures are described on table 3. Most of them were measured between 1997 and 2007. The physico-chemical parameters of seawater for the structures HA7d and HA8b are identical because they are located in the same dock.

SPATIAL VARIABILITY OF THE CORROSION PROCESS

Data pre-processing

The loss of thickness on each location is computed by averaging the three readings corresponding to three positions occupied by the ultrasonic transducer inside the required location (see figure 1). Sometimes, for technical reasons, a single basic measurement is realized in a location. In that case, the loss of thickness which characterizes this loca-tion was removed from the database because the error can be significant.

Moreover, perforations were observed on steel sheet piles. The loss of thickness cannot be measured in that case and the corresponding data were also removed for the statistical analysis.

To analyze the influence of the physico-chemical parameters on the corrosion mechanism in marine environment, the following hypotheses will be verified in the future: -The temporal series of the physico-chemical parameters of the seawater follow a stationarity process (no trend or jump). The hypothesis of the stationarity or non-stationarity can be verified by an appropriate test; -The absence or low influence of the depth on the physico-chemical parameters. We don't have any measure of these parameters along the wharf length; -The homogeneous distribution of the measures of the seawater quality during the various seasons is necessary to make sure that the statistical characteristics of the parameters are not biased. It can easily be verified or corrected.

Analysis of the spatial variability of the corrosion process

The "loss of thickness" due to the corrosion is considered here as a stochastic field indexed by the space at the inspection time. Due to the corrosion process, we consider here that the principal directions are known: x, along the wharf length and z, along the wharf depth.

From the available distances between measurements, the objective is here to analyse if the corrosion is or not a second-order stationarity process in R ² i.e. according to the two coordinates x and z (see figure 6). For that purpose, three conditions are to be verified: -The arithmetic mean of the marginal variables must be constant according to the studied direction; -The variance of the marginal variables must be constant according to the studied direction too. If not, the random process is not stationary. If yes and if the first condition is not verified, the field can be centered and the further analysis is made on the remainder; -The covariance depends only on the distance between points according to the studied direction.

Stationarity according to x

First measurements on in-and out-pans are distinguished for wharves with U shaped piles (HA harbour). In this paper we only consider measurements on out-pans where the corrosion is higher. The variation of the mean value (statistical moment of order 1) and 95% confidence values for wharf HA7d, when one moves horizontally, is plotted on figure 7.

It appears clearly that this statistical value is not constant spatially. The results of the two other wharves lead to the same result. However, for HA8b, due to the fair quantity of measurements, the confidence interval is as large as the variation range.

For BO1, the effect of shape of the structure (cofferdam) is not taken into account at first. Figure 8 presents respectively the standard deviations of readings on out-pans for wharf HA7d. The analysis of the standard deviation leads to the same conclusion as illustrated for wharf HA7d on figure 7.

Measurements don't allow to conclude that the random field is stationary according to x. However, the mean and the standard deviation have been computed from 7 measurements and in that case, the statistical bias is not negligible.

Stationarity according to z

The evolution of the mean of the loss of thickness according to z for two of the three wharves, BO1 and HA7d, is presented on figures 9-11. The results concerning the third one are not presented because the amount of data is not sufficient: only 11 are available where the amount of data is 53 and 4605 for the two others. The trend appears clearly and is conform to the results published in the literature [START_REF] Benaïssa | La corrosion des structures métalliques en mer : types et zones de dégradations[END_REF][START_REF] Clément | Overview of corrosion impact for marine structures : analysis of French structures in several harbours, 1 st International Conference MEDACHS[END_REF], Cetmef 2005[START_REF] Memet | La corrosion marine des structures métalliques portuaires : étude des mécanismes d'amorçage et de croissance des produits de corrosion[END_REF]). The lowest level that can be expected to occur under average meteorological conditions and under any combination of astronomical conditions. For HA7d, due to the small seawater variation in the dock (0.6 m), only two levels are indicated, i.e. High water and low water.

For wharf HA7d, measurements on in and outpans of a U-shape sheet piling are dissociated. Mean of loss of thickness on out-pans on Figure 10.

Coefficient of variation

Mean and standard deviations having the same trend with depth, let us focus now on the evolution of the coefficient of variation with depth. Figure 14 illustrates this evolution. This coefficient of variation is quite constant according to z. For wharf BO1, it reaches 0.47, for wharf HA7d, 0.44 for out-pans and 0.49 for in-spans of sheet piling and 0.64 for wharf HA8b where the quantity of measurements is fair and concerns only out-pans.

Covariance and correlation

Even if the field if not stationary, this assumption is usually made and a lot of papers starts with the plotting of the variogram. The covariance and the coefficient of correlation as functions of the distance h between measurements are plotted on figure 15 along the length x and on figure 18 along the depth z for BO1.

It is clear that no structure of correlation can be deduced according to x, in spite of the small distance h between two measurements of loss of thickness (0.15 to 0.20 m). According to z however, on figure 16 for B01 and on figures 17 for HA07d, the coefficient of correlation seems to decrease with h. Note that the correlation is significant only between very close points (i.e. fair values for h) that are generally in the same area.

For HA8b, the confidence interval is too wide due to the fair quantity of data.

MODELLING OF THE PROCESS OF CORROSION

The aim of the modelling of corrosion process in harbours is to provide predictive models that account for the spatial dependence both in depth and in length and the time variability in a given environment E. We start from the assumption that effects of spatial dependence are decoupled along x and z. A statistical model gives the main trend with z and the trend with x comes from an analysis and a fitting. In a given environment, these trends must account for the type of sheet piling (flat, Z or U-shaped) and eventually, the shape of the structure (wall or cofferdam) that modifies the exposure of some sheet piles. Then the corrosion model of Melchers is extended in case of spatial trend (equation 1) (Melchers, 2003): c(x,z,t,E,θ) = tr(x,z,t,E) + c(z,t,E,θ) (1)

where c(x,z,t,E,θ) is the loss of thickness expressed as a stochastic function indexed by the abscissa along the structure x (expressed in meters), the depth z (in meters

c(z, t, E, θ) = b(z, t, E) fn(z, t, E) + ε( z, t, E,θ) (2)
where fn(z,t,E) denotes the deterministic trend with time at a given depth z. It is fitted on the mean value of experimental data

∑ = = m 1 i 0 i 0 ) E , t , z ( th m 1 ) E , t , z ( th (5)
In equation 6, with an abuse of notation and considering that all readings according to x are independent realizations due to hazard θ i of the same variable (i.e. the spacing between measurements is larger than the correlation distance with x), expression of

) , E , t , z ( th i 0 k θ is suggested: ) t , z , x ( tr ) E , t , z , x ( th ) , E , t , z ( th i k i k i 0 k - = θ (6)
For example, for the wharf BO1, a linear trend is convenient and it corresponds to equation 7:

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - = 2 l x ). z , t ( a ) x ( tr (7)
where 'a' is a deterministic function of time t and depth z describing the slope of the spatial dependence (mm/m) and 'l' the length of the structure where data are available. On figure 18, it is interesting to observe in the case of wharf BO1 that the value of 'a(z)' follows the same evolution with depth as the value of the mean of corrosion loss.

CONCLUSION

Data available for the study of the steel corrosion in marine environment are measurements of residual thickness of steel by ultrasonic non destructive technique at several depths and several locations along the French harbors structures.

The analysis of the stationarity of corrosion process with space is carried out from the statistical analysis of 3 wharves. The conclusion is that the stationarity cannot be proved easily with the available data and that the distance of correlation of these structures is less than 0.2 m.

Moreover, characterization of the spatial variability of the metal loss of thickness of marine structures due to the corrosion is an essential stage to optimize the maintenance plans. The problem consists in integrating the results obtained during the various campaigns of inspection within the decision processes in a most realistic possible way.

The spatial variability of the corrosion cannot be objectively described in a quantitative way by determinist models, so a first level of steel corrosion spatio-temporal modeling in marine environment is suggested, based on the model developed by Melchers.

Now, a statistical analysis of the whole available data is necessary to confirm assumptions to model the spatio-temporal process of corrosion.

Figure 1 .

 1 Figure 1. CETMEF protocol: cardinal points for pile (left) and location for sheet pile (right).

2. 2

 2 Presentation of the corrosion data-baseCorrosion measurements are collected on two types of marine structures: on-pile wharves and on-sheet pile wharves. Structures are located on the French coasts, i.e. Atlantic, Channel and Mediterranean coasts. In this paper, only corrosion measurements associated to steel sheet piles are presented.Almost all the studied structures were built between 1955 and 1985. Only three structures are older than 1955 and none is more recent than 1985. The owners began NDT measurements of residual thicknesses from 1990. Consequently most of the structures are 10 to 40 years old at the inspection time. Figure2presents the distribution of the age of the structures at the first, second, third and more inspection times.

Figure 2 .

 2 Figure 2. Age distribution of the studied structures at inspection time.

Figure 3 .

 3 Figure 3. Harbours and docks distribution.

Figure 4 .

 4 Figure 4. Harbours and docks distribution.

Figure 5 .

 5 Figure 5. Distribution of data (loss of thickness) in the various exposure zones.

Figure 6 .

 6 Figure 6. Study of the stationarity of the corrosion process according to the direction x (a) and the direction z (b).

Figure 7 .

 7 Figure 7. Mean of loss of thickness according to x -HA7d (out-pans measurements).

Figure 8 .

 8 Figure 8. Standard deviation of loss of thickness according to x -HA7d (out-pans).

Figure 9 .

 9 Figure 9. Mean of loss of thickness according to z -BO1.

Figure 10 .

 10 Figure 10. Mean of loss of thickness on out-pans of sheet piling according to z -HA7d.

Figure 11

 11 Figure11illustrates the well-known phenomena on the U shaped piles: out-pans corrode faster than in-pans.

Figure 11 .

 11 Figure 11. Mean of loss of thickness on in and out-pans of sheet piling according to z -HA7d.

Figure 12 .

 12 Figure 12. Standard deviation of loss of thickness according to z -BO1.

Figure 13 .

 13 Figure 13. Standard deviation of loss of thickness on out-pans of sheet piling according to z -HA7d.

Figure 14 .

 14 Figure 14. Coefficient of variation of loss of thickness according to z -BO1.

Figure 15 .

 15 Figure 15. Correlation of loss of thickness according to x -BO1.

Figure 16 .

 16 Figure 16. Correlation of loss of thickness with h according to z -BO1.

Figure 17 .

 17 Figure 17. Correlation of loss of thickness with h according to z -HA07d (in-pans).

  ) and time t (in years). E denotes the corrosion factors and θ the hazard. Deterministic function tr(x,z,t,E) denotes the centered trend of the corrosion field according to x. Stochastic function c(z,t,E,θ) represents the loss of thickness at depth z, time t, in a given environment E. It is based on the model formulated by R. Melchers in equation 2:

.-

  Depending on the type of corrosion, following forms are suggested: -Equation 3 if no presence of bio-corrosion: Equation 4 if presence of bio-corrosion and no possibility to identify the first stage of the model from inspections (inspection results after 10 years): obtained from the 'm' structures in the same environment (see equation 5):

Figure 18 .

 18 Figure 18. Evolution of a(z) and mean of loss of thickness with depth -BO1.

Figure 19

 19 Figure19gives the distribution of ε in the low seawater zone and statistical parameters of ε for the three zones, i.e. tidal, low seawater and immersion zones, are given in table 4.

Figure 19 .

 19 Figure 19. Distribution of ε in the low seawater zone -BO1.

Table 1 .

 1 Characteristics of the selected structures.

	______________________________________________
	Structure ______________________________________________ BO1 HA7d HA8b
	Function	General	General		General
		cargo	cargo		cargo
	Type	Cofferdam Seawall		Seawall
	Sheet piles	Flat	U-sections U-sections
	Width (m)	0.5	0.5		0.5
	Length (m)	758.0	345.0	187.5
	Age (Years)	25	28		26
	Sea	Channel	Channel		Channel
	Dock type	Tidal	Constant	Constant
			seawater	seawater
			level		level
	Seawater
	variation level 9.1 _____________________________________________ 0.6 0.6
	Table 2. Corrosion measurements strategies of the selected
	structures. ______________________________________________
	Structure ______________________________________________ BO1 HA7d* HA8b
	Location of measurements
	on steel sheet piles	-	In & out Out-pans
				pans
	Length of the measured
	part of the structure (m)	758.0 35.0	123.0
	Height of the measured
	part of the structure (m)	3.0	9.6	6.1
	Number of measurements
	following length		5988 71	36
	Mean horizontal spacing
	between measurements (m)	0.15 to 0.5	3.5
			0.2
	Number of measurements
	following depth		7	13	48
	Mean vertical spacing
	between measurements (m)	0.5	0.8	0.1
	Number of loss thickness
	measurements		32238 685	504
	Measurements location in
	several exposure zones** ______________________________________________ T, L, I L, I L, I
	Tidal zone ZT ______________________________________________
	Number of measurements
	following depth		2	-	-
	Number of loss thickness
	Measurements		11073 -	-
	Percentage of
	the total amount of
	measurements ______________________________________________ 34 --
	Low seawater zone ZL ______________________________________________
	Number of measurements
	following depth		1	4	2

Table 3 .

 3 Mean values and ranges of physico-chemical parameters of seawater near the studied structures.

	______________________________________________
	Structure	BO1 _______________	HA7d & HA8b ______________
	Parameter ______________________________________________ Mean Min Max Mean Min Max
	Temperature
	(°C)	13.3	7.2 19.5	13.7	8.1 20.7
	pH	8.0	7.8 8.1	8.1	7.7 8.5
	Conductivity
	(mS/cm)	49.0	46.8 50.5	37.2	33.7 41.4
	Salinity (g/l)	32.9	31.5 33.7	25.4	23.9 27.7
	O2 (mg/l)	8.7	6.4 11.2	8.7	6.9 11.0
	SM* (mg/l)	9.9	4.7 17.7	8.3	3.3 15.0
	NH4 (mg/l)	0.5	0.2 0.9	0.3	0.1 0.6
	NO2 (mg/l)			0.2	0.1 0.3
	NO3 (mg/l)			6.4	2.6 8.8
	PO4 (mg/l) _____________________________________________ 0.4 0.1 0.6
	* SM = Suspended materials.

Table 4 .

 4 Statistical parameters of ε for several exposure zones.

	______________________________________________
	Exposure zone*: ______________________________________________ ZT ZL ZI
	Mean	0.00	0.00	0.00
	Standard deviation	1.21	1.37	1.25
	Min.	-2.89	-3.09	-3.27
	Max.	7.78	7.74	5.66
	Fractile 10 %	-1.16	-1.34	-1.20
	Fractile 90 % _____________________________________________ 1.67 2.00 1.79

* ZT = Tidal zone; ZL = Low seawater zone; ZI = Immersion zone.


Jessica Pinaire

Jérôme Azé

Sandra Bringay

Pascal Poncelet

Christophe Genolini

Paul Landais

Jérôme Az É B

An Innovating Approach to Characterise Patient Flows in Myocardial Infarction

Keywords: Healthcare Trajectories, Prospective Payment System, Myocardial Infarction, Spatio-temporal Pattern Mining, Patient Flows, Longitudinal Data

Introduction

Over the past 20 years, studies related to Myocardial Infarction (MI) have pointed out a shift in gender patterns that showed an increase in both cardiovascular risk and mortality among women [START_REF] Dégano | Twenty-five-year trends in myocardial infarction attack and mortality rates, and case-fatality, in six european populations[END_REF]. In parallel, an increase in the number of hospitalisations has been observed, as has, the cost of health care following a MI, in France, this cost has multiplied by three over the last decade [START_REF] Philippe | Costs of healthcare resource consumption after a myocardial infarction in France: An estimate from a medicoadministrative database (GSB)[END_REF].

In this context, the analysis of patient flows appears timely: 1) to predict patient admissions; 2) to adapt the patient care pathway; and 3) to manage the availability of resources. Usually researchers use both data mining methods [START_REF] Pagnoni | Analysis of patient flows via data mining[END_REF] combined with predictive models based on decision trees to model patient flows and build decision support tools [START_REF] Dart | Analysis of hospitalised patient flows using data-mining[END_REF]. In this article, we characterise the flows of patients with MI through an innovative approach. Inspired by spatio-temporal mining method, we propose to extract patterns from the French Prospective Payment System (PPS). Rather than considering spatiality for extracting patterns, we considered directly the codes from the International Classifi-cation of Diseases 10th Revision (ICD-10). By clustering these trajectories we could automatically identify some trends in time between hospitalisations and costs. Thus these informations extracted enabled to considering health planning strategies.

Section 2 introduces the mining of spatio-temporal patterns and proposes an illustration using PPS data. The patient flow characterisation process is described in Section 3. The Section 4 focuses on the characterisation of patient flows for women aged 65+. A discussion on the results and the limits of the proposal appears in Section 5. Finally, Section 6 concludes the paper.

Spatio-temporal patterns

Spatio-temporal mining aims at extracting set of objects sharing the same behaviour during a period of time. Even if many different patterns can be extracted, in this paper we mainly focus on closed swarm patterns. Let O = {o 1 , ..., o n } be a group of moving objects and a set of timestamps T = {t 1 , ...,t p }. For each object o ∈ O at time t, we have its spatial informations x o j t i , y o j t i . Let min o be a user-defined threshold standing for a minimum number of objects and min t the minimum number of timestamps. Informally, a swarm is a group of moving objects O containing at least min o individuals which are closed each other for at least min t timestamps. A swarm can be formally defined as follows:

Definition 1 (Swarm and Closed Swarm) A pair (O, T ) is a swarm if:

(1) ∀ t i ∈ T, ∃ c s.t. O ⊆ c, c is a cluster; (2) |O| ≥ min o and |T | ≥ min t . A swarm (O, T ) is a closed swarm if: (1) O s.t. (O , T ) is a swarm and O ⊂ O ; (2) T s.t. (O, T ) is a swarm and T ⊂ T .
According to this definition, the first condition is that ∀ t i ∈ T, ∃ c s.t.0 ⊆ c, c is a cluster. In spatio-temporal data this cluster is obtained by grouping the objects that are sufficiently closed according their locations. In this paper rather than location we focus on the different kinds of hospitalisations for patients and we group them together if the share at one time the same hospitalisation reasons.

To illustrate, we consider the events of four patients. Time is divided into timestamps corresponding to one hospitalisation and the ICD-10 codes (I21: acute MI; R07: Chest pain; E14: Diabetes; I20: Angina pectoris; I70: Atherosclerosis) refer to the reasons of hospitalisation. Figure 1 illustrates an example of different trajectories that are followed by patients. For instance, we notice that patient P 2 has been first hospitalised for an acute MI (I21) at time t 0 , then a Chest pain (R07) at time t 1 and finally for Angina pectoris (I20) at time t 2 (see figure 1 (a)). Patients sharing the same code at one time can then be grouped together. Let us now assume that min o = 2 and min t = 2 and we are thus provided by the following swarms: {(P 1 , P 3 ), (0, 1)}, {(P 1 , P 3 ), (1, 2)} and {(P 1 , P 3 ), (0, 1, 2)}. We observe that these swarms are redundant because they can be grouped together in a closed swarm: {(P 1 , P 3 ), (0, 1, 2)}.

PaFloChar: Patient Flow Characterisation Process

In this section we present the patient flow characterisation process which has two main steps. The first one aims at extracting and then sorting spatio-temporal patterns from the PPS data. These patterns will correspond to the flow patterns of patients. The second step cluster the time between hospitalisations and cost trajectories in order to identify trends.

Step 1. Spatio-temporal pattern mining. We found 412 486 MI patients over the period of 2009 to 2014 from the PPS database. Each patient has a sequence of ICD-10 codes which length is equal to the number of stays over these six years. A first filtering is performed in order to remove irrelevant stays, i.e. stays that are not related to cardiopathology. Then, sequences are ordered according to the relative time corresponding to the occurrence of a stay. These final sequences are called the patient trajectories. Finally, we mined closed swarms using the Get Move algorithm [START_REF] Phan | All in one: mining multiple movement patterns[END_REF].

Step 2. Time and Cost Evolution Profiles. We clustered time 2 and cost trajectories with kmlShape [START_REF] Genolini | kmlShape: An Efficient Method to Cluster Longitudinal Data (Time-Series) According to Their Shapes[END_REF]. This method, derived from k-means for longitudinal data (kml), is able to detect curve shapes in order to cluster curves having the same shape. Then, we established the assignment of the flow groups, created in step 1, in the time and tariff clusters.

Experiments

In this section, as an illustration of the process, we present the results related to women aged 65+.

Step 1. We highlighted three groups of patients flows in which first events were Angina pectoris, Ischemic heart disease and MI. Then, the flow was separated into several branches in which vertices were these events and Death. At the third stay, we observed new events: Heart rhythm disorders and Heart failure.

Step 2. We found three clusters of times (see figure 2 (a)). B cluster (medium thick blue curve), M cluster (thin magenta curve) and V cluster (very thick green curve). Curves 3 characterise different trends: B cluster represents patients having short times (less than four months) that increase and then decrease; M cluster represents patients presenting with spaced stays at an early stage, then later more frequent; V cluster represents patients having long sequences. The assignment of group flows in these clusters shows that, in the MI group, most of them (were split into cluster B and V ) have short times at the end of their pathway. We found three clusters of costs (see figure 2 (b)): B cluster represents patients having increasing tariffs over time and decreasing slightly; M cluster represents patients having highly increasing tariffs; V cluster represents patients having first increasing tariff and then decreasing tariffs. The assignment of group flows in these clusters showed that, in the MI group, most patients (were in cluster V ) had initially high tariff which decreased thereafter.

(a) (b)

Discussion

Patient flows. We highlighted three key steps in patient flow patterns: MI, Angina pectoris and Ischemic heart disease. The majority of patients showed signs of recurrence of coronary artery disease in the form of angina pectoris. Many of them experienced MI relapse and/or another manifestation of their ischemic heart disease. These results could be integrated in a decision-making tool. Indeed, this could be useful for a clinician to compare patient profile to similar profiles and warn them about risk of MI relapse.

Times and tariffs profiles. The times profiles provide information on the future hospitalisations related to cardiac pathology. In most cases, after an MI, hospitalisations are increasingly closed in time (on average three months). The majority of patient flow initiated by Angina pectoris or Ischemic heart disease have an upward trend in tariffs (cluster B in figure 2 (b)). In contrast, patient flows initiated by MI largely show a downward trend in tariffs. Moreover, this work raises questions about the rhythms of hospitalisation frequency and the tariff over time.

Limits. This study has several limitations : 1) The choice of the database: PPS is a budget allocation tool, so it presents limits in epidemiology studies [START_REF] Schneeweiss | A review of uses of health care utilization databases for epidemiologic research on therapeutics[END_REF]; 2) The comparison with others studies: Time is mostly investigated [START_REF] Miller | Comparison of Delay Times from Symptom Onset to Medical Contact in Blacks Versus Whites With Acute Myocardial Infarction[END_REF] as a precise event such as readmission for heart failure but less frequently for the more general event of heart disease. Moreover, most direct cost studies take into account emergency cost and drug consumption [START_REF] Blin | Outcomes, health care resources use, and costs in patients with post-myocardial infarction: The horus cohort study in the egb french claims and hospital database[END_REF]. Access to the SNIIRAM (National health insurance system of inter-scheme information) database would allow a similar analysis to be carried out.

Conclusion

We looked for spatio-temporal patterns in the MI patients trajectories. The originality of the approach is that spatiality was assimilated to proximity of pathologies and the temporal aspect was related to the occurrence of a stay. Then we clustered their time and tariff trajectories to determine trends. Such an approach can be used to improve care by providing, for example, an integrated care pathway with scheduled visits proposed in the case of cancer [START_REF] Defossez | Temporal representation of care trajectories of cancer patients using data from a regional information system: an application in breast cancer[END_REF]. In future work, we plan to investigate the co-evolution of the time and cost trajectories in order to establish whether they are related [START_REF] Genolini | Kml3d: a non-parametric algorithm for clustering joint trajectories[END_REF].

Figure 1 .

 1 Figure 1. Patient trajectories. (a) The sequential database with for each patient at each timestamp its hospitalisation code. (b) The different trajectories that when grouping together patients sharing the same code.

Figure 2 .

 2 Figure 2. (a) Times between hospitalisations and (b) Costs kmlShape clusters.

Corresponding Author: Jessica Pinaire, UPRES EA2415, 641 avenue du Doyen Gaston Giraud, 34093 Montpellier, France; E-mail:jessica.pinaire@lirmm.fr.

Time between hospitalisations were calculated as the number of days between the last day of a stay and the first day of the subsequent stay.

The graph reads as follows, at t 0 represents the time between the first stay and the second stay. For instance, the thin magenta curve, has an ordinate equal to 500: it means the time between the first and second stay is about 500 days in this cluster.