Denis Efimov

Emilia Fridman

Converse Lyapunov-Krasovskii theorem for ISS of neutral systems in Sobolev spaces

Keywords: Neutral time-delay systems, Lyapunov-Krasovskii functional, Stabilty

The conditions of existence of a Lyapunov-Krasovskii functional (LKF) for nonlinear input-to-state stable (ISS) neutral type systems are proposed. The system under consideration depends nonlinearly on the delayed state and the delayed state derivative, and satisfies the conditions for the existence and uniqueness of the solutions. The LKF and the system properties are defined in a Sobolev space of absolutely continuous functions with bounded derivatives.

Introduction

Stability analysis for dynamical systems constitutes an important area of research in different domains of science and technology, and especially in the control theory [START_REF] Khalil | Nonlinear Control[END_REF]. For generic dynamical systems, the key tool to study stability is the Lyapunov function method, which for timedelay systems have two extensions based on Lyapunov-Razumkhin functions and Lyapunov-Krasovskii functionals [START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF][START_REF] Hale | Theory of Functional Differential Equations[END_REF][START_REF] Khalil | Nonlinear Control[END_REF][START_REF] Kolmanovskii | Introduction to the Theory and Applications of Functional Differential Equations[END_REF][START_REF] Kolmanovsky | Stability of functional differential equations[END_REF]. These approaches have also their development for systems with inputs, where one of the most popular concepts is ISS [START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF]. It is a well-known fact that existence of a Lyapunov function is necessary and sufficient for asymptotic stability and ISS in the case of ordinary differential equations [START_REF] Khalil | Nonlinear Control[END_REF]. For time-delay systems, the conditions of such an equivalence appeared rather recently (for instance, check [START_REF] Huang | Generalization of Liapunov's theorem in a linear delay system[END_REF][START_REF] Kharitonov | Lyapunov-Krasovskii approach for robust stability of time delay systems[END_REF] for asymptotic stability, and [START_REF] Karafyllis | Stability and Stabilization of Nonlinear Systems[END_REF][START_REF] Karafyllis | Recent Results on Nonlinear Time Delayed Systems, volume 4 of Advances in Delays and Dynamics[END_REF][START_REF] Lin | Lyapunov descriptions of integral-input-to-state-stability for systems with delays[END_REF][START_REF] Lin | Remarks on regularity properties of Lyapunov-Krasovskii functionals[END_REF][START_REF] Mironchenko | Lyapunov characterization of input-to-state stability for semilinear control systems over banach spaces[END_REF][START_REF] Pepe | Converse Lyapunov-Krasovskii theorems for systems described by neutral functional differential equations in Hale's form[END_REF] for ISS case). For the class of nonlinear neutral type time-delay systems, the papers [START_REF] Pepe | Converse Lyapunov-Krasovskii theorems for systems described by neutral functional differential equations in Hale's form[END_REF][START_REF] Pepe | Lyapunov-Krasovskii characterization of the input-to-state stability for neutral systems in Hale's form[END_REF] develop the equivalent conditions of asymptotic stability and ISS in terms of existence of LKF. In these works, the systems in the Hale's form have been analyzed, and an implicit expression of LKF has been proposed. A two steps procedure from [START_REF] Karafyllis | Stability and Stabilization of Nonlinear Systems[END_REF] is used in [START_REF] Pepe | Converse Lyapunov-Krasovskii theorems for systems described by neutral functional differential equations in Hale's form[END_REF][START_REF] Pepe | Lyapunov-Krasovskii characterization of the input-to-state stability for neutral systems in Hale's form[END_REF], which at the last iteration includes an infinite summation over a partition of unity, then the form of dependence of LKF on the state function is complicated and not intuitive. Studying uniform asymptotic stability these results have been extended in [START_REF] Efimov | A note on converse Lyapunov-Krasovskii theorems for nonlinear neutral systems in Sobolev spaces[END_REF] to a different class of neutral time-delay systems (not in the Hale's form) with a coercive LKF given in an explicit form. Differently from [START_REF] Karafyllis | Stability and Stabilization of Nonlinear Systems[END_REF][START_REF] Pepe | Converse Lyapunov-Krasovskii theorems for systems described by neutral functional differential equations in Hale's form[END_REF][START_REF] Pepe | Lyapunov-Krasovskii characterization of the input-to-state stability for neutral systems in Hale's form[END_REF], where the stability has been analyzed in the space W 1,+∞ [-τ,0] (the Sobolev space of continuous functions with essentially bounded derivatives), while also frequently the space W 1,2 [-τ,0] is used [START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF] (the state derivative is square integrable), in [START_REF] Efimov | A note on converse Lyapunov-Krasovskii theorems for nonlinear neutral systems in Sobolev spaces[END_REF] the mathematical treatment has been performed in W 1,1 [-τ,0] (the Sobolev space of continuous functions with integrable derivatives). The technical advantage of such a change consists in the established Lipschitz continuity of the solutions of the neutral type systems with locally Lipschitz continuous righthand side in

W 1,1 [-τ,0] (in W 1,+∞ [-τ
,0] such a property has been proven for the systems in the Hale's form only).

The main contribution of the present work consists in the formulation of equivalent conditions of ISS for neutral type nonlinear time-delay systems with essentially bounded inputs in W 1,1 [-τ,0] . As in [START_REF] Mironchenko | Lyapunov characterization of input-to-state stability for semilinear control systems over banach spaces[END_REF][START_REF] Pepe | Lyapunov-Krasovskii characterization of the input-to-state stability for neutral systems in Hale's form[END_REF], two types of the conditions are given: in terms of existence of an LKF, and in terms of uniform global asymptotic stability of an auxiliary system. A conference version of the paper restricted to the case of asymptotic stability was presented in [START_REF] Efimov | A note on converse Lyapunov-Krasovskii theorems for nonlinear neutral systems in Sobolev spaces[END_REF].

The outline of this paper is as follows. Some preliminary results are introduced in Section 2. The problem statement is given in Section 3. The main result is established in Section 4.

Preliminaries

Denote by R and N the sets of real and natural numbers, respectively, and R + = {s ∈ R : s ≥ 0}.

Definitions of norms and spaces

Denote by

C n [a,b] , [a, b] ⊂ R the Banach space of continuous functions φ : [a, b] → R n with the uniform norm φ [a,b] = sup s∈[a,b] |φ(s)|, where | • | is the standard Euclidean norm in R n . For a Lebesgue measurable function of time d : [a, b] → R m , [a, b] ⊂ R, define the norm d [a,b] = ess sup s∈[a,b] |d(s)|, then d ∞ = ess sup s≥0 |d(s)| and the space of d with d [a,b] < +∞ ( d ∞ < +∞) we further denote as L m [a,b] (L m ∞ ). Denote by W 1,p [a,b] , p ∈ N and W 1,∞ [a,b] the Sobolev spaces of absolutely continuous functions φ : [a, b] → R n , [a, b] ⊂ R, with bounded derivatives having the respective norms φ W 1,p [a,b] = φ [a,b] + b a | φ(s)| p ds 1 p < +∞ and φ W 1,∞ [a,b] = φ [a,b] + φ [a,b] < +∞, where φ(s) = ∂φ(s) ∂s (it is a Lebesgue measurable essentially bounded function for φ ∈ W 1,∞ [a,b] , i.e. φ ∈ L n [a,b] ) 1 .
Lemma 1 [START_REF] Efimov | A note on converse Lyapunov-Krasovskii theorems for nonlinear neutral systems in Sobolev spaces[END_REF] For any φ ∈ W 1,∞ [a,b] and p ∈ N ∪ {+∞} the following inequalities are satisfied:

min{1, (b -a) 1 p -1 } φ W 1,1 [a,b] ≤ φ W 1,p [a,b] ≤ max{1, (b -a) 1 p } φ W 1,∞ [a,b]
.

PROOF.

From the norm definition we deduce:

φ W 1,p [a,b] = φ [a,b] + b a | φ(s)| p ds 1 p ≤ φ [a,b] + b a φ p [a,b] ds 1 p ≤ max{1, (b -a) 1 p } φ W 1,∞ [a,b]
,

1 In [START_REF] Fridman | On input-to-state stability of systems with timedelay: A matrix inequalities approach[END_REF][START_REF] Kolmanovsky | Stability of functional differential equations[END_REF] the norm with p = 2 has been only used for the state space of time-delay systems.

and using integral Jensen's inequality [START_REF] Khalil | Nonlinear Control[END_REF]:

φ W 1,p [a,b] = φ [a,b] + b a | φ(s)| p ds 1 p ≥ φ [a,b] + (b -a) 1 p -1 b a | φ(s)|ds ≥ min{1, (b -a) 1 p -1 } φ W 1,1 [a,b]
that was necessary to prove.

Comparison functions and their properties

A continuous function σ : R + → R + belongs to class K if it is strictly increasing and σ(0) = 0; it belongs to class K ∞ if it is also radially unbounded.

Lemma 2 [START_REF] Karafyllis | Stability and Stabilization of Nonlinear Systems[END_REF] For any α ∈ K there exists a continuous function γ : R + → R + admitting the following properties: γ(0) = 0, γ(s) > 0 for all s > 0, and

γ(s) ≤ α(s), |γ(s) -γ(s )| ≤ |s -s | ∀s, s ∈ R + .
In addition,

lim s→+∞ γ(s) = +∞ provided that α ∈ K ∞ . A continuous function β : R + × R + → R + belongs to class KL if β(•, r) ∈ K and β(r, •
) is a strictly decreasing to zero for any fixed r ∈ R + .

Lemma 3 [START_REF] Sontag | Comments on integral variants of ISS[END_REF] For any β ∈ KL there exist θ 1 , θ 2 ∈ K ∞ such that β(s, t) ≤ θ 1 θ 2 (s)e -t ∀s ≥ 0, t ≥ 0.

Neutral systems under consideration

Consider an autonomous functional differential equation of the neutral type with inputs [START_REF] Kolmanovsky | Stability of functional differential equations[END_REF]:

ẋ(t) = f (x t , ẋt , d(t)), t ≥ 0 (1)
where

x(t) ∈ R n and x t ∈ C n [-τ,0] is the state function, x t (s) = x(t + s), -τ ≤ s ≤ 0, with ẋt ∈ L n [-τ,0] ; d(t) ∈ R m is the external input, d ∈ L m ∞ . The function f : C n [-τ,0] × L n [-τ,0] × R m → R n is
continuous and Lipschitz in the second variable with a constant smaller than 1, ensuring forward uniqueness and existence of the system solutions at least locally in time [START_REF] Kolmanovsky | Stability of functional differential equations[END_REF]. We assume f (0, 0, 0) = 0. For the initial function x 0 ∈ C n [-τ,0] and disturbance d ∈ L m ∞ denote a unique solution of the system (1) by x(t, x 0 , d), which is an absolutely continuous function defined on some time interval [-τ, T ) for T > 0, then x t (x 0 , d) ∈ C n [-τ,0] represents the corresponding state function, and x t (s, x 0 , d) = x(t + s, x 0 , d) for all -τ ≤ s ≤ 0.

Given a continuous functional

V : R + × C n [-τ,0] × L n [-τ,0] → R + define: D + V (t, φ, φ, d) = lim sup h→0 + 1 h [V (t + h, x h (φ, d), ẋh (φ, d)) -V (t, φ, φ)], where x h (φ, d) is a solution of the system (1) for φ ∈ C n [-τ,0] , φ ∈ L n [-τ,0] and d ∈ R m is a constant.

Uniform stability

Denote D = {d ∈ L m ∞ : ||d|| ∞ ≤ 1} and D = {d ∈ R m : |d| ≤ 1} in order
to analyze the behavior of (1) with a bounded input d. Then in the following we summarize the well-known results [START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF][START_REF] Kolmanovskii | Introduction to the Theory and Applications of Functional Differential Equations[END_REF] on stability of (1).

Definition 1 [START_REF] Fridman | On input-to-state stability of systems with timedelay: A matrix inequalities approach[END_REF][START_REF] Lin | Remarks on regularity properties of Lyapunov-Krasovskii functionals[END_REF][START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF] The system (1) is called uniformly globally asymptotically stable (uGAS), if for all

x 0 ∈ W 1,1 [-τ,0] and d ∈ D there exists β ∈ KL such that x t (x 0 , d) W 1,1 [-τ,0] ≤ β( x 0 W 1,1 [-τ,0] , t) ∀t ≥ 0. Instead of W 1,1 [-τ,0] any other space W 1,p [-τ,0] can be used in this definition for p ∈ N ∪ {+∞}.
As it is usually assumed [START_REF] Fridman | On input-to-state stability of systems with timedelay: A matrix inequalities approach[END_REF][START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF], in this case

|x(t, x 0 , d)| ≤ β( x 0 W 1,1 [-τ,0] , t) ∀t ≥ 0.
Recall that in [START_REF] El | Introduction to the theory and application of differential equations with deviating arguments[END_REF], under the assumption that f is Lipschitz in the second variable φ with a constant smaller than 1, it is established that the latter estimate is equivalent to the stability in

W 1,∞ [-τ,0] . Definition 2 A continuous functional V : R + × C n [-τ,0] × L n [-τ,0] → R + is called simple if D + V (t, φ, φ, d) is indepen- dent on φ.
For instance, a locally Lipschitz functional

V : C n [-τ,0] → R + is simple. Definition 3 [6, 16, 18] A continuous functional V : R + × C n [-τ,0] × L n [-τ,0] → R + is called LKF for the system (1) if it is simple, there exist p ∈ N ∪ {+∞}, α 1 , α 2 ∈ K ∞ and α ∈ K such that V is Lipschitz continuous on bounded sets in W 1,p [-τ,0] \ {0}, and for all t ∈ R + , d ∈ D and φ ∈ W 1,p [-τ,0] : α 1 ( φ W 1,p [-τ,0] ) ≤ V (t, φ, φ) ≤ α 2 ( φ W 1,p [-τ,0] ), D + V (t, φ, φ, d) ≤ -α(V (t, φ, φ)).

Note that existence of such a LKF implies that for all

t ∈ R + , d ∈ D and φ ∈ W 1,p [-τ,0] : α 1 (|φ(0)|) ≤ V (t, φ, φ) ≤ α 2 ( φ W 1,p [-τ,0]
),

D + V (t, φ, φ, d) ≤ -α(|φ(0)|),
where α(s) = α(α 1 (s)) is a function from class K, which is the standard LKF formulation used to establish asymptotic stability [START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF].

Theorem 1 [START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF][START_REF] Fridman | On input-to-state stability of systems with timedelay: A matrix inequalities approach[END_REF] If there exists a LKF for the system (1), then it is uGAS.

There exist also some converse results to Theorem 1, see, e.g., [START_REF] Pepe | Converse Lyapunov-Krasovskii theorems for systems described by neutral functional differential equations in Hale's form[END_REF], which are obtained for V : C n [-τ,0] → R + and a special class of f in the Hale's form, and [START_REF] Karafyllis | Stability and Stabilization of Nonlinear Systems[END_REF] for the background framework. Here we will use the following counterpart of Theorem 1 given in [START_REF] Efimov | A note on converse Lyapunov-Krasovskii theorems for nonlinear neutral systems in Sobolev spaces[END_REF] (since in the sequel we have to develop some steps of the proof from [START_REF] Efimov | A note on converse Lyapunov-Krasovskii theorems for nonlinear neutral systems in Sobolev spaces[END_REF], it is presented in the Appendix):

Theorem 2 Let the system (1) be uGAS in the sense of Definition 1 and

f : C n [-τ,0] × L n [-τ,0] × R m → R n be uniformly Lipschitz continuous on bounded sets in W 1,1 [-τ,0]
( i.e., for any closed and bounded subset

Υ ⊂ W 1,1 [-τ,0] there exists L Υ > 0 such that |f (φ, φ, d) -f (ϕ, φ, d)| ≤ L Υ φ -ϕ W 1,1 [-τ,0]
for all φ, ϕ ∈ Υ and d ∈ D). Then there exists a LKF for the system (1) with p = 1.

The expression for a LKF proposed in the proof of Theorem 2, see (4) in the Appendix, has an explicit form (if only negative definiteness of D + V (x 0 , ẋ0 , d) is required), and it is more simple than in [START_REF] Karafyllis | Stability and Stabilization of Nonlinear Systems[END_REF][START_REF] Pepe | Converse Lyapunov-Krasovskii theorems for systems described by neutral functional differential equations in Hale's form[END_REF][START_REF] Pepe | Lyapunov-Krasovskii characterization of the input-to-state stability for neutral systems in Hale's form[END_REF], where a two step procedure for construction of LKF has been proposed.

Robust stability

The ISS property is an extension of the conventional stability paradigm to the systems with external inputs [START_REF] Fridman | On input-to-state stability of systems with timedelay: A matrix inequalities approach[END_REF][START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF][START_REF] Teel | Connections between Razumikhintype theorems and the ISS nonlinear small gain theorem[END_REF].

Definition 4 [START_REF] Fridman | On input-to-state stability of systems with timedelay: A matrix inequalities approach[END_REF][START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF] The system (1) is called practical ISS, if for all

x 0 ∈ W 1,1 [-τ,0] and d ∈ L m ∞ there exist q ≥ 0, β ∈ KL and γ ∈ K such that for all t ≥ 0 x t (x 0 , d) W 1,1 [-τ,0] ≤ β( x 0 W 1,1 [-τ,0] , t) + γ(||d|| [0,t) ) + q.
If q = 0 then (1) is called ISS.

Definition 5

The system (1) is said to possess the practical asymptotic gain (AG) property, if for all

x 0 ∈ W 1,1 [-τ,0]
and d ∈ L m ∞ there exist q ≥ 0 and γ ∈ K such that lim

t→+∞ x t (x 0 , d) W 1,1 [-τ,0] ≤ γ(||d|| ∞ ) + q.
If q = 0 then (1) admits AG property.

Definition 6

The system (1) is said to have the practical global stability (GS) property, if for all

x 0 ∈ W 1,1 [-τ,0] and d ∈ L m ∞ there exist q ≥ 0 and σ 1 , σ 2 ∈ K such that for all t ≥ 0 x t (x 0 , d) W 1,1 [-τ,0] ≤ max{σ 1 ( x 0 W 1,1 [-τ,0]
), σ 2 (||d|| ∞ )} + q.

If q = 0 then (1) admits GS property.

Again, instead of W 1,1 [-τ,0] any other space W 1,p [-τ,0] can be used in these definitions for p ∈ N ∪ {+∞}.

As it follows from the definitions above, a (practical) ISS system has (practical) AG and (practical) GS properties, and for a system in (1) described by an ordinary differential equation (i.e., with τ = 0) the converse implication also holds [START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF].

As it has been observed in [START_REF] Fridman | On input-to-state stability of systems with timedelay: A matrix inequalities approach[END_REF][START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF], a sufficient characterization of ISS property can be introduced for (1):

Definition 7 A continuous functional V : R + × C n [-τ,0] × L n [-τ,0] → R + is called practical ISS LKF for the system (1) if it is simple and there exist p ∈ N∪{+∞}, r ≥ 0, α 1 , α 2 ∈ K ∞ and α, χ ∈ K such that V is Lipschitz continuous on bounded sets in W 1,p [-τ,0] \{0}, and for all t ∈ R + , φ ∈ W 1,p [-τ,0]
and d ∈ R m :

α1( φ W 1,p [-τ,0] ) ≤ V (t, φ, φ) ≤ α2( φ W 1,p [-τ,0]
),

V (t, φ, φ) ≥ max{r, χ(|d|)} =⇒ D + V (t, φ, φ, d) ≤ -α(V (t, φ, φ)).
If r = 0 then V is an ISS LKF.

Theorem 3 [START_REF] Fridman | On input-to-state stability of systems with timedelay: A matrix inequalities approach[END_REF] If there exists a (practical) ISS LKF for the system (1), then it is (practical) ISS with the AG γ

= α -1 1 • χ.
There exist also some converse results to Theorem 3, see e.g. [START_REF] Pepe | Lyapunov-Krasovskii characterization of the input-to-state stability for neutral systems in Hale's form[END_REF], which are also obtained for V : C n [-τ,0] → R + and neutral systems in Hale's form.

Problem statement

The goal of this work is to propose a converse of Theorem 3 for x t ∈ W 1,1 [-τ,0] and for another class of f than in [START_REF] Pepe | Converse Lyapunov-Krasovskii theorems for systems described by neutral functional differential equations in Hale's form[END_REF][START_REF] Pepe | Lyapunov-Krasovskii characterization of the input-to-state stability for neutral systems in Hale's form[END_REF]. In particular, as in Theorem 2 the following hypothesis is accepted in the sequel: Assumption 1 The system (1) is ISS in the sense of Definition 4, and

f : C n [-τ,0] × L n [-τ,0] × R m → R n is Lipschitz continuous on bounded sets in W 1,1 [-τ,0] × R m : for any closed and bounded subsets Υ ⊂ W 1,1 [-τ,0] and D ⊂ R m there exists L Υ,D > 0 such that |f (φ, φ, d) -f (ϕ, φ, δ)| ≤ L Υ,D ( φ -ϕ W 1,1 [-τ,0] + |d -δ|) for all φ, ϕ ∈ Υ and d, δ ∈ D.
Here the boundedness of a subset is understood as boundedness of the least upper bound of the norm for difference of any two elements of the subset.

An example of such a system is given by dynamics with distributed delays in the state derivatives and general delays in the state:

f (φ, φ, d) = F (φ, 0 -τ k(s) φ(s)ds, d)
with a Lipschitz continuous on bounded sets function F : C n [-τ,0] × R 2n → R n and an essentially bounded kernel k : [-τ, 0] → R. An important example covered by Assumption 1 includes retarded system with a pointwise delay:

ẋ(t) = F (x(t), x(t -τ ), d(t)), then f (φ, φ, d) = F (φ(0), φ(0) - 0 -τ φ(s)ds, d)
and it has been observed previously [START_REF] Fridman | New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems[END_REF][START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF] that a LKF in W 1,2 [-τ,0] for this kind of dynamics is usually more efficient than in C n [-τ,0] (especially for robustness analysis).

From the relations between the norms given in Lemma 1 we observe that boundedness of x t in W 1,+∞ [-τ,0] implies immediately a similar property for all other norms. Therefore, a stability analysis performed in the space W 1,1 [-τ,0] seems to be less restrictive, which is the motivation for selection of that space in this problem formulation.

Main results

In this section, first, some preliminary results are established, which clarify the features and significance of the imposed assumptions, and second, a converse result is presented.

Lipschitz continuity of solutions

Under the introduced restrictions we have the following useful property for the solutions of system (1): Proposition 1 Let Assumption 1 be satisfied. Then in (1), for any T > 0, and closed and bounded subsets

Υ ⊂ W 1,1 [-τ,0] and Θ ⊂ L m ∞ , there exists M T,Υ,Θ > 0 such that x t (φ, d) -x t (ϕ, δ) W 1,1 [-τ,0] ≤ M T,Υ,Θ ( φ -ϕ W 1,1 [-τ,0] +||d -δ|| [0,T ) ) ∀t ∈ [0, T ]
for all φ, ϕ ∈ Υ and d, δ ∈ Θ.

PROOF. According to Assumption 1 there are β ∈ KL and γ ∈ K such that (consider also q ≥ 0)

x t (x 0 , d) W 1,1 [-τ,0] ≤ β( x 0 W 1,1 [-τ,0] , t) + γ(||d|| ∞ ) + q for all t ≥ 0, for all x 0 ∈ W 1,1 [-τ,0] and d ∈ L m ∞ .
For any Υ and Θ as above, φ, ϕ ∈ Υ and d, δ ∈ Θ, there exists ρ > 0

such that φ W 1,1 [-τ,0] ≤ ρ, ϕ W 1,1 [-τ,0] ≤ ρ, ||d|| ∞ ≤ ρ and ||δ|| ∞ ≤ ρ. Hence, we have max{ x t (φ, d) W 1,1 [-τ,0] , x t (ϕ, δ) W 1,1 [-τ,0] } ≤ β(ρ, 0) +γ(ρ) + q and |f (x t (φ, d), ẋt (φ, d), d(t)) -f (x t (ϕ, δ), ẋt (ϕ, δ), δ(t))| ≤ L ρ ( x t (φ, d) -x t (ϕ, δ) W 1,1 [-τ,0] + |d(t) -δ(t)|)
for almost all t ≥ 0, where L ρ > 0 represents the Lipschitz constant of f on the set {φ ∈

W 1,1 [-τ,0] : φ W 1,1 [-τ,0] ≤ β(ρ, 0) + γ(ρ) + q} and {d ∈ L m ∞ : ||d|| ∞ ≤ ρ} (it exists due to Assumption 1). For any φ ∈ W 1,1 [-τ,0] and d ∈ L m ∞
we have:

x(t, φ, d) = x(0, φ, d) + t 0 f (x s (φ, d), ẋs (φ, d), d(s))ds
for all t ≥ 0, and for φ, ϕ ∈ Υ and d, δ ∈ Θ:

|x(t, φ, d) -x(t, ϕ, δ)| ≤ |x(0, φ, d) -x(0, ϕ, δ)| + t 0 |f (xs(φ, d), ẋs(φ, d), d(s)) -f (xs(ϕ, δ), ẋs(ϕ, δ), δ(s))|ds ≤ |x(0, φ, d) -x(0, ϕ, δ)| +Lρ t 0 xs(φ, d) -xs(ϕ, δ) W 1,1 [-τ,0] + |d(s) -δ(s)|ds.
Therefore, for t ≥ 0:

xt(φ, d) -xt(ϕ, δ) W 1,1 [-τ,0] = sup -τ ≤s≤0 |x(t + s, φ, d) -x(t + s, ϕ, δ)| + 0 -τ | ẋ(t + σ, φ, d) -ẋ(t + σ, ϕ, δ)|dσ ≤ φ -ϕ W 1,1 [-τ,0] + t 0 |f (xσ(φ, d), ẋσ(φ, d), d(σ)) -f (xσ(ϕ, δ), ẋσ(ϕ, δ), δ(σ))|dσ + 0 -τ | ẋ(t + σ, φ, d) -ẋ(t + σ, ϕ, δ)|dσ ≤ 2 φ -ϕ W 1,1 [-τ,0] +2 t 0 |f (xσ(φ, d), ẋσ(φ, d), d(σ)) -f (xσ(ϕ, δ), ẋσ(ϕ, δ), δ(σ))|dσ ≤ 2 φ -ϕ W 1,1 [-τ,0] +2Lρ t 0 xσ(φ, d) -xσ(ϕ, δ) W 1,1 [-τ,0] + |d(σ) -δ(σ)|dσ ≤ 2 φ -ϕ W 1,1 [-τ,0] + 2Lρt||d -δ|| [0,t) +2Lρ t 0 xσ(φ, d) -xσ(ϕ, δ) W 1,1 [-τ,0]
dσ.

Next, using Gronwall's Lemma [START_REF] Khalil | Nonlinear Control[END_REF] we obtain for all t ∈ [0, T ]:

x t (φ, d) -x t (ϕ, d) W 1,1 [-τ,0] ≤ 2( φ -ϕ W 1,1 [-τ,0] +L ρ T ||d -δ|| [0,T ) )e 2Lρt .
Consequently, take any T > 0 then

x t (φ, d) -x t (ϕ, d) W 1,1 [-τ,0] ≤ M T,Υ,Θ ( φ -ϕ W 1,1 [-τ,0] +||d -δ|| [0,T ) ) ∀t ∈ [0, T ] for M T,Υ,Θ = 2 max{1, L ρ T }e 2LρT .
Thus, under Assumption 1, for any bounded set of initial conditions and inputs, for a compact interval of time, for the solutions of the system (1) there is a kind of local Lipschitz property with respect to the initial conditions and inputs. It is worth to highlight that the result is substantiated in the space W 1,1 [-τ,0] , and a similar conclusion in

W 1,∞ [-τ,0]
was obtained for the system (1) in the Hale's form [START_REF] Pepe | Converse Lyapunov-Krasovskii theorems for systems described by neutral functional differential equations in Hale's form[END_REF][START_REF] Pepe | Lyapunov-Krasovskii characterization of the input-to-state stability for neutral systems in Hale's form[END_REF].

Remark 1 In addition, due to Lemma 1, for any closed and bounded subset

Υ ⊂ W 1,1 [-τ,0] : |f (φ, φ, d) -f (ϕ, φ, δ)| ≤ L Υ,D,p ( φ -ϕ W 1,p [-τ,0] + |d -δ|) for all φ, ϕ ∈ W 1,p [-τ,0] and d, δ ∈ D such that φ W 1,p [-τ,0] ≤ min{1, τ 1 p -1 } and ϕ W 1,p [-τ,0] ≤ min{1, τ 1 p -1 } , where L Υ,D,p = max{1, τ 1-1 p }L Υ,D with L Υ,D
given in Assumption 1 and > 0 is the corresponding norm bound on Υ, i.e. the Lipschitz continuity of f in W 1,p [-τ,0] also follows.

A uniformly GAS system

Further consider an auxiliary system:

ż(t) = f (z t , żt , γ -1 ( 1 2 z t W 1,1 [-τ,0] )δ(t)), t ≥ 0, (2)
where z(t) ∈ R n and z t ∈ C n [-τ,0] is the state as before,

z 0 ∈ W 1,1 [-τ,0]
, γ is the AG function given for (1) in Definition 5 (which exists for (1) under Assumption 1, and we can always assume that γ ∈ K ∞ , then its inverse is well defined), and δ ∈ D is a uniformly bounded input. Obviously, the origin is an equilibrium of (2) since f (0, 0, 0) = 0 in (1). In addition, for any z 0 ∈ W 1,1 [-τ,0] and δ ∈ D the solution z t (z 0 , δ) of ( 2) is defined on some interval of time [0, T max z0 ) and it coincides with the solution x t (z 0 , d) of (1) for

d(t) = γ -1 ( 1 2 z t (z 0 , δ) W 1,1 [-τ,0]
)δ(t) on this interval of time. Let us demonstrate a more strong relation between solutions of the systems (1) and (2) (as in [START_REF] Sontag | New characterizations of input-to-state stability[END_REF], see also propositions 4.3 and 4.4 of [START_REF] Karafyllis | Stability and Stabilization of Nonlinear Systems[END_REF]):

Proposition 2
The following relations are true:

i) If the system (1) is ISS with the AG γ, then the system (2) is uGAS.

ii) If the system (2) is uGAS with the decay β ∈ KL, then the system (1) is ISS with the AG β(2γ(•), 0).

where

z(t) ∈ R n , z t ∈ C n [-τ,0] , δ ∈ D and F (z t , żt , δ(t)) = f (z t , żt , ψ( 1 2 z t W 1,1 [-τ,0]
)δ(t)), is uGAS. Indeed,

θ3( sup s∈[t 0 ,t) ψ( 1 2 zs W 1,1 [-τ,0]
)) ≤ θ3( sup

s∈[t 0 ,t) θ -1 3 ( 1 2 zs W 1,1 [-τ,0] )) = 1 2 sup s∈[t 0 ,t) zs W 1,1 [-τ,0]
as for (2) (and all other arguments of Proposition (2) stay unchanged). Now, take any closed and bounded subset Υ ⊂ W 1,1 [-τ,0] , according to Assumption 1 there exists L Υ,D > 0 such that

|F (φ, φ, δ) -F (ϕ, φ, δ)| = |f (φ, φ, ψ( 1 2 φ W 1,1 [-τ,0] )δ) -f (ϕ, φ, ψ( 1 2 ϕ W 1,1 [-τ,0] )δ)| ≤ L Υ,D ( φ -ϕ W 1,1 [-τ,0] + |ψ( 1 2 φ W 1,1 [-τ,0] )δ -ψ( 1 2 ϕ W 1,1 [-τ,0]
)δ|)

≤ L Υ,D ( φ -ϕ W 1,1 [-τ,0] + |ψ( 1 2 φ W 1,1 [-τ,0] ) -ψ( 1 2 ϕ W 1,1 [-τ,0] )|) ≤ L Υ,D ( φ -ϕ W 1,1 [-τ,0] + 1 2 | φ W 1,1 [-τ,0] -ϕ W 1,1 [-τ,0] |) ≤ 3 2 L Υ,D φ -ϕ W 1,1 [-τ,0]
for all φ, ϕ ∈ Υ and δ ∈ D, where the Lipschitz properties of ψ and

• W 1,1 [-τ,0]
were used on the last two steps. Therefore, F is uniformly Lipschitz continuous on bounded sets in W 1,1 [-τ,0] , then all conditions of Theorem 2 are satisfied and the system (3) admits a LKF V , which is Lipschitz continuous on bounded sets in W 1,1 [-τ,0] \ {0}, and for all δ ∈ D and φ ∈ W 1,1 [-τ,0] :

α 1 ( φ W 1,1 [-τ,0] ) ≤ V (φ, φ) ≤ α 2 ( φ W 1,1 [-τ,0]
),

D + V (φ, φ, δ) ≤ -α(V (φ, φ)) for some α 1 , α 2 , α ∈ K ∞ . Returning to the system (1), for any φ ∈ W 1,1 [-τ,0] and d ∈ R m , let ψ( 1 2 φ W 1,1 [-τ,0] ) ≥ |d|, where ψ(s) = s 1+s inf σ≥s ψ(σ) is a function from class K ∞ , then d = ψ( 1 2 φ W 1,1 [-τ,0] )δ = ψ( 1 2 φ W 1,1 [-τ,0]
)δ for some δ , δ ∈ D, hence, the following implication holds in (1):

V (φ, φ) ≥ χ(|d|) ⇒ φ W 1,1 [-τ,0] ≥ 2ψ -1 (|d|) ⇒ D + V (φ, φ, d) ≤ -α(V (φ, φ)),
where χ(s) = α 2 (2ψ -1 (s)), which implies that V is an ISS LKF for [START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF].

The obtained result shows that the existence of a coercive ISS LKF is necessary and sufficient for ISS of the system (1) in W 1,1 [-τ,0] provided that the Lipschitz continuity property stated in the Assumption 1 is verified. The proposed conditions establish a direct relation between a LKF for the auxiliary system (3) and the required ISS LKF of (1).

Conclusions

The problem of existence of an ISS LKF for nonlinear neutral type time-delay systems is solved considering the ISS property in W 1,1 [-τ,0] space. It is shown that Lipschitz property of f defined in such a space can be transformed to the same property of the solutions of (1), and relations between ISS and uGAS properties of (1) and an auxiliary system (2) are established.

From the decreasing of e -t κ1+t κ2+t it also follows that there exists T x0 > 0 such that

V (x 0 , ẋ0 ) = sup 0≤t≤T x 0 ,d∈D {ς( x t (x 0 , d) W 1,1 [-τ,0] ) κ 1 + t κ 2 + t },
and since

ς( x 0 W 1,1 [-τ,0] ) κ 1 κ 2 ≤ V (x 0 , ẋ0 ) ≤ sup 0≤t≤T x 0 ,d∈D {θ 2 ( x 0 W 1,1 [-τ,0]
)e -t }, then by the definition of T x0 , it has an upper estimate:

T x0 ≤ ln   κ 2 κ 1 θ 2 ( x 0 W 1,1 [-τ,0] ) ς( x 0 W 1,1 [-τ,0] )   . For any 0 < r < R < +∞ define a set Ω r,R = {x 0 ∈ W 1,1 [-τ,0] : r ≤ x 0 W 1,1 [-τ,0] ≤ R}, consequently, there exists a finite T r,R = sup x0∈Ω r,R T x0 , i.e. T r,R = ln κ2 κ1 θ2(R) ς(r) .
Let us check the Lipschitz continuity of V on any bounded and closed subset in W 1,1 [-τ,0] \ {0} (here by {0} we understand

x ∈ W 1,1 [-τ,0] with x W 1,1 [-τ,0] = 0). For any x 0 , x 1 ∈ W 1,1 [-τ,0] \{0} denote T x0,x1 = max{T x0 , T x1 } and consider: |V (x 1 , ẋ1 ) -V (x 0 , ẋ0 )| = | sup 0≤t≤T x 1 ,d∈D {ς( x t (x 1 , d) W 1,1 [-τ,0] ) κ 1 + t κ 2 + t } - sup 0≤t≤T x 0 ,d∈D {ς( x t (x 0 , d) W 1,1 [-τ,0] ) κ 1 + t κ 2 + t }| ≤ sup 0≤t≤T x 0 ,x 1 ,d∈D [ς( x t (x 1 , d) W 1,1 [-τ,0] ) -ς( x t (x 0 , d) W 1,1 [-τ,0] )] κ 1 + t κ 2 + t ≤ sup 0≤t≤T x 0 ,x 1 ,d∈D x t (x 1 , d) W 1,1 [-τ,0] -x t (x 0 , d) W 1,1 [-τ,0] ≤ sup 0≤t≤T x 0 ,x 1 ,d∈D x t (x 1 , d) -x t (x 0 , d) W 1,1 [-τ,0]
,

where on the last step and the step before the Lipschitz properties of the norm

• W 1,∞ [-τ,0]
and the function ς have been utilized, respectively. For any x 0 , x 1 ∈ W 1,1 [-τ,0] \ {0} there exist 0 < r < R < +∞ such that x 0 , x 1 ∈ Ω r,R , then T x0,x1 ≤ T r,R and using Proposition 1 in [START_REF] Efimov | A note on converse Lyapunov-Krasovskii theorems for nonlinear neutral systems in Sobolev spaces[END_REF] (see also Proposition 1) there exists M T r,R ,Ω r,R > 0 such that

x t (x 1 , d) -x t (x 0 , d) W 1,1 [-τ,0] ≤ M T r,R ,Ω r,R x 1 -x 0 W 1,1 [-τ,0] for all t ∈ [0, T r,R ] and all d ∈ D, hence, |V (x 1 , ẋ1 ) -V (x 0 , ẋ0 )| ≤ M T r,R ,Ω r,R x 1 -x 0 W 1,1 [-τ,0]
for all x 0 , x 1 ∈ Ω r,R and any 0 < r < R < +∞, which implies the required Lipschitz continuity of V on bounded sets in W 1,1 [-τ,0] \ {0}. The continuity at the origin follows from the upper estimate:

V (x 0 , ẋ0 ) ≤ κ 1 κ 2 θ 2 ( x 0 W 1,1 [-τ,0] ) that is satisfied for all x 0 ∈ W 1,1 [-τ,0] .
Finally, let us check the decreasing of the LKF V on the trajectories of the system (1) for t > 0: V (x t (x 0 , d), ẋt (x 0 , d))

= sup ) κ 1 + σ κ 2 + σ } = V (x 0 , ẋ0 ), thus, V is strictly decreasing along the trajectories of (1) for all x 0 ∈ W 1,1 [-τ,0] and any d ∈ D. Moreover (this part is absent in [START_REF] Efimov | A note on converse Lyapunov-Krasovskii theorems for nonlinear neutral systems in Sobolev spaces[END_REF]),

V (x t (x 0 , d), ẋt (x 0 , d)) -V (x 0 , ẋ0 ) = sup σ∈[0,T x t (x 0 ,d) ],δ∈D {ς( x σ (x t (x 0 , d), δ) W )

× κ 1 + t + σ κ 2 + t + σ + κ 1 + σ κ 2 + σ - κ 1 + t + σ κ 2 + t + σ } -sup σ≥0,δ∈D {ς( x σ (x 0 , δ) W 1,1 [-τ,0] ) κ 1 + σ κ 2 + σ } ≤ sup
σ∈[0,T x t (x 0 ,d) ],δ∈D {ς( x t+σ (x 0 , δ) W

)

× κ 1 + σ κ 2 + σ - κ 1 + t + σ κ 2 + t + σ },
and applying the Mean Value Theorem we obtain: V (x t (x 0 , d), ẋt (x 0 , d)) -V (x 0 , ẋ0 ) ≤ -sup σ∈[0,T x t (x 0 ,d) ],δ∈D {ς( x t+σ (x 0 , δ) W 1,1

[-τ,0]

)t × κ 2 -κ 1 (κ 2 + σ + δt) 2 } for some δt ∈ (0, t). Finally, )

κ 2 -κ 1 (κ 2 + σ) 2 } ≤ - κ 2 -κ 1 (κ 2 + T x 0 ) 2 ς( x 0 W 1,1 [-τ,0]
) for any x 0 ∈ W 1,1 [-τ,0] and any d ∈ D, which is a negative definite function of V , but probably globally bounded. Now defining a new LKF Ṽ (x 0 , ẋ0 ) = V (x0, ẋ0) 0 ρ(s)ds for a suitably defined function ρ ∈ K ∞ it is possible to obtain another LKF with a negative definite and properly unbounded derivative D + Ṽ (x 0 , ẋ0 , d) for any x 0 ∈ W 1,1 [-τ,0] and any d ∈ D, as desired.

1 [ 1 [) κ 1 + 1 [ 1 [

 11111 σ≥0,δ∈D {ς( x σ (x t (x 0 , d), δ) W 1,{ς( x σ (x 0 , δ) W 1,σ -t κ 2 + σ -t } < sup σ≥t,δ∈D {ς( x σ (x 0 , δ) W 1,{ς( x σ (x 0 , δ) W 1,-τ,0]

D 1 t) κ 2 -κ 1 (κ 2 +

 112 + V (x 0 , ẋ0 , d) = lim sup t→0 + [V (x t (x 0 , d), ẋt (x 0 , d)) -V (x 0 , ẋ0 )] ≤ -lim sup t→0 + sup σ∈[0,T x t (x 0 ,d) ],δ∈D {ς( x t+σ (x 0 , δ) σ + δt) 2 } = -sup σ∈[0,T x 0 ],δ∈D{ς( xσ (x 0 , δ)

Preprint submitted to Automatica

This work was partially supported by the Government of Russian Federation (Grant 08-08), by the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898, by Joint Research Project HetCPS: Ministry of Science & Technology of Israel, CNRS and by Israel Science Foundation (grant No 673/19).

PROOF. i) If the system (1) is ISS, then according to Definition 4

, t -t 0 ) + γ(||d|| [t0,t) ), for all t ≥ t 0 , for some β ∈ KL, γ ∈ K ∞ and all

)δ(t) in the system (2) we get [START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF], for the latter with z t = z t (z t0 , δ) and any z t0 ∈ W 1,1 [-τ,0] , δ ∈ D we obtain:

.

Take t 0 = 0, then from the last inequality 1 2 sup

, 0) for any t ≥ 0, all z 0 ∈ W 1,1 [-τ,0] and all δ ∈ D, which implies global uniform boundedness of the solutions of the system (2). Therefore, T max zt 0 = +∞ for all z 0 ∈ W 1,1 [-τ,0] and all δ ∈ D. Next, select t 0 = t 2 :

sup

, which for t → +∞ (i.e., considering the asymptotic behavior of ( 2)) results in

that can be satisfied for lim

= 0 only. Consequently, the system (2) is uniformly converging to the origin, the uGAS property has been established.

ii) Conversely, let (2) be uGAS, then according to Definition 1:

for some β ∈ KL and all z 0 ∈ W 1,1 [-τ,0] , δ ∈ D, t 0 ≥ 0. Consider the system (1), take any

with some (possibly infinite)

)δ(t) for all t ∈ [0, t 1 ) and, hence, using the properties of (2) we obtain:

) for t ∈ [t 1 , t 2 ) with some t 2 > t 1 (obviously, always t 1 + t 2 > 0 for any

with some t 3 > t 2 , similarly

Next, all these steps can be repeated iteratively if necessary (they cover all possible scenarios), and it is clear that for all t ≥ 0

since by definition β(s, 0) ≥ s for all s ≥ 0, which implies ISS property of (1).

Converse design of LKF

For brevity of exposition the analysis is presented in the space W 1,1 [-τ,0] (we need the result of Proposition 1 formulated in this space) and for ISS case only.

Theorem 4 Let Assumption 1 be satisfied, then there exists an ISS LKF for the system (1).

PROOF. Under the introduced hypothesis and Lemma 3 there are

Recalling Lemma 2, there exists a continuous, positive definite and radially unbounded function ψ : R + → R + admitting the following properties:

Following the arguments of Proposition (2), the system (that is a variant of (2))

Appendix

Under the introduced hypothesis in Theorem 2, and Lemma 3, there are θ 1 , θ 2 ∈ K ∞ such that

)e -t ∀t ≥ 0 for all x 0 ∈ W 1,1 [-τ,0] and d ∈ D, then by recalling Lemma 2, there exists a continuous, positive definite and radially unbounded function ς : R + → R + admitting the following properties:

) and

) κ 1 κ 2 since under introduced restrictions on κ 1 and κ 2 the functions κ1+t

κ2+t and e -t κ1+t κ2+t are strictly increasing and decreasing, respectively. Define ς(s) = s 1+s inf σ≥s ς(σ), which is a function from class K ∞ , then

) for all x 0 ∈ W 1,1 [-τ,0] , and the LKF V (x 0 , ẋ0 ) is coercive admitting lower and upper bounds in terms of functions from the class K ∞ with respect to x 0 W 1,1 [-τ,0] as desired.


Christian Kerbiriou

Yves Bas

Isabelle Le Viol

Romain Lorrillière

Justine Mougnot

Jean François Julien

Romain Lorrilliere

Jean François

Potentiality of the bat pass duration measure for studies dealing with bat activity

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Increasingly threatened worldwide [START_REF] Mickleburgh | A review of the global conservation status of bats[END_REF], bats have the potential to be important biodiversity indicator species, as they are distributed globally and are long-lived species, and it has been suggested that their population trends reflect those of lower trophic level species [START_REF] Jones | Carpenoctem: the importance of bats as bioindicators[END_REF]. They are considered indicators of the response of biodiversity to anthropogenic pressure [START_REF] Jones | Carpenoctem: the importance of bats as bioindicators[END_REF].

We have witnessed an exponential increase in the knowledge in the acoustic identification of bat species [START_REF] Russo | Identification of twenty-two bat species (Mammalia: Chiroptera) from Italy by analysis of time-expanded recordings of echolocation calls[END_REF][START_REF] Obrist | Variability in echolocation call design of 26 Swiss bat species: consequences, limits and options for automated field identification with a synergetic pattern recognition approach[END_REF][START_REF] Barataud | Acoustic Ecology of European Bats, Species identification, study of their habitat and foraging behavior. Biotope, Mèze[END_REF]. In addition to this taxonomic knowledge, the cost of ultrasonic recorders has reduced and has resulted in the development of passive acoustic sensors that are able to record throughout the night. Acoustic detectors have been used to increase inventory completeness in bat assemblage studies (Mac Swiney et al 2008), and these methods have been proposed to form part of a single standardized monitoring protocol [START_REF] Stahlschmidt | Bats as bioindicators -the need of a standardized method for acoustic bat activity surveys[END_REF]. In addition, contrary to capture methods, radio tracking methods, roost surveys, and the use of ultrasonic detectors, all non-intrusive methods, are often the only logistically feasible methods [START_REF] Stahlschmidt | Bats as bioindicators -the need of a standardized method for acoustic bat activity surveys[END_REF]. Thus, since the two last decades, these approaches were widely used by researchers working for environmental consulting firms or government agencies [START_REF] Adams | Do you hear what I hear? Implications of detector selection for acoustic monitoring of bats[END_REF], for example, during the evaluation of development projects.

Acoustic detectors have become increasingly used by academic researchers to investigate the differential use of habitats by bats (Sherwin et al. 2000, Russo andJones 2003) or to test various anthropogenic pressures such as the following: (i) agricultural intensification based on high levels of agrochemicals [START_REF] Wickramasinghe | Bat activity and species richness on organic and conventional farms: impact of agricultural intensification[END_REF]; (ii) non-lethal impacts of wind turbines, such as the disturbance of commuting and migration routes and local habitat loss [START_REF] Hötker | Impacts on biodiversity of exploitation of renewable energy sources: the example of birds and bats -facts, gaps in knowledge, demands for further research, and ornithological guidelines for the development of renewable energy exploitation[END_REF][START_REF] Millon | Bat activity in intensively farmed landscapes with wind turbines and offset measures[END_REF]; or (iii) artificial light at night [START_REF] Stone | Street lighting disturbs commuting bats[END_REF]Azam et al. 2015). Finally, monitoring of bat populations based on nocturnal acoustic standardized recordings are under development in an increasing number of countries, including the UK [START_REF] Russ | The Bats & Roadside Mammals Survey Bat Conservation Trust[END_REF], USA [START_REF] Herzog | Progress Update for Acoustic Monitoring of Summer Bats[END_REF], Ireland [START_REF] Roche | A car-based monitoring method reveals new information on bat populations and distributions in Ireland[END_REF], and France [START_REF] Kerbiriou | Long term trends monitoring of bats, from biodiversity indicator production to species specialization assessment[END_REF].

Within all these studies, the measure used, "bat activity," is based on the number of bat passes. The metrics used for assessing bat activity vary among studies; for example, [START_REF] Hayes | Temporal Variation in Activity of Bats and the Design of Echolocation-Monitoring Studies[END_REF] used the number of files recorded by bat detectors that include echolocation calls, while Tibbel and Kurta (2003) used the number of pulses in all files that were recorded in one night. In acoustic studies using time expansion bat detectors, a bat pass is defined as one or more bat echolocation call during a record of sound at a x10 time expansion [START_REF] Regnery | Tree microhabitats as indicators of bird and bat communities in Mediterranean forests[END_REF]. In these cases, the duration of the record is predefined by the ultra-sound detector (for example, 0.32 s for the Tranquility Transect (David Bale, Courtpan. Design Ltd, Cheltenham, UK), see [START_REF] Roche | A car-based monitoring method reveals new information on bat populations and distributions in Ireland[END_REF][START_REF] Regnery | Tree microhabitats as indicators of bird and bat communities in Mediterranean forests[END_REF][START_REF] Lacoeuilhe | The Influence of Low Intensities of Light Pollution on Bat Communities in a Semi-Natural Context[END_REF]. Other studies calculated bat activity as the number of bat passes per night, and bat pass was defined as a single or several bat calls emitted during a fixed interval (for example 5 s in [START_REF] Millon | Bat activity in intensively farmed landscapes with wind turbines and offset measures[END_REF] study). Relative feeding activity is sometimes measured using a 'buzz ratio' [START_REF] Rowse | The Switch from Low-Pressure Sodium to Light Emitting Diodes Does Not Affect Bat Activity at Street Lights[END_REF], which is the proportion of call sequences that included 'feeding buzzes' (buzzes are signals emitted by bats just before they try to catch insects, or when they drink or before landing). However, the ratio of feeding buzzes in a sample is regularly very low (~1% of whole contacts in the data set of the French bat-monitoring programme).

Curiously, at the time of writing, it appears that the measure of bat pass duration, (i.e., each event (expressed in seconds) of a bat detected within the detection volume of the ultrasonic detector, Fig. 1) has never been used; we expected this measure may contain information about site quality in terms of potentialities for foraging. Because bats are expected to have a more sinuous trajectory and slower velocity when they exhibit foraging behaviour, as opposed to commuting behaviour, between their roost and foraging areas or between two patch of foraging areas (Fig. 2), we thus hypothesize the following: (i) that sites or habitats with high levels of food resources may be characterized by sequences with greater bat pass duration on average than sites with low foraging resources; (ii) that bat pass duration may be lower in the first minutes after sunset when bats commute between their roost to their foraging areas; and (iii) that the pass duration may vary, with a longer pass duration on average during the spring when important energetic demands are imposed by pregnancy [START_REF] Anthony | Night roosting and the nocturnal time budget of the little brown bat, Myotis lucifugus: effects of reproductive status, preys density, and environmental conditions[END_REF][START_REF] Ruedi | Variations de la frequentation de gîtes nocturnes par Myotis daubentoni pendant la période de reproduction. Rôle des précipitations et de la température[END_REF][START_REF] Swift | Activity patterns of Pipistrelle bats (Pipistrellus pipistrellus) in North-east Scotland[END_REF] or in the autumn period when bats accumulate fat before hibernation.

However, acoustic recordings (here the bat pass duration) depend not only on the specificity of the transmitter (here the bats) but also on the receiver (here the detector) and the medium (here the air). For example, the speed of sound is influenced by the density of the medium and temperature. However, temperature and humidity could also indirectly influence the behaviour of bats due to their impact on feeding prey (O' Donnell 2000[START_REF] Ciechanowski | Spatiotemporal variation in activity of bat species differing in hunting tactics: effects of weather, moonlight, food abundance, and structural clutter[END_REF]). In addition, acoustic recordings are expected to be influenced by technical choices such as the following: the microphone of the detector [START_REF] Waters | The influence of bat detector brand on the quantitative estimation of bat activity[END_REF]Walsh 1994, Adams et al. 2012), bat detector height [START_REF] Weller | Variation in Bat Detections due to Detector Orientation in a Forest[END_REF][START_REF] Baerwald | Geographic Variation in Activity and Fatality of Migratory Bats at Wind Energy Facilities[END_REF][START_REF] Collins | Differences in Bat Activity in Relation to Bat Detector Height: Implications for Bat Surveys at Proposed Windfarm Sites[END_REF], Jones 2009), detector orientation [START_REF] Weller | Variation in Bat Detections due to Detector Orientation in a Forest[END_REF], and distance of the signal from the detector [START_REF] Adams | Do you hear what I hear? Implications of detector selection for acoustic monitoring of bats[END_REF]. Moreover, long term acoustic monitoring involves also considering the wear of the microphones. Finally, methods to survey bat activity (mainly, line transects and stationary measurement) are expected to influence acoustic recordings [START_REF] Stahlschmidt | Bats as bioindicators -the need of a standardized method for acoustic bat activity surveys[END_REF].

With the aim to test if bat pass duration was longer in favourable habitats, during seasons with high energetic constraints or during night pic activity, we mobilized two independent data sets (i.e., two protocols' recordings along line transects or stationary recordings) from the French bat-monitoring programme, a large scale acoustic bat survey that has involved numerous volunteers since 2006. These data sets offer the opportunity to simultaneously test multiple factors, such as habitat, season, hour in the night, weather condition, detector characteristics, and ageing of the detector, that are expected to influence acoustic recordings and more particularly, the bat pass duration of various species of bats. In addition, the protocol of this large survey avoids extreme weather conditions (i.e., lower temperatures, strong winds, and rain). Thus, the conditions of the recordings of these data are close to the conditions often selected for comparative studies (see Azam et al. 2015[START_REF] Lacoeuilhe | The Influence of Low Intensities of Light Pollution on Bat Communities in a Semi-Natural Context[END_REF]

MATERIAL AND METHODS

Bat sampling

The data were provided by the French bat-monitoring programme (FBMP) (http://vigienature.mnhn.fr/page/vigie-chiro), a citizen-science programme running since 2006 and coordinated by the French Museum of Natural History (MNHN). The FBMP is based on standardized echolocation recordings [START_REF] Stahlschmidt | Bats as bioindicators -the need of a standardized method for acoustic bat activity surveys[END_REF], where bat calls are detected using 2 types of bat detectors (Tranquility Transect Bat detector, Courtpan Design Ltd, UK; and D240x, Pettersson Elektronik AB) and were recorded on a Zoom H2 digital recorder (Samson Technologies, USA) at a sampling rate of 96 ks/s. Sound was stored on a Secured Card in Waveform Audio File Format, more commonly known as the WAV format.

Each site is monitored twice: once during the period of 15 June to 31 July, when females are expected to give birth and feed their offspring; and second, during the period 15 August to dependent on their reproductive roost. The observers begin their sampling at thirty minutes after sunset, from season to season and year to year. Thus, this sampling occurred during the bat activity peak that begins 30 minutes after sunset and spans less than 3 hours [START_REF] Roche | Development of a Car-Based Bat Monitoring Protocol for the Republic of Ireland[END_REF]. Observers sampled bats only when weather conditions were considered as favourable, i.e., no rain, temperature higher than 12°C, 30 minutes after sunset, and without strong winds (<20km/h). Two different versions are used (road survey by car and count point).

Car transect survey

Volunteer surveyors record bat activity while driving at a constant low-speed (25 ± 5 km/h) along a road circuit. Volunteer observers were asked to choose a road circuit of at least 30 km and located within a 10-km radius around the volunteer's residence. Selecting a circuit met two requirements: the first concerned the safety of the volunteers, because the circuit must be performed at night at low speed; and the second objective was to design a circuit that proportionally crossed, as much as possible, the different habitats present in the area.

Once the circuit was validated by the Museum, the starting plot was randomly chosen. Each circuit was then divided into 10 -2 km transects where bat were recorded, separated by 1 km road portions where recording was not carried out (See ESM 1, Fig. S1-2). Such design allowed to obtain a quite good correlation between proportion of habitat sampled and proportion of habitat existing at the national scale (R² = 0.95). Currently, the database is composed of 160 road circuits representing 1618 different transects (see ESM 1). The car transect survey is one of the three protocols used by the FBMP in order to monitor common bat population (for more details see Azam et al. 2016, Kerbiriou et al. 2018a, andESM 1).

Count point survey

The protocol consists of a 2 km × 2 km square randomly chosen (by the Museum) in a radius of 10 km from the observer's home, within which a minimum of ten points are chosen by the observer, with at least five points being representative of the habitats of the square, the others being located in 'favourable' places for bats, such as riverbanks and wood edges. The ten points of a site were sampled (continuous recording during 6 minutes/point) during the same night. Currently, data have been gathered on 120 squares representing 1272 different points (see ESM 1). The count point survey is the second protocols used by the FBMP in order to monitor common bat population (for more details see Kerbiriou et al. 2018b, and ESM 1 2.3. Biological data, species identification and the measure of bat pass duration 2.3.1. Species identification Volunteers proceeded to species acoustic identification, while final data validation was made by Museum experts. Calls of genus Myotis were pooled in a Myotis spp. group due to their very low occurrence and some identification uncertainties (in this group less than 50% of bat passes were identified at the species level: Myotis daubentonii, Myotis myotis, Myotis mystacinus, Myotis emarginatus, and Myotis nattereri). Twenty-seven experienced volunteers proceeded to the measure of bat pass duration (Fig. 1) using Syrinx software version 2.6 [START_REF] Burt | Syrinx a software for real time spectrographic recording, analysis and playback of sound[END_REF]) for spectrogram analyses with settings provided by the MNHN. Bat passes containing social calls were excluded because the detection radius of social calls is expected to be drastically different from other echolocation calls.

Variables expected to influence bat passes duration

Bat activity is known to vary among habitats [START_REF] Cramel | Habitat use by bats in a Mediterranean ecosystem in Israel, conservation implication[END_REF][START_REF] Russo | Use of foraging habitats by bats in a Mediterranean area determined by acoustic surveys: conservation implications[END_REF][START_REF] Rainho | Summer foraging habitats of bats in a Mediterranean region of the Iberian Peninsula[END_REF][START_REF] Vandevelde | Activity of European common bats along railway verges[END_REF], We hypothesize that in favourable habitats (for example, forest or wetlands), records will contain more bat passes with a hunting pattern, rather than in less favourable habitats (i.e., urbanized or intensive agricultural land), where we expect to record more bat passes with a commuting pattern. As commuting flights are faster and straighter on average than foraging flights, we hypothesized that the duration of bat passes will be greater in favourable foraging habitats. Volunteers involved in the FBMP provided a hierarchical description of the habitat surrounding the sampling point (each point count for the count point protocol and every 400 metres along the transect for the car survey protocol), from which we derived a simplified habitat classification. The Habitat type variable, used hereafter in the modelling, is a categorical variable with 8 classes: continuous urban areas (CU), discontinuous urban areas (DU), arable land (AL), heterogeneous agricultural areas including crops and pasture (HA), coniferous forest (CF), deciduous forest (DF), water courses (WC), and bodies of water (BW).

In addition, foraging behaviour also varies according to the habitat structure; for example, the Myotis group primarily included species considered to be gleaners, which capture the majority of their prey from substrates in cluttered environments [START_REF] Arlettaz | Effect of acoustic clutter on prey detection by bats[END_REF]. Their prey are mainly diurnal Brachycera Diptera and non-volant arthropods, such as weevils, Lepidoptera larvae, harvestmen and spiders [START_REF] Swift | Gleaning as a foraging strategy in Natterer's bat Myotis nattereri[END_REF][START_REF] Dietz | Bats of Britain, Europe and Northwest Africa[END_REF]). In contrast, species of the genera Pipistrellus, Eptesicus and Nyctalus are considered to be aerial hawkers, which forage mostly on flying prey in open spaces [START_REF] Dietz | Bats of Britain, Europe and Northwest Africa[END_REF]Schnitzler et al. 2003, Holderied and[START_REF] Holderied | Echolocation range and wingbeat period match in aerial-hawking bats[END_REF]. We thus built an index of clutter of the habitat (Clutter Index) derived from the hierarchical description of the habitat made by volunteers.

The Clutter Index is an explicit seven-class gradient of habitat structure, ranging from (1) open habitat (i.e., farmland open fields without any trees or bushes) to (7) cluttered habitat (i.e., forests with a dense undergrowth layer) (for more details see Table XX,ESM1). We hypothesize that longer bat pass duration will be found in the foraging environments of species, that for Myotis spp., for example, bat pass duration will be greater in cluttered environments. Note however, that sound propagation could also be biased by the degree of cluttering of a habitat [START_REF] Patriquin | The influence of habitat structure on the ability to detect ultrasound using bat detectors[END_REF], so the net effect of habitat structure on bat pass duration is not easy to predict. Second, we hypothesize that foraging behaviour could vary within the night, with smaller bat pass durations during the period of commuting between the roost and foraging areas (i.e., early night); in addition, we also hypothesized that within a season, bat pass duration could vary according to the variation in energetic demands among seasons [START_REF] Parsons | Swarming activity of temperate zone microchiropteran bats: effects of season, time of night and weather conditions[END_REF]:

we expected to find longer bat pass durations during pregnancy in June and fat accumulation in the autumn. Thus, in our analysis, we take into account Time after sunset (in minutes) and Date (Julian date), and as we expected a non-linear effect, we also included the quadratic effect of these two variables.

Third, weather conditions are known to influence bat prey, i.e., invertebrates; thus, in our analysis, we take into account temperature (°C), humidity (%), cloud cover (% in four classes: 0-25%, 25-50%, 50-75%, and 75-100%), and wind speed (km/h). Note that according to the FBMP protocol, volunteers are invited to avoid carrying out their field survey during extreme weather conditions (see ESM 2 for the distribution of these variables in the dataset).

Fourth, sound propagation is proportional to temperature, air composition (i.e., humidity, see [START_REF] Griffin | The importance of atmospheric attenuation for the echolocation of bats (Chiroptera)[END_REF] and pressure. However, in the Earth's atmosphere, the chief factor affecting sound propagation is the temperature, a variable previously selected for our analyses according to its potential impact on bat prey.

Fifth, the nature of the receiver (i.e., bat detector) is expected to influence the acoustic signal recorded: detection range differs according to the sensitivity of the detector and the directionality of the microphone (see [START_REF] Limpens | Choosing a Bat Detector: Theoretical and Practical Aspects 28[END_REF]. In addition, weatherproofing designs and microphone orientations are expected to impact the quantity and quality of the bat calls [START_REF] Britzke | Effects of orientation and weatherproofing on the detection of bat echolocation calls[END_REF]. However, the survey was carried out (car or point count), and two models of detector are used by the volunteer network: Tranquility Transect and D240x.

According to the type of microphone, electret for D240x and capacitive for the Tranquility Transect, we hypothesized that the Tranquility Transect, with a greater directionality, will record shorter durations of bat passes. In addition, to take into account ageing and wear (1 to 9 years), we included the age of the detector as a variable as well as an interaction effect between the age of detector and model of detector.

Finally, we also take into account the protocol as a source of variation in bat pass duration, hypothesizing that bat pass duration will be greater with the point count protocol. In addition, for the car transect surveys, as the receiver is moving, it also depends on its speed, so we hypothesized a negative relationship between bat pass duration and car speed. Note however, that according to the requirement of the car transect survey (25 km/h ± 5 km/h) and compliance with instructions by volunteers, the variation of the car speed in the data set is relatively small (see ESM 2).

Statistical analysis

Influence of variables on the measure of bat pass duration

For each taxa and method used to survey bat activity (i.e., car transect and point count) we performed general linear mixed models (GLMM function glmmPQL, package R: MASS) using the duration of bat passes as a response variable with a Poisson error distribution according to the long tail pattern of bat passes duration (see ESM 3). The variables potentially influencing bat pass duration, such as habitat (habitat type or clutter index), date, time after sunset, weather conditions (temperature, humidity, cloud cover, and wind), receiver (detector model, age of detector, and the car speed for the car transect survey) were used as fixed effects. We included a quadratic effect for the date and time after sunset, and we used general additive mixed models (GAMM function gamm, package R: mgcv) for visualizing these potentially non-linear effects. To account for possible spatial pseudo-replication involving several measurements taken from the same individual or environment, we included a group effect (named "date-point") for identifying bat passes recorded within the same site at the same date as a random effect. In addition to taking temporal pseudo-replication into account, involving repeated measurements from the same volunteers, we included volunteer identity as one supplementary independent random effect. The fixed effects, except detector model and habitat type (categorical variable), were standardized so that the regression coefficients were comparable in magnitude [START_REF] Schielzeth | Simple means to improve the interpretability of regression coefficients[END_REF]. Before running each GLMM, we systematically evaluated the correlations among explanatory variables using Spearman's rho for quantitative variables [START_REF] Crawley | The R book[END_REF] to detect obvious correlation (See ESM 2).

Second, we performed variance-inflation factors (VIF) on each model [START_REF] Fox | Generalized collinearity diagnostics[END_REF]; all variables had a VIF<3, indicating no problem of multicollinearity in the explanatory variables of our models. Thus, we ran 12 models, one for each bat taxa (n=6) and for each protocol (car transect or point count), structured in the following way:

[bat passes duration] ~ date + date² +time after sunset + time after sunset²+ temperature + wind speed + humidity + cloud cover+ detector model * age of detector + clutter index or habitat type + 1|volunteer +1| date-point

For the six models running on data from the car survey protocol, we added car speed as a fixed effect. According that, clutter index or habitat type are intrinsically correlated variables and could not be simultaneously included in the modelling; thus, we performed separate modelling.

RESULTS

The most common taxa (5 bats species: P. kuhlii, P. pipistrellus E. serotinus, N. leisleri, N. noctula and one genus: Myotis) accounted for 24597 bat passes (Table 1).

We detected relatively few effects of the clutter index: just a significant positive effect on bat pass duration for N. leisleri with data from the car survey (Table 2) and for E. serotinus with data from the count point survey (Table 3). Habitat type was slightly more significant, with two effects detected with data from the car survey (P. pipistrellus and E. serotinus) and three effects with data from the count point survey (P. kuhlii, E. serotinus and N. leisleri) (Table 4).

For P. pipistrellus, we detected that greater bat pass durations were recorded in bodies of water (Fig. 4). For P. kuhlii, smaller bat pass durations were recorded in urban areas. For E. serotinus, greater bat pass durations were recorded in heterogeneous agricultural areas, deciduous forests and water courses, and smaller bat pass durations in continuous urban areas, discontinuous urban areas and arable land (Fig. 4).

We detected a significant effect of time after sunset for three species (P. pipistrellus, E. serotinus and N. noctula, for the latter species, only in the car transect survey). The observed pattern is congruent with our hypothesis: smaller bat pass duration was found in the beginning of the night (Tables 2 &3 Some significant effects of the date have been detected for P. pipistrellus, P. kuhlii, E. serotinus and N. noctula and along with some species or survey type, an additional quadratic effect (Table 2 &3). For the two Pipistrellus species, the fixed effect is positive, while for E. serotinus it is negative. When looking at fixed effects, this seems to be contrasted between species, while when looking at a non-linear effect (see ESM 4.3), species exhibit a similar convex quadratic pattern, with smaller bat pass durations in late July (~200 th day) and an increase in bat pass durations after late August (~240 th day).

Weather conditions (humidity, temperature, cloud cover, and wind) rarely influence bat pass duration, with less than 10% of significant effects. An effect of the bat detector is detected for P. pipistrellus and Myotis spp., with the Tranquility Transect detector recording smaller bat pass durations than the D240x. Ageing and wear of detector are only detected for data from the car survey and for some species, but with some opposite effects (Table 2 & 3), and the interaction between bat detector type and age indicates that when significant, this effect differs among species.

DISCUSSION

When considering the influence of habitat on bat pass durations, the clutter index appears to weakly influence, while we detected more effects of the categorical habitat variables. When a significant effect of habitat is detected, as we hypothesized, greater bat pass durations were recorded in habitats considered as favourable for the species: for E. serotinus, heterogeneous agricultural areas including crops and pastures [START_REF] Catto | Foraging behaviour and habitat use of the serotine bat (Eptesicus serotinus) in southern England[END_REF][START_REF] Robinson | Home range and habitat use by serotine bat, Eptesicus serotinus[END_REF][START_REF] Arthur | Uderstanding bathabitat associations and the effects of monitoring on long-term roost success using a volunteer dataset[END_REF], deciduous forest [START_REF] Russo | Use of foraging habitats by bats in a Mediterranean area determined by acoustic surveys: conservation implications[END_REF][START_REF] Rainho | Summer foraging habitats of bats in a Mediterranean region of the Iberian Peninsula[END_REF],and water courses [START_REF] Ciechanowski | Community structure and activity of bats (Chiroptera) over different water bodies[END_REF][START_REF] Bartonicka | Flight activity and habitat use of four bat species in a small town revealed by bat detectors[END_REF][START_REF] Russo | Use of foraging habitats by bats in a Mediterranean area determined by acoustic surveys: conservation implications[END_REF][START_REF] Kanuch | Relating bat species presence to habitat features in natural forests of Slovakia (Central Europe)[END_REF][START_REF] Rainho | Summer foraging habitats of bats in a Mediterranean region of the Iberian Peninsula[END_REF][START_REF] Arthur | Uderstanding bathabitat associations and the effects of monitoring on long-term roost success using a volunteer dataset[END_REF]; or for P. pipistrellus, bodies of water [START_REF] Nicholls | Habitat selection as a mechanism of resource partitioning in two cryptic bat species Pipistrellus pipistrellus and Pipistrellus pygmaeus[END_REF]Racey 2006, Vandevelde et al. 2014). Congruently, habitats with smaller bat pass duration are, for E. serotinus, habitats that regularly identify as not selected or with lower bat activity: urban areas [START_REF] Davidson-Watts | Differential habitat selection by Pipistrellus pipistrellus and Pipistrellus pygmaeus identifies distinct conservation needs for cryptic species of echolocating bats[END_REF][START_REF] Arthur | Uderstanding bathabitat associations and the effects of monitoring on long-term roost success using a volunteer dataset[END_REF]) and arable land (Davidson-Watts et al. 2006, Nicholls and[START_REF] Nicholls | Habitat selection as a mechanism of resource partitioning in two cryptic bat species Pipistrellus pipistrellus and Pipistrellus pygmaeus[END_REF]. Surprisingly, for P. kuhlii, a species that exhibits a mid-level of bat activity in urban areas [START_REF] Vandevelde | Activity of European common bats along railway verges[END_REF] and that is considered an anthophilous species [START_REF] Tomassini | Cranial size has increased over 133 years in a common bat, Pipistrellus kuhlii: a response to changing climate or urbanization?[END_REF], we recorded a smaller bat pass duration in continuous urban areas. Among the hypotheses, for explaining that clutter index is rarely significant, we can first note that sound transmission can vary among the same type of habitat, i.e., between forest types of deciduous, coniferous, and mixed [START_REF] Patriquin | The influence of habitat structure on the ability to detect ultrasound using bat detectors[END_REF]; second, the sound attenuation effect by vegetation density may differ according to sound frequency studied [START_REF] Patriquin | The influence of habitat structure on the ability to detect ultrasound using bat detectors[END_REF]; and third, we may suspect the opposite effect of the degree of cluttering of habitat on sound propagation (i.e., attenuation) and the selection of clutter habitat by some species, such as Myotis. Another non-exclusive hypothesis could be linked to the roughness of the construction of the clutter index. If we detect a greater effect of habitat type than clutter index, note that we detect effects for only half of the species, again we can hypothesize the accuracy of the habitat classes (only 8 classes) we built that do not allow consideration, for example, of the maturity of the forest [START_REF] Regnery | Tree microhabitats as indicators of bird and bat communities in Mediterranean forests[END_REF], agricultural intensification [START_REF] Wickramasinghe | Bat activity and species richness on organic and conventional farms: impact of agricultural intensification[END_REF], or type of bank in aquatic habitats (Scott et al. 2010). In addition, the species studied are among the most generalist species [START_REF] Lacoeuilhe | Effects of hedgerows on bats and bush crickets at different spatial scales[END_REF].

As expected, we detected a significant effect of time after sunset in bat pass duration for half of the species studied. When significant, the pattern is congruent with our prediction: smaller bat pass durations are recorded in the beginning of the night due to a greater percentage of individuals leaving the colony to reach foraging sites and then exhibit commuting flight. A maximum bat pass duration is detected approximately 1 hour and a half after sunset, which matches the activity pattern of individuals outside the roost during the night, as described by [START_REF] Swift | Activity patterns of Pipistrelle bats (Pipistrellus pipistrellus) in North-east Scotland[END_REF] for P. pipistrellus or by [START_REF] Catto | Activity patterns of the serotine bat (Eptesicus serotinus) at a roost in southern England[END_REF] for E. serotinus.

As predicted, we also detected a variation in bat pass duration within the year for some species; the non-linear effect (see ESM 4.3) showed a convex quadratic pattern with smaller bat pass durations after late July (~200 th day of the year) and an increase of bat pass durations after early September (~240 th day of the year). The longer pass durations observed before the 200 th day of the year match the lactation period: the parturition of P. pipistrellus is known to occur over a ten-day period in late June (~170 th day of the year), lactation lasts 4-5 weeks, and weaning was completed when the adults left the roost approximately 2 weeks later (~210 th day of the year) [START_REF] Swift | Activity patterns of Pipistrelle bats (Pipistrellus pipistrellus) in North-east Scotland[END_REF]Racey 1981, Dietz et al. 2007). [START_REF] Catto | Activity patterns of the serotine bat (Eptesicus serotinus) at a roost in southern England[END_REF] found the lactation period for E. serotinus to be from the end of June to the third week of July (i.e., 180 to 200 th day of the year). [START_REF] Jones | Flight performance, echolocation and foraging behaviour in noctule bats Nyctalus noctula[END_REF] found a similar pattern for N. noctula, where most of the females were lactating between 2 June and 19 July. The increase of bat pass durations after the 240 th day of the year (early September) matches the seasonal weight changes observed in bats and related to fat deposition, particularly brown adipose tissue, which is a prerequisite for hibernation [START_REF] Arévalo | Seasonal variations in the lipid composition of white and brown tissues in the bat Pipistrellus pipistrellus[END_REF]).

We detected relatively few significant effects of weather conditions, but it should be noted that the requirement of the FBMP implies that records are performed only when weather conditions are considered as favourable (see Material and Methods); so, our dataset includes a limited range of weather conditions. An effect of the bat detector was detected for P. pipistrellus and Myotis spp., and as expected, the more directional detector (Tranquility Transect) recorded smaller bat pass durations. Ageing effects and the interactions between bat detector type and the age of detector are not easy to read because the ageing of the detector is partially related to volunteer experience (as volunteers have their own materials).

Note that in these two protocols (car survey transect and count point), the time spent outside by the equipment, and thus exposure to weather such as humidity, is relatively short explaining probably the non-detection of aging of detector.; In a third protocol under development in the FBMP, based on a stationary detector (SM2, SM3, SM4 Wildlife Acoustic, USA) running in the field for several weeks per year, we observed real microphone failures after one to two years of survey.

We found results that confirm the three main hypotheses that we formulated on bat pass duration: (i) longer bat pass durations within selected habitats for foraging; (ii) smaller bat pass durations in the first minutes after sunset; and (iii) longer bat pass durations during periods with important energetic demands. While we did not detect results that clearly reject our hypotheses, we did, however, not simultaneously detect these effects for each species. For P. pipistrellus, and E. serotinus to a lesser extent, our main predictions were found; note that P. pipistrellus is by far the species for which we have the largest number of measures (~8000 and ~10000 measures, respectively, for the count point survey and the car transect survey). This volume of sample appears impressive according to the recorded sample of 4-6 minutes of recording per point, manual identification and manual measure of bat pass duration; however, the recent reduced cost of acoustic recorders has resulted in an exponential development of Passive Acoustic Monitoring (PAM) involving, for example, a third protocol in the FMBP. PAM produces considerable amounts of data, while before we recorded 4-6 minutes per site in the two previous protocols, now these detectors allow recording the entire night; so, regularly, several hundred bat passes could be recorded per night, site and species (see for example [START_REF] Charbonnier | Numerical and Functional Responses of Forest Bats to a Major Insect Pest in Pine Plantations[END_REF], even in habitats with regularly low bat activity, such as agriculture land (see [START_REF] Millon | Bat activity in intensively farmed landscapes with wind turbines and offset measures[END_REF]. In addition to the arrival of a new generation of passive recorders on the market, the development of a software toolbox (Bas et al. in press) allows the detection of every sound event and extracts numerous features, so that sounds could be well characterized during the same process and used to produce species identification and characteristics of the call (buzz, social call, etc.).

Thus, in the near future, we expect that analysing bat pass durations will not be limited by the amount of data.

Our results suggest that bat pass duration is a simple and easy measure that can be used as a position indicator on a gradient between commuting vs. foraging behaviour.

However, in studies and environmental expertise investigating the differential use of habitats by bats, the traditional measure of bat activity (i.e., number of bat passes per time unit) could be, for example, weighted by bat pass duration for giving more weight to events with a potentially greater link to foraging behaviour. Our results obtained by correlative approaches could be validated via field experiments based on 4 microphone array recordings to model flight behaviour [START_REF] Bas | Using acoustic flight path reconstruction to distinguish foraging and commuting behaviour, then predicting it a posteriori from call features. 4th Berlin Bat meeting: Movement Ecology of Bats Mars[END_REF]. Such an approach will allow perform direct relationship between a flight behaviour (commuting vs foraging) and echolocation calls characteristics (including bat pass duration).

  ; Fig. 3; ESM 4.2 & ESM 4.5), and in addition, a quadric effect was detected for P. pipistrellus and E. serotinus (for the latter species, only in the car transect survey), indicating an optimum of approximately 1 hour and a half after sunset (Table 2 & 3; Fig. 3; ESM 4.2 & ESM 4.5).

Figure 1 .Figure 2 .

 12 Figure 1. Bat pass duration

Figure 3 .Figure 4 .

 34 Figure 3. Variation of bat pass duration (scaled values) within early night (expressed as minutes after sunset): (a) P. pipistrellus, and (b) E. serotinus. For GAMM modelling on data from the car transect survey. See the supplementary material for analyses from the count point survey. Rugs on the X-axis indicate values present at least once in the data set.

Table 1 .

 1 Number of bat calls recorded according to type of survey. Bold indicates the taxa studied specifically thereafter.

		Count point	Car transect
		survey	survey
	Pipistrellus kuhlii	805	981
	Pipistrellus pipistrellus	8117	11132
	Eptesicus serotinus	130	713
	Nyctalus leisleri	226	531
	Nyctalus noctula	143	414
	Myotis ssp.	1131	274

Table 2 .

 2 Averaged partial regression coefficient and p-value from GLMM of the fixed effects included in the analysis performed on data from the car transect survey. Bold indicates a Pvalue < 0.05.

		P. pipistrellus	P. kuhli	E. serotinus	N. Leisleri	N. noctula	Myotis ssp
	Clutter index	P=0.102	P=0.183	P=0.311	P=0.024	P=0.089	P=0.967
					β=0.063
	Time after sunset	P<0.001	P=0.151	P=0.001	P=0.426	P=0.198	P=0.582
		β= 0.074		β=0.111
	Time after sunset 2	P=0.006	P=0.004	P=0.002	P=0.910	P=0.702	P=0.908
		β=-0.046	β=-0.066	β=-0.061
	Date	P<0.001	P=0.775	P=0.031	P=0.281	P=0.439	P=0.071
		β=0.033		β= -0.082
	Date 2	P<0.001	P=0.652	P=0.043	P=0.226	P=0.401	P=0.572
		β= 0.081
	Temperature	P=0.599	P=0.722	P=0.001	P=0.059	P=0.179	P=0.625
				β=0.095
	Humidity	P=0.110	P=0.670	P=0.457	P=0.193	P=0.958	P=0.782
	Cloud cover	P=0.121	P=0.798	P=0.240	P=0.790	P=0.953	P=0.793
	Wind	P=0.933	P=0.163	P=0.700	P=0.837	P=0.887	P=0.477
	Bat detector	P<0.001	P=0.751	P=0.177	P=0.163	P=0.468	P=0.851
		D240x>TT
	Age	P=0.018	P=0.055	P=0.436	P=0.068	P=0.007	P=0.019
		β=0.012				β=-0.211	β=-0.026
	Bat detector : Age	P=0.004	P=0.460	P=0.023	P=0.455	P=0.039	P=0.161
		βTT=-0.366		βTT=0.225		βTT=0.285
	Car speed	P<0.001	P=0.280	P=0.857	P=0.275	P=0.051	P=0.905
		β=-0.145

Table 3 .

 3 Averaged partial regression coefficient and p-value from GLMM of the fixed effects included in the analysis performed on data from the count point survey. Bold indicates a Pvalue < 0.05.

		P. pipistrellus P. kuhli	E. serotinus	N. Leisleri	N. noctula	Myotis ssp
	Clutter index	P=0.527	P=0.890	P=0.017	P=0.574	P=0.389	P=0.245
				β=0.163
	Time after sunset	P=0.010	P=0.057	P=0.003	P=0.414	P=0.006	P=0.727
		β=0.059		β=0.391		β=-0.312
	Time after sunset 2	P<0.001	P=0.079	P=0.233	P=0.528	P=0.798	P=0.244
		β=-0.037
	Date	P=0.012	P=0.015	P=0.298	P=0.748	P=0.393	P=0.818
		β=0.052	β= 0.107
	Date 2	P=0.926	P=0.475	P=0.636	P=0.161	P=0.001	P=0.927
						β=-0.538
	Temperature	P=0.214	P=0.849	P=0.892	P=0.858	P=0.166	P=0.569
	Humidity	P=0.011	P=0.063	P=0.285	P=0.022	P=0.349	P=0.547
		β=0.054			β=-0.200
	Cloud cover	P=0.018	P=0.240	P=0.269	P=0.811	P=0.934	P=0.013
		β=-0.052
	Wind	P=0.938	P=0.910	P=0.452	P=0.066	P=0.298	P=0.032
							β=0.087
	Bat detector	P<0.001	P=0.214	P=0.479	P=0.160	P=0.597	P<0.001
		D240x>TT					D240x>TT
	Age	P=0.282	P=0.657	P=0.524	P=0.202	P=0.563	P=0.567
	Bat detector : Age	P<0.001	P=0.260	P=0.986	P=0.012	P=0.731	P=0.011
		βTT=-0.256			βTT=-0.518		βTT=-0.208

Table 4 .

 4 Effects of habitat type on bat call duration. Bold indicates P-value < 0.05; for effects of the other variable that were similar to those found with clutter index analysis, see ESM 4.5 for details.

		P. pipistrellus	P. kuhli	E. serotinus	N. Leisleri	N. noctula	Myotis ssp
	Car transect survey	P<0.0001	P=0.52	P=0.027	P=0.45	P=0.67	P=0.25
	Count point survey	P=0.81	P=0.021	P=0.044	P=0.002	P=0.46	P=0.37

ACKNOWLEDGEMENTS

We thank deeply the Vigie-Nature volunteers for data collection, especially experienced volunteers involve in bat pass duration measures or providing specific sound files for uncommon species: Quentin Amand, Avana Andriamboavonjy, Ronan Arhuro, Mathilde Baradat, Flore Cambon, Julien Cavallo, Robin Derozier, Nicolas Fillol, Gregory Fiquet, Vincent Gibaud, Pascal Guichard, Emmanuel Jacob, Robin Julien, Marine Lauere, Lara Million, Julie Maratrat, Marion Parisot-Laprun, Laura Plichard, Loic Robert, Magali Roche Loic Salaun, Jean Claude Vandevelde, Arthur Vernet, Sophie Wrobel. In addition, we deeply thank two referees for their relevant and useful comments.