Alexander Aleksandrov

Denis Efimov
email: denis.efimov@inria.fr

Sergey Dashkovskiy

On ISS-Lyapunov Functions for Mechanical Systems †

Keywords: Input-to-state stability, Lyapunov function, Mechanical systems

For two canonical models of mechanical systems with disturbances presented by Rayleigh-and Liénard-type equations, several designs of Lyapunov functions are investigated. Under given restrictions, the existence of these functions implies global or local input-to-state stability property for the systems. The proposed stability conditions are constructive, which is demonstrated in applications.

INTRODUCTION

Analysis of stability of dynamical systems is an important and popular subject of research [START_REF] Zubov | Methods of A.M. Lyapunov and Their Applications[END_REF][START_REF] Demidovich | Lectures on Stability Theory[END_REF][START_REF] Rumyantsev | Stability and Stabilization of Motion with Respect to a Part of Variables[END_REF][START_REF] Khalil | Nonlinear Systems[END_REF] . There are different kinds of stability and domains of validity, global or local, that should be characterized in applications. Local conditions can be easier to obtain, while global conditions provide more information about the system, especially in the presence of parametric or signal incertitude.

The most general, necessary and sufficient, conditions of various stability concepts are formulated with the use of the Lyapunov function (LF) method [START_REF] Lin | A Smooth Converse Lyapunov Theorem for Robust Stability[END_REF] . Usually, a LF defines a sort of distance with respect to an object of interest (e.g., invariant set or an equilibrium), and its time derivative should be negative on the trajectories of the studied system. Such a LF is called strict if its derivative is negative definite with respect to the same object, and discovering a strict LF is more difficult but advantageous [START_REF] Malisoff | Constructions of Strict Lyapunov Functions[END_REF] , since it allows to evaluate the rates of convergence, sensitivity and robustness characteristics. The principal weak point of the method is the absence of constructive guidelines for designing these LFs for general nonlinear systems, then different canonical forms are used with established structures of LFs.

One of the most popular concepts of robust stability in the presence of external perturbations and modeling uncertainties is given in the framework of input-to-state stability (ISS) [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF][START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF] . This property implies global asymptotic stability of the origin in the noise-free case, and guarantees asymptotic convergence of all trajectories to a neighborhood of the equilibrium, whose size is proportional to the amplitude of perturbation. The equivalent conditions of ISS are also formulated with the use of strict LFs, and all features of the method are valid in such a case also: local or global analysis, with difficulties in selection of a LF for given nonlinear dynamical system. The choice of LF in ISS analysis is even more tricky than for a conventional global stability, and a LF can be suitable for unperturbed system, but it may fail to quantify the robustness.

Following [START_REF] Dashkovskiy | Practical examples of ISS systems[END_REF] , where the options for ISS-LFs were discussed for one-degree of freedom nonlinear pendulums, in the present work the problem of finding LFs to check global or local ISS property is considered for two classes of multidimensional mechanical systems with exogenous perturbations. Several existing designs of LFs are revisited and generalized in the studied context.

The outline of this work is as follows. After introduction of the preliminaries in Section 2, the problem statement is given in Section 3. The syntheses of LFs for two investigated classes of mechanical systems are presented in sections 4 and 5. The results of applications are summarized in Section 6.

Notation

• ℝ denotes the field of real numbers, and ℝ is the -dimensional Euclidean space, ℝ + = { ∈ ℝ ∶ ≥ 0};

• the Euclidean norm ‖ ⋅ ‖ will be used for vectors;

• for a (Lebesgue) measurable function ) and the set of with the property ‖ ‖ ∞ < +∞ we further denote as  ∞ (the set of essentially bounded measurable functions);

∶ ℝ + → ℝ and [ 0 , 1 ) ⊆ ℝ + define the norm ‖ ‖ [ 0 , 1 ) = ess sup ∈[ 0 , 1 ) ‖ ( )‖, then ‖ ‖ ∞ = ‖ ‖ [0,+∞
• a continuous function ∶ ℝ + → ℝ + belongs to the class  if (0) = 0 and the function is strictly increasing; the function ∶ ℝ + → ℝ + belongs to the class  ∞ if ∈  and it is increasing to infinity; a continuous function ∶ ℝ + × ℝ + → ℝ + belongs to the class  if (⋅, ) ∈  for each fixed ∈ ℝ + and ( , ⋅) is decreasing to zero for each fixed ∈ ℝ + ;

• ∈ ℝ × denotes the identity matrix, diag( ) ∈ ℝ × is a diagonal matrix having ∈ ℝ on the main diagonal;

• for a symmetric matrix ∈ ℝ × , the relations > 0 or ≥ 0 mean that it is positive definite or semi-definite, respectively; max ( ) and min ( ) correspond to the maximal and minimal eigenvalues of .

PRELIMINARIES

For any , ∈ ℝ and > 1 the Young's inequality states:

⊤ ≤ ‖ ‖ + ( -1) || ‖ -1 ,
and its generalized version is given in the following lemma:

Lemma 1. [START_REF] Kellett | A compendium of comparison function results[END_REF]Lemma 17 For any ∈  ∞ and > 0,

≤ -1 ( ) + -1 ( ) , ∀ , ≥ 0.
Consider a nonlinear system: ̇ ( ) = ( ( ), ( )), ≥ 0,

where ( ) ∈ ℝ is the state, ( ) ∈ ℝ is the external input, ∈  ∞ , and ∶ ℝ + → ℝ is a locally Lipschitz (or Hölder) continuous function, (0, 0) = 0. For an initial condition 0 ∈ ℝ and input ∈  ∞ , define the corresponding solutions by ( , 0 , ) for any ≥ 0 for which the solution exists. The theory of ISS is well developed [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF][START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF] :

Definition 1. The system (1) is called ISS, if there are functions ∈ , ∈  such that ‖ ( , 0 , )‖ ≤ (‖ 0 ‖, ) + (|| || [0, ) ) ∀ ≥ 0 for any input ∈  ∞ and any 0 ∈ ℝ . The function is called nonlinear asymptotic gain. The system is called locally ISS if the above inequality is satisfied for || || ∞ ≤ and ‖ 0 ‖ ≤ for some ∈ ℝ + .
Theorem 1. [START_REF] Khalil | Nonlinear Systems[END_REF] The system ( 1) is (locally) ISS if and only if it admits a (local) ISS-LF.

If ( ) = 0 for all ≥ 0, then ISS property is reduced to the conventional global asymptotic stability of the origin, while an ISS-LF becomes a strict LF, whose existence is necessary and sufficient for the asymptotic stability [START_REF] Khalil | Nonlinear Systems[END_REF] .

STATEMENT OF THE PROBLEM

In [START_REF] Dashkovskiy | Practical examples of ISS systems[END_REF] , nonlinear oscillators of the form ̈ + ( ̇ ) + ( ) = were considered for , ̇ , ∈ ℝ, where it was assumed that ( ) > 0, ( ) > 0 for ≠ 0. Sufficient conditions of the existence of global strict LFs (GSLFs) for the case = 0 were obtained, and conditions of ISS were derived for ≠ 0 with the aid of the constructed functions.

In this paper, we will consider a similar problem for more general models of mechanical systems of the following form:

̈ + ( ̇ ) ̇ + ( ) = , ( 2
)
̈ + ( ) ̇ + ( ) = , ( 3
)
where ∈ ℝ and

( ⊤ , ̇ ⊤ ) ⊤ ∈ ℝ 2 is the state vector, ∈ ℝ is the external input, ∈  ∞ ; ∶ ℝ → ℝ + , ∶ ℝ → ℝ + , ∶ ℝ → ℝ
are continuously differentiable functions of their arguments. Here represents the potential energy, while and can be used to characterize the friction profiles. The system (2) is a multi-degree of freedom example of Rayleigh-type equation, which is frequently used for modeling of mechanical, electrical and acoustical systems [START_REF] Rayleigh | The Theory of Sound[END_REF] . The system (3) corresponds to a vector Liénard equation (a popular example is van der Pol oscillator) [START_REF] Andronov | Theory of oscillators[END_REF] .

For the case = 0 it is required to find a GSLF for the systems ( 2) and ( 3) under different additional hypotheses imposed on the functions , and , and next to establish the conditions of ISS with respect to the input .

ISS ANALYSIS FOR THE FIRST SYSTEM

Several designs are given below, which are based on different properties of the functions and .

Linear Potential Forces

In [START_REF] Dashkovskiy | Practical examples of ISS systems[END_REF] , the case with ( ) = for > 0 has been studied, with an additional restriction that ′ ( ) > 0 for all ∈ ℝ. For such a case, the following GSLF was constructed:

( , ̇ ) = 2 2 + ̇ 2 + ( ̇ ) + 1 2 2 ( ̇ ).
To generalize this scenario, consider the system (2) under the following auxiliary hypotheses:

Assumption 1. Let ( ) = 1 2 ⊤
, where = ⊤ > 0, while be twice continuously differentiable and such that

̇ ⊤ ( ̇ ) ̇ > 0, ∀ ̇ ∈ ℝ ⧵ {0}; (4) 2 ( ̇ ) ̇ 2 > 0, ∀ ̇ ∈ ℝ .
The conditions of this assumption imply that the potential energy has minimum at the origin (a prerequisite for stability according to Lagrange-Dirichlet theorem [START_REF] Rouche | Stability Theory by Liapunov's Direct Method[END_REF] ). Then the equation describing dynamics of the system can be rewritten as follows:

̈ + ( ̇ ) ̇ + = ,
and the condition (4) implies that the friction (or velocity) forces are dissipative. Choose a GSLF candidate in the form:

= 1 2 ̇ ⊤ ̇ + 1 2 ⊤ + 1 2 ̇ ⊤ ̇ + 1 2 ( ̇ ) ̇ + ⊤ ( ̇ ) ̇ + , ( 5
)
where the first two terms represent the full energy of the system. The direct computations for = 0 show that 5) is a GSLF for the system (2) under Assumption 1. The additional hypotheses needed to establish ISS in such a scenario are given below:

̇ = -̇ ⊤ ( ̇ ) ̇ - ( ̇ ) ̇ + ⊤ 2 ( ̇ ) ̇ 2 ( ̇ ) ̇ + hence, (
Theorem 2. Consider the system (2) under Assumption 1, then it is locally ISS. Let additionally

0 < + ‖ ̇ ‖ ≤ 2 ( ̇ ) ̇ 2 ≤ + ‖ ̇ ‖ , ∀ ̇ ∈ ℝ ,
where , ∈ ℝ × are symmetric matrices, 0 ≤ ≤ < + 2 and 0 < ≤ , then the system is ISS.

Proof. Clearly, LF ( 5) is positive definite and radially unbounded. The time derivative of the LF (5) calculated for the system (2) has the form:

̇ = -̇ ⊤ ( ̇ ) ̇ - ( ̇ ) ̇ + ⊤ 2 ( ̇ ) ̇ 2 ( ̇ ) ̇ + + ̇ ⊤ ( + ) + ( ̇ ) ̇ + ⊤ 2 ( ̇ ) ̇ 2 .
Due to Assumption 1 (0) ̇ = 0, and using the lower bound imposed on the second derivative of in the theorem we obtain:

̇ ⊤ ( ̇ ) ̇ = ̇ ⊤ ⎛ ⎜ ⎜ ⎝ (0) ̇ + 1 ∫ 0 2 ( ̇ ) ̇ 2 ̇ ⎞ ⎟ ⎟ ⎠ = ̇ ⊤ 1 ∫ 0 2 ( ̇ ) ̇ 2 ̇ ≥ ̇ ⊤ ⎛ ⎜ ⎜ ⎝ 1 ∫ 0 + ‖ ̇ ‖ ⎞ ⎟ ⎟ ⎠ ̇ = ̇ ⊤ ̇ + ‖ ̇ ‖ +2 + 1 , ∀ ̇ ∈ ℝ .
Hence, by applying Young's inequality for = 2:

̇ ⊤ ( + ) = ̇ ⊤ 0.5 -0.5 ( + ) ≤ 1 2 ̇ ⊤ ̇ + ⊤ ( + ) -1 ( + ) , ( ̇ ) ̇ + ⊤ 2 ( ̇ ) ̇ 2 ≤ 1 2 ( ̇ ) ̇ + ⊤ 2 ( ̇ ) ̇ 2 ( ̇ ) ̇ + + 1 2 ⊤ + ‖ ̇ ‖ ,
and next for = +2 :

‖ ̇ ‖ ‖ ‖ 2 = ‖ ̇ ‖ -1 ‖ ‖ 2 ≤ +2 ‖ ̇ ‖ +2 + 2 + ( + 2 -) ‖ ‖ 2 +2 +2- + 2 - +2 +2-
for any > 0, and by taking = +2 +1 +2 for ≠ 0 and = 1 otherwise we derive:

̇ ≤ - 1 2 ̇ ⊤ ( ̇ ) ̇ - 1 2 ( ̇ ) ̇ + ⊤ ( ̇ ) ̇ + + 1 2 ⊤ ( + ) -1 ( + ) + + 2 ( + 2 -) ‖ ‖ 2 +2 +2- + 2 - +2 +2-,
which implies that is an ISS-LF (Definition 2) and the system (2) is ISS (by Theorem 1). For local analysis, a local existence of and with = = 0 follows for any twice continuously differentiable from Assumption 1.

Linear Dissipative Forces

In 9 also the case ( ) = with > 0 was studied, and for this case, the following GSLF was constructed:

( , ̇ ) = 1 2 2 2 + ̇ + ̇ 2 + 2 ∫ 0 ( ) .
This analysis can be generalized by considering again the system (2) under the following hypotheses:

Assumption 2. Let ( ) = 1 2 ̇ ⊤ ̇ , where = ⊤ > 0, while be continuously differentiable, (0) = 0 and such that

( ) > 0, ⊤ ( ) > 0, ∀ ∈ ℝ ⧵ {0}.
This assumption formulates the same properties as Assumption 1, considering nonlinear potential forces and linear dissipative ones (in Assumption 1 the potential forces are linear). Then the system equation can be rewritten as follows:

̈ + ̇ + ( ) = ,
and using the approach proposed in [START_REF] Aleksandrov | The stability and stabilization of non-linear, non-stationary mechanical systems[END_REF] , let us select a LF in the form:

= ℎ 1 2 ̇ ⊤ ̇ + ( ) + 1 2 ⊤ + ⊤ ̇ , ( 6
)
where ℎ > 0 is a design parameter, then

̇ = -̇ ⊤ (ℎ -) ̇ -⊤ ( ) .
Hence, ( 6) is GSLF for a sufficiently big value of ℎ for the system (2) under Assumption 2. Let us demonstrate that a mild modification is needed to get ISS: Theorem 3. Consider the system (2) under Assumption 2, then it is locally ISS. Let additionally

⊤ ( ) ≥ 2 (‖ ‖)‖ ‖, ∀ ∈ ℝ ,
for some ∈  ∞ , then the system is ISS.

Proof. Let ℎ > 0 be selected to ensure that ℎ 2 -> 0,

which is always possible due to Assumption 2. First, let us demonstrate positive definiteness of LF (6), to this end note that

≥ 1 2 ̇ ⊤ ℎ ̇ ,
and the condition of positive definiteness of this quadratic form is that

ℎ -> 0,
which is valid due to the choice of ℎ. Clearly, under the imposed restriction, the LF ( 6) is also radially unbounded. The derivative of this LF calculated for the system (2) under introduced hypotheses takes the form (using Lemma 1):

̇ = -⊤ ( ) -̇ ⊤ (ℎ -) ̇ + ⊤ + ℎ ̇ ⊤ ≤ - 1 2 ⊤ ( ) -̇ ⊤ ℎ 2 - ̇ + -1 (‖ ‖)‖ ‖ + ℎ 2 ⊤ -1 ,
which implies ISS property. And for local ISS property with ‖ ‖ ≤ , ‖ ̇ ‖ ≤ for some > 0 we have

̇ ≤ -⊤ ( ) -̇ ⊤ (ℎ -) ̇ + ‖ ‖ + ℎ ‖ ‖,
which implies locally the desired stability characteristics since ⊤ ( ) > 0 for all ∈ ℝ by Assumption 2.

An extension of the previous cases

The proposed LFs, ( 5) and ( 6), were selected based on physical meaning of forces presented in the system (2) dynamics. Applying the approach developed in [START_REF] Efimov | On analysis of Persidskii systems and their implementations using LMIs[END_REF] , the following generalized sufficient condition of ISS can be formulated in the form of linear matrix inequalities (in such a case both, potential and dissipative, forces can be nonlinear):

Theorem 4. Consider (2) satisfying the hypotheses:

) There exist matrices ∈ ℝ × and functions ∈  ∞ , = 1, 2 such that

( ) ≥ 1 (‖ 1 ‖), ∀ ∈ ℝ , ( ̇ ) ≥ 2 (‖ 2 ̇ ‖), ∀ ̇ ∈ ℝ .
) There exist matrices , ,

∈ ℝ × , 0 ≤ = ⊤ , = ⊤ ∈ ℝ × and 0 < = ⊤ ∈ ℝ × such that ̇ ⊤ ̇ ≤ 2 ̇ ⊤ ( ̇ ) ̇ , ∀ ̇ ∈ ℝ ; ( ) ⊤ ( ) ≤ 2 ⊤ ( ) , ⊤ ≤ 2 ⊤ ( ) , ∀ ∈ ℝ .
) There exist matrices 0 ≤ = ⊤ = 11 12

12 22 ∈ ℝ 2 ×2 , 0 < Φ = Φ ⊤ ∈ ℝ × , Ω ∈ ℝ × , Ψ ∈ ℝ × and scalars > 0,
> 0, > 0, > 0, > 0 for = 1, 2, 3, 1 , 2 ∈ ℝ satisfying the following system of linear matrix inequalities:

- 1 ⊤ 1 1 0 0 2 ⊤ 2 2 > 0, Ξ ≤ 0, Ξ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ -3 + 11 0 2 + 3 -12 - 12 12 11 2 12 -1 + 0 -22 1
-22 22 0 0 -Ψ ⊤ -Ψ -Ψ ⊤ -Ω -Ψ ⊤ Ψ ⊤ 2 ⊤ + 3 ⊤ -12 -22 -Ψ -Ω ⊤ -Ω -Ω ⊤ -2 --Ω ⊤ Ω ⊤ -12 1 ⊤ -22 -Ψ --Ω -2 12 22 Ψ Ω -Φ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .
Then the system (2) is ISS, and it is locally ISS if the properties stated in the parts ( ) and ( ) hold locally.

Proof. Consider a LF candidate:

= ̇ ⊤ ̇ + 2 ( ̇ ) + 2 ( ), (7)
which is positive definite due to the constraint

> 1 ⊤ 1 1 0 0 2 ⊤ 2 2
verified for some 1 , 2 ∈ ℝ and Finsler's lemma.

The time derivative of this LF, under the introduced hypotheses and using the descriptor method, can be written as follows: 0 0

̇ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ̇ ( ) ( ̇ ) ̇ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⊤ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 11 -12 -
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ̇ ( ) ( ̇ ) ̇ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ -2 ( ) ⊤ Ω ⊤ + ̈ ⊤ Ψ ⊤ ̈ + ( ̇ ) ̇ + ( ) - ≤ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ̇ ̈ ( ) ( ̇ ) ̇ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⊤ Ξ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ̇ ̈ ( ) ( ̇ ) ̇ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ -⊤ -̇ ⊤ ̇ + ⊤ Φ ≤ -⊤ -̇ ⊤ ̇ + ⊤ Φ ,
which implies that is an ISS-LF. That was necessary to prove.

Note that ( 6) is in the form of LF (7). This result can be further extended to cover the case of LF ( 5) under corresponding hypotheses on and

2 ( ̇ ) ̇ 2 .

ISS ANALYSIS OF THE SECOND SYSTEM

In [START_REF] Blinov | Estimating the time of motion of certain dynamical systems[END_REF] , the global asymptotic stability of the origin for a scalar Lienard equation was analyzed:

̈ + ( ) ̇ + ( ) = 0,
where , ̇ ∈ ℝ, ( ) > 0 for all ∈ ℝ and ( ) > 0 for all ≠ 0. For this dynamical system a GSLF was suggested:

= ( ̇ + ∫ 0 ( ) ) 2 + 4 ∫ 0 ( ) + ̇ 2 such that ̇ = -2 ( ) ∫ 0 ( ) -2 ( ) ̇ 2 .
The system (3) represents a vector extension of the Lienard equation, for which we will assume the following:

Assumption 3. Let (0) = 0 and
( ) > 0, ⊤ ( ) ( ) > 0, ∀ ∈ ℝ ⧵ {0}, while ( ) + ( ) ⊤ > 0, ∀ ∈ ℝ .
As before, this assumption introduces the necessary requirements on potential and dissipative forces implying stability of the origin. Let us take a candidate LF:

= 1 2 ̇ ⊤ ̇ + 2 ( ) + 1 2 ( ̇ + ( )) ⊤ ( ̇ + ( )), (8)
whose derivative can be presented as

̇ = -⊤ ( ) ( ) -̇ ⊤ ( ) ̇
and, therefore, if (8) is radially unbounded, then it is a GSLF for (3). The conditions of ISS are given below:

Theorem 5. Consider the system (3) under Assumption 3, then it is locally ISS. Let additionally

0 < 4Σ ≤ ( ) + ( ) ⊤ , ⊤ ( ) ( ) ≥ 2 (‖ ( )‖)‖ ( )‖, ∀ ∈ ℝ ,
where Σ ∈ ℝ × is a symmetric matrix and ∈  ∞ , then the system is ISS.

Proof. Under the introduced conditions, the LF candidate ( 8) is radially unbounded ( ( ) possesses such a property as a function of , indeed, using the Mean value theorem we get:

⊤ ( ) = ⊤ (0) + 1 2 ⊤ ∫ 1 0 ( ) + ( ) ⊤ ≥ ⊤ (0) + 2 ⊤ Σ
), and using Lemma 1 its time derivative for the system (3) can be written as follows:

̇ = 2 ̇ ⊤ + ⊤ ( ) -̇ ⊤ ( ) ̇ -⊤ ( ) ( ) ≤ -̇ ⊤ Σ ̇ -(‖ ( )‖)‖ ( )‖ + ⊤ Σ -1 + -1 (‖ ‖)‖ ‖,
which implies that is an ISS-LF in this case. For local analysis with ‖ ‖ ≤ , ‖ ̇ ‖ ≤ for some > 0 we have The equations of the system (2) can be presented as follows with a non-integer elastic connection [START_REF] Cveticanin | Vibration of a two-mass system with non-integer order nonlinear connection[END_REF] : for > 0 with = 1, … , 4, which ensure the needed conditions. The equations of the system (2) can be rewritten in the form:

̇ ≤ (2 + )‖ ‖ -̇ ⊤ ( ) ̇ -⊤ ( ) (
̈ 1 = -1 1 -3 ( 1 -2 ) -4 √ | 1 -2 |sign( 1 -2 ) -1 sign( ̇ 1 ) -3 | ̇ 1 | ̇ 1 + 1 , ̈ 2 = -2 2 -3 ( 2 -1 ) -4 √ | 2 -1 |sign( 2 -1 ) -2 sign( ̇ 2 ) -4 | ̇ 2 | ̇ 2 + 2 .

Coupled Lienard systems

̈ 1 = -1 1 -3 ( 1 -2 ) -1 + 3 + 3 2 1 + 2 2 ̇ 1 -2 1 2 -3 ̇ 2 + 1 , ̈ 2 = -2 2 -3 ( 2 -1 ) -2 + 3 + 3 2 2 + 2 1 ̇ 2 -2 1 2 -3 ̇ 1 + 2 .

CONCLUSION

For two canonical forms of mechanical systems, (2) and (3), several designs of GSLF and ISS-LF are investigated, and the respective stability conditions are formulated. It is demonstrated in the examples that these conditions can be verified in different practical scenarios. An extension to synthesis of LF for finite-time/fixed-time stability analysis can be considered as a direction of future research.

  ) , where = sup ‖ ‖≤ ‖ ( )‖, and the conditions of Assumption 3 are enough for local verification of the properties from Definition 2.for some > 0, > 0 with = 1, … , 4. It is straightforward to verify that the conditions of the part ( ) in Theorem 4 are satisfied for:

2 with 3 ,

 23 Consider coupled systems in the form (3) with = ⊤ ∈ ℝ × and = ⊤ ∈ ℝ × defining the structure of interconnection between subsystems, > 0. Then the conditions of Theorem 5 are verified provided that = , = 1

, passport of goszadanie no. 2019-0898.

APPLICATIONS

Mass-spring systems

The nonlinear oscillations in degrees of freedom systems have been widely studied in the literature [START_REF] Blekhman | Vibrational Mechanics -a General Approach to Solving Nonlinear Problems[END_REF][START_REF] Rivin | Passive Vibration Isolation[END_REF][START_REF] Gendelman | Dynamics of linear oscillator coupled to strongly nonlinear attachment with multiple states of equilibrium[END_REF][START_REF] Kachapi | Analysis of the nonlinear vibration of a two-mass-spring system with linear and nonlinear stiffness[END_REF][START_REF] Rakaric | Approximations for motion of the oscillators with a non-negative real-power restoring force[END_REF][START_REF] Cveticanin | Vibration of a two-mass system with non-integer order nonlinear connection[END_REF][START_REF] Kovacic | Coupled purely nonlinear oscillators: normal modes and exact solutions for free and forced responses[END_REF] due to their numerous applications in various engineering domains (e.g., coupled vibrating systems, elastic beams with springs, vibration in milling machines, micro-electro-mechanical systems (MEMS), suspension in different transports, etc.).

Assumption 1 is naturally satisfied in this kind of dynamics with

, where a symmetric positive definite matrix represents the spring interconnections, while the function defines the admissible friction. Let

where = ⊤ > 0, > 0 and ≥ 2, then it introduces usual, linear in velocities, restoring forces proportional to , with approximations of high-order terms weighted by . Hence,

that verifying conditions of Assumption 1 and Theorem 2 results in = , = + 4( -1) , = 2 , = (6 -4) and = = 2 -2, then the system (2) is ISS.

Example 1. Let = 2 and

where > 0 for all = 1, … , 5. The equations of the system (2) can be rewritten as follows in this case:

and the amplitude of total velocity of the system modifies the friction gain.

The conditions of Assumption 2 can also be easily checked in systems with nonlinear characteristics of the springs. Let ( ) = 1 2 ̇ ⊤ ̇ with = ⊤ > 0 and take

where ∈ ℝ × is a nonnegative definite diagonal matrix, ∈ ℝ -1× is the spring relation matrix between masses, > 0. Hence,

and provided that + ⊤ > 0 the conditions of Assumption 2 and Theorem 3 are satisfied for ( ) = 0.5 min ( + ⊤ ) , therefore, the system (2) is ISS.

Example 2. Let = 2 and

where > 0 for all = 1, … , 5. The equations of the system (2) can be rewritten as follows in this case with a nonlinear stiffness [START_REF] Kachapi | Analysis of the nonlinear vibration of a two-mass-spring system with linear and nonlinear stiffness[END_REF] :

3 ( 2 -1 ) 3 -5 ̇ 2 + 2 . Since LF (6) used in Theorem 3 is an example of LF (7) from Theorem 4, then the above example illustrates also the latter theorem. Note that its important advantage is that non-quadratic functions and can be considered.

Example 3. Note that the sufficient part of Theorem 1 holds true even for a locally Lipschitz continuous ISS-LF , while can be discontinuous under the conventional regularity restrictions [START_REF] Filippov | Stability of differential equations with discontinuous and multi-valued right-hand parts[END_REF] . Let = 2 and


Кatarina Lj Djordjevic
email: katarina.djordjevic@vin.bg.ac.rs

Dragana K Markushev

Slobodanka P Galović

Dragan D Markushev

Jose Ordonez-Miranda

Si plate radius influence on the photoacoustic signal processed by neural networks

Keywords: photoacoustic, semiconductors, artificial neural networks, thermal diffusion, thermal expansion, photothermal, inverse problem

The effect of the sample radius on the total photoacoustic signal processed by neural networks trained with undistorted and distorted signals is carefully analyzed for modulation frequencies from 20 Hz to 20 kHz. This is done for signals generated for a 400-m-thick Si n-type plate, whose radius varies from 2 to 7 mm. It is found that the networks trained with both undistorted or distorted signals yield the best predictions for sample radii between 2 and 3 mm, which is close to the used microphone aperture radius of 1.5 mm. The network trained only with undistorted signals gives the best results for sample radii comparable to the microphone dimensions. The obtained results neural networks in the prediction of Si-plate radius indicate the experimental necessity to use samples with radii slightly over to microphone aperture.

Introduction

The photoacoustic effect is the effect of the sound appearance due to the illumination of a sample with an optical beam whose intensity is time-dependent [START_REF] Rosencwaig | Photoacoustic effect with solids: A theoretical treatment[END_REF][START_REF] Rosencwaig | Theory of the photoacoustic effect with solids[END_REF][START_REF] Mcdonald | Theory of Photothermal and Photoacoustic Effects in Condensed Matter[END_REF][START_REF] Rousset | Influence of thermoelastic bending on photoacoustic experiments related to measurements of thermal diffusivity of metals[END_REF][START_REF] Tam | Applications of photoacoustic sensing techniques[END_REF]. Numerous non-destructive methods for the characterization and imaging of various solids, multilayer structures, microelectronic and photonic devices, soft matter, and biological tissues are based on this effect [START_REF] Vargas | Photoacoustic and related photothermal techniques[END_REF][START_REF] Bialkowski | Photothermal Spectroscopy Methods for Chemical Analysis[END_REF][START_REF] Todorović | Prog. Photothermal Photoacoust. Sci. Technol. -Semicond. Electron. Mater[END_REF][START_REF] Souza Filho | Investigation of doped calcium aluminosilicate glass: A coupling between thermal-expansion and thermal-diffusion models for assessment of nonradiative relaxation time and characteristic diffusion time[END_REF][START_REF] Pichardo | Photoacoustic measurements of the thermal properties of AlyGa1-yAs alloys in the region 0<y<0.5[END_REF][START_REF] Nesic | Developing the Techniques for Solving the Inverse Problem in Photoacoustics[END_REF][START_REF] Pawlak | Simultaneous measurement of thermal conductivity and diffusivity of an undoped Al 0.33 Ga 0.67 As thin film epitaxially grown on a heavily Zn doped GaAs using spectrally-resolved modulated photothermal infrared radiometry[END_REF][START_REF] Galović | Photothermal microscopy: a step from thermal wave visualization to spatially localized thermal analysis[END_REF][START_REF] Somera | Anomalous thermal diffusion in two-layer system: The temperature profile and photoacoustic signal for rear light incidence[END_REF][START_REF] Popovic | Photoacoustic response of a transmission photoacoustic configuration for two-layer samples with thermal memory[END_REF][START_REF] Galović | Theory of photoacoustic effect in media with thermal memory[END_REF][START_REF] Huang | Nanomaterials for Photoacoustic Imaging in Second Near-Infrared Window[END_REF][START_REF] Somer | The thermoelastic bending and thermal diffusion processes influence on photoacoustic signal generation using open photoacoustic cell technique[END_REF][START_REF] Somer | Photoacoustic signal with two heating sources: theoretical predictions and experimental results for the open photoacoustic cell technique[END_REF][START_REF] Balderas-Lopez | Self-normalized photothermal technique for accurate thermal diffusivity measurements in thin metal layers[END_REF][START_REF] Isaiev | Application of the Photoacoustic Approach in the Characterization of Nanostructured Materials[END_REF]. The use of neural networks in photoacoustics has become very popular recently, especially in imaging within biomedical applications [START_REF] Dispirito | Sounding out the hidden data: A concise review of deep learning in photoacoustic imaging[END_REF][START_REF] Nešić | Thermal memory influence on the thermoconducting component of indirect photoacoustic response[END_REF][START_REF] Anas | Robust Photoacoustic Beamforming Using Dense Convolutional Neural Networks[END_REF]. Significant progress has been made, also, in photoacoustics of solids with different types of networks both in material characterization and in the development of experimental procedures [START_REF] Lj | Inverse problem solving in semiconductor photoacoustics by neural networks[END_REF][START_REF] Lj | Photoacoustic Measurements of the Thermal and Elastic Properties of n-Type Silicon Using Neural Networks[END_REF][START_REF] Jordović-Pavlović | The application of artificial neural etworks in solid-state photoacoustics for the recognition of microphone response effects in the frequency domain[END_REF]. Networks have proven to be a very effective tool for predicting the physical characteristics of solid samples [START_REF] Goodall | Predicting materials properties without crystal structure: deep representation learning from stoichiometry[END_REF][START_REF] Hu | Learning Representations of Inorganic Materials from Generative Adversarial Networks[END_REF][START_REF] Tho | Artificial neural network model for material characterization by indentation[END_REF]. Within the framework of multi-parameter fitting in photoacoustics, the networks most often predicted the values of the sample parameters such as thermal diffusion (DT) and thermal expansion (  ) together with the thickness of the sample (l) [START_REF] Markushev | Effects of the photogenerated excess carriers on the thermal and elastic properties of n-type silicon excited with a modulated light source: Theoretical analysis[END_REF][START_REF] Markushev | Experimental photoacoustic observation of the photogenerated excess carrier influence on the thermoelastic response of n-type silicon[END_REF]. These parameters are important because they occur when calculating the temperature distributions of the sample and all components of the photoacoustic signal generated by the periodic illumination of the sample.

It has long been known that the geometric parameters of the sample are very important for the correct interpretation of the photoacoustic signal in the frequency domain [START_REF] Lachaine | Thermal analysis by photoacoustic phase measurements: Effect of sample thickness[END_REF][START_REF] Wang | Relation between photoacoustic signal and sample thickness[END_REF][START_REF] Burbelo | Laser Photoacoustic Diagnostics of Advanced Materials With Different Structure and Dimensions[END_REF]. If the circular plate shape samples are tested, one of such parameters is its radius Rs. It explicitly appears in the equation for calculating the thermoelastic and plasmaelastic components of the photoacoustic signal of a semiconductor, so that its influence on the shape of the total signal through these two components can be extremely significant [START_REF] Markushev | Effects of the photogenerated excess carriers on the thermal and elastic properties of n-type silicon excited with a modulated light source: Theoretical analysis[END_REF][START_REF] Markushev | Experimental photoacoustic observation of the photogenerated excess carrier influence on the thermoelastic response of n-type silicon[END_REF]. In this paper, the analysis of the influence of the semiconductor sample radius changes on all components of the photoacoustic signal, thermodiffusion (TD), thermoelastic (TE), and plasmaelastic (PE), is presented, based on the theoretical model of a composite piston [START_REF] Rosencwaig | Photoacoustic effect with solids: A theoretical treatment[END_REF][START_REF] Rosencwaig | Theory of the photoacoustic effect with solids[END_REF][START_REF] Mcdonald | Theory of Photothermal and Photoacoustic Effects in Condensed Matter[END_REF] that serves to form an appropriate base of photoacoustic signals for neural network training [START_REF] Lj | Inverse problem solving in semiconductor photoacoustics by neural networks[END_REF][START_REF] Lj | Photoacoustic Measurements of the Thermal and Elastic Properties of n-Type Silicon Using Neural Networks[END_REF]. The tested sample is a Si n-type 400 m thick wafer, classified as a plasma-thick samplea sample with a thickness larger than the diffusion length of free carriers [START_REF] Markushev | Effects of the photogenerated excess carriers on the thermal and elastic properties of n-type silicon excited with a modulated light source: Theoretical analysis[END_REF][START_REF] Markushev | Experimental photoacoustic observation of the photogenerated excess carrier influence on the thermoelastic response of n-type silicon[END_REF]. Plasma-thick semiconductor samples have a thermal response of known and characteristic patterns of behavior of the total signal and its components in the modulation frequencies ranging from 20 Hz to 20 kHz [START_REF] Markushev | Experimental photoacoustic observation of the photogenerated excess carrier influence on the thermoelastic response of n-type silicon[END_REF].

These patterns include domination of the TD component at lower, and the TE component at higher frequencies, while the PE component`s influence on the total signal is negligible.

The basic idea of this paper is to determine which type of neural network gives better predictions in the case of a change in the radius of the sample: one that is trained by distorted signals, or by signals undistorted with instrumental deviations [START_REF] Markushev | Photoacoustic signal and noise analysis for Si thin plate: Signal correction in frequency domain[END_REF][START_REF] Popovic | Helmholtz Resonances in Photoacoustic Experiment with Laser-Sintered Polyamide Including Thermal Memory of Samples[END_REF][START_REF] Aleksić | Electro-Acoustic influence of the measuring system on the photoacoustic signal amplitude and phase in frequency domain[END_REF]. This analysis is very important because its results can give the possibility of working with distorted signals, which significantly simplifies and speeds up the process of characterization of the tested samples. Both types of network training signals were obtained theoretically, and the final verification of network efficiency was tested on experimental data. The conclusions of this analysis indicate, in experimental terms, the importance of the choice of sample radius about the dimensions of the microphone as a photoacoustic cell and signal detector. Also, based on the obtained network predictions a conclusion can be drawn on the justification of using approximations in the theoretical model of the composite piston, especially those related to the equality of microphone aperture and sample radii.

Theoretical introduction

The theoretical model of the composite piston presented in this paper is adapted to the experimental setup of an open photoacoustic cell [START_REF] Markushev | Effects of the photogenerated excess carriers on the thermal and elastic properties of n-type silicon excited with a modulated light source: Theoretical analysis[END_REF][START_REF] Markushev | Experimental photoacoustic observation of the photogenerated excess carrier influence on the thermoelastic response of n-type silicon[END_REF][START_REF] Markushev | Photoacoustic signal and noise analysis for Si thin plate: Signal correction in frequency domain[END_REF][START_REF] Nešić | Influence of Thermal Memory on the Thermoelastic Bending Component of Photoacoustic Response[END_REF]. It is assumed that a frequencymodulated light source of wavelength 660nm illuminates the Si n-type plate. Due to the photoacoustic effect of the absorbed modulated light and the sample interaction, a sound is recorded by an open photoacoustic cell in the transmission configuration (Figure 1). The photoacoustic signal detector is assumed to be an electret microphone (ECM 30B, JinInElectronicCo., Ltd.) with a radius of 0 4.9 mm R = , and aperture radius of r0 = 1.5 mm [START_REF] Markushev | Experimental photoacoustic observation of the photogenerated excess carrier influence on the thermoelastic response of n-type silicon[END_REF].

The sample is placed directly on the microphone with points of support that are slightly smaller but close to the radius of the sample s R , so that the microphone at the same time can serve as a measuring cell of minimal volume

2 0 0 c V r l  =
, where lc is the distance from the microphone aperture to its membrane (length of OPC) (see Figure 1). The measured signal in the transmission configuration and in the frequency range from 20Hz to 20 kHz is a sound signal that occurs due to the photoacoustic effect in the sample itself and is usually distorted by the influence of used instruments and the different types of noise. The signal thus measured is called the distorted signal ( ) total Sf . If a well-known signal correction procedure [START_REF] Markushev | Experimental photoacoustic observation of the photogenerated excess carrier influence on the thermoelastic response of n-type silicon[END_REF][START_REF] Markushev | Photoacoustic signal and noise analysis for Si thin plate: Signal correction in frequency domain[END_REF][START_REF] Popovic | Helmholtz Resonances in Photoacoustic Experiment with Laser-Sintered Polyamide Including Thermal Memory of Samples[END_REF][START_REF] Aleksić | Electro-Acoustic influence of the measuring system on the photoacoustic signal amplitude and phase in frequency domain[END_REF] is applied to such a signal, the so-called `true' or undistorted signal ( ) total pf  can be obtained, generated from an illuminated sample. The theoretical model that fully describes the undistorted signals is based on the composite piston model [START_REF] Rosencwaig | Photoacoustic effect with solids: A theoretical treatment[END_REF][START_REF] Rosencwaig | Theory of the photoacoustic effect with solids[END_REF][START_REF] Mcdonald | Theory of Photothermal and Photoacoustic Effects in Condensed Matter[END_REF]. This model explains that the total photoacoustics signal ( ) total pf  of a semiconductor is equal to the sum of its components; thermodiffusion ( )

f p TD  , thermoelastic ( ) f p TE  and plasmaelastic ( ) f p PE  [33,40]: ( ) ( ) ( ) ( ). PE TE TD total f p f p f p f p     + + = (1)
The thermodiffusion component can be written as follows [START_REF] Markushev | Experimental photoacoustic observation of the photogenerated excess carrier influence on the thermoelastic response of n-type silicon[END_REF]41]:

( ) ( )       - = 4 0 0 ТD 2      t i g c e l T T l p t p , (2)
where 2 f  = , f is modulation frequency,  is an adiabatic ratio, 0 p and 0 T are the ambient pressure and temperature, g  is the thermal diffusion length of the surrounding gas, ( ) s Tl (see Appendix I) is the dynamic temperature component at the back sample surface, l is the sample thickness and c l is the length of the cell (microphone). Due to different temperature values on the illuminated and non-illuminated side of the silicon sample, its periodic bending and expansion occur. The effect of bending and spreading is most simply described by cylindrical (z, r) coordinates so that the thermoelastic component ( )

f p TE 
of the photoacoustic signal can be written in the form [START_REF] Markushev | Experimental photoacoustic observation of the photogenerated excess carrier influence on the thermoelastic response of n-type silicon[END_REF][START_REF] Nešić | Influence of Thermal Memory on the Thermoelastic Bending Component of Photoacoustic Response[END_REF]: This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0123041

( ) ( ) /2 4 0 TE T 3 0 /2 3 l s l pR p f zT z dz Vl    - =  , (3)
where ( )

Tz is the temperature distribution in the sample along the z-axes (see Appendix I), T  is the coefficient of thermal expansion, 0 V is the microphone (cell) volume, and s R is the sample radius. In most of the theoretical analyses, the 0 s RR  approximation is taken to simplify the analysis, [START_REF] Markushev | Effects of the photogenerated excess carriers on the thermal and elastic properties of n-type silicon excited with a modulated light source: Theoretical analysis[END_REF][START_REF] Markushev | Experimental photoacoustic observation of the photogenerated excess carrier influence on the thermoelastic response of n-type silicon[END_REF][START_REF] Markushev | Photoacoustic signal and noise analysis for Si thin plate: Signal correction in frequency domain[END_REF][START_REF] Aleksić | Electro-Acoustic influence of the measuring system on the photoacoustic signal amplitude and phase in frequency domain[END_REF].

By modulated illumination of the sample with the light of proper wavelength, free carriers (electrons and holes) are generated, which leads to additional stress and periodic bending of the sample. The influence of phototgenerated carriers on the bending of the sample is described by the plasmaelastic component ( )

f p PE  [33,40]: ( ) ( ) /2 4 0 PE 3 0 /2 3 l s n p l pR p f d z n z d z Vl    - =  , (4) where (
f p PE  .
It must be bear in mind that during the measurement there are a lot of signal distortions due to the influence of the used instruments (microphones and accompanying electronics). These distortions have signal filtering characteristics and are described, in the low-frequency range, by the function of the first-order low-pass filter ( ) e Hf in the form [START_REF] Markushev | Photoacoustic signal and noise analysis for Si thin plate: Signal correction in frequency domain[END_REF][START_REF] Aleksić | Electro-Acoustic influence of the measuring system on the photoacoustic signal amplitude and phase in frequency domain[END_REF]:

( ) 1 2 1 2 (1 ) (1 )
c c e c c Hf i i       = -  +  +  , ( 5
)
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0123041 where 2 f  = , f is the modulation frequency, ( )

1 1 1 1 1 2 c c f R C  - == and
( )

1 2 2 2 2 2 c c f R C  - ==
are the time constants and RC characteristics of the microphone and accompanying electronics (usually sound-card or lock-in).

In the high-frequency range, these distortions are described by the function of the secondorder low-pass filter ( )

a
Hf in the form [START_REF] Markushev | Photoacoustic signal and noise analysis for Si thin plate: Signal correction in frequency domain[END_REF][START_REF] Aleksić | Electro-Acoustic influence of the measuring system on the photoacoustic signal amplitude and phase in frequency domain[END_REF]:

( )

2 2 3 4 2 2 2 2 3 3 3 4 4 4 c c a c c c c c c Hf i i             = + + - + - , (6)
where

k  is the damping factor (k = c3,c4), 3 c
 is the microphone cut-off frequency and 4 c  is the characteristic frequency which depends on the geometry of the microphone body.

The total measured signal ( )

total
Sf can be written as a product of ( )

total pf  , ( ) e Hf i ( ) a
Hf [START_REF] Markushev | Experimental photoacoustic observation of the photogenerated excess carrier influence on the thermoelastic response of n-type silicon[END_REF][START_REF] Markushev | Photoacoustic signal and noise analysis for Si thin plate: Signal correction in frequency domain[END_REF][START_REF] Aleksić | Electro-Acoustic influence of the measuring system on the photoacoustic signal amplitude and phase in frequency domain[END_REF]: ( )

( ) ( ) ( ) ( ) total total e a S f p f H f H f  = . ( 7
a
Hf can be obtained either from the microphone manual or by independent measurements [START_REF] Markushev | Experimental photoacoustic observation of the photogenerated excess carrier influence on the thermoelastic response of n-type silicon[END_REF][START_REF] Markushev | Photoacoustic signal and noise analysis for Si thin plate: Signal correction in frequency domain[END_REF][START_REF] Aleksić | Electro-Acoustic influence of the measuring system on the photoacoustic signal amplitude and phase in frequency domain[END_REF]. By finding them, 7) so the required ( ) total pf  can be obtained [START_REF] Markushev | Experimental photoacoustic observation of the photogenerated excess carrier influence on the thermoelastic response of n-type silicon[END_REF][START_REF] Markushev | Photoacoustic signal and noise analysis for Si thin plate: Signal correction in frequency domain[END_REF], which is further described by the composite piston theory (Eqs.(1-4)).

Neural networks

Our motive for the study of neural networks is their use in the precise characterization of the properties of the tested materials. In the process of supervised learning by connecting the data of input neurons (values of photoacoustic signals), with the data of output neurons (radius), the weights are adjusted to the moment when the output becomes increasingly like the desired data values. In our work with networks so far, we have learned that for the sake of prediction, we need to present undistorted data to the trained network, data that are adapted to the theoretical model used to create the training base. This assumes that during the processing of experimental results, we must also count on the signal correction procedure. If it turns out that signal correction is not required to determine some of the sample parameters (expected in the case of only Rs changes), this will mean significant time savings. Therefore, in this paper, we will try to train networks with distorted (NN1) and undistorted (NN2) photoacoustic signals with the desire to determine the radius of the investigated sample, as a very important experimental parameter.

Our goal is to see if there are differences in the prediction of the radius of the sample of such formed networks and whether, based on that, we can speed up the process of processing experimental data. For the sake of simplicity, and based on the problem we want to solve, we decided that both networks used here have the simplest architecture, the so-called shallow networks: input layer, one hidden layer, and output layer [START_REF] Lj | Inverse problem solving in semiconductor photoacoustics by neural networks[END_REF][START_REF] Lj | Photoacoustic Measurements of the Thermal and Elastic Properties of n-Type Silicon Using Neural Networks[END_REF]. The input layer was determined by the 144 neurons (i = 72 values of the photoacoustic signal amplitude Ai, and i = 72 values of photoacoustic signal phase i characteristic), the hidden layer was determined by 10 neurons, and the output layer was determined by one neuron (the prediction of Rs, Figure 2). The choice of neural network architecture ensures good accuracy and optimal training time. Due to the better precision of the neural network, we adjusted the amplitudes by logarithmic normalization 20log10Ai to obtain the phase values range (Ai represents amplitudes normalized to unit pressure).

Results and discussion

4.1.Signal database formation

Due to the high sensitivity of these components to s R changes in the total photoacoustic signal, we formed a `small` base of 26 undistorted Figure 3&4 clearly shows the instrumental influences as electronic filtering (Eq. ( 5)) at lower (He at f < 100 Hz) and acoustic filtering (Eq. ( 6)) at higher frequencies (Ha at f > 5 kHz) [START_REF] Markushev | Experimental photoacoustic observation of the photogenerated excess carrier influence on the thermoelastic response of n-type silicon[END_REF][START_REF] Markushev | Photoacoustic signal and noise analysis for Si thin plate: Signal correction in frequency domain[END_REF][START_REF] Popovic | Helmholtz Resonances in Photoacoustic Experiment with Laser-Sintered Polyamide Including Thermal Memory of Samples[END_REF][START_REF] Aleksić | Electro-Acoustic influence of the measuring system on the photoacoustic signal amplitude and phase in frequency domain[END_REF]. Electronic filters He reduce amplitudes (Figure 3&4

Network prediction results

The training results of NN1 neural networks trained by the base of distorted (Figure 3) and NN2 trained with the base of undistorted (Figure 4 To evaluate the neural networks` NN1 and NN2`s ability to predict, we formed additional distorted and undistorted signal bases with values from 2.1 mm to 6.9 mm, formed using a step of 0.4 mm. These additional bases are used for network testing. In this way, we Figure 11. Experimental result of a) amplitude I b) phase of distorted (star) and undistorted (line) photoacoustic signal presented to the NN2 and NN1, respectively, for a sample of 400 m thick silicon n-type with a radius of R s = 4.54mm (black), and R s =2.63mm (green). This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0123041

Conclusion

We have shown that the radius of the sample Rs is a very important parameter that can lead to changes in the shape of the amplitude and phase of the total photoacoustic signal and its ( ) ( ) ( ) ( )

2 2 0 2 d 1 exp d g g s s p ε -ε ε Tz Tz βI βz δn z z k ε τ    - = - --     . (A1.1)
The solution ( )

s
Tz to the Eq.( 3) can be given as a sum of thermalization

 - - -     + -   = - +     - - -       , (A1.4) ( ) ( ) ( ) ( ) ( )

Figure 1 .

 1 Figure 1. Simple ideal scheme of the transmission configuration of an open photoacoustic cell with an electret microphone and samples of different radii.

Figure 2 .

 2 Figure 2. Representation of a single-layer neural network used for the analysis of distorted and undistorted photoacoustic signals that depend on the radius of silicon samples.

  photoacoustic signals obtained by Eqs.(1-7). As we said earlier, we assume that the sample is Si n-type plate, 400 m thick. Bases are determined by amplitude-phase characteristics in the

changing

  .a) and raises phases (Figure 3&4.b) at low frequencies. Acoustic filters are characterized by resonant peaks in both amplitudes and phases. The parameters that define He and Ha are not dependent on Rs and are constant during its changes. Therefore, the reasons for changing Rs are the same, and consist of the following: both signals are characterized with a saddle shape in amplitudes, which is more pronounced for 2 mm s R = , while with an increasing radius the saddle shape slowly disappears and at 7 to Rs changes at lower frequencies allows one the possibility of, so-called, signal self-normalization that can contribute to the better performances of network training. On the other hand, the saddle shape of the ( ) typical pattern of plasma-thick silicon samples behavior in the frequency domain (strong TD and TE influence), allowing one to obtain better network parameter prediction. Both mentioned signal characteristics, insensitivity, and saddle shape are the reason why, in this paper, we decided to create a training base with distorted and undistorted signals of the plasma-thick samples, expecting efficient neural network optimization.

Figure 3 .Figure 4 .

 34 Figure 3. The bases of distorted PA signal

Figure 5

 5 Figure 5. Thermodiffusion component

.

Figure 6

 6 Figure 6. Thermoelastic component

Figure 7 Figure 8 .

 78 Figure 7. Plasmaelastic component

Figure 8 .

 8 Figure 8. The tendency of changes in a) amplitudes A and b) phases  of total photoacoustic signals

Figure 9 .

 9 Figure 9. Network performance results: a) NN1 trained on distorted FA signals and b) NN2 trained on undistorted FA signals.

  poorer predictions. The largest (%) relative errors are made by both NN1 and NN2 networks for the radius of the sample in the range (4.0-5.5) mm, ie. about a value equal to the radius of the microphone 0 4.9 mm R = . These results lead to the conclusion that both networks give the best predictions when the value of s R approaches the value of the microphone aperture radius the justification of the approximations Rs = r0 taken here with the experimental configuration of the open cell. In general, the results show that for experimental work it is best to take samples whose dimensions are smaller than the dimensions of the microphone (R0).

  distorted and undistorted signals of a sample that had a radius approximately equal to the microphone radius 0 s RR  , and with a sample radius close to the aperture radius 0 s Rr  . The results of network prediction in the case of the signal from Figure 11 (black for 4the open photoacoustic cell experimental set-up are shown in Table I with an asterisk (*), and in Figure 10 with dots (red for NN1, black for NN2). Based on the experimental results, it can be concluded that the prediction of the experimentally corrected photoacoustic signals by the NN2 network is more accurate. Therefore, it can be recommended that networks trained on undistorted signals (NN2) should be used in determining the Rs of the tested sample only with values smaller than the microphone radius R0.

Figure 10 .

 10 Figure 10. Dependence of relative% error prediction of NN1 (red line) and NN2 (black line) networks on the sample radius for the tests of theoretical results and experimental verification (NN1 (red dot), NN2 (black dot)).

  photoacoustic cell configuration, whose radii are not larger than the dimensions of the microphone should be used.

.

  This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0123041This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0123041This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE ASDOI: 10.1063/5.0123041 This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0123041 NN2 NN1 exp PAS NN2 exp PAS This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0123041

  )

		p nz	is the minority carrier density (see Appendix II), and	n d is the coefficient of
	electronic deformation. Analysing Eq.(3&4) one can see that		p TE	( ) f	and		p PE	( ) f	depends on
	4 s R , meaning that small changes in Rs leads to large changes in		p TE	( ) f	and	( )

Table I :

 I The results of NN1 and NN2 neural networks predictions of sample radius given in the table with relative (%) errors. Asterisk (*) denotes the predictions for experimentally measured PASs.

	s R [mm] NN1	s R [mm]	rel % error, NN1	NN2	s R [mm]	rel % error, NN2
	2.1	2.1356	1.6933		2.1023	0.1086
	2.5	2.4849	0.6048		2.5400	1.5996
	2.9	2.9038	0.1319		2.8572	1.4745
	3.3	3.2037	2.9183		2.8572	0.3332
	3.7	3.8832	4.9527		3.7098	0.2650
	4.1	4.1947	2.3089		4.1427	1.0416
	4.5	4.7968	6.5967		4.5426	0.9470
	4.9	4.8915	0.1740		4.5426	7.2936
	5.3	5.3213	0.4014		5.1965	1.9530
	5.7	5.6535	0.8155		5.6941	0.1028
	6.1	6.0561	0.7190		6.1364	0.5964
	6.5	6.4283	1.1034		6.1364	5.5942
	6.9	6.8216	1.1358		6.8506	0.7156
	4.54*	4.7976	5.6749		4.6644	2.7398
	2.63*	2.6521	0.8315		2.6247	0.2014

To confirm the previous results, both networks presented the experimental results of

This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0123041

Acknowledgments

We are thankful for the financial support of this research by the Ministry of Education, Science and Technology development of the Republic of Serbia, contract number 451-03-09/2021-14/200017.

Autors` declarations Conflict of Interest

The authors have no conflicts to disclose.

Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Appendix I. Periodic temperature component in Si sample

Following the geometry of the problem presented in Figure 1, we are enabled to use the 1D diffusion equation to describe the temperature changes within the investigated Si sample along the heat propagation axis (z-axis) [START_REF] Todorović | Prog. Photothermal Photoacoust. Sci. Technol. -Semicond. Electron. Mater[END_REF][START_REF] Markushev | Effects of the photogenerated excess carriers on the thermal and elastic properties of n-type silicon excited with a modulated light source: Theoretical analysis[END_REF][START_REF] Markushev | Experimental photoacoustic observation of the photogenerated excess carrier influence on the thermoelastic response of n-type silicon[END_REF] Total temperature and its components at the rear (nonilluminated) side of the sample ( ,0 l z l e  -=→ ) can be written as [START_REF] Todorović | Prog. Photothermal Photoacoust. Sci. Technol. -Semicond. Electron. Mater[END_REF][START_REF] Markushev | Effects of the photogenerated excess carriers on the thermal and elastic properties of n-type silicon excited with a modulated light source: Theoretical analysis[END_REF]:

where This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0123041 ( ) ( )

Appendix II. Periodic carrier densities in Si sample

Considering that the used Si membranes (Figure 1) are the low-level injection n-type semiconductors (minority carriers are used to explain carrier dynamics), the dynamic part of the 1D diffusion equation (important for sound wave generation) that explains carrier transport in membranes can be written in the form [START_REF] Todorović | Prog. Photothermal Photoacoust. Sci. Technol. -Semicond. Electron. Mater[END_REF][START_REF] Markushev | Effects of the photogenerated excess carriers on the thermal and elastic properties of n-type silicon excited with a modulated light source: Theoretical analysis[END_REF][START_REF] Markushev | Experimental photoacoustic observation of the photogenerated excess carrier influence on the thermoelastic response of n-type silicon[END_REF]: , and the integration constants A  are defined by [START_REF] Todorović | Prog. Photothermal Photoacoust. Sci. Technol. -Semicond. Electron. Mater[END_REF][START_REF] Markushev | Effects of the photogenerated excess carriers on the thermal and elastic properties of n-type silicon excited with a modulated light source: Theoretical analysis[END_REF][START_REF] Markushev | Experimental photoacoustic observation of the photogenerated excess carrier influence on the thermoelastic response of n-type silicon[END_REF]:


domain: shs sde

text:
Clémence Moreau

Cécile Barnaud

Raphael Mathevet

Raphaël Mathevet

The baseline, a social construction

Introduction

In 1995, the marine biologist Daniel Pauly published his article Anecdotes and the shifting baseline syndrome of fisheries. According to his theory, the baseline, i.e. the state of an ecosystem considered as "natural", evolves over time, without the actors noticing it, insofar as they consider an erroneous baseline: that of previous decades, corresponding to their childhood or the beginning of their professional career. As each generation redefines what is natural and what is not, the standard becomes slippery, and the degradation of ecosystems becomes invisible to the actors. He states that the quantity of fish considered normal in a marine ecosystem tends to decrease with the decreasing age of the researchers (Pauly 1995).

There are several reasons why this short account, written in the first person, has resonated widely in fisheries management and marine ecology. The theory is easy to grasp and its logic seems compelling (Campbell et al. 2009). It also highlights a hitherto imperceptible cause of ecosystem degradation, linked to our own representations. To consider as "natural" an ecosystem that, only yesterday, would have seemed "degraded" to us, is to revise downwards what is desirable in terms of conservation (Maris 2018;Soga and Gaston 2018). The risk, then, is to become increasingly tolerant of ecosystem degradation (Soga and Gaston 2018). Another strength of this theory is its evocative power: it summons images of past abundance, which contrast with a current degraded state; the theory is then mobilizable for conservation programs (Campbell et al. 2009;Alagona et al. 2012). Finally, within the scientific community, the analysis of the shifting baseline syndrome makes it possible to encourage interdisciplinary work, since it requires the combination of historical data from ecology, but also from local knowledge, art or literature (Campbell et al. 2009).

Through this text, Pauly invites us to take a new look at the concept of a baseline, defined as "a state or trajectory of a system used as a comparator" (Maron et al. 2015). Far from being a stable and neutral datum, the baseline is influenced by political agendas, economic realities, scientific knowledge, local knowledge, and even prejudices (Hilding-Rydevik et al. 2017). The baseline can thus be screened by constructivism, according to which we can only access knowledge of reality through our system of representations, individual or collective (Lévy and Lussault 2013).

Considering the baseline as a social construct calls for the exploration of at least three lines of thought that we will develop in this chapter. Firstly, the baseline is not stabilized but evolves over time: this was Pauly's intuition, which has been confirmed but also criticized by numerous studies. Secondly, different elements, not only ecological, but also social, cultural, economic and even political, enter into the construction of the baseline, hence the fact that the baseline may vary according to social groups or individuals, and to contexts. Finally, since the choice of the baseline has strong management implications and there is not necessarily a consensus, it is necessary to debate it. To develop these three lines of thought, we will rely on a review of the literature and will support our remarks with examples taken from a study conducted in the Cévennes National Park (Box 4.1.).

Mont Lozère is a granite massif located in the south of the Massif Central. Its characteristic landscapes (pastures dedicated to sheep and cattle breeding, natural meadows, beech/fir forests, and coniferous plantations), as well as the cultural and ecological richness of the site, have justified various classifications and recognitions: Cévennes National Park in 1970, biosphere reserve of the Unesco Man and Biosphere program (1984), inscription of the Mediterranean agro-pastoral landscapes on the Unesco World Heritage List (2011) (Photo 4.1.) As elsewhere in France and in Europe, the Lozère mountain region is witnessing the "closing of landscapes", i.e. the increase of forest cover in former rangelands and meadows, abandoned or less intensively grazed. In these mountain regions, the expression "landscape closure" implicitly refers to a negative, undesirable phenomenon (Le Floch et al. 2005). One of the structuring postulates of the fight against "landscape closure" is the existence of a baseline landscape, situated in the middle of the 19 th century, when the agricultural exploitation of the territory was at its peak and the landscapes were particularly open. The increase in forest cover is then perceived as a deviation from this model (Marty and Lepart 2001;Le Floch et al. 2005). At the same time, the open landscapes themselves are evolving, in the context of a (moderate) intensification of livestock systems, and there is disagreement on the types of open landscapes that should be preserved or promoted. Certain agricultural practices are indeed on the increase, such as rock removal (which consists of removing large granite boulders to facilitate the mechanical exploitation of plots) or prairie turning (the plowing of permanent prairie or rangeland, which then becomes temporary prairie, plowed and seeded regularly). Insofar as they may impact the character of the park, these two practices are subject to authorization in the core zone of the National Park.

There may therefore be tension between the baseline landscape (extensive agro-pastoral landscapes) and the current landscapes (which are evolving either towards forest or towards more intensive agricultural landscapes). This tension manifests itself in conflicts of interest, or even open conflicts, notably between the agents of the Park and the farmers around certain agricultural practices (rock removal and prairie turning).

To study this tension, we set up a qualitative research device between 2015 and 2019, as part of a PhD thesis in geography, combining a qualitative approach (semi-structured interviews, participant observation) and action research (participatory workshops around a role-playing game) (Moreau 2019). The shifting baseline syndrome theory has met with widespread acceptance in the scientific community. A meta-analysis, conducted in 2019, indicates that 152 publications have been made on the topic, 82% of which are on aquatic ecosystems and 28% on terrestrial ecosystems (Guerrero-Gatica et al. 2019). On the other hand, the appropriation of this theory is still timid in the French-speaking literature. Changes in baseline systems are essentially addressed by the social sciences, which question the representations underlying the development of land-use planning policies (Luginbühl and Terrasson 2013), but without explicitly referring to the shifting baseline syndrome.

In the scientific literature, three types of articles refer to the shifting baseline syndrome: those that reconstruct a past baseline by measuring species abundance, those that focus on representations, and finally those that address the management implications of this syndrome.

The first types of papers aim to assess changes in abundance of species, mostly marine. All of these papers agree on ecosystem degradation, but they mobilize a variety of resources and methods (Pinnegar and Engelhard 2008). Some authors rely on scientific or naturalist data (Baum and Myers 2004 ;McLean et al. 2016). Some authors use scientific or naturalist data, others use fisheries catches and trade data (Guénette and Gascuel 2012;Alleway and Connell 2015). Some studies mobilize less formal sources, such as historical photographs of recreational fishing trophies (McClenachan 2009) or archaeological finds (Drew et al. 2013). Other studies rely on both formal and informal data (Godoy et al. 2010 ;Whipple et al. 2011 ;Fortibuoni et al. 2016). Finally, several works seek to incorporate Pauly's cherished "anecdotes" (Ainsworth et al. 2008;Parsons et al. 2009).

While some studies take a comparative approach, and replicate the same data collection protocol as one conducted several decades earlier (Price et al. 2014), most of these studies simply paint a picture of past ecosystems, without addressing the issue of perceptions. They evoke a shifting baseline syndrome from the baseline without being able to demonstrate it.

The second type of paper focuses on representations of the baseline and its evolution through time. Several studies compare the representation of a "natural" or "good condition" ecosystem across generations (Bunce et al. 2008 ;Turvey et al. 2010;Burbano et al. 2014;Giglio et al. 2014;Katikiro 2014 ;Barbosa-Filho et al. 2020). Researchers have reproduced the same survey protocol as a study conducted in the 1970s (Leeney and Downing 2016). An ethnobotany study compares the ability of different generations to name local species in their local language (Kai et al. 2014). Finally, in the tradition of Pauly, who took himself as an object, some studies focus on the evolution of the baseline in the scientific life of researchers (Muldrow et al. 2020).

These works conclude that there is a change in the baseline between generations, which Papworth (2009) calls generational amnesia. Here again, these articles address the question of representations, without necessarily mobilizing scientific data that would allow us to compare the past state of the ecosystem with the perceived state. Finally, few studies combine a rigorous description of ecological dynamics with an analysis of representations.

Finally, the third type of article focuses on the management consequences of the baseline shift. The shifting baseline syndrome creates an increased tolerance for environmental degradation, resulting in a low level of ambition for conservation or restoration (Soga and Gaston 2018;Guerrero-Gatica et al. 2019). Moreover, if the baseline is not perceived in the same way by all the actors, there may be conflicts between those who implement restoration operations (scientists, managers, technicians) and the users (Guerrero-Gatica et al. 2019). To prevent this syndrome, several avenues are possible: improving knowledge of past ecosystems, encouraging direct experiences with nature or environmental education (Soga and Gaston 2018). Until then, it is necessary to follow the precautionary principle to avoid underestimating the degradation of the environment (Dudgeon 2010). Some authors take a critical look at Pauly's theory. Longitudinal studies of ocean history are few, and studies often mobilize unreliable data; they also resort to extrapolation or focus exclusively on certain species (Campbell et al. 2009;Alagona et al. 2012). The shift in the baseline is also built on assumptions that are not made explicit. The first is the existence of a pristine nature, past and gone, that humans have inexorably degraded. This assumption deserves to be discussed, insofar as it is based on a typically Western separation of humans and nature, according to which marine ecosystems are not populated (Campbell et al. 2009). Many protected sites are more cultural than natural landscapes (Hilding-Rydevik et al. 2017). Moreover, this vision of a past stability contrasts with a dynamic conception of ecosystems, which prevails in ecology today (Campbell et al. 2009). The second unexplained hypothesis is that the past baseline also constitutes a management objective, a desirable future, in a nostalgic approach to conservation ecology. However, Alagona et al. (2012) remind us that if the past can be used as a parable, it cannot be used as the only guide for the future.

All of these reflections invite us to pay close attention to the different representations of the baseline, and to the tensions and disagreements that can exist around the shifting baseline syndrome between the actors of a territory. This is what we highlight in the case study of Mont Lozère (Box 4.2.).

We explored the hypothesis of a shifting baseline syndrome on Mont Lozère, based on 46 individual semi-structured interviews, with actors from the agricultural, forestry, tourism, conservation and hunting communities (2016)(2017), and participant observation by the Scientific Council of the Cévennes National Park (Conseil scientifique du Parc National des Cévennes) (2017), which focused on landscape evolution in relation to agropastoral practices (Moreau, et al. 2019a).

Our results show that among the actors in the field, the dominant representation is that of "landscape closure", experienced as a negative phenomenon, against which it is necessary to fight by "landscape maintenance", i.e. by maintaining "open" landscapes, or even by reopening certain "closed" environments. However, other representations exist, such as that of the foresters, who consider that "we must take advantage of the increase in forest cover", by an ecological or economic development of these landscapes. These actors situate the baseline in a period prior to the agricultural development of the territory, sometimes "the Middle Ages", often in more vague terms. They put forward the "natural" character of the forest, referring as much to the past state of the ecosystem as to a climax. We can therefore speak of a shift in the baseline, insofar as forest landscapes are regaining legitimacy. Indeed, in the past, such normative shifts have already taken place: it is only since the 1970s that the ecological and cultural interest of pastoral landscapes has been put forward (Marty and Lepart 2001).

At the same time, although a large number of actors wish to "maintain open landscapes", they do not all agree on the type of open landscapes to be preserved. For the National Park agents or the tourism actors, the evolution of agricultural landscapes under the effect of rock removal and prairie turning constitutes a "degradation" of heritage landscapes. According to them, there is a significant risk of a shifting baseline syndrome: insofar as these changes are temporary and gradual, they will not necessarily be perceptible to the actors who will gradually get used to them. For other actors, mainly livestock farmers, the evolution of landscapes under the effect of these practices is considered inevitable, even desirable, often described as an "improvement" of the land and an increase in its productive potential, which is in line with the practices of their ancestors who shaped the landscapes to practice livestock farming.

We are thus witnessing two joint phenomena: on the one hand, the tipping of the baseline through a questioning by certain actors of the supremacy of open landscapes as a baseline, and on the other, a shifting baseline syndrome linked to the evolution of open environments under the effect of the intensification of agricultural practices. These two phenomena are partly linked, as the intensification of practices in open environments can be used by some as an argument in favor of a return to a more natural forest environment. In both cases, mobilizing different baselines is a way for actors to support their practices and their own legitimacy to "create landscape" (Hatzfled 2009), whether they are breeders, National Park agents, agronomists or foresters. If we accept that the baseline is a social construct, it remains to be understood how this construction takes place. Soga et al. (2018) identify several causes of shifting baseline syndrome: incomplete data about the natural environment, impoverished experience, and familiarity with the environment. We can therefore identify four factors that are involved in the construction of the baseline: knowledge, memory, experience, and cultural norms.

Regarding the first of these factors, several works illustrate how changes in knowledge have sometimes led to a reconsideration of the baseline. Thus, at the end of the 19 th century, there was a debate among vegetation specialists about the origin of the open landscapes of the Grands Causses: in the 1880s, the French botanist Charles Flahaut argued that the vegetation was a relic of steppe vegetation, before reversing his position at the end of his career, and asserting that the climactic vegetation of the Causses was forest, with overgrazing being able to explain the current landscapes (Marty and Lepart 2001). These scientific reversals have significant consequences for management. For example, Vera et al. (2010) explain that between the 19 th and early 20 th centuries, the degradation of European forests was attributed to the presence of herbivores1 . On the basis of this evidence, two measures were prescribed to return ecosystems to a "natural" state: spontaneous development of vegetation and exclusion of domestic herbivores. However, more recent studies have shown that domestic animals were integrated into forest ecosystems during the Middle Ages. In addition, scientific knowledge can also be mobilized by actors according to their own strategies, for example by conservation organizations (Barnaud et al. 2021) or agricultural unions concerned with maintaining or developing certain practices (Mathevet and Béchet 2020). Finally, alongside scientific knowledge, local knowledge can also be taken into account in the construction of the baseline. In some cases, the erosion of local knowledge can explain the shifting baseline syndrome (Kai et al. 2014). Finally, we can say that knowledge is socially and historically situated: even if the baseline is based exclusively on knowledge, it remains the object of a social construction.

The second factor is related to memory. Papworth et al. (2009) distinguish between different types of baseline shifts: generational amnesia, where new generations are unaware of past biological conditions, and personal amnesia, where people lose memories of their own experiences. The authors also uncover the phenomena of illusory memory, where people relate a past state of the ecosystem that is not confirmed by scientists, and change blindness, where people continue to refer to the baseline without noticing changes that have affected the ecosystem. Alteration and distortion of memory will therefore influence the baseline.

The third factor involves experiences. Keilty et al. (2016) show how everyday life near dams can change their perception, to the point of being integrated into a landscape considered "normal" or even "natural." The role of experience is thus central, and this will have strong consequences. Thus, urbanization implies a loss of experience of nature, and thus a risk of a shifting baseline syndrome increased (Turner et al. 2004). The risk is then to take ordinary nature as a baseline, which is a considerably impoverished nature compared to wild nature (Maris 2018).

The fourth factor concerns cultural norms. In the Grands Causses, for example, it was from the 1970s onwards that the heritage and ecological value of open landscapes was progressively emphasized, whereas these landscapes had previously been considered as degraded landscapes, the baseline being that of a forest climax to be restored. This change in the baseline is closely linked to an evolution in the social representations of farmers. While pastoral societies were considered until the first half of the 20 th century as disrespectful of the natural equilibrium of the environment, they have been gradually recognized in their heritage role since the 1970s (Marty and Lepart 2001). The Cévennes National Park, which was initially conceived as a forestry park, thus made the maintenance of open landscapes the target of its management, which also made it possible to obtain the support of the local population, which was strongly agricultural (Basset 2009). Through this normative shift, the baseline does not refer to a natural landscape but to a cultural one, thus emphasizing the influence of cultural norms.

Finally, biological or evolutionary factors are also mentioned (Keilty 2016) insofar as aesthetic or environmental preferences can be explained by the evolution of humankind in contact with nature (Kaplan 1987;Wilson 2012).

Finally, these factors combine differently depending on the period, the social group, but also on the individuals. Different representations of the baseline thus coexist within a social group, as we have shown in the case of Mount Lozère (Box 4.3.).

As we have seen, the actors of Mont Lozère are opposed in their representations of the landscape dynamics in progress.

A first type of actor, the majority, considers that the "closing of the landscape" constitutes a loss, both ecologically and culturally. According to this representation, the baseline was situated in the middle of the 20 th century, at the peak of the agricultural development of the territory. While the baseline generally refers to a nature untouched by anthropic impact, Mont Lozère is an emblematic example of the development and conservation of cultural landscapes. This can be explained by two main factors. Firstly, cultural norms were progressively put in place from the 1970s onwards, supported by new knowledge, which allowed the recognition of the heritage value of agro-pastoral landscapes. Secondly, collective memory plays an important role, as actors evoke memories and anecdotes experienced by their ancestors.

A second type of actor, mainly public sector foresters but also inhabitants who have come to settle in this region for its wilderness character, sees the increase in forest cover as an enrichment of biodiversity. They insist on the anthropic character of open landscapes, and question the temporality of landscape standards. Indeed, the forest constitutes for them the "natural" landscape of Mont Lozère, whether one looks to the past (Middle Ages, or even before), or whether one projects oneself in the long term of ecological dynamics. For this, they rely on knowledge of the biodiversity associated with forest landscapes, knowledge that is new but in full development, for example around the question of "old forests".

Finally, among the four factors that condition the baseline, we retain three on Mont Lozere: memory, cultural factors, and knowledge (Figure 4.2.). Conversely, evolutionary factors, identified by Keilty et al. (2016), did not emerge in the interviews. Similarly, experience seems to enter into the picture rather little: the inertia of representations is strong, forest landscapes continue to have a negative connotation, even though the inhabitants have sometimes lived their entire lives in a predominantly forested landscape.

Box 4.3. Mont Lozere and the case of cultural landscapes

Debating the baseline

The baseline has a descriptive value: it gives an image of the past state of the ecosystems, and makes it possible to give meaning to the dynamics in progress. But the baseline also has a normative value. Indeed, identifying a baseline is equivalent to choosing as a model a precise point in the history of relations between societies and nature. The baseline refers to the past, but also implicitly, to an "end state", an ideal, or even a desirable future (Luginbühl and Terrasson 2013).

However, in the discourse of actors or in management documents, the baseline is rarely explained or justified, and it often amalgamates an unrecognized and idealized historical state, persistent current states and a future ideal (Donadieu 2002 ;Davodeau and Barraud 2018). This vagueness surrounding the baseline makes it a unifying element in the governance of landscapes, since it serves as a medium for the expression of various, even incompatible, desires and fears (Davodeau and Barraud 2018). Thus, in mountain territories facing "landscape closure", the reference to agro-pastoral landscapes and the need to keep them open are brandished as a unifying element (Peyrache-Gadeau and Perron 2010), being the object of a consensus, at least in appearance (Barnaud and Couix 2020).

An illusion of consensus around the baseline then emerges, which can mask tensions between divergent representations, or make certain points of view inaudible. For example, in the Pyrenees, the consensus on the need to maintain livestock and open landscapes in reality conceals contradictory opinions, for example in favor of rewilding (Barnaud and Couix 2020). This illusion of consensus is sometimes supported by scientific arguments, presented as neutral, whereas knowledge is historically and socially situated, and the mobilization of knowledge by local actors is also done in a strategic perspective (Barnaud et al. 2021).

Finally, the baseline is neither scientific evidence nor necessarily a consensus, but rather a matter of choice. Indeed, the choice of a baseline will determine the feasibility and effort required to achieve conservation goals (Bull et al. 2014). It is therefore necessary to set up discussion arenas, in which the definition of the baseline and the definition of its role in management will be debated at the same time (Mathevet et al. 2015 ;Moreau 2019). The objective is then to recognize the legitimacy of each person to "make the landscape", by means of their practices or the enactment of standards (Hatzfled 2009).

Recognizing the plurality of viewpoints and their legitimacy within a constructivist approach is the basis of many participatory approaches (Barnaud and Mathevet 2015). The role of scientific knowledge in the definition of the baseline is then revised. Without falling into relativism, it is a question of recognizing the existence of other types of knowledge, such as local knowledge (Kai et al. 2014) but also recognizing that scientific knowledge is subject to numerous biases, and that scientists must also make explicit the biases underlying their hypotheses (Etienne 2010; Barnaud et al. 2021).

We have proposed to debate the baseline on Mont Lozère through a participatory approach to modelling (Etienne, 2010) based on a role-playing game (see Box4. 4).

Our qualitative analysis revealed significant divergence of views on Mont Lozère around the question "Open landscapes, yes, but which ones?" (Moreau et al. 2019a).

Debating the baseline through a participatory approach therefore seemed necessary, which we did as part of a SECOCO (Services ECOsystémiques et action COllective, 2015-2019) research project, conducted in partnership with the National Park team. We designed a role-playing game, SECOLOZ, modelling the effects of agricultural practices on landscapes and the ecosystem services they provide. Two workshops were organized on Mont Lozère, with farmers, elected officials and National Park agents (Moreau et al. 2019b). The analysis of the exchanges during the parties and during the debriefing reveals two points of debate on the baseline.

Firstly, participants debated the baseline's degree of naturalness. According to the farmers, the practices of rock removal and prairie turning are part of a long history of shaping landscapes through agricultural practices; there is nothing "natural" about these landscapes. For the Park's agents, the current changes are not of the same nature, due to their magnitude and irreversible nature. According to them, "naturalness" is not to be opposed to the "artificial" character of open landscapes: it is synonymous with biological richness. Thus, permanent or temporary grassland, although both resulting from human action, are in no way equivalent in their eyes in terms of biodiversity.

Secondly, during the game, the Park agents drew the attention of the other players to the evolution of the game board, thus using the game to alert them to the risk of a shift in the baseline on Mont Lozère. On the other hand, the breeders consider that the risk of shifting from the baseline is low, in a conservation context that tends to make landscapes more secure. More generally, the farmers argue that the baseline can be considered as a starting point, but not as a management objective.

We thus proposed a concrete device for debating the baseline. The simplification of reality, the playful aspect and the exchange of roles allowed us to distance ourselves from daily situations. The reflection initiated by the National Park, in which the SECOLOZ roleplaying game was included, led to a modification of the authorization system, in order to promote mutual understanding between the Park and its citizens.

Box 4.4. SECOLOZ, a role-playing game for debating the baseline

Conclusion

Questioning the baseline leads us to collectively define a desirable nature and a desirable future, which is central in a context of global change. This review of the literature on the baseline and its shift, as well as our study of the changing landscapes of Mont Lozère, show the need to support the reflection of territorial actors with critical and integrated approaches, combining the study of social representations and practices, and the study of the ecology of the environments concerned. The material and immaterial dimensions of the baseline must be grasped both in the long term nature of their historical and political trajectory, in the plurality of points of view and the diversity of scholarly and lay knowledge of the evolutionary processes. The ecological, economic, social and institutional transition of the territories is always in search of new development models, new values, new projects for the transformation of the socio-ecological community. Will a better knowledge of the baselines, (favoring a fine knowledge of our interdependencies through dialogue and action) be able to found new trajectories for our territories of life, and thus contribute to the well-being of humans and non-humans?

Box 4. 1 .

 1 Presentation of the case study

Figure 4 . 1 .

 41 Figure 4.1. Agro-pastoral landscapes of Mont Lozère.Credit N. Salliou

Box 4. 2 .

 2 Tilting and shifting of baselines regarding Mt. Lozère 4.3. How is the baseline constructed?

Figure 4 . 2 .

 42 Figure 4.2. Typical citations for the role of different factors identified by Keilty et al. (2016) in the construction of baselines on Mt. Lozere. Scale Invidual Collective Flexibility Hardwired

See Laurent Godet's chapter in this book "Rewilding by the return of the ghosts of the past".

Acknowledgements

We would like to thank all the people interviewed during the work on Mont Lozère, the participants in the role-playing workshops, and the Secoco project team. This work was financed by the Ecoserv Metaprogram, the SAD department of INRAE and the Cévennes National Park.