Gaetano Agazzotti

Madalina Deaconu
email: madalina.deaconu@inria.fr

Antoine Lejay
email: antoine.lejay@univ-lorraine.fr

Long time asymptotic behavior of a self-similar fragmentation equation

Using the Mellin transform, we study a self-similar fragmentation equations whose breakage rate follows the power law distribution, and a particle is split into a fixed number of smaller particles. First, we show how to extend the solution of such equations to mesure-valued initial conditions, by a closure argument on the Mellin space. Second, we use appropriate series representations to give a rigorous proof to the asymptotic behavior of the moments, completing some results known through heuristic derivations.

Introduction

Fragmentation is a phenomenon that appears in a variety of domains in physics, geophysics, biology, ... Although ruled by an apparently simple equation, a wide variety of behaviors can be reached. The self-similar fragmentation is itself a subclass where the solution asymptotically enjoys some scaling property. This class is itself rich enough and can be studied from several point of views, by studying functional analysis [START_REF] Banasiak | Analytic fragmentation semigroups and continuous coagulation-fragmentation equations with unbounded rates[END_REF][START_REF] Banasiak | On the existence of moments of solutions to fragmentation equations[END_REF][START_REF] Cáceres | Rate of convergence to selfsimilarity for the fragmentation equation in 𝐿 1 spaces[END_REF][START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF][START_REF] Gabriel | Long-time asymptotics for nonlinear growth-fragmentation equations[END_REF][START_REF] Mischler | Spectral analysis of semigroups and growthfragmentation equations[END_REF], probability [START_REF] Bertoin | Different aspects of a random fragmentation model[END_REF][START_REF] Bertoin | Asymptotic Laws for Nonconservative Self-similar Fragmentations[END_REF][START_REF] Haas | Regularity of formation of dust in self-similar fragmentations[END_REF], looking for specific solutions [START_REF] Breschi | A note on the self-similar solutions to the spontaneous fragmentation equation[END_REF][START_REF] Ziff | New solutions to the fragmentation equation[END_REF], dimensional analysis [START_REF] Cheng | Kinetics of fragmentation[END_REF], ... or a mix [START_REF] Biedrzycka | Self-similar solutions of fragmentation equations revisited[END_REF].

In this article, we consider the fragmentation equation

𝜕 𝑡 𝑢(𝑡, 𝑥) = 𝜆 2+𝛼 𝑥 𝛼 𝑢(𝑡, 𝜆𝑥) -𝑥 𝛼 𝑢(𝑡, 𝑥), (1)
where 𝛼 ≥ 0 and 𝜆 > 1. We associate with (1) the companion equation 𝜕 𝑡 𝑣(𝑡, 𝑥) = 𝜆 1+𝛼 𝑥 𝛼 𝑣(𝑡, 𝜆𝑥) -𝑥 𝛼 𝑣(𝑡, 𝑥)

with 𝑣(𝑡, 𝑥) := 𝑥 • 𝑢(𝑡, 𝑥). Here, 𝑢(𝑡, 𝑥) represents the concentration of the particles at time 𝑡 at the point 𝑥, while 𝑣(𝑡, 𝑥) represents the mass. The idea is that when a breakage occurs, a particle breaks into 𝜆 daughter's of equal size.

We rewrite [START_REF] Arendt | One-parameter semigroups of positive operators[END_REF] as where 𝜅 is a scaled Dirac delta mass at 1/𝜆.

𝜕
This corresponds to a homogeneous fragmentation, whose solution has some asymptotic self-similar properties. This form is the one found in [START_REF] Doumic | An inverse problem: recovering the fragmentation kernel from the short-time behaviour of the fragmentation equation[END_REF][START_REF] Doumic | Estimating the division rate and kernel in the fragmentation equation[END_REF]. Most often, results related to self-similar fragmentation equations use for 𝜅 a measure With tools from functional analysis [START_REF] Banasiak | Perturbations of positive semigroups with applications[END_REF], we show that there exists an analytic, positive semi-group, and then a mild solution to [START_REF] Adams | On the Linear Ordinary 𝑞-Difference Equation[END_REF] provided that the initial solution belongs to some weighted L 1 space.

We identify functions with a support contained in [0, 1] as an invariant subset of functions. In particular, using some tools from 𝑞-calculus, we give a series expansion of the semi-group that used the 𝑞-binomial coefficients.

Owing to the particular form of the operator, with 𝐶(𝑡, 𝜎) := ∫ +∞ 0 𝑥 𝜎-1 𝑢(𝑡, 𝑥) d𝑥 for 𝜎 ≥ 1, the moment (or Mellin transform) of 𝑢(𝑡, •), the first main result (Theorem 4) of this article is to extend (1) as a measure-valued solution when the initial condition is a measure with support in [0, 1]. Our approach differs from the one in [START_REF] Doumic | An inverse problem: recovering the fragmentation kernel from the short-time behaviour of the fragmentation equation[END_REF] as we only consider solutions in the metric space given by the weak convergence of measures. The idea is to pass to the limit in the space of Mellin transforms, and then to carry back this extension in the original space.

When the initial condition is 𝛿 1 , the Dirac mass at 1, then 𝐶 solves a pantograph equation. We then use this particular property to prove an asymptotic results on 𝐶.

In [START_REF] Cheng | Kinetics of fragmentation[END_REF]Chapter 3], the statistical aspects of homogeneous fragmentation are discussed from the physical point of view. In particular, an ansatz states that

𝑢(𝑡, 𝑥) ≈ 1 𝑠(𝑡) 2 𝜑 ( 𝑥 𝑠(𝑡) ) , as 𝑡 → ∞,
where 𝑠 ′ (𝑡)𝑠(𝑡) -(1+𝛼) = -𝜔 where 𝜔 > 0 is a separation constant. In the present case, we take

𝑠(𝑡) := 1 𝑡 1/𝛼 , 𝜔 := -1 𝛼 , (3)
while 𝜑 is the solution to

𝜔(2𝜑(𝜉) + 𝜉𝜑 ′ (𝜉)) = 𝜆 2+𝛼 𝜉 𝛼 𝜑(𝜆𝜉) -𝜉 𝛼 𝜑(𝜉), (4)
where 𝜉 = 𝑥/𝑠. The function 𝜑 is called a self-similar profile.

In [START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF]Theorem 3.1], it is shown 1 , that a self-similar profile exists and is unique.

In addition [START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF]Theorem 3.2] states that

lim 𝑡→∞ ∫ +∞ 0 𝑦|𝑢(𝑡, 𝑥) -(1 + 𝑡) 2/𝛼 𝜑(𝑦𝑡 1/𝛼 )| d𝑦 = 0.
Another result on the existence of a self-similar profile is given in [START_REF] Biedrzycka | Self-similar solutions of fragmentation equations revisited[END_REF], yet when the measure 𝜅 has a density.

A series of works (see e.g. [START_REF] Cañizo | Spectral gap for the growthfragmentation equation via Harris's theorem[END_REF][START_REF] Cáceres | Rate of convergence to selfsimilarity for the fragmentation equation in 𝐿 1 spaces[END_REF][START_REF] Cáceres | Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations[END_REF][START_REF] Jauffret | Eigenelements of a general aggregationfragmentation model[END_REF][START_REF] Gabriel | Exponential relaxation to self-similarity for the superquadratic fragmentation equation[END_REF][START_REF] Gabriel | Long-time asymptotics for nonlinear growth-fragmentation equations[END_REF][START_REF] Mischler | Spectral analysis of semigroups and growthfragmentation equations[END_REF]) establish exponential rate of convergence toward self-similar solution, yet for the growth-fragmentation equation.

In this setting, the main tools are the ones from the spectral analysis of the problem and the rate of convergence is due to a spectral gap. The presence of a "growth term", equivalent to a transport term, appears crucial to establish a spectral gap.

We establish the long-time asymptotics of the moments when the initial condition is a Dirac mass. Using the above choices (3) and ( 4 1 Actually, this article considers the equation ( 2), but the results can ba adapted to [START_REF] Adams | On the Linear Ordinary 𝑞-Difference Equation[END_REF].

3

We also note that ∫ +∞ 0 𝑥 𝜎-1 𝑡 2/𝛼 𝜑(𝑥𝑡 1/𝛼 ) d𝑥 = 𝑡 (2-𝜎)/𝛼 𝐵(𝜎).

Using appropriate series representation of the Mellin transform, we give a formal proof of this result (Corollary 5 and Propositions 14). In particular, we see that there are different regimes: for 𝜎 < 2, the solution is increasing to infinity, for 𝜎 = 2, 𝐶(𝑡, 2) is constant (this is mass conservation). For 𝜎 > 2, the solution decreases to 0. We also establish that for 𝛼 > 1/2, the self-similar profile is uniquely determined by its moments.

Our tools are based on complex analysis, 𝑞-calculus, appropriate series representation of holomorphic functions, as well as the Karamata theorem. Although Mellin transforms are common to study self-similar fragmentation equations (see e.g. [START_REF] Van Brunt | A Mellin transform solution to a second-order pantograph equation with linear dispersion arising in a cell growth model[END_REF][START_REF] Van Brunt | Cell Division And The Pantograph Equation[END_REF][START_REF] Cheng | Kinetics of fragmentation[END_REF][START_REF] Doumic | An inverse problem: recovering the fragmentation kernel from the short-time behaviour of the fragmentation equation[END_REF][START_REF] Doumic | Estimating the division rate and kernel in the fragmentation equation[END_REF]), and some works use the pantograph equation, [START_REF] Van Brunt | A Mellin transform solution to a second-order pantograph equation with linear dispersion arising in a cell growth model[END_REF][START_REF] Van Brunt | Cell Division And The Pantograph Equation[END_REF], we present alternative series representations that are suitable for finding the appropriate asymptotic behaviors. Besides, our extension to measure-valued solutions by relying on a limiting argument on the moments is also new.

Outline. In Section 2, we construct the solution of the self-similar fragmentation equation using the theory of semi-group, and we identify invariant subspaces. In Section 3, we introduce the equation for the evolution of the moments. In Section 4, we show how to extend the notion of solutions to measure-valued solutions. In Section 5, we study the qualitative behavior of the moments. Finally, in Section 6, we give the asymptotic behavior of the moments.

Solutions of the fragmentation equation

In a first time, we solve the fragmentation equation through semi-groups.

Notation 1 (Space of solutions). We fix some 𝑝 ≥ 0. We set

L 𝑝 := L 1 (R + , 𝑤 𝑝 (𝑥) d𝑥) with 𝑤 𝑝 (𝑥) = 1 + 𝑥 𝑝 , 𝑥 ≥ 0.
We denote by ‖•‖ 𝑝 the norm

‖𝑓 ‖ 𝑝 := ∫ R + |𝑓 (𝑥)|𝑤 𝑝 (𝑥) d𝑥.
We define L + 𝑝 := {𝑓 ∈ L 𝑝 | 𝑓 ≥ 0}, the cone of non-negative functions in L 𝑝 .

Notation 2 (Operators). We set

D 𝜆 𝑓 (𝑥) := 𝑓 (𝜆𝑥), F 𝛼 𝑓 (𝑥) = 𝑥 𝛼 𝑓 (𝑥) and G 𝛼 𝑓 (𝑥) := 𝜆 2 D 𝜆 F 𝛼 𝑓 (𝑥) = 𝜆 2+𝛼 𝑥 𝛼 𝑓 (𝜆𝑥).
We also define 𝐹 (𝑥) := 𝑥 𝛼 .

Since 𝜆 ≤ 1 and 𝑤 𝑝 is non-decreasing,

‖F 𝛼 𝑓 ‖ 𝑝 ≤ ‖𝑓 ‖ 𝑝+𝛼 for 𝑓 ∈ L 𝑝+𝛼 , (5)
‖D 𝜆 𝑓 ‖ 𝑝 = 1 𝜆 ∫ |𝑓 (𝑥)|𝑤 𝑝 ( 𝑥 𝜆 ) d𝑥 ≤ 1 𝜆 ‖𝑓 ‖ 𝑝 for 𝑓 ∈ L 𝑝 , (6)
and

‖ G 𝛼 𝑓 ‖ 𝑝 ≤ ‖D 𝜆 F 𝛼 𝑓 ‖ 𝑝 = 𝜆 1+𝛼 ‖F 𝛼 𝑓 ‖ 𝑝 for 𝑓 ∈ L 𝑝+𝛼 . (7)
From ( 6), D 𝜆 is a bounded operator on L 𝑝 . Besides, D 𝜆 is invertible with inverse D 𝜆 -1 .

Solving the fragmentation equation when 𝛼 = 0

We first consider that 𝛼 = 0, so that (8) becomes

𝜕 𝑡 𝑢(𝑡, 𝑥) = 𝜆 2 𝑢(𝑡, 𝜆𝑥) -𝑢(𝑡, 𝑥). (8)
For any 𝑝 ≥ 0, the operators D 𝜆 and G 𝛼 are bounded in L 𝑝 while F 𝛼 = Id, the identity operator, so that Dom(D 𝜆 ) = Dom(F 𝛼 ) = Dom( G 𝛼 ) = L 𝑝 .

Proposition 1. Fix 𝑝 ≥ 0. For any given initial condition 𝑓 ∈ L 𝑝 , there exists an unique solution 𝑢 ∈ C 1 ([0, 𝑇 ], L 𝑝 ) of (8).

Proof. Since G 0 -F 0 = G 0 -Id is bounded, for any 𝑝 ≥ 0, the semi-group 𝑄 defined by 𝑄(𝑡) := exp(𝑡( G 0 -Id)), 𝑡 ≥ 0 is analytic and generated by ( G 0 -Id, L 𝑝 ). The mild solution of (8) with initial condition 𝑓 ∈ L 𝑝 is given by (𝑡, 𝑥) ↦ → 𝑄(𝑡)𝑓 (𝑥).

Following [START_REF] Melzak | A scalar transport equation[END_REF], this semi-group may be explicitly constructed through the following sequence: for 𝑓 ∈ L 𝑝 , we set

𝑎 0 := 𝑓 and 𝑎 𝑘+1 := G 0 𝑎 𝑘 -𝑎 𝑘 = 𝜆 2 D 𝜆 𝑎 𝑘 -𝑎 𝑘 .
The series

𝑄(𝑡)𝑓 = ∑ 𝑘≥0 𝑡 𝑘 𝑘! 𝑎 𝑘 = ∑ 𝑘≥0 𝑡 𝑘 𝑘! ( G 0 -Id) 𝑘 𝑓 = ∑ 𝑘≥0 𝑡 𝑘 𝑘! (𝜆 2 D 𝜆 -Id) 𝑘 𝑓
converges in L 𝑝 , and gives the semi-group generated by ( G 0 -Id, L 𝑝 ). Moreover, 𝑡 ↦ → 𝑄(𝑡)𝑓 is of class C 1 , with 𝜕 𝑡 𝑄(𝑡)𝑓 = ( G 0 -Id)𝑄(𝑡)𝑓 for any 𝑡 ≥ 0.

Regarding uniqueness, since ( 8) is linear, it is sufficient to consider that 𝑓 = 0. In this case,

‖𝑢(𝑡, •)‖ 𝑝 ≤ (𝜆 -1) ∫ 𝑡 0 ‖𝑢(𝑠, •)‖ 𝑝 d𝑠
so that the Gronwall inequality implies that ‖𝑢(𝑡, •)‖ 𝑝 = 0 for any 𝑡 ≥ 0. This yields the uniqueness.

Solving the fragmentation equation when 𝛼 > 0

We now consider 𝛼 > 0.

Unlike Section 2.1, the operators F 𝛼 and G 𝛼 defined in Notation 2 are not bounded. We define

Dom(F 𝛼 ) := {𝑓 ∈ L 𝑝 | F 𝛼 𝑓 ∈ L 𝑝 } and Dom( G 𝛼 ) := {𝑓 ∈ L 𝑝 | G 𝛼 𝑓 ∈ L 𝑝 }.
With ( 5) and [START_REF] Bertoin | Different aspects of a random fragmentation model[END_REF],

Dom(F 𝛼 ) = Dom( G 𝛼 ) = L 𝑝+𝛼 . (9)
The proof of the next result is straightforward.

Lemma 1. The operator (F 𝛼 , Dom(F 𝛼 )) is closed, densely defined and injective.

The proof of the next result relies on the Hille-Yoshida theorem and [4, Examples 2.8 and 3.9, p. 74, Proposition 2.2, p. 13].

Notation 3. For 𝜇 ∈ R, we set J 𝜇 𝑓 := 𝜇𝑓 for any function 𝑓 .

Lemma 2. For any 𝜇 ≥ 0, the operator (-F 𝛼 -J 𝜇 , Dom(F 𝛼 )) generates the strongly continuous, contractive semi-group (𝑃 (𝑡)) 𝑡≥0 with 𝑃 (𝑡)𝑓 (𝑥) = 𝑒 -𝜇𝑡-𝑡𝐹 (𝑥) 𝑓 (𝑥). Besides, L + 𝑝 is invariant by 𝑃 (𝑡) for any 𝑡 ≥ 0.

Since L 𝑝 is stable under D 𝜆 and D 𝜆 is invertible, Dom( G 𝛼 ) = Dom(F 𝛼 ). Besides, G 𝛼 (L + 𝑝 ) ⊂ L + 𝑝 . Fix 𝑔 ∈ L 𝑝 and 𝜇 > 0. Setting 𝑓 (𝑥) := ∫ +∞ 0 exp(-𝜇𝑡 -𝑡𝐹 (𝑥)
)𝑔(𝑥) d𝑡, we obtain by an integration that for any 𝑥 > 0, (𝜇 + F 𝛼 )𝑓 (𝑥) = 𝑔(𝑥).

In addition, since 𝐹 (𝑥) ≥ 0,

‖𝑓 ‖ 𝑝 ≤ ∫ +∞ 0 ∫ +∞ 0 exp(-𝜇𝑡)|𝑔(𝑥)|𝑤 𝑝 (𝑥) d𝑥 d𝑡 ≤ 1 𝜇 ‖𝑔‖ 𝑝 .
Therefore, 𝜇 + F 𝛼 is invertible with a bounded inverse for any 𝜇 > 0. From [START_REF] Bertoin | Different aspects of a random fragmentation model[END_REF], G 𝛼 is F 𝛼 -bounded.

Lemma 3. Fix 𝑝 > 1 and 𝜇 ≥ 𝜆 -1. There exists an operator (K 𝛼,𝜇 , Dom(K 𝛼,𝜇 )) that extends of ( G 𝛼 -F 𝛼 -J 𝜇 , Dom(F 𝛼 )) and that generates a strongly continuous semi-group of contractions (𝑄 𝜇 (𝑡)) 𝑡≥0 which satisfies the Duhamel equation

𝑄 𝜇 (𝑡)𝑓 = 𝑃 (𝑡)𝑓 + ∫ 𝑡 0 𝑄 𝜇 (𝑡 -𝑠) G 𝛼 𝑃 (𝑠)𝑓 d𝑠, ∀𝑓 ∈ Dom(F 𝛼 ), ∀𝑡 ≥ 0. ( 10
)
Proof. For 𝑓 ∈ Dom( G 𝛼 ) = Dom(F 𝛼 ), since L 𝑝 ⊂ L 1 , ∫ R + G 𝛼 𝑓 (𝑥) d𝑥 = 𝜆 2 ∫ R + D 𝜆 F 𝛼 𝑓 (𝑥) d𝑥 = 𝜆 ∫ R + F 𝛼 𝑓 (𝑥) d𝑥. Similarly, ∫ R + G 𝛼 𝑓 (𝑥)𝑥 𝑝 d𝑥 = 𝜆 1-𝑝 ∫ R + F 𝛼 𝑓 (𝑥)𝑥 𝑝 d𝑥. When 𝑝 > 1, 𝜆 1-𝑝 < 1 since 𝜆 > 1. For 𝑓 ∈ L + 𝑝 ∩ Dom(F 𝛼 ), ∫ R + ( G 𝛼 𝑓 (𝑥) -F 𝛼 𝑓 (𝑥) -J 𝜇 𝑓 (𝑥))𝑤 𝑝 (𝑥) d𝑥 = (𝜆 -1) ∫ R + F 𝛼 (𝑥) d𝑥 + (𝜆 1-𝑝 -1) ∫ R + F 𝛼 𝑓 (𝑥)𝑥 𝑝 d𝑥 -𝜇 ∫ R + 𝑓 (𝑥)𝑤 𝑝 (𝑥) d𝑥 ≤ (𝜆 -1) ∫ R + F 𝛼 𝑓 (𝑥) d𝑥 -𝜇 ∫ R + 𝑓 (𝑥)𝑤 𝑝 (𝑥) d𝑥. ( 11
)
On the other hand, for 𝑝 ≥ 𝛼,

∫ R + F 𝛼 𝑓 (𝑥) d𝑥 ≤ ∫ R + 𝑓 (𝑥)𝑤 𝑝 (𝑥) d𝑥. Therefore, for 𝜇 ≥ 𝜆 -1, ∫ R + ( G 𝛼 𝑓 (𝑥) -F 𝛼 𝑓 (𝑥) -J 𝜇 𝑓 (𝑥))𝑤 𝑝 (𝑥) d𝑥 ≤ 0.
The result follows from [4, Corollary 5.17 From the Duhamel equation ( 10), we have that 𝑄 𝜇 (𝑡)𝑓 ≥ 0 whenever 𝑓 ∈ Dom(K 𝛼,𝜇 ) ∩ L + 𝑝 . Proposition 2. For 𝜇 ≥ 𝜆 -1, the semi-group (𝑄 𝜇 (𝑡)) 𝑡≥0 is positive.

Proof. Fix 𝜈 > 0. We have that

(𝜈 + F 𝛼 + 𝜇) -1 = 1 𝜇+𝜈+𝑥 𝛼 so that G 𝛼 (𝜈 + F 𝛼 + 𝜇) -1 𝑓 (𝑥) = 𝜆 2 (𝜆𝑥) 𝛼 𝑓 (𝜆𝑥) 𝜇 + 𝜈 + (𝜆𝑥 𝛼 )
.

Therefore,

∫ R + | G 𝛼 (𝜈 + F 𝛼 + 𝜇) -1 𝑓 (𝑥)|𝑤 𝑝 (𝑥) d𝑥 ≤ 𝜆 2 ∫ R + |𝑓 (𝜆𝑥)| (𝜆𝑥) 𝛼 𝜇 + 𝜈 + (𝜆𝑥) 𝛼 𝑤 𝑝 (𝑥) 𝑤 𝑝 (𝜆𝑥) 𝑤 𝑝 (𝜆𝑥) d𝑥 ≤ ∫ R + |𝑓 (𝑥)| (𝜆𝑥) 𝛼 𝜓(𝑥) 𝑤 𝑝 (𝑥) d𝑥 with 𝜓(𝑥) := 𝜆 1-𝑝 𝑥 𝛼 𝜇 + 𝜈 + (𝜆𝑥) 𝛼 𝜆 𝑝 + 𝑥 𝑝 1 + 𝑥 𝑝 .
The function 𝜓 is continuous on R + with lim 𝑥→0 𝜓(𝑥) = 0 and lim 𝑥→+∞ 𝜓(𝑥) = 𝜆 1-𝑝 < 1 when 𝑝 > 1. From this, we easily obtain that for 𝜈 large enough, sup 𝑥 𝜓(𝑥) < 1. The result then follows from [4, Lemma 5.12, p. 143]. Proposition 3. For 𝜇 ≥ 𝜆 -1, the operator (K 𝛼,𝜇 , Dom(K 𝛼,𝜇 )) generates an analytic semi-group (𝑄 𝜇 (𝑡)) 𝑡≥0 .

Proof. The semi-group (𝑃 (𝑡)) 𝑡≥0 generated by (-F 𝛼 -𝜇, Dom(F 𝛼 )) is analytic [

Scaling invariance

The self-similar fragmentation presents an invariant behavior.

Proposition 5 (Invariance property). Let 𝑢 be the solution to (1) with initial condition 𝑓 ∈ L 𝑝 . Then 𝑣(𝑡, 𝑥) := 𝑢(𝑡𝛾 𝛼 , 𝑥𝛾) also solves (1) with initial condition 𝑥 ↦ → 𝑓 (𝛾𝑥).

Invariant subspaces

We now exhibit two invariant subspaces.

A Fréchet space

For 0 < 𝑝 < 𝑞, an immediate computation shows that ‖𝑓 ‖ 𝑝 ≤ 2‖𝑓 ‖ 𝑞 for any 𝑓 ∈ L 𝑞 , so that L 𝑞 ⊂ L 𝑝 . We define

F := ⋂ 𝑞>0 L 𝑞 .
When 𝑝 is fixed, with (9), Dom(F 𝑘 𝛼 ) = L 𝑝+𝑘𝛼 . Since the spaces L 𝑝 are decreasing with respect to inclusion,

⋂ 𝑘≥0 Dom(F 𝑘 𝛼 ) = ⋂ 𝑘≥0 L 𝑝+𝑘𝛼 = F. (13)
Fix 𝛾 > 0. We equip F with the norm

‖𝑓 ‖ F,𝛾 := ∑ 𝑘≥0 𝛾 𝑘 ∫ +∞ 0 𝑥 𝑘𝛼 |𝑓 (𝑥)| d𝑥 = ∑ 𝑘≥0 𝛾 𝑘 ∫ +∞ 0 |F 𝑘 𝛼 𝑓 (𝑥)| d𝑥.
This norm is induced by the Fréchet space F equipped with the norms

𝑓 ↦ → ‖𝑓 ‖ 0 + ‖F 𝑘 𝛼 𝑓 ‖ 0 , for 𝑘 = 0, 1, 2, . . . . For any 𝑘 ≥ 0, the norm ‖•‖ 0 + ‖F 𝑘 𝛼 •‖ 0 is equivalent to ‖•‖ 𝑘𝛼 . Lemma 5. For any 𝑓 ∈ F, ‖ G 𝛼 𝑓 ‖ F,𝛾 ≤ 𝜆 𝛾 ‖𝑓 ‖ F,𝛾 , (14)
‖F 𝛼 𝑓 ‖ F,𝛾 ≤ 1 𝛾 ‖𝑓 ‖ F,𝛾 . (15)
Proof. For any 𝑥 > 0,

F 𝑘 𝛼 G 𝛼 𝑓 (𝑥) = 𝜆 2+𝛼 𝑥 (𝑘+1)𝛼 𝑓 (𝜆𝑥). With a change of variable, ‖ G 𝛼 𝑓 ‖ F𝛾 = 𝜆 2+𝛼 ∑ 𝑘≥0 𝛾 𝑘 ∫ +∞ 0 𝑥 (𝑘+1)𝛼 |𝑓 (𝜆𝑥)| d𝑥 = 𝜆 1+𝛼 ∑ 𝑘≥0 𝛾 𝑘 ∫ +∞ 0 𝑥 (𝑘+1)𝛼 𝜆 (𝑘+1)𝛼 |𝑓 (𝑥)| d𝑥 = ∑ 𝑘≥0 𝛾 𝑘+1 𝛾𝜆 𝑘𝛼-1 ∫ +∞ 0 𝑥 (𝑘+1)𝛼 |𝑓 (𝑥)| d𝑥.
Since 𝜆 > 1, for 𝛾 > 𝜆, sup 𝑘≥0 1/𝛾𝜆 𝑘𝛼-1 < 𝜆/𝛾 < 1. This leads to [START_REF] Cañizo | Spectral gap for the growthfragmentation equation via Harris's theorem[END_REF]. Besides,

‖F 𝛼 𝑓 ‖ F,𝛾 = ∑ 𝑘≥0 𝛾 𝑘 ∫ +∞ 0 𝑥 (𝑘+1)𝛼 |𝑓 (𝜆𝑥)| d𝑥 ≤ 1 𝛾 ‖𝜑‖ F,𝛾 ,
which is [START_REF] Cáceres | Rate of convergence to selfsimilarity for the fragmentation equation in 𝐿 1 spaces[END_REF]. Proposition 6. Fix 𝑝 ≥ 0 and 𝛾 > 0. The space F ⊂ L 𝑝 is invariant under the semi-group 𝑄. In addition, 𝑄(𝑡)𝑓 ∈ F for any 𝑓 ∈ L 𝑝 and any 𝑡 > 0. Finally, the semi-group 𝑄 is continuous on (F, ‖•‖ F,𝛾 ).

Proof. Not only F is invariant under the semi-group 𝑄, but as 𝑄 is analytic, for any 𝑝 ≥ 0, 𝑄(𝑓 ) ∈ F for any 𝑓 ∈ L 𝑝 and any 𝑡 > 0.

Besides, ( 14) and [START_REF] Cáceres | Rate of convergence to selfsimilarity for the fragmentation equation in 𝐿 1 spaces[END_REF] show that

G 𝛼 -F 𝛼 is bounded on (F, ‖•‖ F,𝛾 . This implies that 𝑄 is continuous on (F, ‖•‖ F,𝛾 ).
This space F is contained in Dom(F 𝑘 𝛼 ) for any 𝑘 ≥ 0. On the other hand, we can also construct a scale of spaces by extrapolation to extend the solution to distributions. As we focus on another set of spaces, we do not enter the details of such a procedure that may be found e.g. in [22, Sect. 5, Chap. 2].

Functions with compact support

The construction of the above invariant subspace F arises in a general manner in the theory of semi-groups. We now exhibit another invariant space of functions which is specific to the kind of equation we look at.

Notation 5 (Support). When ℎ is a function 𝑓 or a measure 𝜇, its support is denoted by Supp(ℎ).

Using the scaling invariance 2.3, there is no restriction to consider that the support is contained in [0, 1]. Notation 6. We define V as the set of bounded, measurable functions from R + to R with Supp(𝑓 ) ⊂ [0, 1]. We also define V + the subset of non-negative functions in V.

Proposition 7. The space V is invariant by F 𝛼 and G 𝛼 with F 𝛼 (V) ⊂ V and G 𝛼 (V) ⊂ V. Besides, for any 𝑝 ≥ 0, ‖F 𝛼 𝑓 ‖ 𝑝 ≤ ‖𝑓 ‖ 𝑝 , ‖ G 𝛼 𝑓 ‖ 𝑝 ≤ 𝜆 1+𝛼 ‖𝑓 ‖ 𝑝 , ‖F 𝛼 𝑓 ‖ ∞ ≤ ‖𝑓 ‖ ∞ , ‖ G 𝛼 𝑓 ‖ ∞ ≤ 𝜆 2+𝛼 ‖𝑓 ‖ ∞ . Proof. Let 𝑓 ∈ V. Clearly, ‖𝑓 ‖ 𝑝 ≤ 2‖𝑓 ‖ ∞ . Since 𝑓 (𝑥) = 0 for 𝑥 > 1, F 𝛼 𝑓 (𝑥) = 𝑥 𝛼 𝑓 (𝑥) = 0. Since 𝜆 > 1, G 𝛼 𝑓 (𝑥) = 𝜆 2+𝛼 𝑥 𝛼 𝑓 (𝜆𝑥) = 0 when 𝑥 > 1. In addition, F 𝛼 𝑓 (𝑥) ≤ 𝑓 (𝑥) since 𝑥 𝛼 ≤ 1. With (7), ‖F 𝛼 𝑓 ‖ 𝑝 = ∫ 1 0 (1 + 𝑥 𝑝 )F 𝛼 𝑓 (𝑥) d𝑥 ≤ ‖F 𝛼 ‖ 𝑝 𝑓 and ‖ G 𝛼 𝑓 ‖ 𝑝 ≤ 𝜆 1+𝛼 ‖𝑓 ‖ 𝑝 .
The control of the supremum norm ‖•‖ ∞ is also easy to obtain.

Since G 𝛼 -F 𝛼 is a bounded operator, the next result is immediate.

Corollary 1. For 𝑓 ∈ V, 𝑄(𝑡)𝑓 ∈ V for any 𝑡 ≥ 0 with ‖𝑄(𝑡)𝑓 ‖ 𝑝 ≤ exp((1 + 𝜆 1+𝛼 )𝑡)‖𝑓 ‖ 𝑝 and ‖𝑄(𝑡)𝑓 ‖ ∞ ≤ exp((1 + 𝜆 2+𝛼 )𝑡)‖𝑓 ‖ ∞ . Corollary 2. The semi-group 𝑄(𝑡) is positive on (V, ‖•‖ ∞ ).
Proof. We identify continuous functions in V with C([0, 1]), since the support of

𝑓 ∈ V is contained in [0, 1]. The subset C([0, 1]
) is dense in V with respect to the uniform norm so that we have only to show that 𝑄(𝑡)𝑓 ≥ 0 for any non-negative, continuous function 𝑓 .

If 𝑓 is continuous, then

F 𝛼 𝑓 and G 𝛼 𝑓 are continuous. With Proposition 7, G 𝛼 -F 𝛼 is a bounded operator on ( C([0, 1], ‖•‖ ∞ ). Consider 𝑓 ∈ C([0, 1]) and 𝑓 ≥ 0. Let 𝑥 be such that 𝑓 (𝑥) = 0. Then G 𝛼 𝑓 (𝑥) -F 𝛼 𝑓 (𝑥) = 𝜆 2+𝛼 𝑥 𝛼 𝑓 (𝜆𝑥) ≥ 0.
From [2, Theorem 1.3, p. 124], it follows that 𝑄(𝑡)𝑓 ≥ 0 for any 𝑡 ≥ 0.

Remark 1. From the above proof, we have also that 𝑄(𝑡) maps C([0, 1]) to itself.

A monotonic property with respect to 𝛼

Let (𝑄(𝑡; 𝛼)) 𝑡≥0 be the semi-group generated by G 𝛼 -F 𝛼 when the parameter is 𝛼.

Proposition 8. For any 𝑓 ∈ V + and any 𝑡 ≥ 0, 𝛼 ↦ → 𝑄(𝑡; 𝛼)𝑓 is non-increasing.

Proof. We set

K 𝛼 := G 𝛼 -F 𝛼 .
From the properties of the semi-group,

𝑄(𝑡; 𝛼)𝑓 (𝑥) = 𝑓 (𝑥) + ∫ 𝑡 0 𝑄(𝑠; 𝛼)K 𝛼 𝑓 (𝑥) d𝑠.
Fix 𝛽 > 𝛼 and write 𝛾 = 𝛽 -𝛼. Then

( G 𝛽 -F 𝛽 )𝑓 = 𝜆 2+𝛽 𝑥 𝛽 𝑓 (𝜆𝑥) -𝑥 𝛽 𝑓 (𝑥) = 𝜆 2+𝛼 𝑥 𝛼 (𝜆𝑥) 𝛽 𝑓 (𝜆𝑥) -𝑥 𝛾 𝑥 𝛼 𝑓 (𝑥) = K 𝛼 F 𝛾 𝑓. Then for 𝑓 ∈ V + , 𝑄(𝑡; 𝛽)𝑓 -𝑄(𝑡; 𝛼𝑓 ) = ∫ 𝑡 0 𝑄(𝑡 -𝑠; 𝛽)(K 𝛽 -K 𝛼 )𝑄(𝑠; 𝛼)𝑓 d𝑠 = ∫ 𝑡 0 𝑄(𝑡 -𝑠; 𝛽)K 𝛼 (F 𝛾 -J 1 )𝑄(𝑠; 𝛼)𝑓 d𝑠.
The supports of 𝑓 and 𝑄(𝑠; 𝛼)𝑓 are contained in [0, 1]. For a non-negative function 𝑔 with support in [0, 1],

F 𝛾 𝑔(𝑥) = 𝑥 𝛾 𝑔(𝑥) ≤ 𝑔(𝑥)
and then (F 𝛾 -J 1 )𝑔(𝑥) ≤ 0. Hence, 𝑄(𝑡; 𝛽)𝑓 ≤ 𝑄(𝑡; 𝛼)𝑓 .

Expansion of the semi-group

Since (𝑄(𝑡; 𝛼)) 𝑡≥0 is an analytic semi-group, it admits formally the expansion

𝑄(𝑡; 𝛼) = ∑ 𝑘≥0 𝑡 𝑘 𝑘! ( G 𝛼 -F 𝛼 ) 𝑘 . (16)
With [START_REF] Van Brunt | Cell Division And The Pantograph Equation[END_REF], when 𝑓 belongs to F, F 𝑘 𝛼 𝑓 and G 𝑘 𝛼 𝑓 also belong to F for any 𝑘 ≥ 0. With Lemma 5, such expansion is valid for any 𝑓 ∈ F and any 𝑡 > 0, using the norm ‖•‖ F,𝛾 .

In this section, we give an explicit expression involving D 𝜆 , F 𝛼 , as well as 𝑞-binomial coefficients.

Definition 1 (see e.g. [START_REF] Konvalina | A Unified Interpretation of the Binomial Coefficients, the Stirling Numbers, and the Gaussian Coefficients[END_REF][START_REF] Sagan | Combinatorics: the art of counting[END_REF]). For integers 𝑘, 𝑛 with 0 ≤ 𝑘 ≤ 𝑛, and 𝑞 ∈ (0, 1), the 𝑞-binomial coefficient (or Gaussian polynomial ) is

( 𝑛 𝑘 ) 𝑞 := (1 -𝑞 𝑛 ) • • • (1 -𝑞 𝑛-𝑘+1 ) (1 -𝑞) • • • (1 -𝑞 𝑘 ) ,
where we use the convention that an empty product is equal to 1. This means that for 𝑛 = 0, 𝑘 = 0 or 𝑘 = 𝑛,

( 𝑛 𝑘 ) 𝑞 := 1.
) 𝑞 = ( 𝑛 𝑘 ) 𝑞 + 𝑞 𝑛-𝑘+1 ( 𝑛 𝑘 -1 ) 𝑞 . (17)
Moreover, for any 𝑎 ∈ R,

(𝑎; 𝑞) 𝑛 = 𝑛 ∑ 𝑘=0 𝑞 𝑛(𝑛-1)/2 ( 𝑛 𝑘 ) 𝑞 𝑎 𝑘 .
We note the relation

G 𝛼 F 𝛼 𝑓 = 𝜆 𝛼 F 𝛼 G 𝛼 𝑓 (18)
for any 𝑓 ∈ Dom(F 2 𝛼 ).

Proposition 9. For any 𝑘 ≥ 0,

( G 𝛼 -F 𝛼 ) 𝑘 = 𝑘 ∑ 𝑖=0 (-1) 𝑘-𝑖 ( 𝑘 𝑖 ) 𝜆 -𝛼 G 𝑖 𝛼 F 𝑘-𝑖 𝛼 . (19)
Proof. We proceed by induction. For 𝑘 = 0, 1, (19) is easily verified. Let us assume that ( 19) is true for 𝑘 ≥ 1. Then

( G 𝛼 -F 𝛼 ) 𝑘+1 = ( G 𝛼 -F 𝛼 ) 𝑘 ∑ 𝑖=0 (-1) 𝑘-𝑖 ( 𝑘 𝑘 -𝑖 ) 𝜆 -𝛼 G 𝑖 𝛼 F 𝑘-𝑖 𝛼 . = 𝑘 ∑ 𝑖=0 (-1) 𝑘-𝑖 ( 𝑘 𝑘 -𝑖 ) 𝜆 -𝛼 G 𝑖+1 𝛼 F 𝑘-𝑖 𝛼 - 𝑘 ∑ 𝑖=0 (-1) 𝑘-𝑖 ( 𝑘 𝑘 -𝑖 ) 𝜆 -𝛼 F 𝛼 G 𝑖 𝛼 F 𝑘-𝑖 𝛼 .
Applying 𝑖 times the commutation relation [START_REF] Cáceres | Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations[END_REF],

F 𝛼 G 𝑖 𝛼 F 𝑘-𝑖 𝛼 = 𝜆 -𝑖𝛼 G 𝑖 𝛼 F 𝑘+1-𝑖 𝛼 .
Reindexing the sum,

𝑘 ∑ 𝑖=0 (-1) 𝑘-𝑖 ( 𝑘 𝑖 ) 𝜆 -𝛼 G 𝑖+1 𝛼 F 𝑘-𝑖 𝛼 = 𝑘+1 ∑ 𝑖=1 (-1) 𝑘+1-𝑖 ( 𝑘 𝑘 -𝑖 + 1 ) 𝜆 -𝛼 G 𝑖 𝛼 F 𝑘+1-𝑖 𝛼 .
Using the Pascal identity ( 17),

( G 𝛼 -F 𝛼 ) 𝑘+1 = 𝑘 ∑ 𝑖=1 (-1) 𝑘-𝑖 ( 𝑘 𝑖 ) 𝜆 -𝛼 G 𝑖 𝛼 F 𝑘-𝑖 𝛼 + G 𝑘+1 𝛼 + (-1) 𝑘+1 F 𝑘+1 𝛼 .
This completes the proof.

The proof of the next lemma is easily obtained by induction.

Lemma 6. For any 𝜆 > 0, D 𝜆 F 𝛼 = 𝜆 𝛼 F 𝛼 D 𝜆 and for for any 𝑘 ≥ 1,

G 𝑘 𝛼 = 𝜆 2𝑘+𝑘(𝑘+1)𝛼-𝛼 2 F 𝑘 𝛼 D 𝑘 𝜆 = 𝜆 2𝑘+𝑘(𝑘-1)𝛼-𝛼 2 D 𝑘 𝜆 F 𝑘 𝛼 . Corollary 3. The expansion (16) is also given by 𝑄(𝑡; 𝛼) |F = ∑ 𝑘≥0 𝑡 𝑘 𝑘! L 𝑘 F 𝑘 𝛼 ( 20
)
with

L 𝑘 := 𝑘 ∑ 𝑖=0 (-1) 𝑘-𝑖 ( 𝑘 𝑘 -𝑖 ) 𝜆 -𝛼 𝜆 2𝑖+𝛼 𝑖(𝑖-1) 2
The 𝑞-calculus is sometimes seen as a "deformation" of the ordinary calculus. It is also a way to give a meaning to some summation formula for 𝑞 < 1 and then letting 𝑞 converging to 1. In contradistinction, 𝑞 = 0 is equivalent to 𝛼 = 0, so that the situation is much simpler. We illustrate here that our semi-group converges as 𝛼 → 0 to the semi-group generated by 𝛼 = 0. Lemma 7. For any 𝑡 ≥ 0 and any 𝑓 ∈ F, 𝑄(𝑡; 𝛼)𝑓 converges to 𝑄(𝑡; 0)𝑓 as 𝛼 → 0, where 𝑄(𝑡; 0) is given by (2.1).

Proof. It is known that lim 𝑞→1 ( 𝑘 𝑖 ) 𝑞 = ( 𝑘 𝑖 ) .
Hence, when 𝛼 → 0, 𝜆 -𝛼 → 1 and then

lim 𝛼→0 L 𝑘 = 𝑘 ∑ 𝑖=0 (-1) 𝑘-𝑖 ( 𝑘 𝑘 -𝑖 ) 𝜆 2𝑖 D 𝑖 𝜆 = (𝜆 2 D 𝜆 -Id) 𝑘 .
On the other hand, it is easily seen that F 𝛼 converges to Id on F. Therefore, for any 𝑡 ≥ 0,

lim 𝛼→0 𝑄(𝑡; 𝛼) = ∑ 𝑘≥0 𝑡 𝑘 𝑘! (𝜆 2 D 𝜆 -Id) 𝑘 = exp(𝑡(𝜆 2 D 𝜆 -Id)).
This proves the result.

3 Evolution of the moments

Mellin transform

Definition 3 (Mellin transform). The Mellin transform of 𝑓 is

M[𝑓 ](𝜎) = ∫ +∞ 0 𝑥 𝜎-1 • 𝑓 (𝑥) d𝑥
for any 𝜎 ∈ C such that the integral is well defined.

Fix 𝑝 > 0 and define the vertical strip

V(𝑎, 𝑏) := {𝜎 ∈ C | 𝑎 < Re(𝜎) < 𝑏} for 𝑎, 𝑏 ∈ R, 𝑎 < 𝑏. Let 𝑓 ∈ L + 𝑝 . Then 𝜎 ↦ → M[𝑓 ](𝜎)
is well defined on the closure of the vertical strip Cl V(1, 𝑝 + 1), and is analytic for any 𝜎 ∈ V(1, 𝑝 + 1). In particular, for 𝑓 ∈ F, M[𝑓 ](𝜎) is well defined for any 𝜎 with Re(𝜎) = 1 and is analytic in the half-place H 1 .

Since 𝑄(𝑡)𝑓 ∈ F for any 𝑡 > 0, it holds that 𝜎 ↦ → M[𝑄(𝑡)𝑓 ](𝜎) is well defined for any 𝑡 > 0 and any 𝜎 ∈ Cl H 1 . Besides, it is analytic on H 1 . Lemma 8. Assume 𝑓 ∈ L + 𝑝 . Then for any 0 < 𝑟 ≤ 𝑡,

∫ 𝑡 𝑟 M[F𝑄(𝑠)𝑓 ](𝜎) d𝑠 = ∫ 𝑡 𝑟 M[𝑄(𝑠)𝑓 ](𝜎 + 𝛼) d𝑠, (21)
∫ 𝑡 𝑟 M[ G𝑄(𝑠)𝑓 ](𝜎) d𝑠 = 𝜆 2-𝜎 ∫ 𝑡 𝑟 M[𝑄(𝑠)𝑓 ](𝜎 + 𝛼) d𝑠. ( 22
)
If 𝑓 ∈ L 𝑝+𝛼 , then one may set 𝑟 = 0 in (21)- [START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF].

Proof. Let 𝑢 be the function given by Proposition 4, so that 𝑢(𝑡, 𝑥) = 𝑄(𝑡)𝑓 (𝑥) 𝑥-almost everywhere. Then

∫ +∞ 0 𝑥 𝜎-1 ∫ 𝑡 𝑟 F 𝛼 𝑄(𝑠)𝑓 (𝑥) d𝑠 d𝑥 = ∫ +∞ 0 𝑥 𝜎-1 ∫ 𝑡 𝑟 𝑥 𝛼 𝑄(𝑠)𝑓 (𝑥) d𝑠 d𝑥 = ∫ +∞ 0 𝑥 𝛼+𝜎-1
∫ +∞ 0 𝑥 𝜎-1 ∫ 𝑡 0 G 𝛼 𝑄(𝑠)𝑓 (𝑥) d𝑠 d𝑥 = 𝜆 2-𝜎 ∫ 𝑡 0 M[𝑄(𝑠)𝑓 ](𝜎 + 𝛼) d𝑠. (24)
This leads to ( 21) and [START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF].

If 𝑓 ∈ L 𝑝+𝛼 = Dom(F 𝛼 ), then F 𝛼 𝑄(𝑠)𝑓 ∈ L 𝑝 and is bounded for small 𝑠. Therefore, ∫ 𝑡 0 F 𝛼 𝑄(𝑠)𝑓 (𝑥) d𝑠 d𝑥 is well defined and ( 21) and ( 22) are well defined for 𝑟 = 0. Notation 7. We set for 𝜎 ∈ C and a measurable function 𝑓 ∈ L 𝑝 , 𝐶(𝑡, 𝜎; 𝑓, 𝜆) := M[𝑄(𝑡)𝑓 ](𝜎), whenever this quantity is well defined (e.g., for any 𝜎 with Re(𝜎) ≥ 1 if 𝑡 > 0, and

1 ≤ Re(𝜎) ≤ 𝑝 + 1 if 𝑡 = 0).
The next lemma is immediate from the control over the semi-group (𝑄(𝑡)) 𝑡≥0 . It shows that the moments of order 𝜎 remain finite for any time 𝑡 > 0. On this topic, see also [START_REF] Banasiak | On the existence of moments of solutions to fragmentation equations[END_REF]. Proof. This follows from the inequality

|M[𝑔](𝜎)| ≤ ∫ +∞ 0 𝑥 Re(𝜎) |𝑔|(𝑥) d𝑥 ≤ ‖𝑔‖ Re(𝜎)
and ( 12) applied to 𝑔 := 𝑄(𝑡)𝑓 .

Combining ( 21) and ( 22) leads to the following result.

A solution of the moment equation

We now give a setting to consider the solution to [START_REF] Fox | On a Functional Differential Equation[END_REF]. This framework is restrictive so that most initial conditions cannot be considered within it, as the Mellin transform is unbounded in many situations. However, this framework will be used for initial conditions with compact support, and then to extend the solution to measure-valued solutions.

Fix 𝜎 0 > 0. We set S = [𝜎 0 , +∞). Let (U, ‖•‖ U ) be the space U := C(S, R) with

‖𝑔‖ U := sup 𝜎≥2 |𝑔(𝜎)|.
We define the operator A on U by A𝑔(𝜎) = (𝜆 2-𝜎 -1)𝑔(𝜎 + 𝛼) for any 𝜎 ≥ 2.

Since 𝜆 > 1, A is bounded with norm ‖A‖ U→U ≤ max{1, 𝜆 2-𝜎 0 -1}.

For any 𝑔 ∈ U, there exists a unique solution 𝐶 ∈ C(R + × S, R) to

𝐶(𝑡, 𝜎) = 𝑔(𝜎) + ∫ 𝑡 0 A𝐶(𝑠, 𝜎) d𝑠 = 𝑔(𝜎) + (𝜆 2-𝜎 -1) ∫ 𝑡 0 𝐶(𝑠, 𝜎 + 𝛼) d𝑠. ( 26
)
This solution is given by

𝐶(𝑡, 𝜎) = 𝑅(𝑡)𝑔(𝜎) with 𝑅(𝑡) := +∞ ∑ 𝑘=0 𝑡 𝑘 𝑘! A 𝑘 .
The family of linear operators (𝑅(𝑡)) 𝑡≥0 forms a continuous semigroup with the bounded infinitesimal generator (A, U). In addition, ‖𝑅(𝑡)‖ U→U ≤ exp(‖𝐴‖ U→U 𝑡) for any 𝑡 ≥ 0.

4 Initial condition as measure with compact support

Extension of the semi-group

In this section, we consider expanding the notion of solution to the fragmentation equation ( 8) to deal with an initial condition which is a measure. The basic idea is to pass to the limit in the moment equation [START_REF] Gabriel | Exponential relaxation to self-similarity for the superquadratic fragmentation equation[END_REF] with respect to the initial condition, and then then transfer this as a limit in the space of measures. This strategy requires the inversion of the Mellin transform, and in particular to be able to solve the moments problem. As this is not always possible, we restrict ourselves to measures with support contained in [0, 1]. Possibly, this strategy may be applied to a broader class of measures, yet with some restriction on the growth of the moments.

Notation 8 (Sets M and K). We define Recall that V ⊂ F was defined in Notation 6.

M := {𝜇 | 𝜇 is a probability measure, Supp(𝜇) ⊂ [0, 1], 𝜇({0}) = 0}, K := { 𝑓 ∈ V ⃒ ⃒ ⃒ ⃒ ∫ 1 0 𝑥𝑓 (
We equip the set M with the induced topology of weak convergence of measures.

Lemma 10. Let Cl Φ(K) be the closure of Φ(K) in M. Then M = Cl Φ(K).

Proof. For 𝑎 ∈ (0, 1], 𝛿 𝑎 ∈ M. We define when 0 < 𝜖 < 𝑎 the function

𝑓 𝜖 (𝑥) = 1 𝜖(𝑎 + 𝜖/2) 1 [𝑎-𝜖,𝑎] (𝑥).
Therefore, Φ(𝑓 𝜖 ) converges weakly toward 𝛿 𝑎 .

The set M is stable by any barycentric combination. Thus, for any 𝑎 1 , . . . , 𝑎 𝑚 with Hence, the closure of Cl Φ(K) contains ⋃ 0<𝜖<1 M 𝜖 . Finally, let 𝜇 ∈ M. For 𝜖 ∈ (0, 1), we set

0 < 𝑎 𝑖 , 𝑎 1 + • • • + 𝑎 𝑚 = 1, the discrete measure ∑ 𝑚 𝑖=1 𝛿 𝑥 𝑖 belongs to M. Let M 𝜖 := {𝜇 ∈ M | 𝜇([0, 𝜖)) = 0}.
𝜇 𝜖 (d𝑥) := 1 𝜇([𝜖, 1]) 1 𝑥≥1 𝜇(d𝑥) ∈ M 𝜖 .
It is easily shown that 𝜇 𝜖 converges weakly to 𝜇. This proves the result.

We define on M the Mellin transform

N[𝜇](𝜎) := ∫ [0,1] 𝑥 𝜎-2 𝜇(d𝑥) for any 𝜇 ∈ M, so that N[𝜇](2) = 1. We remark that N[𝜇](𝜎) is well defined for any 𝜎 ∈ C with Re(𝜎) ≥ 2, with |N[𝜇](𝜎)| ≤ 1. For 𝑓 ∈ K, N[Φ(𝑓 )](𝜎) = M[𝑓 ](𝜎) for any 𝜎 ≥ 1. ( 27
)
Lemma 11. The set K is invariant under the action of the semi-group 𝑄.

Proof.

For 𝑓 ∈ K ⊂ V ⊂ F, M[𝑓 ](2) = ∫ 1 0 𝑥𝑓 (𝑥) d𝑥 = 1 and M[𝑄 0 (𝑡)𝑓 ](2) = M[𝑓 ](2)
for any 𝑡 ≥ 0 since the mass is preserved. Therefore, K is invariant as K ⊂ V and V is itself invariant.

The space U and the semi-group (𝑅(𝑡)) 𝑡≥0 have been defined in Section 3.2.

Lemma 12. For any 𝑡 ≥ 0, define 𝑃 (𝑡) : Φ(K) → Φ(K) ⊂ M by 𝑃 (𝑡)Φ(𝑓 ) = Φ(𝑄(𝑡)𝑓 ) for any 𝑓 ∈ K.

Then 𝑃 (0) = Id, 𝑃 (𝑡 + 𝑠) = 𝑃 (𝑡)𝑃 (𝑠) for any 𝑠, 𝑡 ≥ 0 and

N∘𝑃 (𝑡) = 𝑅(𝑡) ∘ N, 𝑡 ≥ 0 on Φ(K). (28)
Finally, N(Φ(K)) is invariant by (𝑅(𝑡)) 𝑡≥0 .

Proof. We note first that 𝑃 (𝑡) : Φ(K) → M is well defined, as from Lemma 11, 𝑄(𝑡)(K) ⊂ K for any 𝑡 ≥ 0. Since

𝑃 (𝑡)𝑃 (𝑠)Φ(𝑓 ) = 𝑃 (𝑡)Φ(𝑄(𝑠)𝑓 ) = Φ(𝑄(𝑡)𝑄(𝑠)𝑓 ) = Φ(𝑄(𝑡 + 𝑠)𝑓 ) = 𝑃 (𝑡 + 𝑠)Φ(𝑡),
we have that 𝑃 satisfies the semi-group property. In addition, 𝑃 (0) is the identity operator.

With [START_REF] Gabriel | Long-time asymptotics for nonlinear growth-fragmentation equations[END_REF], for 𝑓 ∈ K and 𝑡 ≥ 0,

N(𝑃 (𝑡)Φ(𝑓 )) = N(Φ(𝑄(𝑡)𝑓 )) = M(𝑄(𝑡)𝑓 ).
With Proposition 10,

M(𝑄(𝑡)𝑓 ) = 𝑅(𝑡) M[𝑓 ] = 𝑅(𝑡) N[Φ(𝑓 )].
We have then established that for any 𝜇 ∈ Φ(K).

N[𝑃 (𝑡)𝜇] = 𝑅(𝑡) N[𝜇]

, 𝑡 ≥ 0 for any 𝜇 ∈ Φ(K),

which is [START_REF] Haas | Regularity of formation of dust in self-similar fragmentations[END_REF].

Since 𝑃 (𝑡)(Φ(K)) ⊂ Φ(K), it follows from (28) that 𝑅(𝑡)(N(Φ(K))) ⊂ N(Φ(K)).

The idea is now to extend 𝑃 (𝑡) to the closure of Cl Φ(K) of Φ(K) in M by taking profit from the fact that N is invertible from a suitable subset of M to a suitable subset of U, continuous with a continuous inverse. Besides, the continuity of 𝑅(𝑡) implies the strong continuity of 𝑃 (𝑡), making (𝑃 (𝑡)) 𝑡≥0 a non-linear semi-group.

Not all probabilities distributions 𝜇 can be recovered from their moments {N[𝜇](𝜎 + 𝑘)} 𝑘≥0 . This is the Stieltjes (for distribution with positive support) or the Hamburger problem [START_REF] Lin | Recent developments on the moment problem[END_REF][START_REF] Widder | The Laplace Transform[END_REF].

Theorem 2 ([24, Example VIII.1.(d), p. 251]). A probability measure 𝜇 with support in [0, 1] is fully determined by its moments, meaning that the knowledge of all its moments is sufficient to characterize the distribution.

This implies the following classical result on convergence [START_REF] Lin | Recent developments on the moment problem[END_REF][START_REF] Takacs | A Moment Convergence Theorem[END_REF].

Theorem 3 (Fréchet-Shohat theorem [START_REF] Lin | Recent developments on the moment problem[END_REF]). Let {𝜇 𝑛 } 𝑛≥0 be a sequence of measures in M. Assume that there exists a sequence {𝑚 𝑘 } 𝑘≥0 for which

M[𝜇 𝑛 ](𝑘 + 2) = ∫ 1 0 𝑥 𝑘 𝜇 𝑛 (d𝑥) ---→ 𝑛→∞ 𝑚 𝑘 for any 𝑘 ≥ 0. ( 30
)
Then there exists a measure 𝜇 ∈ M such that {𝜇 𝑛 } 𝑛≥0 converges weakly to 𝜇 and M[𝜇](𝑘 + 2) = 𝑚 𝑘 , 𝑘 ≥ 0.

On

M[𝜇](𝜎) = 𝜇((0, 1]) = 1 = M[𝜇](2), so that 𝜎 ∈ R ↦ → M[𝜇](𝜎) is continuous on [2, +∞).
In addition, for any 𝜎 > 2,

𝜕 𝑘 𝜎 M[𝜇](𝜎) = ∫ 1 0 ln(𝑥) 𝑘 𝑥 𝜎-2 𝜇(d𝑥) for any 𝑘 ≥ 0.
Since 𝑥 ↦ → 𝑥 𝛾 ln(𝑥) 𝑘 is bounded on [0, 1] for any 𝛾 > 0, 𝑘 ≥ 1, it holds that for some constant 𝐶 𝛾 ,

|𝜕 𝑘 𝜎 M[𝜇](𝜎)| ≤ 𝐶 𝛾 ‖𝑓 ‖ ∞ for any 𝜎 ≥ 2 + 𝛾.
Besides, (- We prove the uniform convergence in two steps: We establish the uniform convergence of N[𝜇 𝑛 ] on [𝜎 0 , +∞) with 𝜎 0 > 2, and then on [2, +∞).

Set 𝐹 𝑛 (𝑥) := 𝜇 𝑛 ([0, 𝑥]) and 𝐹 (𝑥) := 𝜇([0, 𝑥]). For some sequence

{𝑎 𝑛 } 𝑛≥0 in [0, 1], ⃒ ⃒ ⃒ ⃒ ∫ 1 0 𝑥 𝜎-2 𝜇 𝑛 (d𝑥) - ∫ 1 0 𝑥 𝜎-2 𝜇(d𝑥) ⃒ ⃒ ⃒ ⃒ ≤ ⃒ ⃒ ⃒ ⃒ ∫ 𝑎𝑛 0 𝑥 𝜎-2 𝜇 𝑛 (d𝑥) - ∫ 𝑎𝑛 0 𝑥 𝜎-2 𝜇(d𝑥) ⃒ ⃒ ⃒ ⃒ + |𝜇 𝑛 ((𝑎 𝑛 , 1]) -𝜇((𝑎 𝑛 , 1])|. Since 𝐹 (1) = 𝐹 𝑛 (1) = 1, |𝜇 𝑛 ((𝑎 𝑛 , 1]) -𝜇((𝑎 𝑛 , 1])| = |𝐹 𝑛 (𝑎 𝑛 ) -𝐹 (𝑎 𝑛 )|. Fix 𝜎 0 > 2. For any 𝜎 > 𝜎 0 , ⃒ ⃒ ⃒ ⃒ ∫ 𝑎𝑛 0 𝑥 𝜎-2 𝜇 𝑛 (d𝑥) - ∫ 𝑎𝑛 0 𝑥 𝜎-2 𝜇(d𝑥) ⃒ ⃒ ⃒ ⃒ ≤ 𝑎 𝜎-2 𝑛 ≤ 𝑎 𝜎 0 -2 𝑛 .
Since 𝜇({0}) = 0, 𝐹 is continuous at 0 so that there exists some 𝛾 > 0 such that 𝐹 is continuous on [0, 𝛾]. Therefore, as 𝑎 𝑛 → 0, lim 𝑛→∞ 𝐹 (𝑎 𝑛 ) = 0. On the other hand, 𝐹 𝑛 (𝑎 𝑛 ) ≤ 𝐹 𝑛 (𝜂) when 𝑎 𝑛 ≤ 𝜂 < 𝛾 and lim 𝑛→∞ 𝐹 𝑛 (𝜂) = 𝐹 (𝜂) since 𝜂 is a point of continuity of 𝐹 . Therefore, for any 𝜖 > 0, there exists 𝜂 such that 𝐹 (𝜂) ≤ 𝜖, 𝐹 𝑛 (𝜂) -𝐹 (𝜂) ≤ 𝜖 for 𝑛 ≥ 𝑛 0 with 𝑛 0 large enough. Thus, lim 𝑛→∞ 𝐹 𝑛 (𝑎 𝑛 ) = 0. The choice of 𝜂 and 𝑛 0 do not depend on 𝜎.

Letting {𝑎 𝑛 } 𝑛≥0 converging to 0 leads to the uniform convergence on

[M] ⊂ Cl N[Φ(K)].
We then set N -1 (𝑔) = 𝜇. This measure is unique, otherwise, it contradicts Theorem 2 that states that elements of M are uniquely determined by their moments.

From Lemma 12, 𝑅(𝑡)𝑔 𝑛 ∈ N[Φ(K)] and by continuity of

𝑅(𝑡) in U, 𝑅(𝑡)𝑔 ∈ Cl N[Φ(K)] = N[M].
Proposition 11. We define the family 𝑃 = (𝑃 (𝑡)) 𝑡≥0 of operators from M to M by

𝑃 (𝑡) := N -1 ∘𝑅(𝑡) ∘ N for any 𝑡 ≥ 0.
Then 𝑃 is a continuous non-linear semi-group on M. Furthermore, the set Φ(K) is invariant under 𝑃 (𝑡) for any 𝑡 ≥ 0.

By construction, since for any 𝑡 > 0, 𝑅(𝑡) is continuous on U, we obtain that 𝜇 ↦ → 𝑃 (𝑡)𝜇 is also continuous with respect to the weak topology. Since the space of measures is also a metric space, 𝑃 is jointly continuous on (0, +∞) × M [17, Theorem 4].

Measure-valued solutions

If 𝑢 is solution to (1) with the initial condition 𝑓 ∈ K, then (𝑡, 𝑥) ↦ → 𝑥𝑢(𝑡, 𝑥) is solution to (2) with initial condition 𝑥 ↦ → 𝑥𝑓 (𝑥).

Let 𝜇 be a measure in M. Set 𝜑 𝜆 (𝑥) := 𝑥/𝜆. Let 𝜑 𝜆 ♯𝜇 be the pushforward of 𝜇, that is the measure such that for any function 𝜓 ∈ C([0, 1]),

∫ 1 0 𝜓 ( 𝑥 𝜆 ) 𝜇(d𝑥) = ∫ 1 0 𝜓(𝑥)𝜑 𝜆 ♯𝜇(d𝑥).
When 𝜇(d𝑥) = 𝑓 (𝑥)d𝑥, it holds that 𝜑 𝜆 ♯𝜇(d𝑥) = 𝜆𝑓 (𝜆𝑥)1 𝑥≤1 .

For a given measure 𝜇 ∈ M, we define 𝜇 𝑡 = 𝑃 (𝑡)𝜇, which is formally a solution to the equation

𝜕 𝑡 𝜇 𝑡 (d𝑥) = 𝜆 𝛼 𝑥 𝛼 𝜑 𝜆 ♯𝜇 𝑡 (d𝑥) -𝑥 𝛼 𝜇 𝑡 (d𝑥). (31)
For any 𝜓 ∈ C([0, 1]),

𝜕 𝑡 ∫ 1 0 𝜓(𝑥)𝜇 𝑡 (d𝑥) = ∫ 1 0 𝜆 𝛼 𝑥 𝛼 𝜓 ( 𝑥 𝜆 ) 𝜇 𝑡 (d𝑥) - ∫ 1 0 𝑥 𝛼 𝜓(𝑥)𝜇 𝑡 (d𝑥).
Clearly, we could extend by linearity [START_REF] Konvalina | A Unified Interpretation of the Binomial Coefficients, the Stirling Numbers, and the Gaussian Coefficients[END_REF] to any initial condition in the cone Cone(M) := {𝛾𝜇 | 𝛾 > 0, 𝜇 ∈ M} of non-negative measures generated by M.

We also define

M[𝜇](𝜎) := ∫ 1 0 𝑥 𝜎-1 𝜇(d𝑥) = N[𝜇](𝜎 + 1).
The difference with N lies in the index. Let M -1 be the set of measures

𝜇 ∈ M such that M[𝜇](1) < +∞ or equivalently, N[𝜇](2) < +∞. This implies that 𝜎 ↦ → M[𝜇](𝜎) is continuons on [1, +∞). The semi-group (𝑅(𝑡)) 𝑡≥0 may be extended to C([1, +∞)) so that 𝑅(𝑡) M[𝜇] is bounded for any 𝑡 ≥ 0.
For 𝜇 ∈ M -1 , we set R𝜇(d𝑥

) := 1 𝑥 𝜇(d𝑥). It is easily checked that 𝜑♯(R𝜇)(d𝑥) = 1 𝜆 R𝜑♯𝜇(d𝑥).
Theorem 4. For 𝜇 0 ∈ Cone(M -1 ), there exists a unique measure-valued solution

{𝜇 𝑡 } 𝑡≥0 to 𝜕 𝑡 R𝜇 𝑡 (d𝑥) = 𝜆 1+𝛼 𝑥 𝛼 𝜑 𝜆 ♯R𝜇 𝑡 (d𝑥) -𝑥 𝛼 R𝜇 𝑡 (d𝑥), (32)
which is given by 𝜇 𝑡 = 𝑅(𝑡)𝜇 0 . If 𝜇 0 = 𝑓 (𝑥) d𝑥 with 𝑓 ∈ K, then then 𝜇 𝑡 = 𝑐(𝑡, 𝑥) d𝑥 where 𝑐 solves (1) with 𝑐(0, •) = 𝑓 .

Proof. Eq. ( 32) is a rewriting of [START_REF] Konvalina | A Unified Interpretation of the Binomial Coefficients, the Stirling Numbers, and the Gaussian Coefficients[END_REF]. The uniqueness follows from the uniqueness of the moment equations. On the other hand, if 𝑓 ∈ K and 𝜇 0 = 𝑓 (𝑥) d𝑥, then with 𝜈 0 = Ψ(𝑓 ), the solution to (31) has a solution 𝜈 𝑡 ∈ Φ(K) so that 𝜇 𝑡 := R𝜈 𝑡 ∈ Φ(K).

Evolution of the moments -follow up

The above results show that the moment equation is an important tool to study the fragmentation equation. We now pursue our analysis of the moment equation (26).

A differential recursion

Hypothesis 1. For any integer 𝑘, we consider 𝑔 𝑘 > 0. For some 𝑘 0 , we consider 𝛽 𝑘 such that 𝛽 𝑘 < 0 for any 𝑘 > 𝑘 0 , and 𝛽 𝑘 > 0 for 𝑘 < 𝑘 0 , and 𝛽 𝑘 0 ≥ 0.

We consider the differential recursion

𝐶 𝑘 (𝑡) = 𝑔 𝑘 + 𝛽 𝑘 ∫ 𝑡 0 𝐶 𝑘+1 (𝑠) d𝑠. (33)
Hypothesis 2. The family {𝐶 𝑘 } is such that each 𝐶 𝑘 is continuous and non-negative.

Lemma 16. Under Hypotheses 1 and 2, for 𝑘 > 𝑘 0 , each 𝐶 𝑘 is infinitely differentiable at each 𝑡 ≥ 0 and a completely monotonic function on [0, +∞), that is (-1) ℓ 𝜕 ℓ 𝑡 𝐶 𝑘 (𝑡) ≥ 0 for any 𝑡 ≥ 0. Therefore, 𝐶 𝑘 is analytic, is convex and never vanishes.

Proof. For any 𝑘 ≥ 𝑘 0 , since 𝐶 𝑘+1 is continuous, clearly 𝐶 𝑘 is C 1 with -𝜕 𝑡 𝐶 𝑘 (𝑡) = -𝛽 𝑘 𝐶 𝑘+1 (𝑡).
We then deduce that 𝐶 𝑘 is infinitely differentiable and

(-1) ℓ 𝜕 ℓ 𝑡 𝐶 𝑘 (𝑡) = ( ℓ-1 ∏ 𝑖=0 (-𝛽 𝑘+𝑖 ) ) 𝐶 𝑘+ℓ (𝑡) ≥ 0.
By Hypothesis 2, 𝐶 𝑘 is non-negative. If 𝐶 𝑘 vanishes at some time 𝑡 0 , then 𝐶 𝑘 (𝑡) = 0 for any 𝑡 ≥ 𝑡 0 as the sign of its derivative does not change. Being analytic, 𝐶 𝑘 is identically zero. Yet 𝐶 𝑘 (0) = 𝑔 𝑘 > 0 from Hypothesis 1. Thus 𝐶 𝑘 never vanishes.

Since 𝜕 2 𝑡 𝐶 𝑘 (𝑡) = 𝛽 𝑘 𝛽 𝑘+1 𝐶 𝑘+2 (𝑡), 𝑡 ≥ 0, and

𝛽 𝑘 𝛽 𝑘+1 > 0, the function 𝐶 𝑘 (𝑡) is convex. Under Hypothesis 2, 𝐶 𝑘 is non-decreasing if 𝛽 𝑘 > 0, 𝐶 𝑘 is non-increasing if 𝛽 𝑘 < 0 and 𝐶 𝑘 = 𝑔 𝑘 if 𝛽 𝑘 = 0.
Lemma 17. Under Hypotheses 1 and 2, for 𝑘 < 𝑘 0 , lim 𝑡→∞ 𝐶 𝑘+1 (𝑡) > 0. Moreover, 𝐶 𝑘 is increasing and lim 𝑡→∞ 𝐶 𝑘 (𝑡) = +∞.

Proof. Set 𝐴 = lim 𝑡→∞ 𝐶 𝑘+1 (𝑡). Then for some 𝑡 0 ≥ 0, 𝐶 𝑘+1 (𝑡) ≥ 𝐴/2 for any 𝑡 ≥ 𝑡 0 . It follows that

𝐶 𝑘 (𝑡) -𝐶 𝑘 (𝑡 0 ) ≥ 𝛽 𝑘 𝐴 2(𝑡 -𝑡 0 ).
Hence the result.

Lemma 18. Assume Hypotheses 1 and 2. Then for any 𝑘 > 𝑘 0 , 𝐶 𝑘 is decreasing and lim 𝑡→∞ 𝐶 𝑘 (𝑡) = 0.

Proof. Since 𝐶 𝑘+1 is non-negative, 𝐶 𝑘 is non-increasing for any 𝑘 > 𝑘 0 . By Lemma 16, 𝐶 𝑘 does not vanish for 𝑘 > 𝑘 0 and thus 𝐶 𝑘 is decreasing.

Assume that for some 𝑘 > 𝑘 0 , 𝐴 := lim 𝑡→∞ 𝐶 𝑘+1 (𝑡) > 0. Since 𝛽 𝑘 ≤ 0, 𝐶 𝑘 (𝑡) ≤ 𝐶 𝑘 (𝑡 1 ) + 𝛽 𝑘 𝐴(𝑡 -𝑡 1 ), for any 𝑡 ≥ 𝑡 1 . Hence, there exists 𝑡 2 such that 𝐶 𝑘 (𝑡 2 ) = 0. This contradicts Lemma 18.

This implies that lim 𝑡→∞ 𝐶 𝑘 (𝑡) = 0 yet each 𝐶 𝑘 is positive.

Combining Lemma 18 and Lemma 17, we obtain that • for 𝑘 > 𝑘 0 , 𝐶 𝑘 decreases to 0, and is convex,

• for 𝑘 = 𝑘 0 , 𝐶 𝑘 ≥ 𝑔 𝑘 > 0, • for 𝑘 < 𝑘 0 , 𝐶 𝑘 increases to +∞. Besides, if 𝑘 < 𝑘 0 -1, the 𝐶 𝑘 is convex.
Lemma 19. Under Hypothesis 1, there exists at most a unique solution {𝐶 𝑘 } that satisfies Hypothesis 2. Besides, 𝐶 𝑘 is analytic for any 𝑘 ≥ 0.

Proof. For 𝑘 > 𝑘 0 , 𝐶 𝑘 is analytic by Lemma 16. It is uniquely defined by its derivatives at 0, which are related to the {𝑔 𝑘 } and {𝛽 𝑘 }.

For 𝑘 ≤ 𝑘 0 , it is sufficient to iterate ℓ times the development

𝐶 𝑘 (𝑡) = 𝑔 𝑘 + 𝛽 𝑘 𝑔 𝑘+1 𝑡 + 𝛽 𝑘 𝛽 𝑘+1 ∫ 𝑡 0 ∫ 𝑡 1 0 𝐶 𝑘+2 (𝑡 2 ) d𝑡 2 d𝑡 -1,
where ℓ is such that 𝑘 + ℓ > 𝑘 0 . As the integral of analytic functions are analytic, 𝐶 𝑘 is analytic. The long-time behavior of 𝐶 is exponentially decreasing for 𝜎 > 2 and exponentially increasing for 𝜎 < 2.

Behavior of the moments with respect to time

More precise asymptotic behavior are given below in Corollary 5 and Proposition 14.

Expansion of the moments

Since 𝐶(•, 𝜎; 𝜇, 𝜆) is analytic, it has a power series representation. We could apply the Mellin transform to use [START_REF] Doumic | Estimating the division rate and kernel in the fragmentation equation[END_REF]. However, a direct expansion to (25) leads to a power series in a simple form.

Notation 10. The Pochhammer symbol is given in Definition 2. We write 𝜛 𝑘 (𝜎, 𝜆) := (-1) 𝑘 (𝜆 2-𝜎 ; 𝜆 -𝛼 ) 𝑘 , 𝑘 ≥ 1 and 𝜛 0 (𝜎, 𝜆) := 1.

From a straightforward computation,

𝜛 𝑘 (𝜎 + 𝛼, 𝜆) = 𝜛 𝑘+1 (𝜎, 𝜆) 𝜛 1 (𝜎, 𝜆) , (34)
We deduce from [START_REF] Mischler | Spectral analysis of semigroups and growthfragmentation equations[END_REF] that

𝜛 1 (𝜎, 𝜆)𝜛 1 (𝜎 + 𝛼, 𝜆) • • • 𝜛 1 (𝜎 + 𝑘𝛼, 𝜆) = 𝜛 𝑘+1 (𝜎, 𝜆).
If 𝑛 = (2 -𝜎)/𝛼 ∈ N, then 𝜛 𝑚 (𝜎, 𝜆) = 0 for any 𝑚 ≥ 𝑛. In addition, computing 𝜕 𝜆 log 𝜛 𝑘 (𝜎, 𝜆), we obtain

𝜕 𝜆 𝜛 𝑘 (𝜎, 𝜆) = 𝜛 𝑘 (𝜎, 𝜆)𝐷 𝑘 (𝜎, 𝛼, 𝜆) (35)
with 𝐷 𝑘 (𝜎, 𝛼, 𝜆) :=

𝑘-1 ∑ 𝑖=0 (2 -𝜎 -𝑖𝛼) 𝜆 1-𝜎-𝑖𝛼 1 -𝜆 2-𝜎-𝑖𝛼 , (36)
unless 2 -𝜎 -𝑖𝛼 ̸ = 0 for some 𝑖 ∈ {0, . . . , 𝑘 -1}. In the latter case, 𝜛 𝑘 (𝜎, 𝜆) = 0 and then 𝜕 𝜆 𝜛(𝜎, 𝜆) = 0.

Let us end this section with a result regarding the scaling by invariance.

Lemma 20 (Scaling for initial Dirac measure). For any 𝜎 > 0, 𝑡 ≥ 0 and 𝑎 ∈ (0, 1],

𝐶(𝑡, 𝜎; 𝛿 𝑎 , 𝜆) = 𝑎 𝜎-1 𝐶(𝑡𝑎 𝛼 , 𝜎; 𝛿 1 , 𝜆).

for 𝑡 ≥ 0, 𝜎 ≥ 1.

Proof. From [START_REF] Widder | The Laplace Transform[END_REF],

𝐶
F 𝑘 𝑓 ), (40)
for any 𝑡 ≥ 0, 𝑘 ≥ 0. These constants are given by 𝐷 0,1 = -1, 𝐷 1,1 = 𝜆 2-𝜎 , and 𝐷 0,𝑘+1 = -𝐷 0,𝑘 and 𝐷 𝑖,𝑘 = (-1) 𝑘-𝑖 𝜆 𝑖(2-𝜎)-( 𝑖 2 )𝛼

( 𝑘 𝑖 ) 𝜆 -𝛼 . ( 41
)
Proof. We proceed by recursion. The first step 𝑘 = 1 follows from Lemma 21. Let assume that (41) holds for 𝑘 ≥ 1. By using Lemma 21 on the time derivative of ( 40 Using the recursion hypothesis after some reindexing and since

( 𝑖 2 ) = ( 𝑖-1 2 ) + 𝑖 -1, 𝜆 2-𝜎-(𝑖-1)𝛼 𝐷 𝑖-1,𝑘 (𝜆, 𝜎, 𝛼) + 𝜆 -𝑖𝛼 𝐷 𝑖,𝑘 (𝜆, 𝜎, 𝛼) = (-1) 𝑘+1-𝑖 𝜆 𝑖(2-𝜎)-( 𝑖 2 )𝛼 [( 𝑘 𝑖 -1 ) 𝜆 -𝛼 + 𝜆 -𝑖𝛼 ( 𝑘 𝑖 ) 𝜆 -𝛼 ] .
From the Pascal's identity [START_REF] Chernoff | On continuity and smoothness of group actions[END_REF],

𝜆 2-𝜎-(𝑖-1)𝛼 𝐷 𝑖-1,𝑘 (𝜆, 𝜎, 𝛼) + 𝜆 -𝑖𝛼 𝐷 𝑖,𝑘 (𝜆, 𝜎, 𝛼) = 𝐷 𝑖,𝑘+1 (𝜆, 𝜎, 𝛼), (43)
where for all 1 ≤ 𝑖 ≤ 𝑘:

𝐷 𝑖+1,𝑘 (𝜆, 𝜎, 𝛼) := (-1) 𝑘+1-𝑖 𝜆 𝑖(2-𝜎)-( 𝑖 2 )𝛼 ( 𝑘 + 1 𝑖 ) 𝜆 -𝛼 . ( 44
)
By computing the term in 𝑖 = 0 and 𝑖 = 𝑘 + 1, we have easily the desired formula.

A direct consequence that F𝛿 1 = 𝛿 1 in ( 39) is that 𝐶(

with 𝐶(0, 𝜎; 𝛿 1 , 𝜆) = 1.

The pantograph equation has given rise to a large number of studies, see e.g., [START_REF] Fox | On a Functional Differential Equation[END_REF][START_REF] Hall | A functional-differential equation arising in modelling of cell growth[END_REF][START_REF] Kato | The functional-differential equation 𝑦 ′ (𝑥) = 𝑎𝑦(𝜆𝑥) + 𝑏𝑦(𝑥)[END_REF].

Remark 3. Using the scaling [START_REF] Wiśniewolski | On the probabilistic representations of solutions of pantograph equations and triangle coefficients[END_REF] or a direct computation shows that for any 𝑎 > 0, 𝜕 𝑡 𝐶(𝑡, 𝜎; 𝛿 𝑎 , 𝜆) = 𝑎 𝛼 𝜆 2-𝜎 𝐶(𝜆 -𝛼 𝑡, 𝜎; 𝛿 𝑎 , 𝜆) -𝑎 𝛼 𝐶(𝑡, 𝜎; 𝛿 𝑎 , 𝜆) for 𝑡 > 0.

This is why we focus only on the case where the initial measure 𝛿 1 .

We now focus on the long time behavior of 𝑡 ↦ → 𝐶(𝑡, 𝜎; 𝛿 1 , 𝜆), for which a first classification was given in Section 5.2. As this function is analytic, its asymptotic behavior is given by power series, or via Tauberian theorems.

When 𝜎 ̸ = 2, the solution to (45) is given by the power series [25, Eq. (1.5)]

𝐶(𝑡, 𝜎; 𝛿 1 , 𝜆) = ∑ 𝑘≥0 (𝜆 2-𝜎 ; 𝜆 -𝛼 ) 𝑘 𝑘! (-𝑡) 𝑘 , (47)
which is uniformly convergent for any 𝑡 ∈ C. The article [START_REF] Wiśniewolski | On the probabilistic representations of solutions of pantograph equations and triangle coefficients[END_REF] gives some nice probabilistic interpretations of this formula both in term of Brownian motion and Point Processes.

𝐶(𝑡, 𝜎; 𝛿 1 , 𝜆) = (𝜆 2-𝜎 ; 𝜆 -𝛼 ) ∞ ∑ 𝑘≥0 𝜆 𝑘(2-𝜎) (𝜆 -𝛼 ; 𝜆 -𝛼 ) 𝑘 exp(-𝜆 -𝑘𝛼 𝑡), (48)
which is convergent for any 𝑡 ∈ C when 𝜎 > 2, and thus 𝜆 2-𝜎 < 1. Since 𝐶(•, 𝜎; 𝜇, 𝜆) is non-negative, ̂ 𝐶 is completely monotone and thus analytic. For 𝜎 > 2, 𝐶 is bounded so that ̂ 𝐶 exists for any 𝑠 > 0. Since the series (47) is uniformly convergent for any 𝑡 ≥ 0, we may invert integrals and series so that the Laplace transform ̂ 𝐶 exists also when 𝜎 < 2.

Tauberian theorems relate the large-time behavior of 𝐶(•, 𝜎; 𝛿 1 , 𝜆) with the behavior of ̂ 𝐶(𝑠) for 𝑠 close to 0.

Taking the Laplace transform of (47) leads to

 𝐶(𝑠) = ∑ 𝑘≥0 (𝜆 2-𝜎 ; 𝜆 -𝛼 ) 𝑘 (-1) 𝑘 𝑠 𝑘+1 . (49)
This is not sufficient to study the asymptotic behavior of 𝐶(𝑡, 𝜎; 𝛿 1 , 𝜆) for large 𝑡.

Taking the Laplace transform of (48) leads to

 𝐶(𝑠) = 𝐾 ∑ 𝑘≥0 𝜆 𝑘(2-𝜎) (𝜆 -𝛼 ; 𝜆 -𝛼 ) 𝑘 1 𝑠 + 𝜆 -𝑘𝛼 . (50)
Again, there is no clear interpretation. We are then looking for an alternative representation.

Applying the Laplace transform to both sides of (45) leads to

(1 + 𝑠) ̂ 𝐶(𝑠) = 1 + 𝜆 2-𝜎+𝛼 ̂ 𝐶(𝜆 𝛼 𝑠) for 𝑠 > 0,
since 𝐶(0, 𝜎; 𝛿 1 , 𝜆) = 1. We rewrite this equation as

(1 + 𝑞𝑠) ̂ 𝐶(𝑞𝑠) = 1 + 𝜆 2-𝜎+𝛼 ̂ 𝐶(𝑠) with 𝑞 = 𝜆 -𝛼 ∈ (0, 1). (51)
Lemma 24. Assume 𝜎 ≥ 1, 𝜎 ̸ = 2 + 𝛼. Then for any 𝑠 > 0,

 𝐶(𝑠) = 𝐴(𝜎) 1 𝑠 (2-𝜎)/𝛼+1 ∑ 𝑘≥0 𝜆 -𝛼𝑘(𝑘+1)/2 (𝜆 -𝛼 ; 𝜆 -𝛼 ) 𝑘 𝑠 𝑘 + ∑ 𝑘≥0 ( -𝜆 -𝛼 1 -𝜆 2-𝜎+𝛼 ) 𝑘 𝑠 𝑘 ,
for some constant 𝐴(𝜎).

Proof. We use the results of [START_REF] Adams | On the Linear Ordinary 𝑞-Difference Equation[END_REF]. The solution ̂ 𝐶 to (51) is written in a generic way as ̂ 𝐶(𝑠) = 𝑎𝜑(𝑠)𝑃 (𝑠) + 𝑄(𝑠), for a constant 𝑎, where 𝜑(𝑞𝑠) = 𝜑(𝑠), and 𝑃 and 𝑄 are solutions to the homogeneous and inhomogeneous equations 𝑎 0 (𝑞𝑠)𝑃 (𝑞𝑠) + 𝑎 1 (𝑠)𝑃 (𝑠) = 0, 𝑎 0 (𝑞𝑠)𝑄(𝑞𝑠) + 𝑎 1 (𝑠)𝑄(𝑠) = 1 with 𝑎 0 (𝑠) = 1 + 𝑠 and 𝑎 1 (𝑠) = -𝜆 2-𝜎+𝛼 .

Since 𝑃 and 𝑄 are given by power series, and ̂ 𝐶 is analytic, 𝜑 is analytic and it is easily checked that 𝜑 is necessarily constant.

Define 𝜃 := (2 -𝜎 -𝛼)/𝛼, so that

𝑎 0 (0)𝜆 -𝛼𝜃 = 𝜆 2-𝜎-𝛼 = -𝑎 1 (0). Hence, 𝑞 𝜃 = 𝜆 2-𝜎-𝛼 . (52)
A solution of the homogeneous equation is given by

𝑃 (𝑠) = 𝑠 𝜃 (𝑝 0 + 𝑝 1 𝑠 + 𝑝 2 𝑠 2 + • • •)
with 𝑝 0 := 1 (this constant is arbitrary) and

𝑝 𝑘 := 𝑞 𝑘(𝑘+1)/2 (𝑞; 𝑞) 𝑘 . (53)
Due to (51), this choice of 𝑝 0 is arbitrary. Identifying the coefficients of 𝑠 𝑘+𝜃 , we get (𝑝 𝑘 + 𝑝 𝑘-1 )𝑞 𝜃+𝑘 = 𝜆 2-𝜎+𝛼 𝑝 𝑘 .

Owing to (52), this leads to

𝑝 𝑘 = 𝑞 𝑘 1 -𝑞 𝑘 𝑝 𝑘-1
and then to (53). This is immediate by identifying the coefficients.

The same work can be done where 𝑃 and 𝑄 are power series in 1/𝑠. In this case, we obtain (49).

The next corollary precises the results given in Section 5.

as 𝑡 → ∞.

Since 𝐶(𝑡, 2; 𝛿 1 , 𝜆) = 1, it holds that 𝐴(2) = 𝐵(2) = 1. Since 𝐶(𝑡, 2-𝛼; 𝛿 1 , 𝜆) = 1+𝑡, we have also that 𝐵(2 -𝛼) = 1 and 𝐴(2 -𝛼) = 1/2.

Remark 5. Since 𝜎 ↦ → 𝐶(𝑡, 𝜎; 1 , 𝜆) is analytic for any 𝑡 ≥ 0, it follows easily that ̂ 𝐶 is also analytic in 𝜎 and then 𝐴 and 𝐵 are analytic. In particular, 𝐴 and 𝐵 are bounded on any finite interval.

An alternative form form 𝐶(𝑡, 𝜎; 𝛿 1 , 𝜆) is given by [39,

In particular, lim 𝜎→∞ 𝐵(𝜎) = +∞. The support of this measure is not bounded.

Proof. If 𝛼 > 1/2, the growth of 𝐵 is such that the moments problem has a unique solution (see e.g., [START_REF] Lin | Recent developments on the moment problem[END_REF]Theorem 2]).

With (57), the value of (𝐵(𝜎)) 𝜎>𝑠 can be computed from the knowledge of (𝐵(𝜎)) 𝜎∈[𝑠-𝛼,𝑠] . As seen in Remark 5, the later function is bounded.

For any integer 𝑚, there exists and integer 𝑘 = ⌊(𝑚 -𝜎)/𝛼⌋ and 𝜎 ∈ [2 -𝛼, 2] such that 𝜎 + 𝑘𝛼 = 𝑚. By (58) and since the 𝑞-Pochhammer symbol (𝜆 which is sufficient to ensure that 𝐵(𝑚) is the moment of a measure 𝜇 which is determined by its moments.

If the self-similar measure 𝜇 has a support included in [0, 𝐾], then M[𝜇](𝜎) ≤ 𝐶𝐾 𝜎-2 . From (58), the growth of 𝐵 is faster so that the support cannot be contained in a compact. We set 𝑞 := 𝜎-2 𝛼 . With the above notations, 𝐵(𝜎) = 𝐵(𝜎 + 𝑘𝛼) (𝜆 2-𝜎 ; 𝜆 -𝛼 ) 𝑘 (𝑞) 𝑘 .

If 𝜎 > 2, ((𝜆 2-𝜎 ; 𝜆 -𝛼 ) 𝑘 ) 𝑘≥0 has a limit (𝜆 2-𝜎 ; 𝜆 -𝛼 ) ∞ and never vanishes. Therefore, 𝐵(𝜎 + 𝑘𝛼) = (𝑞) 𝑘 (𝜆 2-𝜎 ; 𝜆 -𝛼 ) 𝑘 𝐵(𝜎) = Γ(𝑘) Beta(𝑞, 𝑘) × 𝐵(𝜎) (𝜆 2-𝜎 ; 𝜆 -𝛼 ) 𝑘 .

Using the asymptotic Beta(𝑞, 𝑘) ∼ 𝑘→∞ Γ(𝑞)𝑘 -𝑞 , we obtain (58).

  𝑡 𝑣(𝑡, 𝑥) = -𝐹 (𝑥)𝑣(𝑡, 𝑥) )𝑣(𝑡, 𝑦) d𝑦 𝑦 with 𝐹 (𝑥) := 𝑥 𝛼 and 𝜅(d𝑧) := 𝜆𝛿 1/𝜆 (d𝑧),

  ), we obtain that𝐶(𝑡, 𝜎) = ∫ +∞ 0 𝑥 𝜎-1 𝑐(𝑡, 𝑥) d𝑡 ≈ 𝑠(𝑡) 𝜎-2 ∫ +∞ 0 𝜉 𝜎-1 𝜑(𝜉) d𝜉.Therefore, 𝐶(𝑡, 𝜎) ≈ 𝐵(𝜎)𝑡 2-𝜎 𝛼 for 𝐵(𝜎) := ∫ +∞ 0 𝜉 𝜎-1 𝜑(𝜉) d𝜉. As the mass 𝑚 = ∫ +∞ 0 𝑥 • 𝑢(𝑡, 𝑥) d𝑥 is preserved in the time, we select 𝜑 so that 𝐵(2) = ∫ +∞ 0 𝜉 • 𝜑(𝜉) d𝜉 = 𝑚.

  , p. 147] and [4, Corollary 5.8, p. 139].

Definition 2 . 1 ∏𝑖=0( 1 -

 211 The 𝑞-Pochhammer symbol is for 𝑎 ∈ R and 𝑞 ∈ (0, 1), (𝑎; 𝑞) 𝑘 := 𝑘-𝑎𝑞 𝑖 ) for 𝑘 ≥ 1 and (𝑎; 𝑞) 0 := 1. Theorem 1 ([35, Theorems 3.2.3 and 3.2.4, pp. 79 and 80]). The Pascal identity holds ( 𝑛 + 1 𝑘

  (𝑠, •)](𝜎 + 𝛼) d𝑠 = ∫ 𝑡 𝑟 M[𝑄(𝑠)𝑓 ](𝜎 + 𝛼) d𝑠 (23) Similarly,

Lemma 9 .

 9 If the initial conditions 𝑓 belongs to L Re(𝜎)-1 with Re(𝜎) ≥ 1, then |𝐶(𝑡, 𝜎; 𝑓, 𝜆)| ≤ 𝑒 (𝜆-1)𝑡 ‖𝑓 ‖ Re(𝜎)-1 ≤ 𝑒 (𝜆-1)𝑡 M[|𝑓 |](Re(𝜎)), for any 𝑡 ≥ 0.

Proposition 10 . 1 . 4 (

 1014 Let 𝑓 ∈ L 𝑝+𝛼 . Then (𝑡, 𝜎) ↦ → 𝐶(𝑡, 𝜎; 𝑓, 𝜆) satisfies 𝐶(𝑡, 𝜎; 𝑓, 𝜆) = M[𝑓 ](𝜎) + (𝜆 2-𝜎 -1) ∫ 𝑡 𝑟 𝐶(𝑠, 𝜎 + 𝛼; 𝑓, 𝜆) d𝑠 (25) for any 𝑡 ≥ 0, 𝜎 ≥ 𝑝 + 𝛼 + Corollary Mass conservation). When 𝜎 = 2, 𝐶(𝑡, 𝜎; 𝑓, 𝜆) = 𝐶(0, 𝜎; 𝑓, 𝜆) for any 𝑡 ≥ 0. This means that 𝑡 ↦ → ∫ +∞ 0 𝑥 𝑄(𝑡)𝑓 (𝑥) d𝑥 remains constant.

𝑥) d𝑥 = 1 }

 1 and the map Φ : K → M by Φ(𝑓 )(d𝑥) := 𝑥𝑓 (𝑥) d𝑥 for 𝑓 ∈ K.

Notation 9 .Proposition 12 .

 912 We set M F := {𝜇 | 𝜇(d𝑥) = 𝑓 (𝑥) d𝑥 with 𝑓 ∈ F + }. Let 𝜇 ∈ Cone M -1 or 𝜇 ∈ M F . Then for any 𝜎 ≥ 1, let 𝐶(𝑡, 𝜎; 𝜇, 𝜆) be the solution of the moment equation (26) (resp. (25)) with 𝑔 := M[𝜇](𝜎).Fix some 𝜎 ≥ 1. With 𝐶 𝑘 (𝑡) := 𝐶(𝑡, 𝜎 + 𝑘𝛼; 𝜇, 𝜆) and 𝛽(𝜆) := 𝜆 2-𝜎 -1, {𝐶 𝑘 } 𝑘≥0 solves the differenrial recursion equation.Moreover, 𝛽(2) = 0. We summarize the above results. For any 𝜎 ≥ 1, 𝐶(•, 𝜎; 𝜇, 𝜆) is analytic. In addition, when 𝛼 > 0,• for 𝜎 > 2, 𝐶(•, 𝜎; 𝜇, 𝜆) decreases to 0, and is convex,• for 𝜎 = 2,𝐶(𝑡, 2; 𝜇, 𝜆) = M[𝜇](2) for 𝑡 ≥ 0. • for 𝜎 < 2, 𝐶(•, 𝜎; 𝜇, 𝜆) increases to +∞. Besides, for 𝜎 + 𝛼 < 2, 𝐶(•, 𝜎; 𝜇, 𝜆) is convex and for 2 -𝛼 < 𝜎 < 2, 𝐶(•, 𝜎; 𝜇, 𝜆) is concave, • for 𝜎 = 2 -𝛼, 𝐶(𝑡, 𝜎; 𝜇, 𝜆) = M[𝜇](1) + 𝑡 M[𝜇](2), for 𝑡 ≥ 0. When 𝛼 = 0, (26) is an ordinary differential equation, with 𝐶(𝑡, 𝜎; 𝜇, 𝜆) = exp(𝑡(𝜆 2-𝜎 -1)) M[𝜇](𝜎).

Proposition 13 .

 13 Let 𝜇 ∈ Cone M -1 or 𝜇 ∈ M F , Then 𝐶(𝑡, 𝜎; 𝜇, 𝜆) = +∞ ∑ 𝑘=0 𝜛 𝑘 (𝜎, 𝜆) 𝑡 𝑘 𝑘! M[𝜇](𝜎 + 𝑘𝛼) for 𝑡 ≥ 0, 𝜎 ∈ H 1 .

( 38 ) 1 =

 381 Proof. For the Dirac mass, M[𝛿 𝑎 ](𝜎) = 𝑎 𝜎-1 ,𝐶(𝑡, 𝜎; 𝛿 𝑎 , 𝜆) = +∞ ∑ 𝑘=0 𝜛 𝑘 (𝜎, 𝜆) 𝑡 𝑘 𝑘! 𝑎 𝜎+𝑘𝛼-𝑎 𝜎-1 +∞ ∑ 𝑘=0 𝜛 𝑘 (𝜎, 𝜆) (𝑡𝑎 𝛼 ) 𝑘 𝑘! = 𝑎 𝜎-1 𝐶(𝑡𝑎 𝛼 , 𝜎; 𝛿 1 , 𝜆) as M[𝛿 1 ](𝜎) = 1.6 Asymptotic behavior of the moments We extend the operator F 𝛼 to measures by F 𝛼 𝜇(d𝑥) := 𝑥 𝛼 𝜇(d𝑥). Thus, M[F 𝛼 𝜇](𝜎) = M[𝜇](𝜎 + 𝛼) for any 𝜎 ≥ 1 when 𝜇 ∈ Cone M -1 . Lemma 21. For 𝜇 ∈ Cone M -1 or 𝜇 ∈ M F , 𝜕 𝑡 𝐶(𝑡, 𝜎; 𝜇, 𝜆) = 𝜆 2-𝜎 𝐶(𝜆 -𝛼 𝑡, 𝜎; F 𝛼 𝜇, 𝜆) -𝐶(𝑡, 𝜎; F 𝛼 𝜇, 𝜆),

𝐷

  𝑖,𝑘 (𝜆, 𝜎, 𝛼)𝜆 2-𝜎-𝑖𝛼 𝐶(𝜆 -(𝑖+1)𝛼 𝑡, 𝜎; F 𝑘+1 𝑓 ) -𝑘 ∑ 𝑖=0 𝐷 𝑖,𝑘 (𝜆, 𝜎, 𝛼)𝜆 -𝑖𝛼 𝐶(𝜆 -𝑖𝛼 𝑡, 𝜎; F 𝑘+1 𝑓 ). (42)

Remark 4 .

 4 Assume that for some integer 𝑛, 𝜎 + 𝑛𝛼 = 2. In particular, 𝜎 < 2. Then𝐶(𝑡, 𝜎; 𝛿 1 , 𝜆) = 𝑛 ∑ 𝑘=0 (𝜆 2-𝜎 ; 𝜆 -𝛼 ) 𝑘 𝑘! (-𝑡) 𝑘 ,as (𝜆 2-𝜎 ; 𝜆 -𝛼 ) 𝑘 = 0 for any 𝑘 ≥ 𝑛. Hence,𝐶(𝑡, 𝜎; 𝛿 1 , 𝜆) ∼ 𝑡→∞ (-1) 𝑛 (𝜆 2-𝜎 ; 𝜆 -𝛼 ) 𝑛 𝑛! 𝑡 2-𝜎 𝛼 .This could have also been deduce from the moment equation.An alternative form to (47) is given by [25, Eq. (1.6)] or[START_REF] Zhang | Analytical study of the pantograph equation using Jacobi theta functions[END_REF] Proposition 2.1]

Notation 11 (

 11 Laplace transform). The Laplace transform of 𝐶 is denoted by  𝐶(𝑠) := ∫ +∞ 0 exp(-𝑠𝑡)𝐶(𝑡, 𝜎; 𝜇, 𝜆) d𝑡 for 𝑠 > 0.

Assume 𝑎 1 ̸ = - 1 , 1 -

 111 that is 𝜎 ̸ = 2 + 𝛼. A solution of the inhomogeneous equation is given by𝑄(𝑠) = 𝑞 0 + 𝑞 1 𝑠 + 𝑞 2 𝑠 2 + 𝜆 2-𝜎+𝛼) 𝑘 .

Corollary 6 .

 6 When 𝛼 > 1/2, there exists a unique measure 𝜇 such that𝐵(𝜎) = ∫ +∞ 0 𝑥 𝜎-1 𝜇(d𝑥).

Notation 12 ( 1 ∏𝑗=0(

 121 Rising factorial or Pochhammer symbol). We set(𝑥) 𝑛 := 𝑛-𝑥 + 𝑗),that is (𝑥) 𝑛 is the rising factorial, or Pochhammer symbol. It satisfies(𝑥) 𝑛 = Γ(𝑥 + 𝑛) Γ(𝑥) = Γ(𝑛) Beta(𝑥, 𝑛) ,where Beta is the Beta function.Proof. When 𝜎 ̸ = 2, we deduce from (57) that 𝐵(𝜎) = 𝐵(𝜎 + 𝑘𝛼)

Corollary 7 .

 7 Using the scaling property (Lemma 20), it holds that 𝐶(𝑡, 𝜎; 𝛿 𝑎 , 𝜆) ∼ 𝑡→∞ 𝑎𝐵(𝜎)𝑡 2-𝜎 𝛼 .

  [START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF], Theorem 4.6, p. 101]: Since for any 𝜈 ∈ C, (𝜈 + 𝜇 + F 𝛼 ) -1 𝑓 (𝑥) = 𝑓 (𝑥)/(𝜈 + 𝜇 + 𝑥 𝛼 ) for 𝑥 > 0, and |𝜈 + 𝜇 + 𝑥 𝛼 | ≥ ℑ(𝜈), this implies that Notation 4 (Semi-group 𝑄). The semi-group (𝑄 0 (𝑡)) 𝑡≥0 is denoted by (𝑄(𝑡)) 𝑡≥0 .

	The next proposition summarizes our results.
	Proposition 4. For 𝑓 ∈ L 𝑝 and 𝑇 > 0, there exists a solution 𝑢 ∈ C([0, 𝑇 ], L 𝑝 ) ∩
	C 1 ((0, 𝑇 ], Dom(F 𝛼 )) to (1), and then to (8). If 𝑓 ∈ L + 𝑝 , then 𝑢(𝑡, •) is non-negative
	for any 𝑡 ∈ [0, 𝑇 ].
	Proof. From [4, Theorem 2.29, p. 39 and Theorem 2.40, p. 40], for any 𝑡 > 0,
	𝑡 ↦ → 𝑄 𝜇 (𝑡)𝑓 is infinitely differentiable for any 𝑓 ∈ L 𝑝 . Thus, there exists a
	measurable function 𝑢(𝑡, 𝑥) on [𝜖, 𝑇 ] × R * + such that for any 𝑘 ≥ 0, 𝜕 𝑘 𝑡 𝑢(𝑡, 𝑥) exists
	almost everywhere, is absolutely continuous and 𝜕 𝑘 𝑡 𝑢(𝑡, •) = 𝜕 𝑘 𝑡 𝑄 𝜇 (𝑡)𝑓 for any
	𝑡 ∈ [0, 𝑇 ].
	∫ R +	|(𝜈 + 𝜇 + F 𝛼 ) -1 𝑓 (𝑥)|𝑤 𝑝 (𝑥) d𝑥 ≤	‖𝑓 ‖ 𝑝 ℑ(𝜈)	.
	By Proposition 2, the resolvent of ( G 𝛼 -F 𝛼 -𝜇, Dom(F 𝛼 )) is positive so that by
	the Arendt-Rhandi theorem ([3] or [5, Theorem 1.1]), (𝑄 𝜇 (𝑡)) 𝑡≥0 is an analytic
	semi-group.
	We may now remove the restriction on 𝜇.
	Lemma 4. For any 𝑟 ∈ R and 𝜇 ≥ 𝜆 -1, set 𝑄 𝑟 (𝑡) := exp((𝜇 -𝑟)𝑡)𝑄 𝜇 (𝑡) for 𝑡 ≥ 0.
	Then (𝑄 𝑟 (𝑡)) 𝑡≥0 is generated by (K 𝛼,𝑟 , Dom(K 𝛼,𝑟 )). This semi-group is positive and
	for 𝑓 ∈ L 𝑝 ,
	‖𝑄 𝑟 (𝑡)𝑓 ‖ 𝑝 ≤ exp((𝜆 -1 -𝑟)𝑡)‖𝑓 ‖ 𝑝 , ∀𝑡 ≥ 0.	(12)

  By identifying elements of M 𝜖 with measures on the compact [𝜖, 1], we know that any element of M 𝜖 may be approximated by discrete probability measures on [𝜖, 1] (See e.g. [10, Example 8.1.6, vol. 2]).

  the converse, if {𝜇 𝑛 } 𝑛≥0 converges weakly to 𝜇, then M[𝜇 𝑛 ](𝜎) converges to M[𝜇](𝜎) for any 𝜎 ≥ 2. In particular, (30) holds with 𝑚 𝑘 = M[𝜇](𝑘 + 2).

	Lemma

13. For 𝜇 ∈ M, 𝜎 ∈ R ↦ → M[𝜇](𝜎) is continuous on [2, +∞). Besides, this function is completely monotonic on (2, +∞). In particular, it is decreasing, convex and never vanishes. It can be extended analytically on the half-plane H 2 . Remark 2. It follows from the proof that if 𝜇({0}) > 0, then lim 𝜎→2 M[𝜇](𝜎) < 1 and then M[𝜇] is discontinuous at 𝜎 = 2. Proof. As for 𝜇 ∈ M, 𝜇({0}), we obtain from the monotone convergence theorem that 𝜎 ↦ → M[𝜇](𝜎) is continuous, decreasing on (2, +∞. Besides, lim 𝜎→2,𝜎>2

  1) 𝑘 𝜕 𝑘 𝜎 M[𝜇](𝜎) ≥ 0. This means that M[𝜇](𝜎) is completely monotonic on (2, +∞). The properties of M[𝜇] then follow from [37, Theorem IV.3a and Theorem IV.3a, p. 146-147]. Lemma 14. Let {𝜇 𝑛 } 𝑛≥N with 𝜇 𝑛 ∈ M. It holds that: (i) If 𝜇 𝑛 converges weakly to 𝜇 ∈ M then N[𝜇 𝑛 ] converges uniformly to N[𝜇] on [2, +∞). (ii) If N[𝜇 𝑛 ] converges uniformly to some function 𝑔 on [2, +∞) then there exists 𝜇 ∈ M such that 𝑔 = N[𝜇] and 𝜇 𝑛 converges weakly to 𝜇. Proof. We assume the weak convergence of 𝜇 𝑛 ∈ M to 𝜇 ∈ M. Since Supp(𝜇 𝑛 ) ∈ [0, 1], and 𝜇({0}) = 0, M[𝜇 𝑛 ](𝜎) converges to N[𝜇](𝜎) for any 𝜎 ≥ 2.

  Regarding (ii), the Fréchet-Shohat theorem implies the weak convergence of 𝜇 𝑛 to some measure 𝜇 with compact support on[0, 1]. Since N[𝜇 𝑛 ] = 2, then 𝑔(2) = 1 so that 𝜇is a probability measure. It remains to prove that 𝜇({0}) = 0. With Remark 2, this happens if and only if N[𝜇] is right-continuous at 𝜎 = 2. Since {N[𝜇 𝑛 ]} 𝑛≥0 converges uniformly on [2, +∞), N[𝜇] is continuous at 𝜎 = 2 and then 𝜇({0}) = 0. Let 𝑔 ∈ Cl N[Φ(K)]. This implies that there exists a sequence {𝑓 𝑛 } 𝑛≥0 such that 𝑓 𝑛 ∈ K and 𝑔 𝑛 = N[Φ(𝑓 𝑛 )] converges uniformly to 𝑔. With 𝜇 𝑛 = Φ(𝑓 𝑛 ) and Lemma 14 there exists 𝜇 ∈ M such that 𝜇 𝑛 converges weakly to 𝜇 and N[𝜇] = 𝑔. Thus, 𝑔 ∈ N[M]. Hence, Cl N[Φ(K)] ⊂ N[M]. Conversely, if 𝑔 = N[𝜇] with 𝜇 ∈ M, there exists a sequence {𝜇 𝑛 } 𝑛≥0 with 𝜇 𝑛 ∈ Φ(K) such that 𝜇 𝑛 converges wealky to 𝜇. By Lemma 14 again, this proves that N

	Lemma 15. It holds that
	Cl N[Φ(K)] = N[Cl Φ(K)] = N[M],
	and N is invertible and bicontinuons from M to N[M]. Finally, N[M] is invariant
	under the semi-group (𝑅(𝑡)) 𝑡≥0 .
	Proof.

N[𝜇 𝑛 ] to N[𝜇] on any interval [𝜎 0 , +∞) with 𝜎 0 > 2. Since 𝜎 ↦ → N[𝜇 𝑛 ](𝜎) is non-increasing for each 𝑛, and 𝜎 ↦ → N[𝜇](𝜎) is continuous, it follows from the Dini theorem that the convergence of N[𝜇 𝑛 ] toward N[𝜇] is uniform on [0, 𝜎 0 ].

  (𝑡, 𝜎; F 𝛼 𝜇, 𝜆) = 𝜕 𝑡 𝐶(𝑡, 𝜎; 𝜇, 𝜆) = 𝜛 1 (𝜎, 𝜆) 𝜛 1 (𝜎, 𝜆)𝐶(𝑡, 𝜎; F 𝛼 𝜇, 𝜆).Since 𝜛 1 (𝜎, 𝜆) = 𝜆 2-𝜎 -1, we obtain[START_REF] Zhang | Analytical study of the pantograph equation using Jacobi theta functions[END_REF] and then the result. Lemma 22. There exists a family of constants 𝐷 𝑖,𝑘 (𝜆, 𝜎, 𝛼) with 0 ≤ 𝑖 ≤ 𝑘 such that

	With (34),
	+∞ ∑ 𝑘=0 M[𝜇](𝜎+(𝑘+1)𝛼) = 𝜕 𝑘 𝜛 𝑘 (𝜎, 𝜆) 𝑡 𝑘 𝑘! 𝑘 𝑡 𝐶(𝑡, 𝜎; 𝑓 ) = ∑ 𝐷 𝑖,𝑘 (𝜆, 𝜎, 𝛼)𝐶(𝜆 -𝑖𝛼 𝑡, 𝜎;
		𝑖=0
		+∞ ∑ 𝑘=0	𝜛 𝑘 (𝜎, 𝜆)	𝑡 𝑘 𝑘!	M[𝜇](𝜎 + (𝑘 + 1)𝛼).
	Besides,
	𝜕 𝑡 𝐶(𝑡, 𝜎; 𝜇, 𝜆) =	+∞ ∑ 𝑘=0	𝜛 𝑘+1 (𝜎, 𝜆)	𝑡 𝑘 𝑘!	M[𝜇](𝜎 + (𝑘 + 1)𝛼).

  •, 𝜎; 𝛿 1 , 𝜆) solves a pantograph equation. Lemma 23. For any 𝑡 ≥ 0, 𝜎 ≥ 1, 𝜕 𝑡 𝐶(𝑡, 𝜎; 𝛿 1 , 𝜆) = 𝜆 2-𝜎 𝐶(𝜆 -𝛼 𝑡, 𝜎; 𝛿 1 , 𝜆) -𝐶(𝑡, 𝜎; 𝛿 1 , 𝜆)

  2. It gives the asymptotic behavior of 𝐶(𝑡, 𝜎; 𝛿 1 , 𝜆) for large 𝑡 is an immediate consequence of a Tauberian theorem [37, Theorem 4.3, p. 192 and Corollary 4.4b, p. 194]. Corollary 5. Assume 𝛼 > 0 and 𝜎 < 2 + 𝛼. Then 𝐶(𝑡, 𝜎; 𝛿 1 , 𝜆) ∼ 𝑡→∞ 𝐵(𝜎)𝑡

				2-𝜎 𝛼	(54)
	with	𝐵(𝜎) :=	2-𝜎 𝛼 + 1 ( 2-𝜎 Γ 𝛼 + 2	) 𝐴(𝜎),

  Theorem 1.2].Proposition 14. Assume that 𝛼 > 0 and 𝜎 > 2. Then there exists a constant 𝐵(𝜎) such that 𝐶(𝑡, 𝜎; 𝛿 1 , 𝜆)∼ 𝑡→∞ 𝐵(𝜎)𝑡 𝜆 -𝛼 ) ∞ (𝜆 -𝛼 ; 𝜆 -𝛼 ) ∞ Proof. Eq. (57) follows from the L'Hospital rule applied to 𝑡 ↦ → 𝐶(𝑡, 𝜎; 𝛿 1 , 𝜆) and 𝑡 ↦ → 𝑡 (2-𝜎)/𝛼 , since 𝜕 𝑡 𝐶(𝑡, 𝜎) = (𝜆 2-𝜎 -1)𝐶(𝑡, 𝜎 + 𝛼).The relation (57) is not sufficient to reconstruct the function 𝐵. Actually, let 𝑃 be any 𝛼-periodic function, then 𝜎 ↦ → 𝐵(𝜎)𝑃 (𝜎) also solves (57). Lemma 25. For any 𝜎 > 2, 𝐵(𝜎 + 𝑘𝛼) ∼ 𝑘→∞ Γ(𝑘) (𝜆 2-𝜎 ; 𝜆 -𝛼 ) 𝑘 𝑘

	Proposition 15. For 𝜎 ̸ = 2,
					𝐵(𝜎) = 𝛼	𝜆 2-𝜎 -1 2 -𝜎	𝐵(𝜎 + 𝛼).	(57)
												𝜎-2 𝛼	𝐵(𝜎) Γ ( 𝜎-2
													2-𝜎 𝛼 .
	Proof. It holds that
	𝐶(𝑡, 𝜎; 𝛿 1 , 𝜆) =	1 𝛼 ln 𝜆	(𝜆 𝜎-2 ; 1 𝑡 (𝜎-2)/𝛼
		×	∑ 𝑘≥0	𝜆 -𝛼𝑘(𝑘+1)/2 (𝜆 -𝛼 ; 𝜆 -𝛼 ) 𝑘	𝜓	(	𝜎 -2 𝛼	+ 𝑘,	𝜋 2 𝛼 ln(𝜆)	, -	ln 𝑡 𝛼 ln 𝜆	)	(-1) 𝑘 𝑡 𝑘	(56)
	with	𝜓(𝑢, 𝑣, 𝑤) :=	∑	Γ(𝑢 + 2𝑖𝜋𝑣𝑘) exp(2𝑖𝑘𝜋𝑤).
						𝑘≥0
	This concludes the proof.

𝛼

) .

  2-𝜎 ; 𝜆 -𝛼 ) 𝑘 converges as 𝑘 → ∞,

	log	1 𝑚	𝐵(𝑚) 1/2𝑚 ∼	1 2𝑚	log Γ	(⌊	𝑚 -𝜎 𝛼	⌋)	-log 𝑚.
	With the Stirling formula, when 𝛼 ̸ = 1/2,
			log	1 𝑚	𝐵(𝑚) 1/2𝑚 ∼	(	1 2𝛼	) -1	log 𝑚.
	If 𝛼 = 1/2, log 1 𝑚 𝐵(𝑚) 1/2𝑚 ∼ -1. If 𝛼 > 1/2, this proves that
				lim sup

𝑚→∞

𝑚 -1 𝐵(𝑚) 1/2𝑚 < +∞,


Valessa Valentim Viana

Jérémie Kreiss

Marc Jungers
email: marc.jungers]@univ-lorraine.fr

Valessa V Viana

Adaptive input redundancy of polynomial parameter-dependent systems

come

Introduction

The output of a system might not be uniquely generated by one specific input. In contrast, two or more different input functions may produce the same output trajectory. This concept is known as input redundancy [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF][START_REF] Serrani | Output regulation for over-actuated linear systems via inverse model allocation[END_REF][START_REF] Kreiss | Input redundancy: Definitions, taxonomy, characterizations and application to over-actuated systems[END_REF]. An input redundant system exhibits several degrees of freedom concerning the selection of inputs responsible for generating the desired output. These degrees of freedom can be leveraged to meet additional requirements beyond the main control task, as investigated in the context of the control allocation problems [START_REF] Harkegard | Efficient active set algorithms for solving constrained least squares problems in aircraft control allocation[END_REF][START_REF] Oppenheimer | The Control Handbook[END_REF]. The system inputs can be allocated with the aim of, for example, minimizing the total energy required, strengthening the resistance to failure, or promoting one input over another to reduce power consumption [START_REF] Johansen | Control allocation-a survey[END_REF].

The characterization of input redundancy of the system plays an important role in how the control allocation problem is defined. These techniques use the redundancy information of the system to state the control allocation problem. Therefore, an inadequate characterization of input redundancy can result in a control allocation strategy that will potentially demonstrate limitations due to the lack of full access to the redundancy of the system [START_REF] Kreiss | Input redundancy: Definitions, taxonomy, characterizations and application to over-actuated systems[END_REF].

In this context, [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF] proposed an input redundancy characterization for linear time-invariant (LTI) systems. The concepts of strong and weak input redundancy were introduced along with their characterization based on the null space dimension of the input matrices and the steady state plant transfer function matrix, respectively. Subsequently, [START_REF] Serrani | Output regulation for over-actuated linear systems via inverse model allocation[END_REF] proposed a new characterization of these concepts under the assumption that the system is right-invertible, strictly proper, and minimal. Geometric control theory tools began to be used in this context. More recently, [START_REF] Kreiss | Input redundancy: Definitions, taxonomy, characterizations and application to over-actuated systems[END_REF] proposed a reformulation of the input redundancy definition based on the idea that an output can be generated by at least two different inputs from the same initial condition. A formal framework is proposed to support this idea, along with a tractable and complete characterization of this concept for LTI systems using geometric control tools. This paper aims to provide a unified definition among the various ones proposed in the literature. See [START_REF] Kreiss | Input redundancy: Definitions, taxonomy, characterizations and application to over-actuated systems[END_REF] for more details on the differences between the mentioned works.

In the last decades, systems whose operation is influenced by parameters have gained significant importance in the field of control theory. These parameters that affect the system may be responsible for representing non-deterministic parameters, unmodeled dynamics, or non-linearities. Consequently, this representation is recognized as a more realistic model compared to the LTI representation. In the context of the characterization of input redundancy for parameterdependent systems, [START_REF] Kreiss | Robust input redundancy for uncertain systems[END_REF] introduced the notion of robust input redundancy for systems subject to unknown parameters. This concept means that the input redundancy property and the degrees of freedom associated are not parameterdependent. [START_REF] Kreiss | Robust input redundancy for uncertain systems[END_REF] also provided a characterization of this concept by using geometric control tools.

Differently from [START_REF] Kreiss | Robust input redundancy for uncertain systems[END_REF], in this paper, we deal with parameter-dependent systems in which the parameter is accessible for measurement, i.e., the parameter is known. In this context, an input redundancy that depends on the parameter can be considered. A parameter-dependent system might not be robust input redundant but still manifests some input redundancy that may depend on the parameter values. For instance, a system may present input redundancy properties for some values of parameters but not for others. Therefore, a more comprehensive notion of input redundancy for parameter-dependent systems emerges, in which the redundancy adapts according to the parameter.

In this paper, we aim to introduce a new definition of input redundancy in the context of parameter-dependent systems, which is called adaptive input redundancy. Obtaining a complete characterization of this concept is a difficult problem. Therefore, the core idea of the paper is to propose a practical approach to verify the adaptive input redundancy of linear parameter-dependent systems. The strategy is developed for systems with polynomial dependence on the parameter, where the parameter is subject to variations in time. The geometric concepts of controlled invariant subspaces are used to meet the results. In particular, the contributions of the article are: (i) the extension of the concept of input redundancy to a new one that ex-ploits the parameter to have a more general notion of input redundancy for parameter-dependent systems, (ii) the development of practically tractable sufficient conditions to check if a polynomial parameter-dependent system is adaptive input redundant.

The paper is organized as follows. In Section II, we present the system description and the adaptive input redundancy concept. We recall some tools from the geometric control theory for parameter-dependent systems in Section III. Section IV presents the Smith Normal form and some mathematical preliminaries related to parameter-dependent subspaces. A technical result and the main results related to the adaptive input redundancy are presented in Section V. Section VI provides a numerical example to highlight the applicability of our strategy. Finally, in Section VII, some concluding remarks are presented. Notation. Let R, R + , N, and N * be, respectively, the set of (i) real numbers, (ii) real positive numbers including zero, (iii) natural numbers including zero, and (iv) natural numbers without zero. For a matrix M , M ⊤ means its transpose. 0 n×m stands for the zero matrix of n rows and m columns, and I n stands for the identity matrix of dimension n. The notation diag N i=0 M i denotes the block diagonal matrix where M i is the i-th block. Im{V } is the vector space spanned by the columns of matrix V . The inverse map of set V by an application B is given by B -1 V := {u ∈ R m ; Bu ∈ V}, where B is not necessarily invertible. The kernel of a parameter-dependent matrix is given by ker{B(θ)} := {u ∈ R m ; B(θ)u = 0}, which depends on the parameter θ. For polynomial functions β 1 (•) and β 2 (•), the notation β 1 | β 2 means that β 2 is divisible by β 1 . The term y ua denotes the output trajectory produced by the system when excited by an input trajectory u a . For time functions, the notation u a ̸ = u b means that ||u a (τ, θ(τ )) -u b (τ, θ(τ ))||dτ > 0. In order not to make notations heavy, depending on the context, θ will denote sometimes a time function and sometimes just a real value.

System description and input redundancy

Consider a linear parameter-dependent system:

ẋ(t) = A(θ(t))x(t) + B(θ(t))u(t), y(t) = C(θ(t))x(t) + D(θ(t))u(t), (1)
where x(t) ∈ R n is the state with initial condition x(0) := x 0 ∈ R n , u(t) ∈ R m is the input, y(t) ∈ R p is the output of the system, θ(t) ∈ Θ ⊂ R is a known scalar time-varying parameter, and Θ a bounded set. The parameter θ(t) is piecewise continuous with a finite number of discontinuities in a finite length interval. Moreover, A(θ(t)), B(θ(t)), C(θ(t)), D(θ(t)) are matrices of appropriate dimension depending polynomially on the parameter θ(t). Since the system has linear dynamics on the state x(t), the system matrices are polynomial on θ(t), Θ is bounded, and t → θ(t) is piecewise continuous, the existence and uniqueness of the solutions of system (1) are ensured [START_REF] Khalil | Nonlinear systems[END_REF].

In this work, we would like to investigate the input redundancy of parameter-dependent systems, represented by system [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF]. The parameter affects the input redundancy of the system, which means that the redundancy adapts according to the parameter. Therefore, based on the definitions given in [START_REF] Kreiss | Input redundancy: Definitions, taxonomy, characterizations and application to over-actuated systems[END_REF] and [START_REF] Kreiss | Robust input redundancy for uncertain systems[END_REF], we introduce the definition of adaptive input redundancy.

Definition 1. The parameter-dependent system (1) is adaptive input redundant if there exists an output y that can be produced by, at least, two distinct parameter-dependent inputs, for some function θ and some initial condition, i.e., there exist θ : t → θ(t), x 0 ∈ R n , and two distinct input functions u a ̸ = u b , where u a : (t, θ) → u a (t, θ) and u b :

(t, θ) → u b (t, θ), such that y ua (t, θ(t), x 0 ) = y u b (t, θ(t), x 0 ) for all t ≥ 0.
This paper aims to provide tractable conditions for which a system is adaptive input redundant, i.e., when the input redundancy of the system depends on the parameter.

Background on geometric control theory

The fundamental concepts for characterizing input redundancy originate from the geometric control theory [START_REF] Trentelman | Control theory for linear systems[END_REF][START_REF] Wonham | Linear multivariable control: a geometric approach[END_REF].

The controlled invariant and output invisible subspaces have been one of the main tools used in the problem [START_REF] Serrani | Output regulation for over-actuated linear systems via inverse model allocation[END_REF][START_REF] Kreiss | Input redundancy: Definitions, taxonomy, characterizations and application to over-actuated systems[END_REF][START_REF] Kreiss | Robust input redundancy for uncertain systems[END_REF]. Then, let us recall this concept of the geometric control theory for parameter-dependent systems. We can mention the definitions of generalized and generalized adaptively controlled invariant and output invisible subspaces [START_REF] Kreiss | Robust input redundancy for uncertain systems[END_REF][START_REF] Bhattacharyya | Generalized controllability, (A,B)invariant subspaces and parameter invariant control[END_REF][START_REF] Bokor | Invariant subspaces for LPV systems and their applications[END_REF][START_REF] Viana | On the computation of controlled invariant and output invisible subspaces for parameter-dependent systems[END_REF]. Both subspaces are independent of the parameters. The differences between them rely on the fact that the input trajectories making the subspace invariant and output invisible must be independent of the parameters in the generalized subspaces, whereas they are allowed to depend on them for the generalized adaptively ones. The generalized subspaces can be considered as a particular case of the generalized adaptively ones. Here we are interested in inputs that possibly depend on the parameter, so we focus on the generalized adaptively subspaces. The definition of such a subspace is presented in the following.

Definition 2. [13]

A subspace V ⊆ R n is a generalized adaptively controlled invariant and output invisible subspace if for any x 0 ∈ V, there exists an input function u(t, θ(t)), where θ : R + → R ρ , such that x u (t, θ(t), x 0 ) ∈ V and y u (t, θ(t), x 0 ) = 0, for all θ(t) ∈ R ρ and t ≥ 0.

For system (1), a subspace V ⊆ R n is called generalized adaptively controlled invariant and output invisible if there exists a matrix F : Θ → R m×n such that, for all θ ∈ Θ ⊂ R,

(A(θ) + B(θ)F (θ))V ⊆ V, (C(θ) + D(θ)F (θ))V = 0. ( 2
)
The set of these subspaces admits a maximal element which is called here V * that is the largest generalized adaptively controlled invariant and output invisible subspace. Characterizing this subspace is not an easy task. Some works have proposed practical techniques for computing this subspace [START_REF] Bhattacharyya | Generalized controllability, (A,B)invariant subspaces and parameter invariant control[END_REF][START_REF] Bokor | Invariant subspaces for LPV systems and their applications[END_REF][START_REF] Balas | Invariant subspaces for LPV systems and their applications[END_REF][START_REF] Basile | On the robust controlled invariant[END_REF]. Recently, [START_REF] Viana | On the computation of controlled invariant and output invisible subspaces for parameter-dependent systems[END_REF] proposed a practical way to compute this subspace for parameter-dependent systems depending polynomially on the parameter and under the assumption that the friend also depends polynomially on θ. This work also developed an iterative algorithm to compute V * and a procedure to compute a friend of this subspace. Note that a matrix F (θ) verifying ( 2) is not necessarily unique. Accordingly, we denote here F(V) as the set of all the friends, depending on θ, of the subspace V.

For the remaining part of the paper, we will consider the following assumption.

Assumption 1. We assume that the largest generalized adaptively controlled invariant and output invisible subspace is already known.

Remark 1. It is important to highlight that to use the technique from [START_REF] Viana | On the computation of controlled invariant and output invisible subspaces for parameter-dependent systems[END_REF] to compute the largest subspace V * , Θ has to be a compact set since [START_REF] Viana | On the computation of controlled invariant and output invisible subspaces for parameter-dependent systems[END_REF] develops its results based on the assumption that the parameters lie in a compact set.

Preliminaries

This section presents some concepts and relations that will be useful to derive the results. First, we recall the Smith normal form. After, we present a practical way of computing the kernel of parameter-dependent matrices using the Smith normal form. To finish, we also present a practical way to compute the inverse map of a subspace V by a parameterdependent matrix.

Smith normal form

The Smith normal form can be derived for matrices with polynomial entries as matrices of system (1) [START_REF] Bourlès | Systèmes linéaires -De la modélisation à la commande[END_REF].

Definition 3. [17] Let θ ∈ R. Consider a matrix θ → W (θ) ∈ R n×m of
normal rank r, this matrix is equivalent to a matrix in the following form:

S(θ) =        β 1 (θ) 0 • • • 0 0 0 β 2 (θ) • • • 0 0 . . . . . . . . . . . . . . . 0 0 • • • β r (θ) 0 0 0 • • • 0 0 n-r×m-r        , (3)
where β i (θ) for i = {1, . . . , r} are non-null polynomial functions on θ such that

β i | β i+1 , 1 ≤ i ≤ r -1. S(θ) is called the Smith normal form of W (θ).
The Smith normal form of W (θ) can be obtained after a combination of a finite number of elementary operations applied over the rows and columns of W (θ). Moreover, nonsingular applications P : R → R n×n and Q : R → R m×m can be defined, such that

P (θ)W (θ)Q(θ) = S(θ). ( 4
)
This form is valid for matrices that depend polynomially only on one scalar parameter, which allows to use Euclidian division of polynomials, corresponding with the framework of this paper, where Θ ⊂ R.

Kernel of a parameter-dependent matrix

Computing the kernel of a parameter-dependent matrix is not an easy task since we have an infinite number of values for the parameter, so we have to analyze the kernel of such a matrix for each parameter value. An alternative and more efficient way of computing such a kernel is to use the Smith normal form.

Lemma 1. Consider a matrix W (θ) ∈ R n×m . Define its associated Smith normal form by S(θ) as in (3), Q(θ), and P (θ) such that (4) holds. Then,

ker{W (θ)} = Q(θ)ker{S(θ)}, (5)
such that

ker {S(θ)} = Im diag r i=1 ∆ i 0 r×m-r 0 n-r×r I n-r×m-r , (6)
∆ i = di ji=1 ∆ i,ji , (7)
and

∆ i,ji = 1, if θ = δ i,ji 0, otherwise, (8)
where δ i,ji are the roots of each β i (θ) such that β i (δ i,ji ) = 0 for i = {1, . . . , r} and j i = {1, . . . , di }, with di equals the degree of β i (θ).

Proof. From the Smith normal form, W (θ

) = P -1 (θ)S(θ)Q -1 (θ), then ker{W (θ)} = {u ∈ R m , P -1 (θ)S(θ)Q -1 (θ)u = 0}. (9) Since Q(θ) is non-singular, by applying a change of variables z = Q -1 (θ)u, we obtain the set {z ∈ R m , P -1 (θ)S(θ)z = 0} which is equiva- lent to ker{S(θ)} = {z ∈ R m , S(θ)z = 0}, then Q -1 (θ)u ∈ ker{S(θ)}.
Therefore, back in the original variable u, the kernel of the parameter-dependent matrix can be represented as

ker{W (θ)} = Q(θ)ker{S(θ)}. (10)
Furthermore, since matrix S(θ) has some diagonal terms and zeros elsewhere as shown in (3), its kernel can be obtained from the values of θ such that the diagonal term β i (θ) = 0 for i = {1, . . . , r}, which are the roots of each polynomial β i (θ) in the diagonal. Therefore, the direction related to each non-zero diagonal will be activated when θ meets one of the roots of the polynomial β i (θ), and the directions related to the zeros will be always active. Then, the kernel of S(θ) can be defined as in [START_REF] Johansen | Control allocation-a survey[END_REF], where ∆ i represents a function dependent on θ that will be equal to one only when θ meets one of the roots of the correspondent diagonal term, activating the direction of the kernel, otherwise is equal to zero.

Remark 2. Note that, in (8), δ i,ji is equal to θ only when δ i,ji is a real value root since θ ∈ R. Hence, when there is a complex value root δ i,ji , the total number of roots that can be equal to θ is less than the degree of β i (θ). If β i has two or more equal roots, then only one indicator (8) should be defined for this specific root value.

Remark 3. Since β i+1 is divisible by β i , all β j with j ≥ i depending on θ has a common root, then a common ∆ i,ji . Thus, when θ = δ i,ji such that ∆ i,ji = 1, all ∆ i 's in the kernel diagonal will be one.

Remark 4. It is important to highlight that although ∆ i is defined as a sum of terms as in [START_REF] Kreiss | Robust input redundancy for uncertain systems[END_REF], it can only be zero or one since when a ∆ i,ji is one, all the others in the sum are zero.

Some properties of the parameter-dependent kernel

In order to analyze the kernel of a parameter-dependent matrix, let us define L(θ) such that Im{L(θ)} = ker{W (θ)}.

Note that obtaining such a basis is now systematically possible by using Lemma 1. One can show that the structure of such a matrix is given by

L(θ) = ∆ 1 L 1 (θ) • • • ∆ r L r (θ) L(θ) , (11)
where L i (θ), for i = 1 to i = r, and L(θ) depend polynomially on θ. All these matrices may be empty. It follows that both the dimension and directions of such a kernel may depend on the parameter θ. We can derive three different cases regarding the kind of dependency w.r.t. θ which are summarized in the three following points:

• Independence: Some directions are possibly independent of θ, i.e., constant directions with a constant dimension. In this case, L(θ) contains constant columns.

• Isolated dimension drop: When directions depend on the value of θ, such directions can be time-varying if θ varies in time. The respective basis adapts according to the parameter value. The directions depend continuously on the parameter θ. For some isolated values of θ, some directions may vanish or be a linear combination of the others, leading to drops in the kernel dimension. This is represented by the columns of matrix L(θ) that depend on θ.

• Isolated dimension increase: Differently from the second case, some directions are only active for a finite number of parameter values. It means that when θ takes some specific values, a direction appears in the kernel. In Lemma 1, this is represented by the indicator ∆ i in [START_REF] Kreiss | Robust input redundancy for uncertain systems[END_REF]. Therefore, it influences the kernel dimension which can change when such directions appear or not. This is represented by the columns ∆ i L i (θ), for i = 1 to i = r, of L(θ) in [START_REF] Bhattacharyya | Generalized controllability, (A,B)invariant subspaces and parameter invariant control[END_REF].

To conclude, even with an unusual geometric structure due to the parameter dependence that leads to changes in the subspace dimension, the kernel of a parameter-dependent can be efficiently computed by obtaining the Smith normal form of the matrix and by defining the kernel of matrix S(θ) which is given in (6).

Parameter-dependent inverse map

In the following, for a subspace V ⊆ R n with V ⊥ as an orthonormal basis of the orthogonal complement V ⊥ of V (i.e. Im{V ⊥ } = V ⊥ ), we define

W V (θ) = V ⊥ B(θ) D(θ) . ( 12
)
Similar to the kernel of parameter-dependent matrices, computing the inverse map of a subspace by a parameterdependent matrix (B -1 (θ)V), is not straightforward. Therefore, we also use the Smith normal form and the result from Lemma 1 to help us in computing it.

Lemma 2. Consider a subspace V ⊆ R n . The following relation holds:

B -1 (θ)V ∩ ker {D(θ)} = ker {W V (θ)} , (13)
where W V (θ) is given by [START_REF] Bokor | Invariant subspaces for LPV systems and their applications[END_REF].

Proof. Let V ⊥ be the orthogonal complement of V. Let V and V ⊥ be a basis of V and V ⊥ , respectively, so that Im{V } = V and Im{V ⊥ } = V ⊥ . First, we know that

B -1 (θ)V = {u ∈ R m , B(θ)u ∈ V}. Then, since V is a basis of V, B -1 (θ)V = {u ∈ R m , ∃z ∈ R q , B(θ)u = V z}.
Multiplying at the left the expression B(θ)u = V z by V ⊤ ⊥ , we obtain V ⊤ ⊥ B(θ)u = V ⊤ ⊥ V z = 0, then it follows that, as shown in [START_REF] Basile | Controlled and Conditioned Invariants in Linear System Theory[END_REF]Property 3.1.3],

B -1 (θ)V = {u ∈ R m , V ⊤ ⊥ B(θ)u = 0} = ker{V ⊤ ⊥ B(θ)}.
Therefore, B -1 (θ)V ∩ ker{D(θ)} = ker{V ⊤ ⊥ B(θ)} ∩ ker{D(θ)} which can be rewritten as ker {W V (θ)}, then we can conclude that relation (13) holds.

Therefore, the subspace B(θ) -1 V * ∩ ker{D(θ)} can be practically computed since we can compute the kernel of a parameter-dependent matrix by Lemma 1 using the Smith normal form.

adaptive input redundancy

Here, we present a technical result that will be helpful for the adaptive input redundancy result.

Lemma 3. Let V * ⊆ R n , F : Θ → R m×n ∈ F(V * ), and L : Θ → R m×q such that Im{L(θ)} = Q(θ)ker{S(θ)}, ( 14
)
where Q(θ) and S(θ) are the matrices of the Smith normal form of matrix W V * (θ) defined in [START_REF] Bokor | Invariant subspaces for LPV systems and their applications[END_REF], and ker {S(θ)} is defined in [START_REF] Johansen | Control allocation-a survey[END_REF]. Let x 0 (θ) ∈ V * and u be an input function of the form:

u(t, θ) = F (θ)x(t) + L(θ)w(t), (15)
for some function w, then the output resulting from u and x 0 is zero.

Proof. Consider the input in the form (15), the system trajectories have the following dynamics,

ẋu (t, θ, x 0 ) = (A(θ) + B(θ)F (θ))x u (t, θ, x 0 ) + B(θ)v(t, θ), y u (t, θ, x 0 ) = (C(θ) + D(θ)F (θ))x u (t, θ, x 0 ) + D(θ)v(t, θ), where v(t, θ) = L(θ)w(t). Since B -1 (θ)V * ∩ ker{D(θ)} = ker{W V * (θ)} by Lemma 2 and ker{W V * (θ)} = Q(θ)ker{S(θ)} by Lemma 1, then Im{L(θ)} = B -1 (θ)V * ∩ ker{D(θ)}. It follows that v(t, θ) ∈ B -1 (θ)V * ∩ ker{D(θ)}. ( 16
)
Therefore, B(θ)v(t, θ) ∈ V * and D(θ)v(t, θ) = 0, ∀t ∈ R + . Thus, we can conclude that y u (t, θ,

x 0 ) = C(θ)x u (t, θ, x 0 ) + D(θ)u(t) = 0, ∀t ∈ R + .
Now, the main result of the paper is presented in Theorem 1, where we develop conditions to verify the adaptive input redundancy of parameter-dependent systems in the form (1).

Theorem 1. The parameter-dependent system (1) is adaptive input redundant if there exists a parameter θ ∈ Θ, such that dim B -1 (θ)V * ∩ ker{D(θ)} > 0, [START_REF] Kailath | Linear systems[END_REF] or equivalently,

dim (Q(θ)ker{S(θ)}) > 0, (18)
where Q(θ) and S(θ) are the matrices of the Smith normal form of matrix W V * (θ) defined in [START_REF] Bokor | Invariant subspaces for LPV systems and their applications[END_REF], and ker {S(θ)} is defined in [START_REF] Johansen | Control allocation-a survey[END_REF].

Proof. Let us consider two trajectories of system (1), x a and x b , related to two inputs u a and u b , respectively, and a parameter θ(t) = θ ∈ Θ, ∀t ≥ 0. We consider two trajectories starting from the same initial condition x a (0, θ(0

)) = x b (0, θ(0)) = x 0 ∈ R n .
The error between the two trajectories satisfies the following dynamics:

ẋa (t, θ) -ẋb (t, θ) =A(θ)(x a (t, θ) -x b (t, θ))+ B(θ)(u a (t, θ) -u b (t, θ)), y a (t, θ) -y b (t, θ) =C(θ)(x a (t, θ) -x b (t, θ))+ D(θ)(u a (t, θ) -u b (t, θ)),
for all t ≥ 0, where the initial condition x a (0, θ(0))x b (0, θ(0)) = 0. In view of input redundancy, we have to impose y a (t, θ(t)) = y b (t, θ(t)) for all t ≥ 0, leading to y a (t, θ(t)) -y b (t, θ(t)) = 0 for all t ≥ 0, which is the output of the error dynamics. Then, since x a (0, θ(0))-x b (0, θ(0)) = 0 ∈ V * , applying Lemma 3, we have that the output trajectory of the error, y a (t, θ(t)) -y b (t, θ(t)), is zero if

u a (t, θ) -u b (t, θ) = F (θ)(x a (t, θ) -x b (t, θ)) + L(θ)w(t),
for some function w. Hence, if there exists a parameter θ such that Im {L(θ)} does not have a zero dimension thus system (1) is adaptive input redundant. By Lemma 2, and Lemma 3, Im {L(θ)} = Q(θ)ker{S(θ)} = B(θ) -1 V * ∩ ker{D(θ)}, then these subspaces do not have a zero dimension either, and conditions ( 17) and ( 18) are equivalent.

Some properties of adaptive input redundancy

Theorem 1 establishes conditions to verify if a parameterdependent system is input redundant. This is represented by the dimension of B(θ) -1 V * ∩ker{D(θ)}, sometimes called the input unobservability subspace, being greater than zero. This subspace is derived from the kernel of a parameterdependent matrix. As discussed in Section 4.2, the kernel of a parameter-dependent matrix presents an unusual geometric structure that can be described by three distinct characteristics. The input unobservability subspace of an adaptive input redundant system follows the same geometric structure.

The input unobservability subspace plays a role not only in verifying if a system is adaptive input redundant but also in the ability to compute the invisible inputs through the L(θ) matrix. This matrix is related to unobservable inputs since the difference between the two distinct inputs leading to the same output is:

u a (t) -u b (t) = F (θ)(x a (t) -x b (t)) + L(θ)w(t).
Therefore, the different features in the input unobservability subspace lead us to a matrix L(θ) that has the same structure as the basis defined in [START_REF] Bhattacharyya | Generalized controllability, (A,B)invariant subspaces and parameter invariant control[END_REF]. Hence, an adaptive input redundant system can have invisible inputs from the output that depend polynomially on the parameter or exist only for specific parameter values.

Numerical Example

Consider the following parameter-dependent system,

ẋ(t) = -1 0 θ 3 (t) -1 x(t) + 1 0 θ(t) θ(t) 2θ(t) -θ 2 (t) u(t), y(t) = 0 1 x(t), (19)
where 0 ≤ θ(t) ≤ 1 for all t ≥ 0. To verify the adaptive input redundancy of this system we need first to compute the largest generalized adaptively controlled invariant and output invisible subspace of this system, denoted V * . Since θ lies in a compact set, we can apply the strategy proposed in [START_REF] Viana | On the computation of controlled invariant and output invisible subspaces for parameter-dependent systems[END_REF] to compute V * of system (19), as mentioned in Remark 1. Therefore, by applying the algorithm proposed in [START_REF] Viana | On the computation of controlled invariant and output invisible subspaces for parameter-dependent systems[END_REF], the largest generalized adaptively subspace is given by V * = [1 0] ⊤ R and one of the friends of the subspace is

F (θ) = 0 0 θ 0 0 0 ⊤ .
Now, to verify the input redundancy by applying Theorem 1, we have to compute the subspace Q(θ)ker{S(θ)}, where Q(θ) and S(θ) are the matrices of the Smith normal form of the matrix W V * (θ) in [START_REF] Bokor | Invariant subspaces for LPV systems and their applications[END_REF], where V ⊥ is the basis of the orthogonal complement of V * , and ker {S(θ)} is defined in [START_REF] Johansen | Control allocation-a survey[END_REF]. Since V * = [1 0] ⊤ R, a basis of this subspace is V = [1 0] ⊤ and a basis of the orthogonal complement of

V * is V ⊥ = [0 1] ⊤ . Therefore, W V * (θ) = θ 2θ -θ 2 0 0 0 .
The Smith normal form of W V * (θ), is given by P -1 S(θ)Q -1 (θ), where S(θ) = θ 0 0 0 0 0 , P = I 2 , and

Q(θ) =   1 -2 θ 0 1 0 0 0 1   .
Now that we have the matrix in the Smith normal form, we can determine the kernel of S(θ). First, we should identify the roots of the polynomials in each diagonal component of S(θ). In this case, we have only one polynomial in the diagonal that is θ, then the root of this diagonal term is δ 1,1 = 0, and we can define the indicator ∆ 1 = ∆ 1,1 , where

∆ 1,1 = 1, if θ = 0 0, otherwise. (20)
Then, from [START_REF] Johansen | Control allocation-a survey[END_REF],

ker {S(θ)} = Im      ∆ 1 0 0 0 1 0 0 0 1      , such that Q(θ)ker{S(θ)} = Im      ∆ 1 -2 θ 0 1 0 0 0 1      . (21)
The subspace Q(θ)ker{S(θ)} has a dimension greater than zero, then system (19) is adaptive input redundant. This subspace presents constant, θ-dependent, and ∆ i -dependent directions. Hence, we can define a matrix

L(θ) =   ∆ 1 -2 θ 0 1 0 0 0 1   ,
such that the first column is only different from zero when θ = 0, the second column is constant, and the third column depends on θ.

In order to verify the adaptive input redundancy of system (19) concerning a parameter θ, let us present practical simulations. We are going to analyze two different scenarios where the parameter influences the input redundancy, which are given by the ∆ 1 -dependent and θ-dependent directions of the subspace (21). The main objective is to understand how the parameter influences the input redundancy in each of these cases. Then, we define distinct inputs, a parameter function, and an initial condition x 0 = [1 1] ⊤ in each scenario. For the first scenario, let us consider only the ∆ 1dependent direction. Then, we define a parameter function:

θ(t) = 0, if 0 ≤ t < 2, 1, if t ≥ 2, s.t. ∆ 1 (t) = 1, if 0 ≤ t < 2, 0, if t ≥ 2, (22)
and three distinct inputs:

u a (t) = F (θ)x a (t), u b (t) = F (θ)x b (t) + ∆ 1 0 0 ⊤ sin(t), u c (t) = F (θ)x c (t) + 1 0 0 ⊤ sin(t). (23)
For 0 ≤ t < 2, the term [∆ 1 0 0] ⊤ sin(t) in input u b (t) is equal to the term [1 0 0] ⊤ sin(t) in u c (t), then inputs u b (t) and u c (t) are equal and different of u a (t), as verified in Fig 1 .  Since the term [∆ 1 0 0] ⊤ sin(t) is in the ∆ 1 -dependent direction of (21), by Theorem 1, the outputs generated by the three inputs should be equal for 0 ≤ t < 2, and this is verified in Fig 2 . On the other hand, for t ≥ 2, the parameter takes the value of one, so ∆ 1 becomes zero. The term [∆ 1 0 0] ⊤ sin(t) is not anymore equal to [1 0 0] ⊤ sin(t), thus input u b (t) is now different from u c (t) and it is equal to u a (t). Then, y a (t) = y b (t) for t ≥ 2, while y c (t) has a different trajectory, as we verify in Fig 2 . This happens because [1 0 0] ⊤ sin(t) is not in the ∆ 1 -dependent direction of (21) for a θ(t) ̸ = 0. Thus, the two outputs y a (t) = y b (t) for all t ≥ 0.

For the second scenario, let us consider the θ-dependent direction. Then, we define a parameter function:

θ(t) =      0.2, if 0 ≤ t < 2, 0.2(t -2) + 0.2, if 2 ≤ t < 6, 1, if t ≥ 6, (24)
and three distinct inputs:

u a (t) = F (θ)x a (t), u b (t) = F (θ)x b (t) + θ 0 1 ⊤ sin(t), u c (t) = F (θ)x c (t) + 0.2 0 1 ⊤ sin(t). (25)
For 0 ≤ t < 2, the term [θ 0 1] ⊤ sin(t) in input u b (t) is equal to the term [0.2 0 1] ⊤ sin(t) in u c (t), then inputs u b (t) and u c (t) are equal and different of u a (t), as shown in Fig 3 . The term [θ 0 1] ⊤ sin(t) is in the θ-dependent direction of (21). Hence, by Theorem 1, the outputs generated by the three inputs should be equal for 0 ≤ t < 2, as verified in Fig 4 . On the other hand, for t ≥ 2, the term [θ 0 1] ⊤ sin(t) is not anymore equal to [0.2 0 1] ⊤ sin(t). However, it is still in the θ-dependent direction of (21). Then, y a (t) still equals y b (t), but y c (t) has a different trajectory since [0.2 0 1] ⊤ sin(t) is not in one of the directions of (21) for a θ(t) ̸ = 0.2. This is verified in Fig 4. In this case, we also have y a (t) = y b (t) for t ≥ 0.

To conclude, in the two presented scenarios, we verified that there are different inputs leading to the same output trajectory for some parameter θ(t) and for the same initial condition, confirming that the system is adaptive input redundant. Furthermore, we verify that the parameter influences the invisible inputs from the output so that an implicit or explicit dependence on the parameter may appear.

Conclusion

In this paper, we introduced the concept of adaptive input redundancy for parameter-dependent systems. In the context where the parameter of the system is known, the input redundancy of the system can be characterized as dependent on the parameter. Hence, we have provided a sufficient condition to verify the input redundancy of parameterdependent systems. This condition is described by the dimension of the subspace B -1 (θ)V * ∩ker{D(θ)} being greater than By considering that the system matrices depend polynomially on the parameter and using its Smith normal form, we provide a practical way to compute this parameterdependent subspace. So, a tractable condition to verify adaptive input redundancy is obtained. It is important to highlight that since we consider subspaces that do not depend on the parameter, it restrains us from providing a complete characterization of input redundancy for this type of system. However, we can deal with all the previous results from the literature as particular cases. Future work should explore the possibility of considering parameter-dependent subspaces so that a complete characterization of adaptive redundancy can be provided.

Figure 1 :

 1 Figure 1: Input trajectories (23).

Figure 2 :

 2 Figure 2: Output trajectories for inputs in given (23) and θ(t) given in (22).

Figure 3 :

 3 Figure 3: Input trajectories (25).

Figure 4 :

 4 Figure 4: Output trajectories for inputs in given (25) and θ(t) given in (24).