text:
Franca Somers

Clément Roos

Francesco Sanfedino

Samir Bennani

Valentin Preda

A µ-analysis based approach to probabilistic delay margin analysis of uncertain linear systems*

Monte Carlo and deterministic µ-analysis are well known tools in aerospace industry for Validation and Verification (V&V) purposes. On the one hand, Monte Carlo analysis can provide an accurate measure of probability for sufficiently frequent phenomena. Nevertheless, it is often timeconsuming and may fail to compute worst-case stability margins and performance levels. Deterministic µ-analysis, on the other hand, cannot miss worst-case scenarios. But in the absence of a measure of probability, a control system may be invalidated on the basis of extremely rare events. About twenty years ago, probabilistic µ-analysis was therefore introduced to bridge the analysis gap between both techniques. It combines efficient µbased algorithms with a branch-and-bound scheme to explore the whole uncertainty domain, also considering probability distributions on the uncertain parameters. In this context, this paper describes a novel algorithm for probabilistic delay margin analysis, which provides upper and lower bounds on the probability of an uncertain system violating a desired delay margin.

I. INTRODUCTION

Time delays occur in almost every control engineering problem, and generally have a significant impact on system behavior. Most studies, including this work, focus on constant time delays. A wide variety of analysis techniques are available in both the time and frequency domains.

Time domain techniques mainly use Lyapunov-Krasovskii functionals or Lyapunov-Razumikhin functions [START_REF] Gu | Stability of time-delay systems[END_REF]. These approaches are based on the classical Lyapunov theory, and aim at constructing nontraditional energy-storage functions/functionals for time-delay closed-loop systems. Much literature deals with deterministic linear time-invariant (LTI) problems in presence as well as absence of uncertainty, see e.g. [START_REF] Kharitonov | Robust stability analysis of time delay systems: a survey[END_REF], [START_REF] Gouaisbaut | Delay-dependent robust stability of time delay systems[END_REF]. The use of Lyapunov function(al)s usually leads to criteria that can be expressed in terms of Linear Matrix Inequalities (LMI) and solved with dedicated solvers.

Frequency domain approaches are generally based on the small-gain theorem and use µ-based techniques [START_REF] Zhou | Robust and optimal control[END_REF] or Integral Quadratic Constraints (IQC) [START_REF] Megretski | System analysis via integral quadratic constraints[END_REF]. First, µ-analysis makes use of the structured singular value µ to study robust stability and performance problems accounting for the structure of the uncertainties in linear models. Examples of application to uncertain time-delay systems can be found in [START_REF] Huang | Robust stability of uncertain time-delay systems[END_REF] and [START_REF] Lescher | Robust stability of time-delay systems with structured uncertainties: a µ-analysis based algorithm[END_REF]. Then, IQC analysis can be interpreted as the combination of (scaled) small gain techniques, mainly µ-analysis, and positivity/passivity techniques which study the interconnection of a linear operator with a non-linearity ("Lur'e problem"). It is used to study robust stability and performance properties of uncertain, time-varying and nonlinear systems. Some applications of IQC to robustness analysis of time-delay systems can be found in [START_REF] Megretski | System analysis via integral quadratic constraints[END_REF] and [START_REF] Jun | IQC robustness analysis for time-delay systems[END_REF].

Most frequency domain approaches require to replace the constant delay element e -sτ , where s is the Laplace variable and τ the considered time delay in seconds, with a rational expression. This step has a major influence on the conservatism of the analysis. In some studies, time-delay elements are eliminated by covering their value sets with unit disks, which leads to overly conservative results [START_REF] Zhang | Stability of time-delay systems: equivalence between Lyapunov and scaled small-gain conditions[END_REF]. Other approaches replace the exponential term by Padé approximations [START_REF] Zhang | Design of wide-area damping control robust to transmission delay using µ-synthesis approach[END_REF], Taylor series expansions [START_REF] Hanta | Rational approximation of time delay[END_REF] or the Laguerre formula [START_REF] Lam | Analysis on the Laguerre formula for approximating delay systems[END_REF]. . . The accuracy of these approximations can often be improved by introducing more complexity. In contrast, an exact representation of the time delay effects is proposed in [START_REF] Lescher | Robust stability of time-delay systems with structured uncertainties: a µ-analysis based algorithm[END_REF]. The replacement function has the same properties (unit gain and phase varying linearly with frequency) as the actual time delay. The price to pay is the introduction of a parameter with frequency dependent bounds.

With this literature review in mind, let us now describe the context and the motivations of this work. Due to their simplicity, Monte Carlo (MC) simulations [START_REF] Landau | A guide to Monte Carlo simulations in statistical physics[END_REF] have long been the preferred Verification and Validation (V&V) approach in the industry. They can provide an accurate measure of probability for sufficiently frequent phenomena. Nevertheless, they are often time-consuming and may fail to compute worst-case stability margins and performance levels. In contrast, deterministic simulation-free techniques such as µ and IQC analysis have reached a good level of maturity. Unlike MC simulations, rare scenarios are no longer missed, but their probability of occurrence is not measured. Analysis results can therefore be overly conservative [START_REF] Khatri | Guaranteed bounds for probabilistic µ[END_REF] and a control system may be invalidated on the basis of very rare and therefore extremely unlikely events [START_REF] Marcos | Stochastic µ-analysis for launcher thrust vector control systems[END_REF]. So there is a real need to develop new methods to fill the gap between MC simulations and worst-case analysis. The work presented in this paper seeks precisely to make a step in this direction.

However efficient and accurate they may be, LMI-based time-domain approaches seem difficult to implement at the present time when it comes to be fast while analyzing realistic systems characterized by high-dimensional models with multiple uncertainties. On the other hand, frequencydomain approaches such as µ and IQC analysis have shown their ability to handle some real-world systems. IQC analysis has the advantage of being able to deal with uncertainties, time-varying parameters and non-linearities at the same time. But it can be conservative and usually requires to use an LMI solver, which significantly impacts the computational time as already highlighted above. On the opposite, µanalysis is restricted to the analysis of uncertain systems, but guaranteed and very accurate bounds on stability margins and performance levels can usually be obtained very quickly [START_REF] Roos | A detailed comparative analysis of all practical algorithms to compute lower bounds on the structured singular value[END_REF]. And probabilistic µ-analysis has even been identified for many years as a serious candidate to the analysis of rare events [START_REF] Zhu | Soft vs. hard bounds in probabilistic robustness analysis[END_REF], [START_REF] Khatri | Guaranteed bounds for probabilistic µ[END_REF]. A significant effort has been made in recent years to propose new probabilistic µ tools, which are both versatile and computationally efficient [START_REF] Roos | A new step towards the integration of probabilistic µ in the aerospace V&V process[END_REF], [START_REF] Somers | Probabilistic gain, phase & disk margins with application to AOCS validation[END_REF]. The objective of this paper is now to focus on delay margin. Considering the aforementioned literature review on timedelay systems and V&V needs, it seems relevant to propose a µ-analysis based approach to probabilistic delay margin analysis, able to compute tight bounds on the probability of delay margin violation for linear systems in the presence of parametric uncertainties with given probability distributions.

The paper is organized as follows. The considered problem is first stated in Section II. Section III gives a brief overview to the proposed solution. The main theoretical results are then detailed in Section IV, as well as a practical algorithm. The potential of this algorithm is finally evaluated in Section V.

II. PROBLEM STATEMENT

Let us consider the following continuous-time uncertain LTI system:

ẋ = A(δ)x + B(δ)u y = C(δ)x + D(δ)u (1)
The real uncertain parameters δ = (δ 1 , . . . , δ N ) are bounded and without loss of generality normalized, so that the whole set of admissible uncertainties is covered when In this work, system (1) and Fig. 1 describe the control loop opened at the place where the delay margin should be computed. The closed-loop interconnection is therefore recovered by applying a unit negative feedback between y and u, i.e. by setting G(s) = 1 in Fig. 2.

δ ∈ B δ = [- 1

Fig. 2: Feedback loop for delay margin analysis

Using these notations, probabilistic delay margin analysis can be formalized as follows:

Problem 2.1: Compute the probability P ϕ ∆,f (M (s)) that the delay margin is smaller than a given threshold ϕ when ∆ ∈ B ∆ for the negative feedback loop obtained by connecting y to u in Fig. 1. So, P ϕ ∆,f (M (s)) is the probability that the interconnection of Fig. 2 is not stable for all G(s) = e -τ s , τ ∈ [0 ϕ]. A control system can then be rejected or validated depending on whether P ϕ ∆,f (M (s)) does or does not exceed a given threshold ϵ. A practical approach is presented in Section IV to compute tight bounds on P ϕ ∆,f (M (s)) for Single-Input Single-Output (SISO) systems, i.e. when u ∈ R and y ∈ R. The global underlying idea is now summarized in Section III.

III. OVERVIEW OF THE PROPOSED SOLUTION

During the past 5 years, a significant effort has been put in the development of probabilistic µ theory and its implementation in the Stochastic Worst Case Analysis Toolbox (STOWAT) [START_REF] Roos | A new step towards the integration of probabilistic µ in the aerospace V&V process[END_REF]. Stability and H ∞ performance were studied first [START_REF] Thai | Probabilistic µ-analysis for stability and H∞ performance verification[END_REF], [START_REF] Biannic | Advanced probabilistic µ-analysis techniques for AOCS validation[END_REF], followed by gain/phase/disk margins [START_REF] Somers | Probabilistic gain, phase & disk margins with application to AOCS validation[END_REF]. The novelty of this paper lies in the introduction of a delay margin analysis algorithm to solve Problem 2.1, and to enrich this set of probabilistic µ-based tools.

Probabilistic µ-analysis builds upon classical µ. A preliminary step is therefore to transform the interconnection of Fig. 2 into an LFR, which requires to replace the exponential delay G(s) = e -τ s with a rational function Φ(α). A simple but exact representation is proposed in Section IV-A, in the sense that it has the same properties (unit gain and phase varying linearly with frequency) as the actual time delay. This is the first main contribution of the paper.

From an algorithmic point of view, probabilistic µ-analysis combines classical µ-based tools with a branch-and-bound (B&B) algorithm to explore the whole uncertainty domain B δ . The delay margin is first computed at the center of B δ , i.e. for ∆ = 0. If it is larger (resp. smaller) than the desired threshold ϕ, it is then checked whether the delay margin requirement is satisfied (resp. violated) on the entire domain B δ using sufficient conditions involving µ upper bound computations (see Propositions 4.1 and 4.3 in Sections IV-B and IV-C). If this cannot be guaranteed, B δ is finally partitioned into smaller boxes and this process is repeated until each box has guaranteed sufficient/insufficient margin, or is small enough to be neglected (see Section IV-D). Guaranteed upper and lower bounds on the exact probability of delay margin violation P ϕ ∆,f (M (s)) are finally obtained, based on the probability distributions of the uncertain parameters δ, thus solving Problem 2.1.

A major difficulty with respect to the gain/phase/disk margins considered in [START_REF] Somers | Probabilistic gain, phase & disk margins with application to AOCS validation[END_REF] is that the aforementioned exact rational representation of the delay function introduces a parameter with frequency-dependent bounds that adds to the uncertainties already present in ∆. An algorithm is proposed in Section IV-B to take this dependence on frequency into account, which is not possible with state-of-the-art µ-based tools. This is the second main contribution of the paper. As explained in Section IV-C, this exact representation can for the moment only be used for the delay margin satisfaction test, and its applicability to the violation test remains an open issue currently under investigation. A second-order Padé approximation is therefore introduced in Section IV-A and used in a more classical way to perform the delay margin violation test in Section IV-C.

IV. PROBABILISTIC DELAY MARGIN A. Interconnection for delay margin analysis

The negative feedback loop Fig. 2

G(s) = e -τ s (2)
and the nominal closed-loop uncertain system is obtained for τ = 0, i.e. G(s) = 1. G(s) does not have a rational dependence on τ . Therefore, it should first be replaced with a rational expression to be able to build an LFR and apply robustness tools such as µ-analysis. Two expressions are used: one to check whether the delay margin is larger than ϕ over an entire box D ⊂ B δ (satisfaction test, see Section IV-B) and the other to check whether it is lower than ϕ (violation test, see Section IV-C).

The transformation used for the delay margin satisfaction test is based on the same idea as in [START_REF] Lescher | Robust stability of time-delay systems with structured uncertainties: a µ-analysis based algorithm[END_REF], but it is much simpler. G(s) has a unitary gain and introduces a frequencydependent phase shift equal to -τ ω at the frequency ω. The delay margin is thus guaranteed to be larger than ϕ if the system remains stable for any phase shift in the interval I = [-min(ϕω, 2π) 0] at the frequency ω. In this context, G(s) can be replaced with any rational expression Φ(α) such that |Φ(α)| = 1, and whose phase ∠Φ(α) covers the interval I when α covers [0 α max (ω)]. The following choice is made:

Φ(α) = 2jα 2 -2(1 + j)α + 1 -2jα 2 -2(1 -j)α + 1 (3)
for which:

α max =          1 -β -1 + β 2 2 if ω < π/ϕ 0.5 if ω = π/ϕ 1 -β + 1 + β 2 2 if ω > π/ϕ (4)
and β = tan -min(ϕ ω,2π)

2

. To get equation ( 4), just note that:

∠Φ(α) = 2 arctan 2α(α -1) 1 -2α (5)
which gives for α = α max :

tan ∠Φ(α max ) 2 = β = 2α max (α max -1) 1 -2α max (6)
and finally 2α 2 max + 2(β -1)α max -β = 0. The motivation for this choice of Φ(α) is that ∠Φ(α) and α max (ω) are simple functions with a smooth behavior, as shown in Fig. 3. 2, where G(s) is replaced with Φ(α), into that of Fig. 4, where ∆ m = αI 2 , α ∈ [0 α max (ω)], contains a single parameter with frequency dependent bounds. Note that a rational function Φ(α) of degree 1 in α would lead to a simpler LFR, where α is not repeated in ∆ m . This is however not possible here. α would indeed tend to ±∞ to allow ∠Φ(α) to cover [-2π 0], but the µ-based tools used in the sequel require the ∆ block of the LFR to be bounded. Fig. 4: Transformation of the interconnection of Fig. 2 For the delay margin violation test, the exponential term (equation 2) is replaced by a Padé approximation, whose order is chosen based on a balance between accuracy, complexity and validity. Indeed, the higher the order, the better the accuracy, but the larger the number of repetitions of τ in the resulting LFR. Moreover, the negative phase shift caused by the approximation should be less in absolute value than the one induced by the true time delay for all frequencies, as clarified in Section III.C, and this property may not be satisfied for all orders. In view of these elements, replacing G(s) by a second-order Padé approximation:

Ψ(s) = (τ s) 2 -6τ s + 12 (τ s) 2 + 6τ s + 12 (7)
appears to be a good compromise. It has indeed been observed in practice that the first-order approximation is not accurate enough, whereas the second-order one is usually sufficient, as shown in Section V. Moreover, the second-order Padé approximation satisfies the condition on the phase. The use of a approximation does not raise any technical difficulty, provided that the condition on the phase is satisfied. To equivalently transform the interconnection of Fig. 2, where G(s) is replaced with Ψ(s), into that of Fig. 4, where ∆ m = τ I 2 , τ ∈ [0 ϕ], an approach similar to the one used for first order Padé approximations in [START_REF] Zhang | Design of wide-area damping control robust to transmission delay using µ-synthesis approach[END_REF] is applied.

B. Checking stability margin satisfaction on a box

Let us consider a box D ∈ B δ and the exact characterization (3) of the delay function. The interconnection N (s)diag (∆, αI 2 ) , ∆ ∈ D ∆ , of Fig. 4 is normalized and equivalently replaced by Ñ (s)diag ( ∆, αI 2 ), ∆ ∈ B ∆ . Checking whether the delay margin requirement is satisfied on the entire box D can then be done using the necessary and sufficient condition of Proposition 4.1.

Proposition 4.1: The delay margin is larger than ϕ on a given box

D ⊂ δ if only if the interconnection Ñ (s) - diag ( ∆, αI 2 ) is stable ∀ ∆ ∈ B ∆ and ∀α ∈ [0 α max (ω)].
Proposition 4.1 requires solving a non-standard µ-analysis problem, where the bound on α is frequency-dependent. A classical approach is to reduce the whole frequency range to a finite grid

(ω k ) k . α is then normalized at each ω k , i.e. α = αmax(ω k ) 2 (δ m + 1), δ m ∈ [-1 1], resulting in a fully normalized interconnection Ñ (s) -diag ( ∆, δ m I 2 ). It is finally checked if µ( Ñ (jω k ))
≤ 1 for all k (due to NP-hardness, an upper bound µ( Ñ (jω k )) is computed in practice, see [START_REF] Young | Computing bounds for the mixed µ problem[END_REF]). But this strategy is doomed to fail. When the box D gets closer to the limit between the domains of delay margin satisfaction and violation, the peak value of µ( Ñ (jω)) tends to 1 and the risk of missing critical frequencies where µ > 1 increases. Adding more frequencies to the grid can be tempting, but the computational time quickly becomes prohibitive. Moreover, several tests have shown that all frequencies where µ > 1 always end up being missed, thus leading to the erroneous claim that the delay margin requirement is met. A better approach uses the Hamiltonianbased algorithm of [START_REF] Roos | Efficient computation of a guaranteed stability domain for a high-order parameter dependent plant[END_REF], which computes a guaranteed µ upper bound over an entire frequency interval. A three-step strategy is proposed here, as detailed in Algorithm 1. The method of [START_REF] Roos | Efficient computation of a guaranteed stability domain for a high-order parameter dependent plant[END_REF] is first applied at a given frequency ω k assuming that all uncertainties have constant bounds, which leads to an initial interval [ω min ω max ] including ω k on which µ( Ñ (jω)) ≤ 1 (step 1). But α has an increasing frequency-dependent upper bound, as shown in Fig. 3 (right). Therefore, ω min is an admissible lower bound on the validity interval for the considered delay margin problem, but it can be further decreased (step 2). On the opposite, ω max is too large and is reduced by means of a dichotomic search (step 3). Algorithm 1 is repeated until the union of all validity intervals covers the whole frequency range, which guarantees that the delay margin is larger than ϕ, or until an ω k is found such that µ( Ñ (jω k )) > 1. In the latter case, either the delay margin is not larger than ϕ, or the µ upper bound is too conservative to prove the converse.

Algorithm 1 Validity interval [ω min ω max ] around ω k
Step 1 -Initialization:

1) select an unevaluated frequency ω k and normalize α at ω k to get the fully normalized interconnection

Ñ (s) -diag ( ∆, δ m I 2 ), ∆ ∈ B ∆ , δ m ∈ [-1 1] 2) check whether µ( Ñ (jω k )) ≤ 1 3) if not then STOP else use [23] to compute an initial interval [ω min ω max ] around ω k on which µ( Ñ (jω)) ≤ 1 Step 2 -Lower bound improvement: initialization: set ω test = ω k while ω test -ω min > ϵ do 1) set ω test = ω min 2
) normalize α at ω test and compute Ñ (s) 3) use [START_REF] Roos | Efficient computation of a guaranteed stability domain for a high-order parameter dependent plant[END_REF] to compute an interval [ω 1 ω 2 ] around ω test on which µ( Ñ (jω)) ≤ 1 4) set ω min = ω 1 end while Step 3 -Upper bound improvement:

initialization: set ω low = ω k and ω high = ω max while ω high -ω low > ϵ do 1) set ω test = (ω high +ω low ) 2
2) normalize α at ω test and compute Ñ (s)

3) check if the D and G scaling matrices associated to the µ upper bound of step 1 are still valid at ω low 4) if valid then

• use [START_REF] Roos | Efficient computation of a guaranteed stability domain for a high-order parameter dependent plant[END_REF] to determine the validity interval [ω 1 ω 2 ] • if ω 2 > ω test then set ω max = ω test and ω low = ω test else set ω high = ω test end while

C. Checking stability margin violation on a box

The Padé approximation [START_REF] Lescher | Robust stability of time-delay systems with structured uncertainties: a µ-analysis based algorithm[END_REF] of the delay function is now used. The interconnection

N (s) -diag (∆, τ I 2 ) , ∆ ∈ D ∆ , τ ∈ [0 ϕ], of Fig. 4 is normalized and equivalently replaced by Ñ (s) -diag ( ∆, δ m I 2 ), ∆ ∈ B ∆ , δ m ∈ [-1 1], i.e. τ = ϕ 2 (δ m +1). Proposition 4.
2 then provides a sufficient condition to check whether the delay margin requirement is violated on the entire box D ⊂ B δ . Proposition 4.2: The delay margin is lower than ϕ on a given box

D ⊂ B δ if ∀ ∆ ∈ B ∆ , ∃ δm ∈ [-1 1] such that the interconnection Ñ (s) -diag ( ∆, δm I 2 ) is unstable.
⊂ B δ if ∃ δm ∈ [-1 1] such that the interconnection Ñ (s)-diag ( ∆, δm I 2 ) is unstable ∀ ∆ ∈ B ∆ .
As suggested in [START_REF] Somers | Probabilistic gain, phase & disk margins with application to AOCS validation[END_REF], δm is determined by studying the stability of the nominal interconnection Ñ (s)diag (0 p×p , δ m I 2 ). A finite number of values (δ k m ) k which grid the interval [-1 1] are considered, and the one which moves a pole of the interconnection the farthest in the right half-plane is selected to be δm . As δm remains constant in Proposition 4.3, it can be integrated into Ñ (s) to form a reduced normalized interconnection Ñr (s)-∆, where Ñr (s) is unstable. It is then sufficient to check whether this reduced interconnection remains unstable ∀ ∆ ∈ B ∆ . This can be done easily by determining with standard µ-based tools if µ( Ñr (jω)) ≤ 1 on the whole frequency range.

Unlike the delay margin satisfaction test of Section IV-B, some conservatism is introduced here. First, the Padé approximation ( 7) is used instead of the exact delay (2). This is admissible, since its phase is smaller in absolute value than the one of G(s) for all frequencies. But unlike Proposition 4.1, this makes the condition of Proposition 4.2 non-necessary. Higher-order Padé approximations can be used to mitigate this conservatism, at the price of a slight increase in the computational time. Second, δm is fixed to a constant value in Proposition 4.3, which makes the condition even more non-necessary. But as the size of D decreases along the iterations of the B&B algorithm (see Section IV-D), it is reasonable to think that there is more and more chance that a single value of δm will fit.

Remark 4.1: The exact characterization (3) cannot be used as for the satisfaction test. Indeed, the upper bound α max (ω) on α is frequency-dependent, and tends to 0 when ω → 0. So δm in Proposition 4.3 could not be constant and would necessarily depend on ω. As a result, the state-state matrices of Ñr (s) would also be frequency-dependent, thus preventing to use the Hamiltonian-based approach of [START_REF] Roos | Efficient computation of a guaranteed stability domain for a high-order parameter dependent plant[END_REF] in Algorithm 1.

D. Algorithmic issues

The conditions for determining whether a given delay margin requirement is satisfied (Proposition 4.1) or violated (Proposition 4.3) on an entire box are embedded into the same B&B algorithm as in [START_REF] Somers | Probabilistic gain, phase & disk margins with application to AOCS validation[END_REF], which is not reproduced here for the sake of brevity. B δ is divided into smaller and smaller boxes until each box has guaranteed sufficient/insufficient delay margin, or has a probability lower than a user-defined threshold p min . Before that, a preliminary stability analysis should be performed with Algorithm 1 of [START_REF] Thai | Probabilistic µ-analysis for stability and H∞ performance verification[END_REF], leading to:

B δ = D s ∪ D s ∪ D su (8)
where D s , D s and D su are the domains of guaranteed stability, guaranteed instability and undetermined stability respectively. The delay margin can indeed only be evaluated for stable systems, so the application of the delay margin algorithm is restricted to D s , leading to:

D s = D m ∪ D m ∪ D mu (9)
where D m , D m and D mu are the domains of guaranteed delay margin satisfaction, guaranteed delay margin violation and undetermined delay margin, with corresponding probabilities p(D m ), p(D m ) and p(D mu ). Combing ( 8) and ( 9), the final partitioning of B δ can be obtained as:

B δ = D m ∪ D m ∪ D mu ∪ D s ∪ D su (10)
Guaranteed bounds on the exact probability P ϕ ∆,f (M (s)) of delay margin violation are finally obtained, which solves Problem 2.1:

p(Dm) ≤ P ϕ ∆,f (M (s)) ≤ p(Dm) + p(Dm u ) = p(Ds) -p(Dm)
V. NUMERICAL RESULTS The proposed algorithm has been implemented in the STOWAT and is validated here on a simple satellite model with two uncertainties, adapted from [START_REF] Zhou | Robust and optimal control[END_REF]. The low number of uncertainties indeed allows the results to be represented graphically, which enhances clarity. The satellite is represented as a symmetric cylinder spinning around the symmetry axis z with a constant angular rate Ω. The angular rates ω x and ω y around the x and y axes are controlled using torques T x and T y . Let I x , I y = I x and I z be the inertia of the satellite with respect to the x, y and z axes respectively. The system rotational motion can be described by:

T x = I x ωx -ω y Ω(I x -I z ) T y = I x ωy -ω x Ω(I z -I x ) (11)
Uniformly distributed uncertain parameters δ 1 ∈ [-0.5, 2.5] and δ 2 ∈ [0, 2] are introduced, leading to:

ωx ωy = 0 a -a 0 ω x ω y + δ 1 0 0 δ 2 u x u y (12)
where u x = Tx Ix , u y = Ty Ix . Two measures ν x , ν y are available:

ν x ν y = 1 a -a 1 ω x ω y (13)
and a static controller K is applied:

u x u y = -K ν x ν y = - 1 0 0 1 ν x ν y (14)
It is assumed in the sequel that a = 10. A preliminary probabilistic stability analysis is performed on the uncertain closed-loop system ( 12)-( 14) using Algorithm 1 of [START_REF] Thai | Probabilistic µ-analysis for stability and H∞ performance verification[END_REF]. The first channel of the control loop is then opened, the second one remaining closed. An LFR is obtained as in Fig. 1, where u = u x and y = ν x . A probabilistic delay margin analysis is finally performed with ϕ = 0.2 using the proposed algorithm.

The B&B algorithm stops when p(D) ≤ p min = 1/5000% for all D ∈ L. It can be determined in a few seconds that: 1) 48.1% of the uncertainty domain [-0.5 2.5] × [0 2] is guaranteed to be stable in closed loop, 2) the probability of delay margin violation satisfies P ϕ ∆,f (M (s)) ∈ [26.4% 28.2%]. Results are presented in Fig. 5 with the following color code:

• green: the delay margin is guaranteed to be ≥ ϕ,

• red: the delay margin is guaranteed to be < ϕ,

• blue: the delay margin is undetermined,

• orange: instability is guaranteed,

• gray: stability is undetermined. A comparison is made with a classical grid-based approach:

• magenta: the delay margin is guaranteed to be ≥ ϕ,

• yellow: the delay margin is guaranteed to be < ϕ,

• black: the system is unstable.

Both analyses are in good agreement, but the STOWAT provides guaranteed results on the whole uncertainty domain, and not only for a finite number of points.

However, the second-order Padé approximation can sometimes lead to some conservatism. If the same analysis is performed with ϕ = 0.4, a substantial part of the uncertainty domain remains undetermined, as seen in Fig. 6 (blue region in the area δ 1 ∈ [0 1] and δ 2 ∈ [0 0.75]). A comparison with a grid-based approach indeed shows that in this region, the system is unstable when perturbed by a true time delay (upper plot), but remains stable when perturbed by a secondorder Padé approximation (lower plot). The conservatism in this region cannot be reduced by changing the stopping criterion of the B&B algorithm, but probably by considering higher-order Padé approximations, at the price of a slight increase in the computational time.

VI. CONCLUSION

An algorithm for probabilistic delay margin analysis of SISO systems has been outlined and integrated into the STOWAT. Its capabilities and limitations have been presented through the application to a demonstrative benchmark. Follow-up research will focus on replacing the Padé approximation-based violation test with an exact computational method, as it is already the case for the margin satisfaction test, so as to reduce conservatism.

Fig. 1 :

 1 Fig. 1: Linear Fractional Representation (LFR)

  is first built from the LFR of Fig. 1. G(s) represents a time-delay τ ∈ [0 ϕ]:

Fig. 3 :

 3 Fig.3: Φ(α) and α max (ω) used for the satisfaction test Standard matrix manipulations based on the Redheffer star product allow to equivalently transform the interconnection of Fig.2, where G(s) is replaced with Φ(α), into that of Fig.4, where ∆ m = αI 2 , α ∈ [0 α max (ω)], contains a single parameter with frequency dependent bounds. Note that a rational function Φ(α) of degree 1 in α would lead to a simpler LFR, where α is not repeated in ∆ m . This is however not possible here. α would indeed tend to ±∞ to allow ∠Φ(α) to cover [-2π 0], but the µ-based tools used in the sequel require the ∆ block of the LFR to be bounded.

Proposition 4 . 3 . 4 . 3 :

 4343 2 cannot be directly evaluated using standard µ-based tools and is therefore replaced by Proposition 4.Proposition The delay margin is lower than ϕ on a given box D

Fig. 5 :

 5 Fig. 5: Delay margin analysis with ϕ = 0.2s

the OSIP 4000134724 supported by ESA.


Geoffrey Scozzaro

Clara Buire

Daniel Delahaye

Aude Marzuoli
email: aude.marzuoli@gmail.com

Optimizing air-rail travel connections: A data-driven delay management strategy for seamless passenger journeys

Keywords: Multimodal transport network, Air-rail, Delay management, Passenger metrics, Integer linear programming

In the world of modern travel, where multimodal trips are becoming increasingly common, flight or train delays can jeopardise passengers' journeys by threatening connections. To address this issue, we present a delay management strategy on a multimodal network that involves seamless collaboration between air and rail transportation stakeholders. The objective is to minimise the total delay experienced by passengers at their final destination by rescheduling flights and trains at a tactical level. The decision whether to hold a train or a flight for connecting passengers depends on the available re-accommodation options. We propose an integer linear programming formulation of the problem at the network level, considering real-world constraints such as train station and airport capacities, minimum aircraft turnaround time, and flight slot adherence. To demonstrate the potential of this approach, we dive into a data-driven case study covering 496 airports, including three major hubs and 72 train stations across Europe. We simulate an incident on the French rail network that causes significant delays at Paris-CDG station. The results show that delaying 5% of departing flights at Paris-CDG airport by 13 minutes on average could reduce the number of stranded passengers at the airport by 71%. Such a decrease translates into a 40% reduction in the total delay experienced by passengers at their destination. This work highlights the potential benefits of air-rail integration, and the importance of information sharing between stakeholders to improve passenger journey reliability.

I. INTRODUCTION

Improving air passengers' door-to-door journey is one of the European Commission's main objectives for 2050 [START_REF]Flightpath 2050. Europe's vision for aviation. Report of the high-level group on aviation research[END_REF]. In addition, in the context of climate change and increasing airport congestion, a new paradigm has emerged in which trains are replacing short-haul flights to relieve airport congestion and reduce passengers' carbon footprint. In such a case, coordination between trains and flights is necessary to maintain a high level of service for passengers. To support the development of airground system integration, in 2020, the European Commission launched several research projects under the SESAR research and innovation program, such as MODUS [START_REF] Paul | Modelling and assessing the role of air transport in an integrated, intermodal transport system[END_REF], X-D2D [START_REF] Bagamanova | Extended atm for seamless travel (x-team d2d)[END_REF], IMHOTEP [START_REF] Mota | Creating the future airport passenger experience: Imhotep[END_REF], TRANSIT [START_REF] Bueno | Transit final project results report[END_REF], SYN+AIR [START_REF] Undertaking | SYN+AIR -Synergies between transport modes and air transportation[END_REF]. In 2023, new projects focusing on multimodality started, such as MAIA * : these authors equally contributed to this paper [START_REF]MAIA-Multimodal Access for Intelligent Airports[END_REF], MultiModX [START_REF]New EU SESAR project, MultiModX, kicks off to enhance multimodal transport networks[END_REF] or SIGN-AIR [START_REF]SIGN-AIR-Implemented SynergIes, data sharing contracts and Goals between transport modes and AIR transportation[END_REF]. When considering multimodal journeys, one of the key challenges is managing delays efficiently. Such events could cause passengers to miss their connections, resulting in significant delays at their final destination. To protect passengers, Regulation 261 [START_REF]Regulation (ec) no 261/2004[END_REF] imposes airlines to re-accommodate or compensate them in case of delay. However, there is no equivalent of such regulation for multimodal transportation services [11]. Such a service might, in practice, be one key lever to encourage people to use the train as a feeder mode, especially for non-frequent travellers [START_REF] Chiambaretto | Measuring the willingness-to-pay of air-rail intermodal passengers[END_REF].

Here, we envision a scenario where airlines and rail service providers would cooperate and communicate actively. In this scenario, passengers travel using an air-rail integrated ticketing system, submitted to similar compensation laws defined in Regulation 261. This collaborative environment would enable transportation providers to be promptly notified if a train or a flight carrying connecting passengers is delayed. In addition, transport operators would share the costs of re-accommodating stranded passengers, incentivising them to minimise delays. In this context, we present a tactical rescheduling of flights and trains to limit the impact of delays on passengers. More specifically, we propose to solve an extension of the delay management problem originally introduced by Schobel [START_REF] Schöbel | A model for the delay management problem based on mixed-integer-programming[END_REF] with the aim of minimising the total delay experienced by passengers in an air-rail transportation network. This problem relies on deciding, in a transport network, whether a connecting vehicle (e.g. train, bus, plane, etc.) should wait for connecting passengers who are delayed on their first leg. We propose an extension of the problem to include both airside and ground side constraints. In addition, we use a data-driven approach to evaluate the passenger reallocation time in the case of a missed connection. In the following, we refer to this problem as the Air Rail Delay Management Problem (ARDMP). We address the ARDMP through the Western Europe air-rail network case study, with a model including 496 airports and 72 railway stations. We apply the proposed delay management strategy considering a disruption occurring on the rail network.

This paper is structured as follows. Section II presents a state-of-the-art related to the delay management problem on the ground and air sides. Sections III and IV introduce the ARDMP and its mathematical model, respectively. Section V describes the European case study considered, and Section VI presents the results.

II. LITERATURE REVIEW

Previous works have studied the Delay Management (DM) problem on the ground side. Schobel [START_REF] Schöbel | A model for the delay management problem based on mixed-integer-programming[END_REF] is the first to formulate the problem of deciding whether or not to delay a vehicle in a public transport system to wait for transferring passengers. She proposes a mixed-integer formulation to minimise the total delay of passengers at their final destination. She assumes that passengers' delay is equal to the delay of their train, if they catch it, or to a constant delay otherwise. Gatto et al. [START_REF] Gatto | The computational complexity of delay management[END_REF] show that even restricted versions of this problem are NP-complete. Later, Schobel [START_REF] Schöbel | Capacity constraints in delay management[END_REF] considers track capacity constraints for a railway system. Dollevoet et al. [START_REF] Dollevoet | Delay management with rerouting of passengers[END_REF] propose integrating passenger rerouting into the DM process. The same authors consider station capacity constraints and track re-allocation in [START_REF] Dollevoet | Delay management including capacities of stations[END_REF]. For a review of DM problem handling, the reader can refer to [START_REF] König | A review on railway delay management[END_REF].

On the airside, Santos et al. [START_REF] Santos | Airline delay management problem with airport capacity constraints and priority decisions[END_REF] are the first to propose a version of the delay management problem applied to the airline. Montlaur and Delgado [START_REF] Montlaur | Flight and passenger delay assignment optimization strategies[END_REF] consider the problem of balancing airport capacity at a hub airport by assigning delays to departing flights at a pre-tactical level and to arriving flights at a tactical level. They test different strategies to minimise either flight or passenger delays, considering connecting passengers and turnaround constraints. Delgado et al. [START_REF] Delgado | Hub operations delay recovery based on cost optimisation-dynamic cost indexing and waiting for passengers strategies[END_REF] propose a delay recovery strategy at a hub airport through gate delays to wait for delayed connecting passengers and a dynamic cost index to recover from such delays. Delgado et al. [START_REF] Delgado | Network-wide assessment of atm mechanisms using an agent-based model[END_REF] propose an agent-based model for handling air traffic delays through 4D trajectory adjustment to reduce costs and delays for connecting passengers.

Collaboration between air and ground transportation systems received a growing interest these last years. Li et al. [START_REF] Li | Determinants of partnership levels in air-rail cooperation[END_REF] present an overview of actual collaboration between airlines and train service providers to create an integrated airrail service for passengers. Laplace et al. [START_REF] Laplace | META-CDM: Multimodal, efficient transportation in airports and collaborative decision making[END_REF] present the META-CDM project, which aims to involve ground transport stakeholders into the airport Collaborative Decision Making (CDM) to improve passenger door-to-door journeys. In this context, studies on multimodal recovery solutions in case of massive disruptions show promising results in mitigating the impact of such events on passengers (see for instance [START_REF] Dray | Air transportation and multimodal, collaborative decision making during adverse events[END_REF]- [START_REF] Marzuoli | Improving disruption management with multimodal collaborative decision-making: A case study of the asiana crash and lessons learned[END_REF]). Regarding air-rail coordination mechanisms, Buire et al. [START_REF] Buire | Airrail timetable synchronization for a seamless passenger journey[END_REF] suggest strategically synchronising air and rail timetables to ensure smooth passenger transfers. However, they do not consider delays and their impact on the passenger journey. Scozzaro et al. [START_REF] Scozzaro | Flight rescheduling to improve passenger journey during airport access mode disruptions[END_REF] propose flight rescheduling at the tactical level to mitigate the impact of airport access mode disruptions on passengers. They consider airside constraints such as terminal capacity, maximum runway throughput or minimum passenger connecting time. Their work focuses on a single airport and does not consider reactionary delay. Here, we propose a tactical delay management strategy at the network level, combining the works of [START_REF] Buire | Airrail timetable synchronization for a seamless passenger journey[END_REF] and [START_REF] Scozzaro | Flight rescheduling to improve passenger journey during airport access mode disruptions[END_REF]. This study is the first to address the delay management problem in a long-haul multimodal network, combining constraints on the air and ground sides. We extend the original version of the problem developed by Schobel [START_REF] Schöbel | A model for the delay management problem based on mixed-integer-programming[END_REF]. We take into account real operational constraints, such as airport and railway station capacities, Air Traffic Flow Management (ATFM) slot adherence or even minimum aircraft turnaround time. We also consider the reallocation time for passengers who miss their connections.

III. AIR-RAIL DELAY MANAGEMENT PROBLEM

This section introduces the total passenger delay metric and then defines the ARDMP.

A. Total passenger delay

As explained by Cook et al. [START_REF] Cook | Passenger-oriented enhanced metrics[END_REF], flight delays do not necessarily capture the actual delays experienced by passengers. The situation is similar for train delays, which can lead to missed connections and potentially late arrivals at the final destination. We therefore introduce the total passenger delay as the sum of the delays experienced by passengers when arriving at their final destination. To compute passenger delays, we define the following three groups of passengers:

• on-time passengers: passengers who catch their flight/train; their delay is equal to the delay of the flight/train; • reallocated passengers: passengers who miss their connections due to a delay on the first leg, they are consequently reallocated to another flight/train going to the same destination within the same day; • stranded passengers: passengers who miss their connections and without reallocation option (no seat available or no more flight/train going to the same destination within the day). The delay of reallocated passengers is computed as follows.

For each flight and train, we consider the direct alternative, enabling the passengers to arrive at their destination with the smallest possible delay. This alternative can be either a train or a flight, and, in this study, we only consider direct alternatives for the sake of simplicity. The delay of reallocated passengers corresponds to the difference between the arrival time of the new flight/train at the destination and the initial one. Regarding stranded passengers, we assume they will be re-accommodated to the same flight on the next day at the same departure time, thereby experiencing a 24-hour delay. The objective of the ARDMP is therefore to reschedule flights and trains at the tactical level to minimise the total passenger delay.

B. ARDMP description

To address the ARDMP problem, a one-day time window is considered. In the event of disruptions on the ground or air sides leading to train or flight delays, we assume that service providers are notified ahead of time about the affected vehicles and their expected delays for the remainder of the day. For instance, consider a power outage on a railway network between 6 am and 8 am, causing delays for several trains throughout the day due to a domino effect. We assume that the rescheduling of trains and flights can occur once operators anticipate delays caused by the incident, such as when power is restored at 8 am. The key challenge is deciding whether a vehicle should wait for connecting delayed passengers. For example, consider a flight of 100 passengers scheduled to leave at 9 am, with 10 passengers connecting from a previous train. Due to the disruption, these passengers arrive at the boarding gate 10 minutes after the scheduled boarding time. There are two options: depart on time or delay the flight. On the one hand, if there is another flight to the same destination in three hours, departing on time will result in a total passenger delay of 3×60×10 = 1800 minutes. On the other hand, if the flight waits for the delayed passengers, the total passenger delay will only be 100×10 = 1000 minutes. In this situation, the aircraft should wait for the connecting passengers. However, if only five passengers were connecting, it would be better to depart on time.

This study considers an air-rail transportation network covering a specific region. The time scope is discretised into hminute time steps. Air and rail networks can be represented by graphs, where nodes correspond to airports and train stations, and links model flight and train legs that are operated between stations. Each flight or train has expected departure and arrival times. Passengers' itineraries on this multimodal network are known, including possible transfers between modes. Assuming that delays arise on several trains or flights during the day, ARDMP problem consists in assigning tactical delays to other trains and flights so as to minimise the total passenger delay. The following operational constraints are taken into account:

IV. ARDMP MODEL

This section describes the optimization model of the ARDMP. Data, decision variables, constraints and the objective function are detailed below. The model, which aims to minimise the total passenger delay, therefore reads:

Sets and Parameters

N a /N r index
min l∈L n D l d l + l ′ ∈C l n l ′ ,l d l ′ ,l , (1)
subject to:

t D l = h(k D l -1) l ∈ L (1a) t A l = t D l + IV T l l ∈ L (1b) k A l = k D l + IV T l h l ∈ L (1c) k D l ≤ s + M x D l,s l ∈ L, s ∈ S (1d) Ns s=0 x D l,s ≤ k D l l ∈ L (1e) k A l ≤ s + M x A l,s l ∈ L, s ∈ S (1f) Ns s=0 x A l,s ≤ k A l l ∈ L (1g) t D l ≤ t D,0 l + ∆ l ∈ L (1h) t D l ≤ t D,0 l + ∆ atfm l ∈ L atfm (1i) t D l -t D,0 l = d l l ∈ L (1j) t D l2 -t A l1 ≥ T AT l1,l2 (l 1 , l 2 ) ∈ P air (1k) t D l2 -t A l1 = dw l1,l2 (l 1 , l 2 ) ∈ P rail (1l) o n,0 = o 0 n n ∈ N r (1m) o n,s = o n,s-1 + l∈L A n (x A l,s-1 -x A l,s ) + l∈L D n (x D l,s-1 -x D l,s ) n ∈ N r , s ∈ S (1n) o n,s ≤ o max n n ∈ N r , s ∈ S (1o) s+w-1 τ =s l∈L A n x A l,τ -x A l,τ +1 ≤ y A,w,s n n ∈ N a , w ∈ W, s ∈ S ′ (1p) s+w-1 τ =s l∈L D n x D l,τ -x D l,τ +1 ≤ y D,w,s n n ∈ N a , w ∈ W, s ∈ S ′ (1q) t D l ′ -t A l ≥ MCT l ′ ,l (l, l ′ ) ∈ C p l (1r) t D l ′ -t A l + M y l ′ ,l ≥ MCT l ′ ,l l ∈ L, l ′ ∈ C l (1s) d l ′ ,l ≥ y l ′ ,l r l l ∈ L, l ′ ∈ C l (1t) d l ′ ,l ≥ d l l ∈ L, l ′ ∈ C l (1u) k D l , k A l ∈ S l ∈ L (1v) t D l , t A l ∈ {0, . . . , h(|S| -1)} l ∈ L (1w) d l ∈ {0, h, . . . , ∆} l ∈ L (1x) d l ′ ,l ∈ {0, h, . . . , r l } l ∈ L, l ′ ∈ C l (1y) y l ′ ,l ∈ {0, 1} l ∈ L, l ∈ C l (1z) x A l,s , x D l,s ∈ {0, 1} l ∈ L, s ∈ S (1aa) o n,s ∈ {0, . . . , o max n,s } n ∈ N r , s ∈ S. (1ab)
Constraints (1a) and (1b) link the time slot to the actual auxiliary time variable for the departure and arrival time of leg l, respectively. Constraints (1c) calculate the arrival time slot of leg l based on its departure time slot. Constraints (1d) and (1e) fix the values of the binary variables x D l,s . Similarly, the constraints (1f) and (1g) fix the values of the binary variables x A l,s . Constraints (1h) limit the maximum departure delay for each leg l. Constraints (1i) ensure ATFM slot adherence. The actual delay is calculated by constraints (1j). The turnaround time constraints and the train dwell time constraints are given by (1k) and (1l), respectively. Constraints (1m) to (1o) fix the train station occupancy and ensure that it does not exceed the number of tracks at each train station. The maximum arrival and departure flight movements per time window are limited with constraints (1p) and (1q), respectively. Constraints (1r) ensure that passenger minimum connecting times for priority flights are maintained. Constraints (1s) fix the value of variables y l ′ ,l that characterise if passengers connecting between legs l ′ and l miss their connection. Constraints (1t) and (1u) fix the reallocation delay between flights l ′ and l to d l if passengers have their connection, and to the reallocation delay r l , otherwise. Finally, constraints (1v)-(1ab) define the definition domain of the decision variables.

V. WESTERN EUROPE CASE STUDY

This section focuses on the Western European case study. It first outlines the data used and the assumptions made. The modelling approach for passenger transfers is then presented. Finally, it describes in detail the post-processing procedure for reallocating passengers, which is crucial for accurately assessing the total passenger delay. Table I presents the characteristics of the case study considered. This case study focuses on the historical day of December 4, 2019 when the French National Railway Providers (SNCF) went on strike. We gather initial flight schedules [START_REF] Eurocontrol | Aviation Data for Research[END_REF] from the 18 largest airports in France, Germany, and Spain, including three major hub airports: Frankfurt Airport (FRA), Madrid-Barajas Airport (MAD), and Paris-Charles de Gaulle Airport (CDG). Throughout this day, 10,407 flights were operated, with 593 departures scheduled at CDG, as illustrated in Figure 1. We also consider the train schedules associated with each hub airport and their respective connecting airports, involving 646 train legs [START_REF] Sncf | Horaire des TGV[END_REF]- [START_REF] Deutschebahn | Open data portal[END_REF]. The train schedules were collected on each operator website. It is important to note that we could not access the actual train delay data or the number of train cancellations. Therefore, we simulate the disruption by randomly delaying 30% of trains arriving at CDG-High Speed Rail train station. The delay times were randomly selected using a uniform distribution ranging from a minimum delay, denoted as t min , to a maximum delay, denoted as t max .

A. Network characteristics and data

Air Traffic Flow Management (ATFM) delays were not considered here. Therefore, delayed flights are only those impacted by the proposed rescheduling algorithm. We assume that the maximum delay assignable to a flight is 30 minutes and that 25% of flights at each main hub airport were subject to ATFM slot adherence. This percentage is arbitrarily fixed and can be tuned by a final user, depending on airport characteristics. The maximum assignable delay for flights subject to ATFM slot adherence is set to 10 minutes.

Additionally, we allowed train tactical rescheduling to ensure compliance with the maximum train station capacity, which may have been compromised due to the initial train delays and disruptions caused by the strike. Lastly, we assume that all information regarding train delays and connecting passengers is fully known before rescheduling. Therefore, we employ a one-iteration process to reschedule all legs operated from the morning until the end of the day.

B. Modelling passenger transfers

In order to model passenger transfers between modes, the same method proposed by [START_REF] Buire | Air-rail timetable synchronization: Improving passenger connections in Europe within and across transportation modes[END_REF] is used. In a nutshell, the method consists in modelling intramodal (air-air) and intermodal (train-air) passenger connections at a hub airport, using a Constraint Programming approach [START_REF] Rossi | Handbook of constraint programming[END_REF]. Based on airport modal share and flight passenger volume, the method generates passenger volume that transfers between two scheduled legs. We limit the study by considering only passengers with at most two legs in their total journey. The number of connecting passengers simulated with the proposed methodology is presented in Table II.

C. Passenger reallocation procedure

Similarly to the rebooking procedure proposed by [START_REF] Ball | Total delay impact study: a comprehensive assessment of the costs and impacts of flight delay in the united states[END_REF], we propose the following passenger reallocation procedure. In the mathematical model, we assume that passengers will be accommodated on the next flight to the same destination if they miss their scheduled flights. However, each aircraft has a finite capacity, defined by the number of seats it can offer. To overcome this limitation, we present a post-processing method that effectively reallocates stranded passengers to alternative flights, considering each aircraft capacity. Since we do not know the actual number of seats available, we assume an 80% load factor for each aircraft. For example, if an original flight carried 50 passengers, we assume 50× 100 80 -50 ≈ 12 available seats. We extend this reallocation approach to direct trains as an alternative method, again assuming an 80% load factor for each train.

The reallocation process follows a systematic sequence. We consider the chronological list of passengers who have missed their flights and a corresponding set of feasible direct alternatives for each individual. These alternatives are ranked according to the delay they cause at the passenger final destination. For each passenger, we offer the best available rerouting option (in terms of delay). In the case where the best alternative flight/train is full, we select the second best option, and so on. When no re-accommodation option is available, the passenger is stranded and subject to a 24-hour delay. Note that the reallocation procedure is operated after the rescheduling, i.e. flight and train delays are considered to select the best reallocation options for passengers who miss their flights.

VI. RESULTS

Computations are performed using an AMD Ryzen 5 4500U CPU and 16 GB RAM laptop. The resolution of the optimization problem formulation is made with the MIP solver Gurobi, version 9.1.2 [START_REF] Optimization | Gurobi Optimizer Reference Manual[END_REF]. The computation time is 23 seconds.

A. Passengers gain

Figure 2 displays the distribution of buffer time for passengers transferring from a train to a flight at CDG airport. The transfer buffer time equals the difference between the actual passenger transfer time and the minimum required connection time. We only display buffer times of passengers who would have missed their flight based on the original schedule but can still make it on time if the flight is delayed. We do not show passengers who arrive before the initial departure time or who arrive more than 30 minutes after the initial departure time. The figure shows a significant increase in passenger connections with a 0-minute buffer time after rescheduling. A Figure 2. Distribution of passenger transfer buffer times before and after rescheduling. Buffer times are calculated by subtracting the minimum connecting time from the actual passenger transfer time. A negative buffer time indicates that passengers do not have enough time to transfer, caused by a delay on their first leg. This graph only shows passengers who could recover their initial flights thanks to the rescheduling (i.e. missing their flights by 30 minutes or less before rescheduling). 0-minute buffer time corresponds to a transfer time equal to the minimum connection time required for passengers to catch their flight. Consequently, the delay management strategy allows 484 of the 1221 passengers who initially missed their flights to arrive on time for boarding. The rescheduling does not induce buffer time strictly larger than 0 minutes for these passengers as this would delay the on-time passengers and, therefore, increase the total passenger delay.

Figure 3 depicts the total delay experienced by passengers before and after optimization. The main difference between the initial and the optimized schedule lies in the number of stranded passengers. Indeed, 614 passengers have no reallocation option before optimization and should wait until the next day to reach their final destination. After optimization, the number of stranded passengers is reduced by 71% and the total passenger delay by 55%. Indeed, the algorithm prioritises these passengers if the flight can wait since the cost of a missed connection is large. However, the maximum flight delay authorised to wait for passengers is 30 minutes (or 10 minutes for priority flights that need to respect their departure slots). Hence, some passengers might not have their connections if the required time to make the connection is above that limit. Therefore, several passengers remain stranded even after the rescheduling. Finally, the total delay experienced by direct passengers departing from CDG is 12810 minutes, resulting in an average passenger delay of 0.3 minutes. As a result, the rescheduling has a minimal impact on on-time passengers. Indeed, passengers who miss their connections at the end of the day are more likely to be stranded without reallocation options until the next day. Hence, rescheduling gives higher priority in waiting for them.

On average, due to the rescheduling, all flights across Europe experience a delay of 0.04 minutes, while the departing flights at CDG experience a delay of 0.84 minutes. The proposed rescheduling plan delays 5% of the departing flights at Paris-CDG airport by 13 minutes on average. To put this into perspective, Table III shows the characteristics of the actual delays experienced by flights during the historical operating day in question. As per the table, departing flights at CDG were operated with an average delay of 11 minutes. Therefore, our proposed rescheduling approach seems reasonable compared to the actual delays the airport has to deal with during a typical operating day.

Figure 5 shows a map of delayed flights and the magnitude of these delays. The colour and the width correspond to the departure time and the delay assigned to the flight, respectively. More specifically, a darker colour indicates that the flight's departure time is later in the day, and the greater the width, the greater the delay. It can be seen that long-haul flights are generally those with the highest assigned delay. The colour of these flights also indicates that they are scheduled in the morning. In fact, these long-haul flights tend to have a daily frequency compared to short-haul flights. The re-routing time for passengers who miss their connections is, therefore, 24 hours. On the other hand, delays on short-haul flights are generally assigned in the evening, when passengers have no more opportunities for re-routing. It can also be observed that a few flights are delayed due to network propagation. These delays occur because the turnaround time initially planned by the airlines between a delayed flight at CDG and the following flight is small. It can be observed that the rescheduling delays an evening flight by 30 minutes departing from the US due to minimum turnaround time constraints and the assumption of constant in-vehicle time. However, in practice, the previous flight operated by the aircraft could have been speeding up to recover from its departing delay, reducing the impact of the proposed delay management strategy. Such action is especially true for long-distance flights and could be included in further work.

Finally, as mentioned above, exogenous ATFM delays were not considered in this study. However, the proposed rescheduling strategy could deal with these delays by rescheduling flights to wait for connecting passengers and reduce station congestion. Taking these exogenous delays into account would have an impact on the rescheduling solution, as ATFM delays of departing flights could already reduce the number of passengers missing their rail/air connections.

VII. CONCLUSION AND FUTURE WORKS

Europe's investment in different multimodal research projects underlines the need for collaboration between air and ground transport stakeholders, to provide passengers with reliable journeys. Such air-rail integration would not only improve passenger experience but also allow airlines and airports to have accurate information about passenger connections. This could create a win-win situation for the stakeholders involved, by boosting passenger demand while limiting extra expenses for the service providers. In this context, we have presented a delay management strategy tailored to a large integrated air-rail network. We simulated a disruption occurring on the French railway network which led to passengers missing their connections at CDG airport. The results highlight the effectiveness of our mitigation strategy, demonstrating its ability to reduce passenger delays by 55%, while only delaying 5% of departure flights at CDG airport. By considering the entire network, our delay management strategy creates new flight and train schedules that satisfy operational constraints such as station capacities and minimum aircraft turnaround times at other airports throughout the day. This rescheduling approach limits delay spread by identifying which flights may propagate delays.

The research conducted in this study contributes to enhancing the passenger experience when travelling across a multimodal long-distance network. Further research on the operator rescheduling cost and passenger preferences should be conducted to implement the proposed delay management strategy. This extension will ultimately lead to better acceptance among transportation stakeholders and an improved passenger travel experience. Analysing a potential airside disruption would provide valuable insights into how the rescheduling differs, based on constraint differences on the air and railway sides. Another interesting extension would be to consider dynamic cost indexing, as proposed by [START_REF] Delgado | Hub operations delay recovery based on cost optimisation-dynamic cost indexing and waiting for passengers strategies[END_REF], which relaxes the constant travel time assumption and allows aircraft and trains to speed up to recover from delays. Finally, the rescheduling process should consider ATFM constraints such as en-route capacity and airside delays.

1 )

 1 the number of trains scheduled to stop at each train station cannot exceed the number of tracks at this station (train-station capacity constraint), 2) the number of airport departure and arrival movements, operated at each time step and at each hour is limited (airport capacity constraint), 3) a minimum turnaround time between two flights operated by the same aircraft is considered, 4) train dwell time at the station remains the same as in the initial schedule, 5) the train and flight travel times remain unchanged (the vehicle maintains its scheduled speed), 6) a Minimum Connection Time (MCT) is ensured for passengers who are not directly affected by the disruption (i.e. whose first leg is on time), to let them enough time to transfer between their first and second legs, 7) the departure time of flights subject to an ATFM slot must happen within a [-5,10]-minute interval around the scheduled departure time (ATFM slot adherence), 8) a maximum tactical delay of 30 minutes can be assigned to flights or trains (except for flights under ATFM slot).

n

  number of tracks at train station n, n ∈ N r y A,w n runway arrival capacity for a window of length w, w ∈ W , (i.e., the maximum number of arrival flights that could be scheduled within the hw minutes window), n ∈ N air y D,w n runway departure capacity for a window of length w,, w ∈ W , n ∈ N air . MCT l ′ ,l minimum connection time to connect from leg l ′ to leg l, l ∈ L, l ′ ∈ C l Input data T D l initial scheduled departure time of leg l, l ∈ L T A l initial scheduled arrival time of leg l, l ∈ L o 0 n the initial number of trains at train station n, n ∈ N r IV T l in-vehicle time of leg l, l ∈ L T AT l1,l2 minimum turnaround time between legs l 1 and l 2 , (l 1 , l 2 ) ∈ P air dw l1,l2 stop time between legs l 1 and l 2 , (l 1 , l 2 ) ∈ P rail n D l volume of passengers using l as a direct connection, l ∈ L n l ′ ,l volume of passengers transferring from leg l ′ to leg l, l ∈ L, l ′ ∈ C l r l reallocation delay for passenger missing their connection with leg l, l ∈ L Main decision variables k D l index of slot at which leg l is scheduled to depart, l ∈ L Auxiliary decision variables k A l index of slot at which leg l is scheduled to arrive l ∈ L t D l new scheduled departure time of l, l ∈ L t A l new scheduled arrival time of l, l ∈ L d l delay associated to leg l, l ∈ L d l ′ ,l delay experienced by passenger connecting from leg l to leg l ′ , l ∈ L, l ′ ∈ C l y l ′ ,l binary, indicates whether the connection from leg l ′ to leg l is feasible or not, l ∈ L, l ′ ∈ C l x D l,s binary, indicates whether leg l is scheduled to depart after s, s ∈ S x A l,s binary, indicates whether leg l is scheduled to arrive after s, s ∈ S o n,s the number or trains stopped at n at slot s, n ∈ N r

Figure 1 .

 1 Figure 1. Illustration of flights operated on December 4, 2019, on Western Europe (source: [35]).

Figure 3 .

 3 Figure 3. Total passenger delays before and after rescheduling, stacked by passenger types (on-time passengers, reallocated passengers and stranded passengers).

Figure 4 .

 4 Figure 4. Distribution of total vehicle delays per hour. Flight and train delays are displayed in blue and orange, respectively. The hatched bars represent reactionary delays.

Figure 5 .

 5 Figure 5. Visualisation of post-rescheduling flight delays. The linewidth and the colour-coding system indicate the delay magnitude and the departure time of the day, respectively. Dotted lines correspond to flights departing from CDG airport, plain lines to other flights.

TABLE I .

 I CASE STUDY CHARACTERISTICS

	Case study description
	Case Study	Western European Transport Network
	Number of airports	496
	Number of train stations	72
	Number of flights	10407 (593 from CDG)
	Number of trains	646 (66 to CDG station)
	Airports with limited capacity	18 largest airports in France, Germany,
		and Spain
	Airport with connecting pas-	CDG, FRA, MAD
	sengers
	Train stations with limited ca-	3 stations, each associated with a hub
	pacity
	Train schedule data source	GTFS data
	Flight schedule data source	OAG
	Minimum aircraft turnaround	45 min
	time (TAT)
	Disruption scenario characteristics
	Date	4 December 2019
	Considered events	French railway company on strike
	Disruption duration	From 00:00 to 23:59
	Train delay percentage	30% of trains are late at CDG
	Train delay duration (min)	X ∼ U (30, 90)
	Train cancellation	Not considered
	Flight/Train travel time	Constant
	Priority flights	25% of flights need to comply with
		their ATFM slots at main airport hubs
	ATFM delays	Not considered
	Maximum priority flight delay	10 min
	Maximum flight delay	30 min

TABLE II .

 II NUMBER OF CONNECTING PASSENGERS PER AIRPORT. A DIS-TINCTION IS MADE BETWEEN TRAIN-AIR CONNECTIONS AND AIR-AIR CONNECTIONS.

	Airport	Connection	Number of connecting
		type	passengers
	CDG	air-air	30638
		train-air	18022
	FRA	air-air	41599
		train-air	12101
	MAD	air-air	22277
		train-air	8168

TABLE III .

 III ACTUAL FLIGHT DELAY ON DECEMBER 4 TH 2023 (IN MIN-UTES). (SOURCE: EUROCONTROL)

		CDG	All
	Average actual flight delay	11.0	6.1
	Maximum actual flight delay	120	1310

ACKNOWLEDGMENT

This study has been supported by SESAR 3 Joint Undertaking within the project TRANSIT under grant agreement No 893209 and, within the project SIGN-AIR under grant agreement No 101114845.


Reading

Thank you very much for the invitation and the organization, I'm delighted to present this lecture here in Cluj. Let's start with a definition of commons.

[Slide 2, concepts) Rouglhy spealing, Commons can be defined from two perspectives, the ownership and the type of good. It is a particular kind of ownership, neitheir private nor public/state ownership, but it belongs to the community. Considering the type of goods, of asset, commons are rival assets used by communities, by its members individually or collectively. In other words, behind the form of ownership, the main issue is accessing resources, more precisly rival resources. In economics, rivalry is a property of a good whose consumption by one agent reduces the quantity of the good available to other agents. Rivalry concerns the amount of resources, which are limited to a certain extend. For example, fishermen taking fish from the sea limits the amount taken by others. Nowadays, the environment in its whole can be considered as a common good.

Hence, the main issues are to control the access to the goods, to regulate conflicts and increase the efficency of resource allocation. This regulation can take different forms and is a great part of the community political life. Generally, common are managed jointly or with specific management. Regarding the medieval small communities I'm studying, commons can be grazing land or collective facilities such as irrigation canals.

This phenomenon has different names, structures and operating methods, and relates to various resources in different times and places. Commons refers to a complex and evolving reality that has been subject to various interpretations in humanities and especially in history. For a long time, the dominant historiography regarded the commons as a form of archaism. In recent decades, however, they have been reinterpreted and re-evaluated. Recent medieval historiography has entered into a fruitful dialogue with modernists in order to place commons in a long-term perspective, to study how they were created, managed and evolved, and to show their fundamental role in the history and functioning of communities, taking into account the contributions of other social sciences under the influence of Anglo-Saxon research.

Sorry for this quite long and conceptual definition, it is necessary to study commons handle by the small communities I'm working on.

[slide 2, map of the region and a circle on the map of Europe] I am studying political life in the small communities of the lower Rhône valley, in the Avignon region, between the end of the twelfth and the beginning of the fifteenth centuries. These small communities were part of a context of urban expansion and forms of local autonomy from the early thirteenth century, followed by a takeover by the princely powers fifty years later. The region was situated between different political areas: to the west, the Kingdom of France; to the south-east, the County of Provence, which became a possession of the Angevins in the mid-thirteenth century; and to the north, the Comtat Venaissin, which passed from the hands of the Counts of Toulouse to those of the Papacy in 1274. In this paper, I'm concentrating on Saint-Rémy de Provence, a small community north of the Alpilles. The Counts of Provence were also lords of the town, giving them a strong hold over the community. They held half the rights to the castrum, while the Church held the other half.

Between the end of the 12th century and the middle of the 13th century, small communities gradually appeared in the sources. The inhabitants of Saint-Rémy are mentioned for the first time in 1198. Over the course of a century, the communities gradually began to structure themselves around tax issues and the demarcation of their territory, particularly around common pastures.

[slide 3] My aim here is to show how the historiographical revision of the commons is very useful for studying the relationship between these communities and their environment. I will begin by outlining the evolution of historiography on the subject, and then show what this allows me to study in relation to Saint-Rémy de Provence.

The commons in historiography

Economic and juridical approaches during the 19th and 20 th centuries

[slide 4]
The commons refers both to the question of collective ownership or collective use, or even civic use, and to both resources and the common nature of management. The law has focused on the legitimacy, origin and legal instruments of this other form of ownership.

Consequently, the law can study the issue from two angles: from a private law perspective, as a particular relationship between a private subject and an object, or from a public law perspective, as a way of possessing and managing an asset or resource belonging to the State or public authority.

The development of bourgeois law and liberalism, based on free individual initiative as the sole and full manifestation of man and the bearer of benefits for the community and the State, led to a negative interpretation of collective property, which had to be liquidated, both for the bourgeoisie and in academic circles. However, from the 19th century onwards, both positivism and romanticism were critical of bourgeois property and took an interest in collective property. (p. 580)

In the 1870s and 1880s, the European historiographical debate centred on the anteriority of collective property over individual property, with the work of the Englishman Mayne, the Belgian Laveleye and Fustel de Coulanges in France. This debate took place in the context of socio-political developments at the time, with the emergence of scientific socialism and the defence of bourgeois ideology. The debate then turns to the relationship between public property and Roman and/or Germanic law as it was known throughout the Middle Ages. Ennio Cortese shows that under the Carolingians, 'strategic' public assets such as roads, bridges, ports, salt works, etc. gradually came under the authority and ownership of the king. Under Frederick II, these assets became regalia and demanium (public property).

[Slide 5]

The historiography of the commons is marked by the publication of an article by Garrett Hardin in 1968, The Tragedy of the Commons. It refers to a thought experiment: a pasture is used in common by farmers. It is always in their interest to add an extra animal, because they will earn income from it after fattening. However, each addition has a negative effect: the grass is shared between more animals, so each one puts on less weight. But because this effect is spread across all the farmers' animals, and the benefit of adding an animal is individual, it is always profitable to add an animal. But as the pasture becomes more and more populated, it is over-exploited and then destroyed. Hardin concludes that common ownership of a resource is, in essence, incompatible with its sustainability. For him, the solution is limited to a closed alternative: only private appropriation or centralised management (by the state, for example) can ensure sustainable exploitation. Common property is the excluded third party in this binary logic, which will do much to ensure the success of the tragedy. The reasoning will be used just as much by the advocates of state action as by the zealots of the free market, whose solitary confrontation it enshrines.

"The process set in motion by the lack of government is inescapable and ignores values and awareness. Even if he is aware of the deterioration, it is always in the farmer's interest to add another animal, and he will do so. (p. 1319) It is in this sense that we must understand the term tragedy, in the Greek sense.

New approaches

Hardin sees the commons as open-access spaces with no institutional regulation by communities, which is historically unfounded. Hardin is part of the liberal tradition and, as well as criticising the commons, he points out the deficits of government, particularly in the United States, in relation to parks and land: both the "socio-ecological entities" (pp. 1318-1319) appropriated by overly permissive states (the central state in the United States on the question of national parks or federal land in the West), and resources that are not appropriated but are freely accessible, such as the high seas or the atmosphere. This 1968 article was written against a backdrop of debate about the causes of the environmental crisis: for some, the origins of the crisis lie in the Judeo-Christian matrix, for others in capitalism, and finally in consumerism. The hypothesis on which the article is based is that of an actor, *homo oeconomicus*, driven entirely by rational choices. We can draw a parallel here with game theory and the stowaway strategy. Fabien Locher speaks of an 'individualistic and agonistic social cosmology' (p. 1320).

Hardin has never hidden his neo-Malthusian commitment: the animals put down by breeders are also the children. The Population Bomb was published the same year by someone close to him.

Against this approach, Elinor Ostrom has shown the importance of structures built by local players, the importance of the long term, the optimal exploitation of resources, and the limitation of irresponsible appropriation without compensation. Elinor Ostrom stresses the importance of finding a variety of institutional solutions for different problems in changing contexts. She is part of an Anglo-Saxon way of thinking that is empiricist, liberal, antiideological, individualist and neo-institutional. It presents various cases of local selfgovernment for the exploitation of different resources, analysing institutional structures, construction methods, relationships with the environmental context and the resource itself, and the weight of external factors. The joint exploitation of grazing land or irrigation systems is more effective and works better than the examples of fishing. Success depends on good practice or the characteristics of the resource. → the most important is the definition of 7 characteristics necessary for the proper functioning of commons according to Ostrom.(p. 588)

The question Elinor Ostrom asks is: why do some communities manage to "devise rules and [rely on] social norms to enforce sustainable resource use, while others do not?" [@sethi2016] It identifies a series of principles necessary for sustainable resource management. It stresses the importance of letting stakeholders decide between themselves, and the factors that facilitate trust. The importance, too, of sanctions for deviant behaviour. Elinor Ostrom calls these principles polycentric games, between the individual (1, 2), the collective (3, 4, 5) and the State. → Ricardo Rao stresses the importance of crossing political and economic history. In the Italian cities of the late Middle Ages, the

Reconsidering Saint-Rémy's commons in the light of new historiography

The importance of commons in the political life of the community My approach was initially purely political, to study the organisation and gradual structuring of communities. In the sources, these communities take the political form of an universitas. According to Michaud-Quantin's definition, this is a "legally constituted group with the express or tacit authorisation of the competent superior", in this case the inhabitants, the members of a territorial community, which means that they had representatives in court and a common treasury. The use of the term is tantamount to asserting the existence (or recognition) of a "genuine community possessing as such an institutional consistency that ensures a real collective life for all its members". However, such a community does not erase the social antagonisms that may exist within it. The importance of these conflicts in the history of communities has long been well documented. In the case of rural communities in particular, the work carried out by Joseph Morsel's team has clearly shown that the creation of these communities should not be understood as opposition to the power of the princely administration, but rather as their integration into the exercise of political domination, in this case by the administration of the Counts of Provence.

The development of the Angevin administration severely limited the political possibilities of the universitas. For small communities such as Saint-Rémy, the management of the commons was a central aspect of their activity, not only because of the practical issues involved in managing resources and conflicts, but also because it justified their political existence.

The approach in terms of the commons makes it possible to bring together political, economic, social and environmental history, and to build a bridge between the institutional history of small communities and their economic life, which has been separated for too long. This has led me to take a new look at the problems of defining territory and sharing the commons. I analysed the question of ownership, whether collective or individual, and conflicts over access to resources. In this way, I was able to integrate the study of universitas into new issues: the dynamics of community formation depend not only on problems of taxation and defence, but also on the management of rural life.

The regulation of these communal relationships provides an opportunity to examine the internal workings of the community, the universitas. This applies in particular to a commune to the east of the castrum.

[slide : map of Saint-Rémy] Prior to 1325, the community purchased the palud from the Count of Provence, in exchange for a cens of 100 pounds. This common was appropriated by the inhabitants, who sought to share the land concerned. However, the division took time, probably due to internal dissension. It wasn't until 1335 that the community got serious about sharing out the land. On this occasion, it produced communal statutes. Statutes are normative documents that communities of inhabitants produce or receive, providing for internal organisation in a certain relationship to the lord or princely administration. The production of these statutes helps us to understand their internal workings: it is notable that the most detailed procedure preserved in the sources concerns the commons.

On 3 March 1335, the baile of the court of Saint-Rémy, the representative of the administrative authority, convened the representatives of the inhabitants at their request. This was followed by a series of eight articles, organising the appointment and specifying the mandate of the representatives, in fact the notables who held all the positions: these were twelve "exacteurs de taille", a restricted "council" of six members, who were also procuratores of the universitas, including a nobleman, a knight and two notaries, one of whom belonged to the Porcelet family 1 -who had dominated the universitas since the end of the THIRTEENTH CENTURY e . Of these six, four were exactors and five were syndics or procuratores in the first half of the XIV e century. From 1 er April, these representatives produced nineteen articles, which were proclaimed on 17 May, thirteen of which concerned provisions that I would describe as environmental, in other words regulating the relationship between the inhabitants and their environment, around conflicts of ownership and rural use. The relationship between these communities and the non-human environment is built through the appropriation of resources and territory and the regulation of withdrawals. To manage conflicts, the community controls access to this environment, distinguishing between the nature of the areas, the different activities and the animals that can access them. Most of the articles concern the palud (or palun), which was not shared between the inhabitants in 1335. The 1335 statutes were only valid for a short time, until the beginning of October2 , which may be explained by negotiations or restrictions imposed by the Count of Provence.

The care taken in the drafting process, given its duration, suggests that these statutes may have been more important to the community's representatives than to the community itself. The fact that the drafting process was spread over three months, as well as the numerous meetings mentioned, suggests possible negotiations, between residents or with the administration. The representatives took the opportunity to organise a small council, and their mandate was limited to producing the 1335 statutes and planning the division of the rent for the following year and future statutes. In 1336, new statutes were enacted at the time of the division of the palud, which had been granted by the royal court since the early 1320s in return for the payment of a cens. In addition to confirming the role of the council, the 1336 statutes included articles from the previous year, but not without amendments and additions. For example, charcoal burning in the Alpilles was henceforth prohibited, which meant that this area was wooded and belonged to the inhabitants. Individual use was no longer possible.

Regulate or dismantle the commons?

The Saint-Rémy statutes of 1336 are the most important element in addressing this issue. It is a rotulus preserved in the Bouches du Rhône departmental archives, and is extremely long and detailed. It contains the legal procedure for sharing the palud, the new communal statutes and the result of the sharing, with the names of all those present. The division in 1336 was unequal, favouring the great families at the head of the universitas, such as the Porcelets.

The statutes of 1335 distinguished between a divided part, in which the inhabitants appropriated the resources, and an undivided part, where the resources were shared and access to them regulated. Once the division was made the following year, the stakes changed. The community organised the access roads to the palud (article 5), restricted entry to ploughing and pack animals only (article 6), limited grazing to certain areas in the autumn (article 7), and prohibited the lighting of fires, presumably in view of the risk of fire caused by burning (article 10). The inhabitants organised the private management of former shared areas for new use, in return for an annual fee to the Counts of Provence, in the form of long leases.

The intended use is agricultural, ploughing rather than grazing, and may have led to the gradual drying out of this wetland. The other paluds of Saint-Rémy were probably not shared out, as they were intended for grazing and belonged to the Count of Provence. The community regulated the use of the environment by organising the area and controlling access and use, both internally and externally. Conflicts were reduced through private and collectively controlled appropriation, as was the extension of agricultural land.

Little is known about these commons: they do not belong to the community, even though their members use them individually to graze their flocks. There is therefore little mention of them in the archives of Saint-Rémy, or even in those of the Counts of Provence. They were generally located on the edge of the community's territory, and were a source of conflict with neighbouring communities.

One of the few mentions of the communal areas can be found in the communal statutes of 1335. In the statutes of 1335, articles 1 and 3 mention the pastures, communal areas and defens, in which the grazing of foreign herds or the cutting of grass is prohibited, following a dispute and a lawsuit against a nobleman from outside Saint-Rémy who was grazing his herds there. Grazing and mowing in the unshared palud were subject to the payment of a fee, which was higher for non-inhabitants. This distinction was made as a result of the territorial demarcation procedures introduced in the 1260s, following conflicts over access to grazing land between neighbouring communities. The aim of controlling access to grazing land was to prevent overgrazing and the depletion of resources.

Overgrazing was a real problem in Provence. In 1345, the administration of the Counts of Provence launched an investigation into overgrazing, which was destroying the pastures of rural communities, particularly in the eastern part of the region. Is this a case of the tragedy of the commons? Yes, because of the over-exploitation of resources, but no, because in Hardin's theory there is no form of regulation, which is not the case here.

After the Black Death of 1348, the sources regularly show the low productivity of this area, which the inhabitants use to request a reduction in the cens.

See Martin Aurell, Une Famille de la noblesse provençale au Moyen âge, lesPorcelet, Avignon, Aubanel, 1986.

quod omnia ista durent hinc ad quindenam post festum Sancti Michaelis (AC Saint-Rémy FF 1C).