Daniele Astolfi

Swann Marx
email: swann.marx@ls2n.fr.

Vincent Andrieu

Christophe Prieur
email: christophe.prieur@gipsa-lab.fr.

Global exponential set-point regulation for linear operator semigroups with input saturation

In this paper the problem of set-point regulation of linear operator semigroups is considered, in presence of saturation of the control input. It is shown how global exponential stability can still be obtained by adding an anti-windup loop to the integral action. As first contribution towards the solution of this regulation problem, we give some new results related to the characterization of global exponential stability of the equilibrium of a subclass of semi-linear dynamical systems.

I. INTRODUCTION

Recent years have witnessed a growing interest in the problem of feedback control of dynamical systems modeled by partial differential equation or more generally with state evolving in an infinite-dimensional space. Among the problems that have been studied, the output regulation (as considered e.g., in [START_REF] Paunonen | Internal model theory for distributed parameter systems[END_REF]) holds a great place because of its importance in applications. Hence, many results have been obtained in this field and in this context, integral actions are essential to obtain set-point regulation (see [START_REF] Pohjolainen | Robust multivariable PI-controller for infinite dimensional systems[END_REF] or more recently [START_REF] Coron | PI controllers for 1-D nonlinear transport equation[END_REF]- [START_REF] Terrand-Jeanne | Adding integral action for open-loop exponentially stable semigroups and application to boundary control of PDE systems[END_REF]). However, when applying these regulation methods in the case of saturated controls if the global regulation can be preserved, the global exponential regulation can be lost (see, e.g., [START_REF] Sontag | An algebraic approach to bounded controllability of linear systems[END_REF] in the context of finite-dimensional systems). In this note, inspired by so-called anti-windup methods (see, e.g., [START_REF] Grimm | Antiwindup for stable linear systems with input saturation: an LMI-based synthesis[END_REF]- [START_REF] Romanchuk | Some comments on anti-windup synthesis using LMIs[END_REF]), we provide an explicit solution to the regulation problem, by an appropriate choice of an integral action, while obtaining the global exponential stabilization of the equilibrium.

Before solving this regulation problem, we give some new results concerning global exponential stability conditions for nonlinear infinite-dimensional systems composed of a linear operator to which is added a globally Lipschitz and bounded function. These type of systems are a subclass of semi-linear systems. The proposed sufficient conditions and necessary conditions extend the corpus of existing results for such a class of systems, e.g. [START_REF] Curtain | Introduction to infinite-dimensional systems theory: a state-space approach[END_REF]- [START_REF] Mironchenko | Input-to-state stability of infinitedimensional systems: recent results and open questions[END_REF].

The rest of the article is organized in two main parts. In Section II, we characterize the property of global exponential stability for a class of semi-linear systems. These results are then applied in Section III to the particular problem of set-point regulation in case of saturated control input. Conclusions are drawn in Section IV. Notation: Given two Hilbert spaces X 1 , X 2 , L(X 1 , X 2 ) denotes the class of linear bounded operators from X 1 to X 2 , and we define L(X ) := L(X , X ). Given a real number L > 0, the saturation function sat L : R → R and the dead-zone function dz L : R → R are defined as

sat L (s) :=    L if s ≥ L , s if |s| ≤ L , -L if s ≤ -L , (1)
dz L (s) := s -sat L (s) . (2)

II. GLOBAL EXPONENTIAL STABILITY CONDITIONS

FOR SOME SEMI-LINEAR SYSTEMS

A. Preliminaries

Let X be a Hilbert space equipped with a scalar product ⟨•, •⟩ X , from which one can deduce a norm ∥•∥ X . We consider in this section semi-linear systems of the form

d dt x = Fx + Gψ L (Kx) := F ψ (x) (3)
in which F : D(F) ⊂ X → X is a linear operator which is an infinitesimal generator of a strongly continuous semigroup denoted (e tF ) t≥0 , G : R → X and K : X → R are bounded linear operators, and ψ L : R → R is a function satisfying the following assumption.

Definition 1. Given a positive real number L, ψ L : R → R is a generalized saturation function if it satisfies the following properties

1) ψ L (0) = 0. 2) |ψ L (s ′ ) -ψ L (s ′′ )| ≤ |s ′ -s ′′ | for any s ′ , s ′′ ∈ R. 3) sup s∈R |ψ L (s)| ≤ L.
In particular, ψ L is a continuous globally bounded function with Lipschitz constant equal to 1. For instance, the saturation function defined in (1) satisfies all properties of Assumption 1. We stress that the Lipschitz constant is selected equal to 1 to ease the presentation of the following section, but we could remove the knowledge assumption on the value of the Lipschitz constant.

Next, we study the solution to system (3). The generalized saturation function being globally Lipschitz, applying [START_REF] Curtain | Introduction to infinite-dimensional systems theory: a state-space approach[END_REF]Theorem 11.1.5], the following well-posedness result follows. Note that we refer the reader to [START_REF] Curtain | Introduction to infinite-dimensional systems theory: a state-space approach[END_REF]Definition 11.1.3] and [START_REF] Curtain | Introduction to infinite-dimensional systems theory: a state-space approach[END_REF]Definition 11.1.2] for the definitions of mild and classical solutions, respectively. Proposition 1. For any x 0 ∈ X (resp. x 0 ∈ D(F)), there exists a unique mild (resp. classical) solution

x ∈ C(R + ; X ) (resp. x ∈ C 1 (R + ; X ) ∩ C(R + ; D(F))
, that we will denote by x(t) := e tF ψ (x 0 ), to (3).

In this first section, our aim is to study the exponential stability of the origin of (3). This notion of stability is defined as follows.

Definition 2. The origin is said to be globally exponentially stable for system (3) if there exist positive real numbers c and λ such that the following inequality holds

∥e tF ψ (x 0 )∥ X ≤ ce -λt ∥x 0 ∥ X (4)
for all (x 0 , t) in X × R + .

It has to be noticed that in this property it is required that the positive real number c and λ are uniform. In the following part of this section a necessary condition and two sufficient conditions are given to obtain this global exponential stability property.

B. A necessary condition for global exponential stability

We start by stating the following necessary condition. Even though the result is probably known in the finitedimensional context, we haven't found any proof of it. Note that this condition is in the same spirit of some other results related to feedback stabilization with saturated control, e.g. [ Let v ∈ X satisfying ∥v∥ X = 1. Note that, by definition of the norm, for all µ > 0 and all t ≥ 0

∥e tF v∥ X = 1 µ ∥e tF µv∥ X , ≤ 1 µ ∥e tF ψ (µv)∥ X + 1 µ ∥e tF ψ (µv) -e tF µv∥ X .
With inequality (4), this implies by Definition 2 and equation ( 5) that for all µ > 0 and all t ≥ 0

∥e tF v∥ X ≤ ce -λt ∥v∥ X + 1 µ t 0 e (t-s)F Gψ L e sF ψ µv ds X ,
which gives, using the fact that ∥v∥ X = 1, the Cauchy Schwarz inequality and item 3) of Definition 1

∥e tF v∥ X ≤ ce -λt + L µ t 0 ∥e (t-s)F ∥ L(X ) ∥G∥ L(R,X ) ds.
Letting µ going to infinity yields, for all t ≥ 0

∥e tF v∥ X ≤ ce -λt , (6)
for all v in X such that ∥v∥ X = 1. By linearity of the semigroup (e tF ) t≥0 it holds, for all w in X , 1 ∥w∥ X ∥e tF w∥ X = ∥e tF w ∥w∥ X ∥ X and using ( 6) with v = w ∥w∥ X , we deduce that F is an infinitesimal generator of an exponential strongly continuous semigroup. 2

C. Sufficient conditions for global exponential stability

Sufficient conditions exist to prove the asymptotic stability from separate properties on the linear operator and on the semilinear map. See, e.g., [START_REF] Aksikas | Asymptotic behaviour of contraction nonautonomous semi-flows in a Banach space: Application to first-order hyperbolic PDEs[END_REF]Theorem 4] (see also [START_REF] Dus | On L ∞ stabilization of diagonal semilinear hyperbolic systems by saturated boundary control[END_REF], [START_REF] Hayat | Global exponential stability and Input-to-State Stability of semilinear hyperbolic systems for the L 2 norm[END_REF]) for exponentially stable linear operators and dissipative semilinear maps; [START_REF] Aksikas | Asymptotic behaviour of contraction nonautonomous semi-flows in a Banach space: Application to first-order hyperbolic PDEs[END_REF]Theorem 5] for an other sufficient condition based on the comparison of the decay rate of the linear operator and the Lipchitz constant of the semilinear map; [START_REF] Jayawardhana | Infinitedimensional feedback systems: the circle criterion and input-tostate stability[END_REF] for conditions based on the circle-criterion for Lur'e type systems. In our context, we give sufficient conditions based on Lyapunov consideration. Before presenting the next result, we provide the definition of coercive Lyapunov functionals for linear systems, that will be used all along the sequel (see [START_REF] Mironchenko | Input-to-state stability of infinitedimensional systems: recent results and open questions[END_REF]Definition 2.11] for the definition of coercive Lyapunov functionals for more general systems). Definition 3. Let F : D(F) ⊂ X → X be a linear operator generating a strongly continuous semigroup. A functional V (x) := ⟨Px, x⟩ X , with P ∈ L(X ) a self-adjoint operator, is said to be a coercive Lyapunov functional for F if there exist positive constants α and λ satisfying

α∥x∥ 2 X ≤ V (x) , ∀ x ∈ X , (7) ⟨Fx, Px⟩ X + ⟨Px, Fx⟩ X ≤ -λ∥x∥ 2 X , ∀ x ∈ D(F). (8)
Inequality [START_REF] Terrand-Jeanne | Adding integral action for open-loop exponentially stable semigroups and application to boundary control of PDE systems[END_REF] and the assumption P ∈ L(X ) imply in particular that V (•) is equivalent to the usual norm ∥ • ∥ X . Note that the existence of non-coercive Lyapunov functional (possibly implying a non-equivalence between V (•) and ∥ • ∥ X ) is a particular feature of infinitedimensional systems, as discussed in [START_REF] Mironchenko | Non-coercive Lyapunov functions for infinite-dimensional systems[END_REF].

In the following, two sufficient conditions for the exponential stability of the origin of systems of the form (3) are stated. The first condition is based on a "small-gain control property" which extends the result [8, Theorem 2.4] presented in the case of finite-dimensional linear systems in the particular case in which the function ψ L is a saturation function. It is a trivial extension of the fact that exponential stability is robust to small Lipschitz perturbations in the dynamics when there exists a coercive Lyapunov functional. Proposition 3. Consider system (3) with the function ψ L satisfying Definition 1. Suppose that there exists a coercive Lyapunov functional V for F given by, for all x ∈ X , V (x) := ⟨Px, x⟩ X for some P ∈ L(X ). Then, for all bounded linear operator K : X → X satisfying ∥K∥ L(X ,R) < λ 2∥PG∥ X the origin of (3) is globally exponentially stable.

Proof. We consider classical solutions to (3), i.e. we consider initial condition x 0 in D(F) and we will prove that the result of Proposition 3 for such solutions. One can deduce the same result for any initial x 0 ∈ X by applying a standard density argument (see e.g., [17, Lemma 1]).

Consider the Lyapunov functional V (x) := ⟨Px, x⟩ of the statement. Its derivative along classical solutions to (3) yields, for all x in D(F)

d dt V (x) =⟨Fx, Px⟩ X + ⟨Px, Fx⟩ X + ⟨Gψ L (Kx), P x⟩ X + ⟨Px, Gψ L (Kx)⟩ X , ≤ -λ∥x∥ 2 X + 2∥x∥ X ∥PG∥ X |ψ L (Kx)| , (9)
where in the second step we used the property [START_REF] Sontag | An algebraic approach to bounded controllability of linear systems[END_REF]. Let ε > 0 be such that ε < λ 2∥PG∥ X -∥K∥ L(X ,R) . Such ε > 0 exists by assumption on the bound of K in Proposition 3. Using Definition 1 (point 1. and 2.) and the definition of ε, one obtains that, for all x in X , |ψ

L (Kx)| ≤ |Kx| ≤ λ 2∥PG∥ X ∥x∥ X -ε∥x∥ X . We deduce with (9) d dt V (x) ≤ -2ε∥PG∥ X ∥x∥ 2 X ∀x ∈ D(F).
Using the coercivity of V and the boundedness of the linear operator P, we obtain the desired result. Indeed, P being coercive, V (•) is equivalent to the usual norm of X . Using this equivalence together with the Grönwall's inequality, one can deduce that the origin of ( 3) is globally exponentially stable (in the sense of Definition 2).

2 The second sufficient condition is based on the existence of a common coercive Lyapunov functional between the operators F and F +GK. For the finite-dimensional case in which the function ψ L is a saturation function this result is well known, see, e.g. [START_REF] Romanchuk | Some comments on anti-windup synthesis using LMIs[END_REF].

Proposition 4. Consider system (3) with the function ψ L satisfying Definition 1. Suppose there exists a common coercive Lyapunov functional V for F and F + GK, namely there exists a functional V defined as V (x) := ⟨Px, x⟩ X , for all x in X , satisfying [START_REF] Terrand-Jeanne | Adding integral action for open-loop exponentially stable semigroups and application to boundary control of PDE systems[END_REF] and

⟨Fx, Px⟩ X + ⟨Px, Fx⟩ X ≤ -λ∥x∥ 2 X ⟨(F + GK)x, Px⟩ X + ⟨Px, (F + GK)x⟩ X ≤ -λ∥x∥ 2 X (10)
for all x ∈ D(F), for some λ > 0. Then the origin of (3) is globally exponentially stable.

Proof. Similarly to the proof of Proposition 3, we consider here classical solutions to (3), i.e. we consider initial condition x 0 in D(F), and we will prove that the result of Proposition 4 for any initial condition x 0 in D(F). One can deduce the same result for any initial condition x 0 ∈ X by applying a standard density argument (see e.g., [START_REF] Marx | Cone-bounded feedback laws for m-dissipative operators on Hilbert spaces[END_REF]Lemma 1]).

Define F K := F + GK and let α : R → R be the function defined as α(0) = 0 and

α(s) := 1 - ψ L (s) s , ∀s ̸ = 0,
Note that with Points 1 and 2 of the definition 1, it yields that α(s) ∈ [0, 1] for all s. With such a notation, the system (3) can be rewritten as

d dt x = α(Kx)Fx + (1 -α(Kx))F K x. (11)
Now consider the coercive Lyapunov functional given in the statement V (x) := ⟨Px, x⟩ X with P ∈ L(X ) satisfying ( 7) and [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF]. By compactly writing α = α(Kx), we compute its derivative along classical solutions to (3), i.e. equivalently [START_REF] Romanchuk | Some comments on anti-windup synthesis using LMIs[END_REF]. Using [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF], we obtain, for all x in D(F),

d dt V (x) = α⟨Fx, P x⟩ X + α⟨P x, Fx⟩ X + (1 -α)⟨F K x, P x⟩ X + (1 -α)⟨P x, F K x⟩ X ≤ -αλ∥x∥ 2 X -(1 -α)λ∥x∥ 2 X ≤ -λ∥x∥ 2 X .
The proof concludes by following the same arguments used in the proof of Proposition 3. 2

III. SET-POINT REGULATION

A. Problem statement and assumptions

In this section we consider a class of controlled linear systems with saturated control. A slightly modified notation is used. In particular, we consider H as a Hilbert space equipped with a norm ∥ • ∥ H and a scalar product ⟨•, •⟩ H . We consider systems which are described as

d dt ξ = Aξ + B sat L (u) + w, y = Cξ, (12)
in which A : D(A) ⊂ H → H is an infinitesimal generator of a C 0 -semigroup denoted t → e tA , the operator B : R → H is linear and bounded, the operator C is in 1 L(D(A), R), u in R is the control input, y in R is the measured output, w in H is an unknown constant disturbance vector, and the saturation function sat L is defined as in (1) for some L > 0. Given a constant reference y ref ∈ R, we are interested in the design of a feedback law for system [START_REF] Curtain | Introduction to infinite-dimensional systems theory: a state-space approach[END_REF] so that there exists a unique equilibrium which is exponentially stable and

lim t→∞ y(t) = y ref , (13)
by means of an integral action. Note that the regulation problem could be defined without requiring existence and uniqueness of equilibrium leading to weaker assumptions. It is however beyond the scope of this paper.

To this end, we follow the set of conditions established below (similarly to [START_REF] Terrand-Jeanne | Adding integral action for open-loop exponentially stable semigroups and application to boundary control of PDE systems[END_REF]). Comments are postponed at the end of the section.

Assumption 1. The operator A generates a strongly continuous semigroup (e tA ) t≥0 , and moreover there exist a coercive and self-adjoint operator Q ∈ L(X) and a positive constant ν such that for all ξ ∈ D(A)

⟨A ξ, Q ξ⟩ H + ⟨Q ξ, A ξ⟩ H ≤ -2ν∥ξ∥ 2 H . (14)
Then, following [START_REF] Pohjolainen | Robust multivariable PI-controller for infinite dimensional systems[END_REF], we assume (up to a reparametrization) the following.

Assumption 2. The condition CA -1 B = 1 holds.

Following classical finite-dimensional strategies, we extend the system [START_REF] Curtain | Introduction to infinite-dimensional systems theory: a state-space approach[END_REF] with an integral action [START_REF] Astolfi | Integral action in output feedback for multi-input multi-output nonlinear systems[END_REF] and an anti-windup mechanism [START_REF] Grimm | Antiwindup for stable linear systems with input saturation: an LMI-based synthesis[END_REF]- [START_REF] Romanchuk | Some comments on anti-windup synthesis using LMIs[END_REF]. In particular, we propose the following regulator

d dt z = Cξ -y ref -µ dz L (u), u = kz, (15)
with z in R, the function dz L defined as in (2), and µ, k positive parameters to be selected so that the solutions to the closed-loop system ( 12), ( 15) converge exponentially to an equilibrium on which the regulation objective ( 13) is attained. We complete this section by introducing the following compact form. In particular, recalling the definition of the deadzone function in (2), the extended closed-loop system [START_REF] Curtain | Introduction to infinite-dimensional systems theory: a state-space approach[END_REF], [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF] reads as

x = ξ z , d dt x = Fx + G sat L (Kx) + Γ, ( 16
)
1 Since A is an infinitesimal generator of a C 0 -semigroup, its domain D(A) is a Hilbert space equipped with the graph norm ∥ξ∥ D(A) = ∥ξ∥ H + ∥Aξ∥ H . with F, G, L defined as

F := A 0 C -µk , G = B µ , K = 0 k , ( 17
)
and Γ = (w, y ref ) ⊤ . The state space is defined by X = H × R and it is equipped with the norm ∥x∥ X = ∥(ξ, z)∥ X = ⟨ξ, ξ⟩ H + |z| 2 , with the associated scalar product ⟨(ξ 1 , z 1 ), (ξ 2 , z 2 )⟩ X = ⟨ξ, ξ⟩ H +z 1 z 2 . Moreover, we define the domain of the operator F given by D(F) = {(ξ, z) ∈ H × R | Aξ ∈ H}, which is equipped with the usual graph norm.

Remark 1. Assumption 1 establishes that the semigroup generated by A is exponentially stable. This assumption is necessary in finite dimension in view of Proposition 2 and the lower triangular structure of the operator F. Note that exponential stability could be obtained without assuming coercivity of the operator Q. However,it is not straightforward to obtain exponential stability for the closed loop system that we study, because it is nonlinear due to the saturation operator. Following [START_REF] Mironchenko | Non-coercive Lyapunov functions for infinite-dimensional systems[END_REF], it may be possible to remove this coercivity assumption. It is left for future research. Assumption 2 could be also relaxed as CA -1 B ̸ = 0, which is necessary for the design of an integral action feedback design (see, e.g., [START_REF] Astolfi | Integral action in output feedback for multi-input multi-output nonlinear systems[END_REF] for the finite-dimensional case and [START_REF] Terrand-Jeanne | Adding integral action for open-loop exponentially stable semigroups and application to boundary control of PDE systems[END_REF] for the infinitedimensional case). Here we select its value set to 1 to ease the rest of the presentation.

B. Main results

The next result ensures that there exists a unique equilibrium point to [START_REF] Marx | Stability analysis of dissipative systems subject to nonlinear damping via Lyapunov techniques[END_REF]. This existence result relies mainly on Assumption 2. In addition to this existence, we provide a condition ensuring that, if the perturbation w and the reference y ref are sufficiently small, then at the equilibrium the regulation objective Cξ • = y ref is satisfied, despite the perturbation w. Lemma 1. There exists a unique equilibrium point (ξ • , z • ) ∈ D(F) to [START_REF] Marx | Stability analysis of dissipative systems subject to nonlinear damping via Lyapunov techniques[END_REF]. In other words, there exists a unique solution (ξ • , z • ) ∈ D(F) to the following set of equations:

0 = Aξ • + B sat L (kz • ) + w 0 = Cξ • -y ref -µ dz L (kz • ) . (18)
Moreover, let Ω ⊂ H × R be defined as

Ω = {(w, y ref ) ∈ H × R : |CA -1 w + y ref | ≤ L}.
Then, for all (w, y ref ) ∈ Ω, one has Cξ • = y ref .

Proof. The first line of ( 18) can be equivalently written as:

ξ • = -A -1 B sat L (kz • ) -A -1 w. (19)
Plugging this identity in the second line of (18) leads to

-CA -1 B sat L (kz • ) -y ref -CA -1 w -µ dz L (kz • ) = 0.
Using Assumption 2 and the definition of the dead-zone function, one finds the equivalent equation

f (kz • ) = -y ref -CA -1 w, (20)
with f defined as

f : s ∈ R → µs -(µ -1) sat L (s) ∈ R.
By definition of the saturation function, f can be equivalently rewritten as:

f : s → f (s) =    µs -(µ -1)L if s ≥ L , s if |s| ≤ L , µs + (µ -1)L if s ≤ -L . (21
) Note that once one is able to prove that there exists a unique z • ∈ R solving [START_REF] Hayat | Global exponential stability and Input-to-State Stability of semilinear hyperbolic systems for the L 2 norm[END_REF], the result of Lemma 1 follows, thanks to [START_REF] Dus | On L ∞ stabilization of diagonal semilinear hyperbolic systems by saturated boundary control[END_REF]. If µ = 1, the result follows easily, and we have even an explicit solution given by kz • = -y ref -CA -1 w. Suppose then that µ ̸ = 1. Note that f is continuous. Moreover, lim s→+∞ f (s) = +∞ and lim s→-∞ f (s) = -∞. Then, according to the intermediate value theorem, and since f is increasing, for every s ∈ R, there exists a unique s ∈ R such that f (s) = s. Setting s = kz • and s = -y ref -Cw, there exists a unique solution to [START_REF] Mironchenko | Input-to-state stability of infinitedimensional systems: recent results and open questions[END_REF]. Now, we prove the second part of Lemma 1. The definition of the set Ω and the condition [START_REF] Hayat | Global exponential stability and Input-to-State Stability of semilinear hyperbolic systems for the L 2 norm[END_REF] gives |f (kz • )| ≤ L. As a consequence, in view of the definition of the function f given in [START_REF] Mironchenko | Non-coercive Lyapunov functions for infinite-dimensional systems[END_REF], we obtain f (kz • ) = kz • and |kz • | ≤ L. Furthermore, using the definitions of sat L and dz L in (1), (2), the second line of equation ( 18) reads 0 = Cξ • -y ref concluding the proof.

2

Our aim now is to transform (16) into a system in the form (3). To do so, we introduce the change of coordinates

ξ → ξ := ξ -ξ • , z → z := z -z • ,
where the pair (ξ • , z • ) is given by Lemma 1. The system [START_REF] Marx | Stability analysis of dissipative systems subject to nonlinear damping via Lyapunov techniques[END_REF] reads in these new coordinates:

d dt ξ = A ξ + B[sat L (kz + kz • ) -sat L (kz • )] d dt z = Cξ -µ[dz L (kz + kz • ) -dz L (z • )]
or equivalently, selecting x := ( ξ, z),

d dt x = F x + Gψ L (Kx), (22)
where

ψ L (Kx) := sat L (K(x + x • )) -sat L (Kx • ).
System [START_REF] Astolfi | Integral action in output feedback for multi-input multi-output nonlinear systems[END_REF] is in the form (3) and moreover the function ψ L so defined satisfies Definition 1 (picking 2L instead of L). Well-posedness of system [START_REF] Astolfi | Integral action in output feedback for multi-input multi-output nonlinear systems[END_REF] follows from direct application of Proposition 1. Finally, we can show that an appropriate selection of the parameters µ, k allows to apply Proposition 4 in order to show that the origin of system ( 22) is globally exponentially stable. To this end, we need to introduce a new operator that will be denoted as M. In particular, since A generates a strongly continuous semigroup which origin is exponentially stable, it admits an inverse, A -1 which belongs to L(H, D(A)) (see e.g., [START_REF] Chill | Semi-uniform stability of operator semigroups and energy decay of damped waves[END_REF]Theo. 2.1]). As a consequence, C being A-bounded, the operator M = CA -1 belongs to L(H, R). Note that such an operator M is solution to the following equation

MA ξ = Cξ ∀ξ ∈ D(A) . (23)
Proposition 5. Suppose Assumption 1 and 2 hold. For any µ > 0 and k ∈ (0, k ⋆ ) with k ⋆ given by

k ⋆ = min ν µ∥M∥ 2 L(H,R) , ν ∥(Q + M ⋆ M)B∥ 2 L(H,R) , ( 24
) the origin of ( 22) is globally exponentially stable.

Proof. The main idea of the proof of this proposition consists in showing the existence of a common Lyapunov functional for the operators F and for F + GK, which is the context of Proposition 4. To this end, let us define the operator P in L(X ) as

P(ξ, z) = Q + M ⋆ M -M ⋆ ⋆ 1 ξ z , (25)
with Q given by Assumption 1 and M defined by [START_REF] Chill | Semi-uniform stability of operator semigroups and energy decay of damped waves[END_REF].

We compute now, for all x = (ξ, z) in D(F),

⟨Fx, Px⟩ X = ⟨PFx, x⟩ X , = QA M ⋆ µk ⋆ -µk x, x X , (26)
where we used the fact that M ⋆ MAξ -M ⋆ Cξ = 0 in view of [START_REF] Chill | Semi-uniform stability of operator semigroups and energy decay of damped waves[END_REF]. Then, using [START_REF] Aksikas | Asymptotic behaviour of contraction nonautonomous semi-flows in a Banach space: Application to first-order hyperbolic PDEs[END_REF], inequality (26) gives

⟨Fx, Px⟩ X + ⟨Px, Fx⟩ X ≤ - ∥ξ∥ H |z| ⊤ S 1 ∥ξ∥ H |z| with S 1 = 2 νI -µk∥M∥ L(H,R) ⋆ µk .
Finally, it can be verified that for any k ∈ (0, k ⋆ ), with k ⋆ smaller than the first value in (24), there exists

ε 1 > 0 such that S 1 ≥ ε 1 I from which we obtain ⟨Fx, Px⟩ X + ⟨Px, Fx⟩ X ≤ -ε 1 ∥x∥ 2 X . (27)
Then, denoting

F K = F + GK = A kB C 0
and recalling that MB = 1 by Assumption 2, we compute for all x = (ξ, z) in D(F)

⟨F K x, Px⟩ X = ⟨PF K x, x⟩ X , = QA k(Q + M ⋆ M)B ⋆ -k x, x X . With Assumption 1, it implies ⟨F K x, Px⟩ X + ⟨Px, F K x⟩ X ≤ - ∥ξ∥ H |z| ⊤ S 2 ∥ξ∥ H |z| ⊤ with S 2 = 2 -νI k∥(Q + M ⋆ M)B∥ L(H,R) ⋆ -k .
As in the previous case, it can be verified that for any k ∈ (0, k ⋆ ), with k ⋆ smaller than the second value in (24), there exists ε 2 > 0 such that S 2 ≥ ε 2 I. Hence, we obtain ⟨F K x, Px⟩ X + ⟨Px, F K x⟩ X ≤ -ε 2 ∥x∥ 2 X . Combining the last inequality with (27), we conclude that P is a common coercive Lyapunov functional for the operators F and F + GK. As a consequence, the conditions [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF] of Proposition 4 are satisfied and the origin of system ( 22) is globally exponentially stable. 2

Remark 2. From condition (24) we obtain that if µ ∈ 0,

∥(Q+M ⋆ M)B∥ 2 L(H,R) ∥M∥ 2 L(H,R)
, then k ⋆ is independent of µ, allowing for a larger interval for the choice of k.

Finally, in view of the previous result, we state the following result which combines the result of Proposition 5 with the properties of the equilibria of the closed-loop dynamics established in Lemma 1. In summary, global exponential set-point regulation is obtained.

Corollary 1. Consider the controller (15) and select µ, k according to Proposition 5. For any w, y ref ∈ Ω, with Ω defined in Lemma 1, the closed-loop system (12), (15) admits a unique equilibrium (ξ • , z • ) which is globally exponentially stable and satisfies y = Cξ • = y ref .

IV. CONCLUSION

In this paper, we have first given some necessary and sufficient conditions for the global exponential stability of some subclass of semi-linear abstract Cauchy problems. This class of system is composed of a linear generator to which is added a nonlinear function which is globally Lipschitz and globally bounded. This has allowed us to tackle the set-point regulation problem with saturated control input. Indeed, employing an antiwindup design strategy, we have shown how to obtain global exponential set point regulation for infinitedimensional linear systems. The case of boundary control is currently under inverstigation.

His work has been partially supported by MIAI@Grenoble Alpes (ANR-19-P3IA-0003)

This research was partially supported by the French Grant ANR ODISSE (ANR-19-CE48-0004-01), and by MIAI@Grenoble Alpes (ANR-19-P3IA-0003).


Marina Chauvet
email: marina.chauvet@uca.fr

Arthur Monjot

Cécile Lepère

High diversity of microsporidian parasites and new planktonic hosts in freshwater and marine ecosystems

Microsporidia are a large group of obligate intracellular eukaryotic parasites related to Fungi primarily known as parasites of vertebrates and invertebrates. They are well described as parasites of organisms of interest (e.g., edible fish and crustaceans, honeybees, bioindicators such as daphnia, humans) on which they can have an important impact (e.g., reduced survival or fecundity and sex ratio distortion). However, their diversity in aquatic environments, especially in marine ecosystems, has been greatly understudied since they are not targeted by classical eukaryotic primers used in metabarcoding studies. Moreover, little is known about their hosts among protists or microzooplankton and therefore about their impact on the trophic food web functioning. In this work, we sampled 15 different sites across marine and freshwater environments, size-fractioned the samples and used microsporidian specific primers associated with metabarcoding to study the microsporidian diversity (and the associated spatial variation). Co-occurrence networks as well as Tyramide Signal Amplification-Fluorescent In Situ Hybridization were used to link potential hosts (planktonic eukaryotes < 150 μm) and Microsporidia diversity. Our analysis unraveled a large microsporidian diversity which was widely divergent between the two environments studied. In both of them, an important part of this diversity was not affiliated to a genus, suggesting an important reservoir of new microsporidian species and thus new hosts among planktonic eukaryotes. Co-occurrence networks and fluorescence microscopy showed for the first time associations between Microsporidia and dinoflagellates in freshwater and marine environments.

Introduction

Parasitism has been described as one of the oldest and most ubiquitous interactions in nature [START_REF] Cavalier-Smith | Kingdom Protozoa and Its 18 Phyla[END_REF], however the diversity of parasites and their effects on the aquatic trophic food web functioning have long been neglected [START_REF] Marcogliese | Food webs: a plea for parasites[END_REF]. In the last decades, high throughput sequencing allowed the description of an increasing number of parasites (e.g., [START_REF] Singer | Protist taxonomic and functional diversity in soil, freshwater and marine ecosystems[END_REF]) and revolutionized our understanding of aquatic food webs in term of trophic interactions as well as of energy and biomass flows [START_REF] Lafferty | Parasites in food webs: the ultimate missing links[END_REF][START_REF] Kagami | Mycoloop: chytrids in aquatic food webs[END_REF]). Among these parasites, Microsporidia (obligate intracellular eukaryotic parasites related to Fungi [START_REF] Karpov | Morphology, phylogeny, and ecology of the aphelids (Aphelidea, Opisthokonta) and proposal for the new superphylum Opisthosporidia[END_REF])) were known in aquatic ecosystems only through the search of infectious agent in organisms of interest (e.g., edible fish and crustaceans, bioindicators, humans, etc.). These host-targeted studies show that Microsporidia can have a considerable impact on host populations that are of great importance in freshwater ecosystems, by causing a decrease in the survival and fecundity rates of daphnia for example [START_REF] Ebert | Some parasites of Daphnia, In Ecology, Epidemiology, and Evolution of Parasitism in Daphnia[END_REF]. However, they were missing from metabarcoding studies focusing on microbial eukaryotes diversity [START_REF] Bass | Diverse Applications of Environmental DNA Methods in Parasitology[END_REF] especially due to their divergence from other eukaryotes (rapid evolution and compaction of their genome [START_REF] Keeling | Microsporidia: biology and evolution of highly reduced intracellular parasites[END_REF]) making the primers unable to detect them [START_REF] Hadziavdic | Characterization of the 18S rRNA Gene for Designing Universal Eukaryote Specific Primers[END_REF]Hugerth et al. 2014). As a result, the overall microsporidian diversity in aquatic (freshwater and marine) ecosystems is largely unknown and thus their impact on ecosystem functioning is neglected.

In 2018, a targeted metabarcoding study (i.e. using specific primer for Microsporidia) presented the first description of the microsporidian diversity in various environments including freshwater and marine samples [START_REF] Williams | Group-specific environmental sequencing reveals high levels of ecological heterogeneity across the microsporidian radiation[END_REF], but this allowed only a limited analysis due to the low number of operational taxonomic units (OTUs) obtained. This study nevertheless allowed the identification of a new diversity across the microsporidian tree and encouraged further work. Indeed, two others metabarcoding studies focusing on lacustrine ecosystems followed and allowed the discovery of an unsuspected diversity of Microsporidia [START_REF] Dubuffet | A phylogenetic framework to investigate the microsporidian communities through metabarcoding and its application to lake ecosystems[END_REF]Chauvet et al. 2022a). In these studies, between 40% and 50% of the uncovered diversity couldn't be affiliated to a genus suggesting that the microsporidian phylum hide a large amount of undiscovered diversity. These studies also provided temporal variation data and highlighted new microsporidian planktonic hosts in a small size fraction (i.e. <150µm) (Chauvet et al. 2022a,b). Moreover, few Microsporidia are currently described as parasites of protists or microzooplankton [START_REF] Murareanu | Generation of a Microsporidia Species Attribute Database and Analysis of the Extensive Ecological and Phenotypic Diversity of Microsporidia[END_REF], even though they are major components of food webs.

Altogether, the previous studies did not allow a broad comparison of the diversity of Microsporidia across aquatic ecosystems and thus lead to an incomplete picture of their diversity. Although [START_REF] Dubuffet | A phylogenetic framework to investigate the microsporidian communities through metabarcoding and its application to lake ecosystems[END_REF] and Chauvet et al. (2022a) showed an in-depth description of the microsporidian diversity, these studies were limited to two lakes (Aydat and Pavin, Massif Central, France).

Moreover, the microsporidian diversity in marine ecosystems is probably still underestimated as it has been neglected so far. Here, spatial variation of the microsporidian diversity (canonical (or longbranch) Microsporidia [START_REF] Bass | Clarifying the Relationships between Microsporidia and Cryptomycota[END_REF])) was assessed through targeted metabarcoding in nine different freshwater and six marine ecosystems. Co-occurrence networks and fluorescence microscopy were eventually used to identify potential new microsporidian hosts among microeukaryotes.

Materials and methods

Study sites and sampling procedure

Water samples were collected in France from different freshwater lakes (n = 9) and marine environments (n = 6; Mediterranean Sea and Atlantic Ocean) (Fig. 1) between 18/06/21 and 27/09/21 (Supporting Information Table S1). Then, 2.5 to 22 litres of water were sampled at the surface with a Niskin bottle or along the euphotic zone using an integrated sampler IWS (HYDRO-BIOS) (Supporting Information Table S1).

For most of the samples we collected the 0.8-150 μm fraction to concentrate potentially infected planktonic hosts except for the Brest and Roscoff samples where we were able to collect the >3 μm fraction (Supporting Information Table S1). Part of the microbial biomass of each size-fraction was collected on 0.8-μm-pore-size polycarbonate filters (Millipore) at a very low vacuum to prevent cell damage (<15 kPa) and stored at -80°C until nucleic acids extraction. For some samples (Supporting Information Table S1), the rest of the microbial biomass was fixed with paraformaldehyde (final concentration, 2%), incubated at 4°C for 24 h and collected on 0.8-μm-pore-size polycarbonate filters (Millipore; pressure <15 kPa). Filters were preserved by dehydration in an ethanol series (50, 80, and 100% ethanol for three min each) and stored at -20°C until tyramide signal amplification-fluorescent in situ hybridization (TSA-FISH) staining.

Nucleic acids extraction, amplification and sequencing

Filters prepared for DNA extraction were processed as described in [START_REF] Dubuffet | A phylogenetic framework to investigate the microsporidian communities through metabarcoding and its application to lake ecosystems[END_REF]. Amplification of the V1-V3 region of the microsporidian gene coding for the SSU rRNA was performed using the universal microsporidian V1F and 530R primers set [START_REF] Zhu | Small Subunit rRNA Sequence of Enterocytozoon bieneusi and Its Potential Diagnostic Role with Use of the[END_REF][START_REF] Baker | Small Subunit Ribosomal DNA Phylogeny of Various Microsporidia with Emphasis on AIDS Related Forms[END_REF]. Even though other recently developed primers have shown their efficiency (i.e. CM-V5F CM-V5R, ~200 bp, [START_REF] Trzebny | A new method of metabarcoding Microsporidia and their hosts reveals high levels of microsporidian infections in mosquitoes (Culicidae)[END_REF], this set of primers amplify a fragment of ~400 bp and has shown its efficiency in the detection of aquatic Microsporidia [START_REF] Dubuffet | A phylogenetic framework to investigate the microsporidian communities through metabarcoding and its application to lake ecosystems[END_REF]Chauvet et al. 2022). Each PCR was performed in a total volume of 30 μL containing 6 μL of 5x Green GoTaq Flexi buffer, 2.4 μL of 25 mM MgCl2, 0.18 μL of GoTaqG2 Flexi DNA Polymerase (Promega, 5 U/μL), 0.6 μL of 10 mM each dNTP, 0.3 μL of 50 mg/mL BSA and 0.6 μL of each 10 μM primer. The amplification conditions consisted of initial denaturation at 95 °C for 5 min followed by 35 cycles for 1 min at 95 °C, 1 min 30 s at 62 °C and 1 min 30 s at 72 °C, and a final elongation of 7 min at 72 °C. Amplification of the V4 region of total eukaryotes (excluding Microsporidia) was performed using the universal primer 515F [START_REF] Caporaso | Global patterns of 16S rRNA diversity at a depth of millions of sequences per sample[END_REF]) and the eukaryotic primer 951R [START_REF] Mangot | Short-term dynamics of diversity patterns: evidence of continual reassembly within lacustrine small eukaryotes[END_REF]. Each PCR was performed in the mix described above. The amplification conditions consisted of initial denaturation at 95 °C for 5 min followed by 33 cycles for 1 min at 95 °C, 1 min at 55 °C and 1 min 30 s at 72 °C, and a final elongation of 10 min at 72 °C.

The amplicons were purified and concentrated using the QIAquick Gel Extraction kit (Qiagen).

Samples were then sequenced on a MiSeq Illumina instrument (2x250 bp) (Microsynth AG).

Amplicon analysis and taxonomic affiliation

Sequencing data were processed using the PANAM2 (Phylogenetic Analysis of Next-generation AMplicons v2 -https://github.com/meb-team/PANAM2) pipeline [START_REF] Taib | Phylogenetic Affiliation of SSU rRNA Genes Generated by Massively Parallel Sequencing: New Insights into the Freshwater Protist Diversity[END_REF]. Sequences with ambiguous 'N' bases, those below 200 bp, chimeras or sequencing errors in the primers were eliminated. The sequences were then assembled (overlap of 50 bp minimum between forward and reverse reads, maximum 2 mismatches in the overlap) and clustered into OTUs with a similarity threshold of 99% for Microsporidia [START_REF] Dubuffet | A phylogenetic framework to investigate the microsporidian communities through metabarcoding and its application to lake ecosystems[END_REF]) and 95% for other eukaryotes (Mangot et al. 2013;[START_REF] Debroas | Overview of freshwater microbial eukaryotes diversity: a first analysis of publicly available metabarcoding data[END_REF]. The representative sequences of each OTU were affiliated using a custom curated reference database (Freshwater-database: https://github.com/panammeb/Freshwaterdatabase). Affiliations to microsporidian groups and genera were considered as valid when the identity scores were above 85% and 94.5%, respectively [START_REF] Dubuffet | A phylogenetic framework to investigate the microsporidian communities through metabarcoding and its application to lake ecosystems[END_REF]. As clustering threshold used here (especially for Microsporidia) may generate OTUs resulting from sequencing errors, we only kept for the final analysis OTUs representing more than 0.005% (Bokulich et al. 2013) of the total reads. Data were normalized using the SRS (Scaling with Ranked Subsampling) method [START_REF] Beule | Improved normalization of species count data in ecology by scaling with ranked subsampling (SRS): application to microbial communities[END_REF].

Phylogenetic analyses were performed to show the placement of the sequences obtained in this study among the different microsporidian clades (Supporting Information Fig. S1,S2). We focused our analyses on abundant OTUs (100 most abundant of each clade, when more than 100 OTUs were identified) and on the five major clades of this study (I, III, IV1, IV2 and V). After an alignment using MAFFT algorithm [START_REF] Katoh | MAFFT version 5: improvement in accuracy of multiple sequence alignment[END_REF], ML trees were obtained using the best predicted model of evolution according to the BIC score (general time-reversible (GTR) model with gamma-distributed rate heterogeneity (G)) with 200 bootstrap replications and partial deletion (85%), using MEGA 7 [START_REF] Kumar | MEGA7: Molecular Evolutionary Genetics Analysis Version 7.0 for Bigger Datasets[END_REF].

Diversity and correlation analyses

Diversity analyses were performed using R (version 3.6.3, R Development Core Team 2012). As microsporidian and eukaryotic communities' composition were largely different between marine and freshwater samples, the correlation analysis was done on each dataset. To reduce the number of tested hypotheses and focus the analysis on 'abundant OTUs', only those representing more than 0.05% of the reads were retained, representing 302 eukaryotic OTUs and 253 microsporidian OTUs in freshwater and 237 eukaryotic OTUs and 268 microsporidian OTUs in marine ecosystems.

Spearman's ρ correlations were calculated, and to decrease false discovery rate due to multiple testing we used Benjamini-Hochberg corrections [START_REF] Benjamini | Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing[END_REF]. A correlation was considered as valid when ρ > 0.8 and the associated p-value < 0.01. Correlations were then visualized with chord diagrams using the circlize package (version 0.4.14) on R (R Development Core Team 2012; [START_REF] Gu | circlize implements and enhances circular visualization in R[END_REF].

Tyramide Signal Amplification-Fluorescent In Situ Hybridization

TSA-FISH was performed using a Microsporidia specific probe (USP01) as described in Chauvet et al. (2022a). This probe is suitable for the detection and observation of aquatic Microsporidia, being able to hybridize (in silico) with 100% of the microsporidian sequences obtained during recent metabarcoding studies [START_REF] Dubuffet | A phylogenetic framework to investigate the microsporidian communities through metabarcoding and its application to lake ecosystems[END_REF]Chauvet et al. 2022a) as well as all the sequences obtained in the current study. In silico, this probe does not hybridize with any sequence other than microsporidian. Hybridization conditions were determined on a target species (Encephalitozoon cuniculi) and a non-target species (Anncaliia algerae) having one mismatch with the probe sequence.

The probe was also tested on two other Microsporidia in culture: Nosema ceranae and Nematocida parisii. Observations were performed using a Zeiss Axio Imager 2 microscope (Carl Zeiss).

Results and discussion

In this work, we sampled 15 different sites across freshwater (n = 9) and marine (n = 6) environments (Fig. 1, Supporting Information Table S1), size-fractioned the samples and used microsporidian specific primers to unravel microsporidian diversity which once associated with total eukaryotic diversity allow the identification of potential host-parasite associations.

After quality filtering, clustering at 99% and cleaning steps, a total of 1 853 microsporidian OTUs were obtained from 2 944 084 SSU rRNA gene sequencing reads (1 045 560 reads after normalization). OTUs obtained in this study belonged to the canonical (or long-branch [START_REF] Bass | Clarifying the Relationships between Microsporidia and Cryptomycota[END_REF])) Microsporidia and were all affiliated at least to a clade. Among the 10 clades defined for canonical Microsporidia [START_REF] Dubuffet | A phylogenetic framework to investigate the microsporidian communities through metabarcoding and its application to lake ecosystems[END_REF], our OTUs were distributed in seven clades: I, II, III, IV1, IV2, V, VIII (Fig. 1, Supporting Information Fig. S1,S2). 63.9% of the OTUs were affiliated to a group and 9 groups were identified. Less than half (45.5%) of the OTUs were affiliated to a genus among the 20 genera identified, demonstrating that a large part of the microsporidian diversity remains to be described.

Microsporidian diversity across freshwater environments

So far, few studies have investigated the global diversity of Microsporidia in freshwater ecosystems using high-throughput sequencing (i.e. metabarcoding). However, these studies [START_REF] Williams | Group-specific environmental sequencing reveals high levels of ecological heterogeneity across the microsporidian radiation[END_REF][START_REF] Dubuffet | A phylogenetic framework to investigate the microsporidian communities through metabarcoding and its application to lake ecosystems[END_REF]Chauvet et al. 2022a) allowed the identification of a huge and new microsporidian diversity but did not allow a broad comparison of this diversity across different freshwater ecosystems.

Here, we studied this diversity in nine different freshwater sites: eight lakes and one temporary marsh (Fig. 1, Supporting Information Table S1). The freshwater microsporidian diversity was largely represented by clade IV2, accounting for over 87% of the total reads (Fig. 2a). This clade is highly diverse and is known to parasitize a wide range of hosts such as insects, nematodes, crustaceans and fish, as well as humans, and its members have been identified in terrestrial, freshwater, and marine environments [START_REF] Dubuffet | A phylogenetic framework to investigate the microsporidian communities through metabarcoding and its application to lake ecosystems[END_REF]. Euplotespora binucleata and Globosporidium paramecii [START_REF] Fokin | Euplotespora binucleata n. gen., n. sp. (Protozoa: Microsporidia), a parasite infecting the hypotrichous ciliate Euplotes woodruffi, with observations on microsporidian infections in ciliophora[END_REF][START_REF] Yakovleva | The first case of microsporidiosis in Paramecium[END_REF], which are among the few microsporidian species known (SSU sequences available) to parasitize protists (i.e. ciliates), are also found in this clade. Clades IV1 and I were the next most abundant representing 6% and 5% of the OTUs, respectively. The first one mainly includes species described as insects' parasites in terrestrial environments [START_REF] Park | Revisiting the phylogeny of microsporidia[END_REF], including also human-infecting species. Clade I, on the other hand, is commonly described as a freshwater clade, since it is mainly found in freshwater crustaceans and insects in various environments such as lakes, rivers, ponds, etc. [START_REF] Andreadis | Ultrastructural characterization and comparative phylogenetic analysis of new microsporidia from Siberian mosquitoes: Evidence for coevolution and host switching[END_REF][START_REF] Vossbrinck | Phylogeny of the Microsporidia[END_REF]. The other clades were less than 1% abundant (Fig. 2a). These findings partly overlap those from [START_REF] Dubuffet | A phylogenetic framework to investigate the microsporidian communities through metabarcoding and its application to lake ecosystems[END_REF] and Chauvet et al. (2022a) in which clade IV2 and I were the most abundant. However, in these two studies clade IV1 was not well represented (~2%), whereas it was the second most abundant clade in the present dataset, largely due to its overrepresentation in the Estivadoux sample (Fig. 1,3). Among the sampled freshwater ecosystems which are mostly lakes, the Estivadoux is quite different, it is indeed a temporary marsh that can be completely dried during summer. The typical hosts of clade IV1 are mainly insects of terrestrial environments, it therefore seems consistent to retrieve this clade with a higher relative abundance in this sample.

The Arlequin group (clade IV2) was the most abundant (49% of the reads), this group being largely represented by the Crispospora genus (40% of the reads), as already found in previous studies [START_REF] Dubuffet | A phylogenetic framework to investigate the microsporidian communities through metabarcoding and its application to lake ecosystems[END_REF]Chauvet et al. 2022a) (Fig. 1). It's noteworthy that the only one OTU shared by the nine freshwater samples was affiliated to the genus Crispospora and was actually the most abundant OTU in the whole dataset. In this genus only one species is described (Crispospora chironomi) and known to parasite chironomid larvae (Chironomus plumosus) in freshwater ecosystems [START_REF] Tokarev | Life cycle, ultrastructure, and molecular phylogeny of Crispospora chironomi g.n. sp.n. (Microsporidia: Terresporidia), a parasite of Chironomus plumosus L. (Diptera: Chironomidae)[END_REF]. Moreover, a recent study suggests that a new Crispospora species might be able to parasitize Kellicottia rotifer in lake environments (Chauvet et al. 2022b). Also in agreement with previous studies, Pseudoberwaldia and Conglomerata (Aquatic outgroup; clade I) were the next most abundant genera (Fig. 2a). These two genera are, each, described by only one species, both of which are known to be parasites of cladocerans in freshwater [START_REF] Vávra | Molecular and structural assessment of microsporidia infecting daphnids: The "obtusa-like" microsporidia, a branch of the monophyletic Agglomeratidae clade, with the establishment of a new genus Conglomerata[END_REF][START_REF] Vávra | Establishment of a new microsporidian genus and species, Pseudoberwaldia daphniae (Microsporidia, Opisthosporidia), a common parasite of the Daphnia longispina complex in Europe[END_REF]. Some groups and/or genera were recorded with a significant relative abundance only in two or three samples as the Orthosomella-like group (clade IV2), mainly found in samples from lakes Chambon, Saint-Gervais and Aydat. This group is so far only described as parasite of insects and nematodes in terrestrial environments [START_REF] Nishikori | A new microsporidium Percutemincola moriokae gen. nov., sp. nov. from Oscheius tipulae : A novel model of microsporidia-nematode associations[END_REF]. Conglomerata (Aquatic outgroup; clade I) was mainly found in lakes Chambon, Pavin and Chauvet, and Pseudoberwaldia (Aquatic outgroup; clade I) in lakes Bourdouze and Chambon (Fig. 3). These sites may be particularly rich in daphnia (cladoceran), which is the only described host of the latter two genera. Surprisingly, the Vittaforma genus (Arlequin group; clade IV2) has been mainly found in lake Bourdouze and not in the recreative lakes (Aydat, Chambon, Leman, Bourget), whereas it is currently described through a single species that is a human parasite [START_REF] Silveira | Vittaforma corneae N. Comb. for the Human Microsporidium Nosema corneum Shadduck, Meccoli, Davis & Font, 1990, Based on its Ultrastructure in the Liver of Experimentally Infected Athymic Mice[END_REF].

It therefore appeared that the microsporidian diversity varied across freshwater sampling sites even at a high taxonomic rank (i.e. clade level (Fig. 1,3,4)). In addition, a surprising result compared to previous studies was the significant proportion of clade IV2 OTUs not affiliated to any group (33%), especially in samples from lakes Bourget and Leman where these OTUs accounted for 83.2% and 75%, respectively (Fig. 3). These two lakes could represent an important reservoir of new microsporidian genera and species and thus potential new hosts. Altogether, these results illustrated the importance to increase the sampling effort in freshwater to unravel the full extent of the microsporidian diversity in these environments.

Extent of the marine microsporidian diversity

To date, the overall diversity of Microsporidia in marine ecosystems has received little attention. Indeed, to our best knowledge, only [START_REF] Williams | Group-specific environmental sequencing reveals high levels of ecological heterogeneity across the microsporidian radiation[END_REF] have initiated the study of the global microsporidian diversity in the marine environment, through coastal, estuary and rock pool samples, however with a low number of OTUs. Here, we studied this diversity in six different marine sites close to the coast: three bays, one estuary and two coastal ecosystems (Fig. 1, Supporting Information Table S1). The marine microsporidian diversity was largely dominated by clade IV2 and III, accounting for 49% and 40% of the total reads (Fig. 2b), the latter being a clade commonly described as 'marine', but which includes species parasite of crustaceans and fish, in both marine and freshwater environments [START_REF] Vossbrinck | Phylogeny of the Microsporidia[END_REF][START_REF] Park | Revisiting the phylogeny of microsporidia[END_REF] Poulin 2021 (clade V in these studies = clade III here)). This clade also includes a protist parasite, the species Hyperspora aquatica hyperparasite of paramyxids [START_REF] Stentiford | Hyperspora aquatica n.gn., n.sp. (Microsporidia), hyperparasitic in Marteilia cochillia (Paramyxida), is closely related to crustacean-infecting microspordian taxa[END_REF]. Clades IV1, V and I followed with 7%, 3% and 2% of the reads, respectively. Clade II was the least abundant with 0.1% of the reads and was identified only in Corsica (Fig. 2b). Due to the low number of OTUs (50) obtained by [START_REF] Williams | Group-specific environmental sequencing reveals high levels of ecological heterogeneity across the microsporidian radiation[END_REF] the comparison with our data is limited. However, we can mention that higher proportion of clade III was recorded in our samples but lower proportions of clades I, V and especially II and no OTUs belonging to clade IX was found. The latter clades (I, II, V and IX) are known to be parasites of terrestrial and/or freshwater hosts including crustaceans, insects, nematodes, annelids, bryozoans, and humans [START_REF] Vossbrinck | Phylogeny of the Microsporidia[END_REF][START_REF] Dubuffet | A phylogenetic framework to investigate the microsporidian communities through metabarcoding and its application to lake ecosystems[END_REF], it is thus not surprising to found them with a low relative abundance here. Clades IV2 and III on the other hand are the ones with the most members found in marine ecosystems [START_REF] Park | Revisiting the phylogeny of microsporidia[END_REF].

The Ameson-like group (clade III) and Arlequin group (clade IV2) were the most abundant with 38% and 35% of the total reads, respectively (Fig. 2b). Three samples were mostly represented by the Ameson-like group (Roscoff, Corsica and Arcachon), while two other samples were dominated by the Arlequin group (Brest and La Rochelle) (Fig. 3).

Three genera were particularly abundant: Crispospora (Arlequin group), Ameson and Nadelspora (Ameson-like group), accounting for 31%, 23% and 15% of the reads, respectively (Fig. 2b).

Crispospora has never been identified in hosts living in marine environments, and neither chironomids nor rotifers are abundant in these systems, thus raising the hypothesis of one or many other hosts for the Crispospora genus. Moreover, the proportion of Crispospora identified here in the two ecosystems studied, both in terms of OTUs and abundance, suggests that a significant diversity remains to be discovered within this genus where only one species is currently described. This relative abundance may also reveal significant host diversity, as ~80% of microsporidian species are specialists [START_REF] Murareanu | Generation of a Microsporidia Species Attribute Database and Analysis of the Extensive Ecological and Phenotypic Diversity of Microsporidia[END_REF].

The Ameson genus includes different species all described as parasitizing arthropods in the marine environment. The hosts described are mainly crustaceans (crab, lobster, shrimp) and one species is described as an insect parasite [START_REF] Murareanu | Generation of a Microsporidia Species Attribute Database and Analysis of the Extensive Ecological and Phenotypic Diversity of Microsporidia[END_REF]. Nadelspora is known to parasite the crab Metacarcinus magister through the only described species: Nadelspora canceri [START_REF] Olson | Nadelspora canceri N. G., N. Sp., an Unusual Microsporidian Parasite of the Dungeness Crab, Cancer Magister[END_REF].

Other genera identified in these samples were much less abundant such as Sporanauta (clade IV1), Octosporea and Pseudoberwaldia (Aquatic outgroup; clade I), each representing about 0.5% of the reads (Fig. 2b). Sporanauta and Pseudoberwaldia are described by one species each as parasites of nematodes and cladocerans, respectively (Ardila-Garcia and Fast 2012; [START_REF] Vávra | Establishment of a new microsporidian genus and species, Pseudoberwaldia daphniae (Microsporidia, Opisthosporidia), a common parasite of the Daphnia longispina complex in Europe[END_REF].

Octosporea genus, on the other hand, includes many species that are parasitic on different hosts, all of which belong to the arthropods. Octosporea species are mainly described as insect parasites (Diptera and Hemiptera) and are also found in some cladocerans (Daphnia) and amphipods (Gammarus) [START_REF] Murareanu | Generation of a Microsporidia Species Attribute Database and Analysis of the Extensive Ecological and Phenotypic Diversity of Microsporidia[END_REF].

In these marine samples, a higher proportion of the diversity was common between the samples compared to freshwater ecosystems as 402 OTUs were shared by the six marine samples; 84% of these OTUs were affiliated to Crispospora. Despite these shared OTUs, some samples could be quite different from each other considering some clades/groups/genera (Fig. 3). The sample from Brest, for example, had only one OTU belonging to clade III. This was surprising since this was the second most abundant clade in our study and the clade in which the highest number of marine Microsporidia are found.

Microsporidian community composition in aquatic ecosystems in relation to their hosts

Although marine samples contained less reads than freshwater samples, all diversity index showed a more diverse community in marine sites (Supporting Information Fig. S3,S4). The marine sample from La Rochelle for example was the one with the highest number of OTUs (1 157) but the fewest reads (71 122). 63.4% of the OTUs were common to both ecosystem types (the majority belonging to the Crispospora genus, or being unaffiliated beyond clade IV2), 16.1% and 20.5% were specific to marine and freshwater ecosystems, respectively. Considering only OTUs richness, ecosystem types were indistinguishable, whereas differences were identified when considering the relative abundance of OTUs (Fig. 4). Most of the OTUs (82%) found only in freshwater samples belong to the clade IV2, whereas OTUs specific to marine samples were mostly found in clade III (72%) (Supporting Information Fig. S2b,S2D). Only one OTU (the most abundant of the whole dataset), affiliated to Crispospora (clade IV2; Arlequin group), was shared by all the samples. Some OTUs were shared by different samples (Supporting Information Fig. S5), but all the 15 samples showed specific OTUs (on average 1% of the OTUs). The affiliation was better in marine samples, as 58.4% of OTUs were affiliated to a group and 38.9% to a genus in freshwater versus 62.0% and 53.6% in marine samples suggesting that a large part of the diversity is unknown. It therefore seems essential to continue the sampling effort across aquatic ecosystems to unravel the global microsporidian diversity (without the bias of targeting hosts).

At the clade level, the major differences were the absence of clade VIII in the marine samples and the very low relative abundance (0.003%) of clade III in freshwater whereas it represents 40% of the microsporidian abundance in marine samples (Fig. 2a,2b). Species belonging to clade VIII are described as parasites of insects (mainly Diptera), affecting larvae that are present in freshwater [START_REF] Murareanu | Generation of a Microsporidia Species Attribute Database and Analysis of the Extensive Ecological and Phenotypic Diversity of Microsporidia[END_REF]). Clade III is mostly described as parasitizing fishes and crustaceans in marine and freshwater environments and is sometimes described as the 'marinosporidia' clade [START_REF] Vossbrinck | Phylogeny of the Microsporidia[END_REF]) even if [START_REF] Williams | Group-specific environmental sequencing reveals high levels of ecological heterogeneity across the microsporidian radiation[END_REF] showed the inconsistency in the classification based on the ecological niche that Microsporidia and/or their hosts inhabit. However, the primers used in this study target a wide microsporidian diversity, except in clade III (data not shown). The environmental diversity of Microsporidia affiliated to clade III is therefore likely underestimated in the two types of ecosystems.

Other differences at the genus level (apart from the genera belonging to clade III) were notable between the two types of environments such as the significantly higher relative abundance of Conglomerata (Aquatic outgroup; clade I) in freshwater (Supporting Information Fig. S6) and the absence of Octosporea (Aquatic outgroup; clade I) as well as Mrazekia and Helmichia (both in Arlequin group; clade IV2) in freshwater and marine samples, respectively (Fig 2a,2b,3). Some of these results are consistent with the type of host described for these genera, such as Conglomerata which is currently described as parasitizing only the freshwater Daphnia pulex [START_REF] Vávra | Molecular and structural assessment of microsporidia infecting daphnids: The "obtusa-like" microsporidia, a branch of the monophyletic Agglomeratidae clade, with the establishment of a new genus Conglomerata[END_REF]).

These differences nevertheless involve genera found at low relative abundance, not allowing us to clearly establish their real absence in the second ecosystem and to draw conclusions on these differences.

The environmental variables studied here (salinity and temperature), trophic states, ecosystem type, size fraction, and geographical location, do not seem to explain the distribution of the microsporidian communities (Fig. 4, Supporting Information Table S1). Although the number of parameters considered here are low, it has also never been shown that abiotic parameter can directly influence Microsporidia in their environment. The diversity observed was rather in agreement with the hosts described so far, at least in terms of habitat, even though some genera have been surprisingly identified in an environment that does not harbor their (often single) described hosts. Ultimately, it appears that microsporidian communities are probably driven by the dynamic of their hosts and not directly influenced by environmental variables. Most of the Microsporidia identified in this study are so far known to parasite large hosts such as fish, crustaceans, and insects. It therefore appears that the Microsporidia found in our study can infect smaller and so far, unknown hosts and/or that their freeliving forms, the spores, actively circulate in the water column via ingestion/filtration by planktonic organisms of different sizes. Approximately 80% of the microsporidian species described (with host data) are specialists (i.e. described as infecting only a single host), the remaining species are generalists with some capable of infecting more than ten hosts [START_REF] Murareanu | Generation of a Microsporidia Species Attribute Database and Analysis of the Extensive Ecological and Phenotypic Diversity of Microsporidia[END_REF], some having the ability to infect both vertebrate and invertebrate [START_REF] Stentiford | Microsporidia -Emergent Pathogens in the Global Food Chain[END_REF]. The hypothesis of a second host is already suggested for Pseudoberwaldia [START_REF] Vávra | Establishment of a new microsporidian genus and species, Pseudoberwaldia daphniae (Microsporidia, Opisthosporidia), a common parasite of the Daphnia longispina complex in Europe[END_REF]. Our results thus also suggest that a significant proportion of Microsporidia could infect more than one host, especially in the overlooked microeukaryotic compartment.

We studied the diversity of Microsporidia and planktonic eukaryotes (< 150 μm) in parallel (Fig. 5).

The eukaryotic communities identified were quite different between the two types of ecosystems studied (Fig. 5, Supporting Information Fig. S7). The freshwater diversity was mainly represented by dinoflagellates (16%), Chlorophyta (11%), ciliates (9%) and rotifers (5%), whereas marine diversity was mainly composed of arthropods (29%), dinoflagellates (14%), Chlorophyta (10%) and Bryozoa (6%) (Fig. 5).

As for Microsporidia, some sites showed a specificity in their eukaryotic diversity that could explain the distribution of the microsporidian communities. Lakes Leman and Bourget, for example, were particularly rich in rotifers compared to other lakes (Fig. 5), these rotifers belonging mainly to the Brachionidae and Notommatidae. Similarly, dinoflagellates were most abundant in lake Saint Gervais and Bryozoa in the Brest sampling point. To identify new host-parasite associations, OTUs representing more than 0.05% of reads in both datasets were used to reveal co-occurrences. Only positive associations were considered as it has already been showed that parasitism is retrieved as a positive link between the partners in co-occurrence network despite the detrimental impact of the parasite on the host [START_REF] Weiss | Correlation detection strategies in microbial data sets vary widely in sensitivity and precision[END_REF]. Moreover, potential associations were searched by ecosystem types since it is better to have homogeneous environments to identify biotic interactions [START_REF] Faust | Open challenges for microbial network construction and analysis[END_REF].

In freshwater, 205 potential associations were identified between 32 microsporidian OTUs and 14 eukaryotic OTUs, mostly involving metazoans (platyhelminths, gastrotrichs and arthropods) and alveolates (dinoflagellates, apicomplexans, and ciliates) (Fig. 6a). These correlations largely involved OTUs unaffiliated beyond the clade (IV1, IV2 and I) or the group (Arlequin and Aquatic outgroup), as well as two genera: Cystosporogenes and Crispospora (both in the Arlequin group; clade IV2). In marine environments, 128 potential associations were identified between 69 microsporidian OTUs and 35 eukaryotic OTUs (Fig. 6b). As well as for freshwater, a lot of potential associations involved Microsporidia OTUs unaffiliated beyond the clade (V, IV2, IV1, II and III) or the group (Arlequin, Orthosomella-like, Enterocytozoonidae, Cucumispora-like and Aquatic outgroup), and some genera: Crispospora, Ameson, Obruspora, Nadelspora, Vittaforma, Sporanauta and Octosporea. Eukaryotic OTUs involved belong mostly to the alveolates (dinoflagellates, apicomplexans, and ciliates).

Metazoans (cnidarians, arthropods, annelids, mollusks, and Bryozoa) and Fungi (Ascomycota, Basidiomycota, chytrids) were also involved in these co-occurrences (Fig. 6b). Metazoans represent most of the described microsporidian host. Among them, arthropods, platyhelminths, gastrotrichs, cnidarians, annelids, mollusks, and Bryozoa (identified here in the co-occurrences) have already been described as hosts [START_REF] Murareanu | Generation of a Microsporidia Species Attribute Database and Analysis of the Extensive Ecological and Phenotypic Diversity of Microsporidia[END_REF]). Similarly, the potential associations identified with alveolates were previously described for some, as Microsporidia have been shown to parasitize apicomplexans and ciliates although these descriptions are anecdotal and generally only morphological (e.g., only two microsporidian SSU sequences parasites of ciliates [START_REF] Fokin | Euplotespora binucleata n. gen., n. sp. (Protozoa: Microsporidia), a parasite infecting the hypotrichous ciliate Euplotes woodruffi, with observations on microsporidian infections in ciliophora[END_REF][START_REF] Yakovleva | The first case of microsporidiosis in Paramecium[END_REF]). In contrast, there are currently no descriptions of Microsporidia parasitizing dinoflagellates. Fungi, also retrieved in these co-occurrences in potential association with Microsporidia, have also never been described as their hosts. In both ecosystems, some correlations were also found with phytoplanktonic organisms (Fig 6a,6b), whereas to date, infection by Microsporidia in phytoplankton has never been recorded. Interestingly, some of the potential associations identified here were also retrieved in co-occurrence networks involving short-branch Microsporidia [START_REF] Doliwa | Identifying Potential Hosts of Short-Branch Microsporidia[END_REF].

Although co-occurrences network analyses should be interpreted with caution [START_REF] Carr | Use and abuse of correlation analyses in microbial ecology[END_REF][START_REF] Blanchet | Co-occurrence is not evidence of ecological interactions E. Jeffers[END_REF]) they are increasingly used to identify potential associations within microorganisms. The method used here has its limitations, as other tools are more efficient and adapted to such datasets (e.g., [START_REF] Kurtz | Sparse and Compositionally Robust Inference of Microbial Ecological Networks C. Von Mering[END_REF] but often require more samples. However, this correlation-based method can show comparable efficiency to other more sophisticated methods [START_REF] Hirano | Difficulty in inferring microbial community structure based on co-occurrence network approaches[END_REF][START_REF] Faust | Open challenges for microbial network construction and analysis[END_REF]. However, associations between Microsporidia and eukaryotes suggested by co-occurrences network should be considered as working hypothesis and need to be confirmed by other approaches allowing the observation and the identification of both partners (e.g., fluorescence microscopy or host/parasite genomic analysis).

Using TSA-FISH with a universal microsporidian probe (USP01, Chauvet et al. 2022a) we confirmed the Microsporidia-Kellicottia (rotifer) association already observed in another lake (Chauvet et al. 2022a,b), here in lake Chauvet. We also identified microsporidian infections in dinoflagellates (probably belonging the genus Ceratium) both in freshwater and marine ecosystems (Fig. 7, Supporting Information Fig. S8). Indeed, these infections were observed in lakes Leman (prevalence, 49% of infected individuals), Bourget (53%) and Saint Gervais (20%), and in the La Rochelle (38%) marine sample. These observed Microsporidia-dinoflagellates associations in both ecosystems significantly expand the host range of Microsporidia and their impact on food web functioning. Indeed, dinoflagellates are relatively abundant in freshwater and marine environments, and show a diversity of trophic modes (autotrophy, heterotrophy or mixotrophy). Thanks to this diversity, dinoflagellates are among the major primary producers (with diatoms), they can as well be grazers, and have symbiotic or parasitic relationships, while others are known to be important toxin producers [START_REF] Carty | Dinoflagellates[END_REF].

Conclusions

In conclusion, this study highlighted for the first time a huge microsporidian diversity in marine samples, with an important part (46%) of the OTUs not affiliated to a genus suggesting an important reservoir of new microsporidian species and thus potential new hosts, especially among clade IV2. This diversity was also widely divergent from the diversity identified in freshwater. Interestingly, most of the identified genera are described as parasites of organisms much larger than the studied size fraction (<150 μm). It therefore appears that these Microsporidia infect smaller and, so far, unknown hosts such as dinoflagellates both in marine and freshwater ecosystems. These findings also greatly expand the host range of Microsporidia which were never described in association with phytoplanktonic hosts. Also, the proportion of the Crispospora genus raises questions about their hosts and impact especially in marine environments and should be the subject of further research.

Altogether, these results indicate that the microsporidian diversity and host spectrum could be therefore more important than previously described, and their role in the functioning of aquatic ecosystems and especially marine ecosystems is undoubtedly largely unknown.

Figure legends

  Community richness and diversity indices (Richness observed, Chao1, Shannon and Simpson) were used to infer the taxa richness of Microsporidia. These estimators were computed with the phyloseq package (version 1.30.0, McMurdie and Holmes 2013). Principal coordinates analysis (PCoA) ordination based on a Bray-Curtis dissimilarity matrix (Bray and Curtis 1957; Paliy and Shankar 2016) was computed using ape (version 5.6-2, Paradis and Schliep 2019) and vegan (version 2.5-7, Oksanen et al. 2020) packages. An Upset plot was done to show the sets and intersections of OTUs between samples with the UpSetR package (version 1.4.0, Conway et al. 2017).

Fig. 1 .

 1 Fig. 1. Sample collection sites. Samples were collected in France from freshwater lakes (green stars):

Fig. 3 .

 3 Fig. 3. Relative abundances of the most abundant (>0.1% of the total reads) microsporidian genera

Fig. 4 .

 4 Fig. 4. PCoA based on Bray-Curtis distances of the microsporidian OTUs richness (left) and relative

Fig. 5 .

 5 Fig. 5. Relative abundances of the most abundant (>1% of the total reads) eukaryotic phyla across

Fig. 6 .

 6 Fig. 6. Diagrams showing the positive correlations between microsporidian OTUs (bottom) and

Fig. 1

 1 Fig. 1

  Fig.3

  Fig. 4

Polymerase Chain Reaction. Journal of Infectious Diseases 168: 1570-1575. doi:10.1093/infdis/168.6.1570

Acknowledgements

The authors are very grateful towards the LIENSs, EPOC and CARRTEL labs for their hospitality and help with the sampling. We especially want to thank Hélène Agogué, Philippe Pineau, Yves Brizard, Yolanda Del Amo, Laurent Letord, Anne Bohi, Isabelle Domaizon, Jean-Christophe Hustache and the French Oceanographic Fleet. We also gratefully acknowledge Raffaele Siano (Ifremer), Fabienne Rigaut-Jalabert (Station Biologique de Roscoff) and the captain and crew of the Neomysis research ship for the sampling in Brest and Roscoff. We also thank Simon Lafarge for facilitating the access of the private lake Chauvet. We thank the SOMLIT program (CNRS/INSU, a monitoring network of the Research Infrastructure ILICO), ROME project and OLA observatory (OLA-IS, AnaEE-France, INRAE of Thonon-les-Bains, CIPEL, Rimet et al. 2020) for collecting samples and granting access to some metadata. Finally, we would like to thank Jonathan Colombet, Marie Diogon, Maxime Fuster, Karine Massé, Jean-Christophe Pointud and Agnès Vellet for their help with sampling and filtration. The work of MC was supported by a PhD fellowship funded by the 'Ministère de l'Enseignement Supérieur, de la Recherche et de l'Innovation'.

Data availability statement

All data and material are available in this publication and additional information online (Supporting Information). The data have been deposited with links to BioProject accession number PRJNA922474 in the NCBI BioProject database.

Competing Interests

The authors declare they have no competing interests.

Author contributions

MC and CL designed the study. MC, AM, and CL performed sampling and experiments. MC and CL wrote the manuscript. All authors have read and approved the manuscript.