Arthur Perodou
email: arthur.perodou@ims-bordeaux.fr

Christophe Combastel
email: christophe.combastel@ims-bordeaux.fr

Senior Member, IEEE Ali Zolghadri
email: ali.zolghadri@ims-bordeaux.fr

Critical-Time Analysis of Cyber-Physical Systems subject to Actuator Attacks and Faults

Keywords: resilience, cyber-physical systems, security, quadratic constraints

A novel quantitative criterion, namely critical time, is investigated to characterize the degree of resilience of controlled cyber-physical systems. Resilience is defined as system's ability to contain the maximal impact of anomalies and recover to a nominal mode. Anomalies are understood as any kind of attack or fault that leads to abnormal behavior of the controlled system. The critical time is the maximal time-horizon for which a system is considered to be safe after the occurrence of an anomaly. An increase of critical time will leave more time for defense mechanisms, including human operators, to detect and mitigate anomalies. While most of the literature focuses on the impact part of resilience, this criterion is tied with the recovery part. In this work, it is shown how the computation of the critical time can be done for discrete-time LTI models. To achieve this, sufficient conditions in the form of iterative LMI-based algorithms are established. A numerical example is provided to illustrate the theoretical results.

I. INTRODUCTION

An open challenge in cyber-physical systems (CPS) security is to find an appropriate trade-off between the desired security level and the satisfaction of a performance level of practical interest. In the last decade, the System and Control community has developed numerous methods to address this issue [START_REF] Dibaji | A Systems and Control Perspective of CPS Security[END_REF], [START_REF] Chong | A Tutorial Introduction to Security and Privacy for Cyber-Physical Systems[END_REF]. A relevant concept that has emerged is resilience. Resilience is the system's ability to contain the maximal impact of anomalies and recover to a nominal mode [START_REF] Dibaji | A Systems and Control Perspective of CPS Security[END_REF], [START_REF] Rieger | Resilient Control Systems: Next Generation Design Research[END_REF], [START_REF] Zhu | Game-Theoretic Methods for Robustness, Security, and Resilience of Cyberphysical Control Systems[END_REF], where an anomaly is understood here as any difference between the behaviors of a system and a suitable operating model, resulting for instance from an attack or a fault.

There exist several criteria to quantify the resilience of a system [START_REF] Dibaji | A Systems and Control Perspective of CPS Security[END_REF], [START_REF] Chong | A Tutorial Introduction to Security and Privacy for Cyber-Physical Systems[END_REF]. For instances, the anomalies effect on the cost function of an optimal control problem is considered in [START_REF] Zhu | Game-Theoretic Methods for Robustness, Security, and Resilience of Cyberphysical Control Systems[END_REF]; in [START_REF] Teixeira | Strategic stealthy attacks: The output-to-output l2-gain[END_REF], the attacks are additionally constrained to be stealthy; in [START_REF] Teixeira | Secure Control Systems: A Quantitative Risk Management Approach[END_REF], the maximum perturbation on the state trajectory is evaluated for stealthy attacks with minimum resources. While these works focus on the impact part of the concept of resilience, additional resilient metrics tied to the recovery part are proposed in [START_REF] Wei | Resilient Industrial Control System (RICS): Concepts, Formulation, Metrics, and Insights[END_REF], such as the required time to recover to normal operation, but no explicit computing methods are provided. In this work, we investigate and compute a new resilience criterion, namely critical time.

The critical time is the maximal time-horizon for which a system is considered to be safe after the occurrence of an anomaly, that is the system is not in a critical state and is still able to recover to a normal mode. The underlying motivation is that an increase of critical time allows more time for defense mechanisms, including human operators, to detect and mitigate anomalies. In particular, this criterion covers the detection and reconfiguration delays classically used in the fault-management literature [START_REF] Hwang | A Survey of Fault Detection, Isolation, and Reconfiguration Methods[END_REF], [START_REF] Zhang | Bibliographical Review on Reconfigurable Fault-Tolerant Control Systems[END_REF].

In this paper, the critical time will be computed for any actuator anomaly that is allowed by the physical limitations of the actuators. This includes for instance deny-of-service (DoS) or false-data injection attacks on actuators. To achieve this, we will take advantage from the framework of quadratic constraints (QC). This framework has led to numerous methods for system analysis [START_REF] Megretski | System Analysis via Integral Quadratic Constraints[END_REF], [START_REF] Scorletti | Improved Efficient Analysis for Systems with Uncertain Parameters[END_REF], [START_REF] Scherer | Stability Analysis by Dynamic Dissipation Inequalities: on merging Frequency-Domain Techniques with Time-Domain Conditions[END_REF] and synthesis [START_REF] Iwasaki | Generalized KYP Lemma: Unified Frequency Domain Inequalities with Design Applications[END_REF], [START_REF] Perodou | Frequency Design of Interconnected Dissipative Systems: a Unified LMI Approach[END_REF], [START_REF] Perodou | Frequency Design of Lossless Passive Electronic Filters: A State-Space Formulation of the Direct Synthesis Approach[END_REF]. However, to the best of our knowledge, no analog problems to critical time computation were considered using QCs. An important benefit of the QC framework is its special link with linear matrix inequality (LMI) optimization [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF].

The contribution of this paper is considered to be twofold: (i) Firstly, a new criterion for evaluating the resilience of a system is proposed and (ii) Secondly, the critical-time computation problem is tackled for a discrete-time linear time-invariant (LTI) system under actuator anomalies subject to actuator limitations. By using the QC framework, it is shown that this leads to solve iterative algorithms involving LMI feasibility problems.

The paper is organized as follows. First, brief reminders on QCs are provided in Section II. In Section III, the critical-time computation problems, with pre-specified or free initial state, are explicitly formulated. In Section IV and Section V, it is then shown how the computation of the critical time can be done for discrete-time LTI models. Finally, an illustrative example is provided in Section VI, while Section VII provides some concluding remarks.

II. PRELIMINARIES

A. Notations

Lower (upper) case letters are used for vectors (matrices). N denotes the set of natural numbers, Z the set of integers, k 1 , k 2 the integers between, and including, k 1 and k 2 , R n×m the set of real-valued matrices of size n × m and D n the set of diagonal real-valued matrices of size n × n. For the sake of brevity, z k1,k2 := z k1 . . . z k2 is used. I n and 0 n×m are respectively the identity matrix of R n×n and the zero matrix of R n×m . The subscripts are omitted when obvious from the context. X stands for transpose of X while M > (resp. ≥) 0 denotes positive (semi-) definiteness. trace(M ) is the sum of the diagonal elements of M and diag(M ) is their concatenation in a column vector. The sign ⊗ represents the Kronecker product. Bold characters denote either explicit decision variables in a design problem or optimization variables in an optimization problem.

B. Background

In this work, constraint sets will be defined using QCs. Definition 1 (Quadratic constraint): A vector z ∈ R n is said to satisfy a quadratic constraint φ,

with φ = φ ∈ R (n+1)×(n+1) , if z 1 φ z 1 ≥ 0 holds. Define φ := Q s s r , where Q = Q ∈ R n×n , s ∈ R n×1
and r ∈ R. Notice that if Q ≤ 0, the set of all vectors z ∈ R n that satisfy the previous QC is convex. In particular, the resulting set is inside an ellipsoid if Q < 0 and in a halfspace if Q = 0. Finally, observe the following lemma. Lemma 1: Let z ∈ R be a real scalar. Then:

z ∈ [z min , z max ] ⇔ z 1 -1 z c z c z 2 r -z 2 c z 1 ≥ 0
where z c := (z min + z max ) /2 and z r := (z max -z min ) /2. An important result in the QC framework is the so-called S-procedure.

Lemma 2 (S-procedure for quadratic constraints [START_REF] Jönsson | A lecture on the S-procedure[END_REF]): Consider the following quadratic functions: ∀z ∈ R n , σ q (z) = z 1 Q q s q s q r q z 1 , q = 0, 1, . . . , N

where Q q = Q q ∈ R n×n , s q ∈ R n×1 and r q ∈ R. Then (ii) ⇒ (i).

(i) The constraint σ 0 (z) ≥ 0 holds for all z ∈ R n such that σ q (z) ≥ 0, q = 1, . . . , N . (ii) There exist τ q ≥ 0, q = 1, . . . , N , such that

Q 0 s 0 s 0 r 0 - N q=1 τ q Q q s q s q r q ≥ 0

III. PROBLEM STATEMENT A. Critical-time definition

Consider a controlled cyber-physical system subject to general anomalies modeled as:

x k+1 = Ax k + Bu k + E 1 k α k + D 1 k d k y k = Cx k + E 2 k α k + D 2 k d k where x k ∈ R n , u k ∈ R m ,
and y k ∈ R l are respectively the state, the control input and the measured output signals evaluated at time k. The anomaly α k ∈ R mα stands for either attacks or faults. The vector d k ∈ R w models additive noise or unknown disturbance, and may include model uncertainties. In a similar fashion to [START_REF] Hwang | A Survey of Fault Detection, Isolation, and Reconfiguration Methods[END_REF], this linear time-varying model covers actuator, sensor and component anomalies.

Moreover, assume that the signals u, α and d are constrained by S u ⊆ R m , S α ⊆ R mα and S d ⊆ R w such that:

∀k ∈ Z, u k ∈ S u α k ∈ S α d k ∈ S d
In addition, assume that a safety set S s , that is a set of states x for which the system is considered to be safe, is provided. The safety set S s may be derived from physical constraints or the validity domain of a linearized model. Finally, without loss of generality, assume that the anomalies appear from initial time k = 0. Definition 2 (Critical-time): The critical time k c is the maximal time-horizon k f such that, after the occurrence of an anomaly, the system is safe over the time-window I(k f ) := 0, k f . Using previous notations:

k c = max {k f |∀k ∈ I(k f ), ∀u k ∈ S u , ∀α k ∈ S α , ∀d k ∈ S d ,
x k ∈ S s } There are several expected benefits from availability of the critical-time criterion. First, one can evaluate the time that can be allowed for defense mechanisms. Moreover, by considering all states that additionally satisfy some performance criteria S p ⊆ S s , one may improve the resilience of the system by driving it to the state that has the maximum critical-time, while guaranteeing a performance level. Finally, it is expected to provide a relevant trade-off between anomalies with opposite impacts, as for instance in the case of DoS attacks and attacks by upper saturation.

In the sequel, two problems are formulated and solved: 1) the computation of the critical-time for a system in a given state x 0 and 2) the computation of the critical-time when the initial state x 0 is to be selected among a set of admissible states that satisfy some performance constraints S p .

B. Problem formulation

This paper focusing on actuator anomalies, m α = m and ∀k ∈ Z,

E 1 k = B D 1 k = D E 2 k = 0 D 2 k = 0
are considered, where D ∈ R n×w is a known matrix. Then, by defining the contaminated input u a := u + α, the controlled system under actuator anomalies is described as:

(Σ) :

x k+1 = Ax k + Bu a k + Dd k y k = Cx k (1)
In addition, due to physical limitations, such as saturation or slew rate, assume that the abnormal input u a is constrained by a given set S a ⊆ R m as follows: ∀k ∈ N, u a k ∈ S a . Remark 1: Under previous assumptions, a significant class of anomalies are considered, such as actuator additive faults, DoS attacks or attacks by saturation. By considering the augmented system of (Σ) interconnected with a controller by a communication network, this may also include abnormal control inputs, resulting for instance from measurement falsification, or from direct attacks on the controller. Moreover, assume that the sets S p , S s , S a and S d are defined using multiple quadratic constraints such as:

S p := ∩ np h=1 {x 0 ∈ R n , σ p h (x 0 ) ≥ 0} (2) S s := ∩ ns i=1 {x ∈ R n , σ si (x) ≥ 0} (3) S a := ∩ na j=1 u a ∈ R m , σ aj (u a ) ≥ 0 (4) S d := ∩ n d g=1 d ∈ R w , σ dg (d) ≥ 0 (5)
where

σ p h (x 0 ) := Cx 0 1 Q p h s p h s p h r p h Cx 0 1 σ si (x) := x 1 Q si s si s si r si x 1 σ aj (u a ) := u a 1 Q aj s aj s aj r aj u a 1 σ dg (d) := d 1 Q dg s dg s dg r dg d 1
The problem of computing the critical-time of a system for a given initial state x 0 is now explicitly formulated.

Problem 1 (Critical-time computation for a given x 0 ): GIVEN

• a system (Σ) modeled by (1) with initial state x 0 ∈ R n ,

• a safety set S s defined by (3),

• an abnormal input set S a defined by ( 4),

• a disturbance set S d defined by ( 5), FIND the maximal time-index k f such that system (1) is safe over the time-window I kf = 0, k f for all abnormal inputs of S a and disturbances of S d :

max kf ∈N k f subject to ∀k ∈ 0, k f , ∀u a k ∈ R m , ∀d k ∈ R w , ∀i = 1, . . . , n s , σ si (x k ) ≥ 0 (6) ∀j = 1, . . . , n a , σ aj (u a k ) ≥ 0 (7) ∀g = 1, . . . , n d , σ dg (d k ) ≥ 0 (8) hold.
A second interesting problem is the computation of the critical-time of a system for which the initial state x 0 is not specified but assumed to belong to the performance domain S p . The resulting state may then be used as a target state for resilient control.

Problem 2 (System critical-time computation): GIVEN S s , S a and S d as in Problem 1 and

• a system (Σ) modeled by (1),

• a performance set S p defined by (2) FIND x 0 ∈ S p that maximizes the time index k f such that the system (1) is safe over the time-window I kf = 0, k f for all abnormal inputs of S a and disturbances of S d :

max x0∈R n , kf ∈N k f subject to ∀k ∈ 0, k f , ∀u a k ∈ R m , ∀d k ∈ R w , (6) -(8), ∀h = 1, . . . , n p , σ p h (x 0 ) ≥ 0 (9) hold.
IV. CRITICAL-TIME COMPUTATION WITH SPECIFIED x 0

In this section, Problem 1 is tackled. To achieve this, the sub-problem of checking if the state x k f is into the safety set S s , over all abnormal inputs of S a and disturbances of S d , for a fixed time-index k f is first considered. Then, an algorithm is proposed to find the maximum k f , providing an under-estimate of the critical time k c : k f ≤ k c .

A. Fixed time-index k f

Assume that x 0 ∈ R n is given and k f ∈ N is fixed. Define P k and G k as: P 0 = 0, G 0 = 0 and for all k ≥ 1,

P k = A k-1 B . . . AB B G k = A k-1 D . . . AD D
Using the recursive equation of (1), x k f is computed as:

x k f = A k f x 0 + P k f u a 0,k f -1 + G k f d 0,k f -1 (10)
This property enables to solve an alternative version of Problem 1, with fixed k f , as shown in next lemma. Lemma 3: Let (Σ) be a system defined by (1) with a given state x 0 . Consider the associate sets S s , S a and S d given by ( 3)- [START_REF] Teixeira | Strategic stealthy attacks: The output-to-output l2-gain[END_REF]. Assume that the time-index k f ∈ N is fixed.

Then (i) ⇒ (ii). (i) ∀i = 1, . . . , n s , ∀j = 1, . . . , n a , ∀g = 1, . . . , n d , ∃T i,j ∈ D k f ≥ 0, ∃Θ i,g ∈ D k f ≥ 0, N k f Q si s si + Q si A k f x 0 (s si + Q si A k f x 0 ) r si + 2s si A k f x 0 N k f - na j=1   T i,j ⊗ Q aj 0 diag (T i,j ) ⊗ s aj 0 0 0 diag (T i,j ) ⊗ s aj 0 trace(T i,j r aj )   - n d g=1   0 0 0 0 Θ i,g ⊗ Q dg diag (Θ i,g ) ⊗ s dg 0 diag (Θ i,g ) ⊗ s dg trace(Θ i,g r dg )   +   0 0 0 0 0 0 0 0 (A k f x 0 ) Q si A k f x 0   ≥ 0 (11)
where

N k := P k G k 0 0 0 1 .
(ii) The system (Σ) with initial state x 0 is safe at time k f for all past abnormal inputs of S a and disturbances of S d :

∀k ∈ 0, k f -1 , ∀u a k ∈ R m , ∀d k ∈ R w , (7) -(8), ∀i = 1, . . . , n s , σ si (x k f ) ≥ 0 hold.
Proof: See App. A. Remark 2: Condition (i) is a feasibility optimization problem with LMI constraints. It is then solvable in polynomial-time. This computational advantage comes at the price of the conservatism of condition (i), that implies but is not equivalent to condition (ii). This results from the application of the S-procedure (see [START_REF] Iwasaki | Generalized KYP Lemma: Unified Frequency Domain Inequalities with Design Applications[END_REF], [START_REF] Jönsson | A lecture on the S-procedure[END_REF], and the references therein, for a detailed discussion).

B. Critical-time computation

In order to compute the critical-time k c , Algorithm 1 is proposed. Based on the sufficient condition provided by Lemma 3, the time index k f is incremented until there is no more solution to the LMI feasibility problem. An underestimate of the critical-time k c is so obtained and ensures that the system is safe at least until the obtained result k * f . Theorem 1: Let (Σ) be a system defined by (1) with a given state x 0 . Consider the associate sets S s , S a and S d given by ( 3)- [START_REF] Teixeira | Strategic stealthy attacks: The output-to-output l2-gain[END_REF]. Denote k * f as the solution returned by Algorithm 1. Then the system (Σ) with initial state x 0 is safe over the time-window I(k * f ) = 0, k * f and for all abnormal inputs of S a and disturbances of S d .

Proof: By Lemma 3, (Σ) is safe for each k ∈ I(k * f ).

Algorithm 1: Critical-time computation with fixed x 0

1 k f ← 0 2 f ← true 3 while f do 4 k f ← k f + 1 5 f ← isF easible( 6 ∃ T i,j ∈ D k f ≥ 0, i = 1, . . . , n s , j = 1, . . . , n a , 7 ∃ Θ i,g ∈ D k f ≥ 0, i = 1, . . . , n s , g = 1, . . . , n d , 8
such that (11) hold. )

9 /* true if a solution is found /* 10 k f ← k f -1 11 return k f V. SYSTEM CRITICAL-TIME COMPUTATION
In this section, Problem 2 is tackled. Analogous to Section IV, the sub-problem of finding an initial state x 0 ∈ S p such that x k ∈ S s , k ∈ 0, k f , over all abnormal inputs of S a and disturbances of S d , for a fixed time-index k f is first considered. Then, an algorithm is proposed to find the maximum k f , providing an under-estimate of the system critical-time k c , as well as a related state x 0 .

A. Fixed time-index k f

First, the problem of finding an initial state x 0 for a fixed time-index k f is addressed.

Lemma 4: Let (Σ) be a system defined by [START_REF] Dibaji | A Systems and Control Perspective of CPS Security[END_REF]. Consider the associate sets S p , S s , S a and S d given by (2) - [START_REF] Teixeira | Strategic stealthy attacks: The output-to-output l2-gain[END_REF]. Assume that

∀h = 1, . . . , n p , Q p h ≤ 0 and r Qp h := rank(Q p h ) ≥ 1 ∀i = 1, . . . , n s , Q si ≤ 0 and r Qs i := rank(Q si ) ≥ 1
Factorize each Q p h and Q si such as

Q p h := V p h Σ p h V p h Q si := V si Σ si V si where Σ p h < 0 ∈ D r Qp h , V p h ∈ R r Qp h ×l , Σ si < 0 ∈ D r Qs i and V si ∈ R r Qs i ×n . In addition, assume that k f ∈ N is fixed. Then (i) ⇒ (ii). (i) ∃x 0 ∈ R n subject to ∀h = 1, . . . , n p , r p h + 2s p h (Cx 0 ) (V p h Cx 0 ) V p h Cx 0 -Σ -1 p h ≥ 0 (12) ∀i = 1, . . . , n s , ∀j = 1, . . . , n a , ∀g = 1, . . . , n d , ∀k ∈ 0, k f , ∃T i,j,k ∈ D k ≥ 0, ∃Θ i,g,k ∈ D k ≥ 0, N k,si   Q si s si + Q si A k x 0 0 (s si + Q si A k x 0 ) r si + 2s si A k x 0 0 0 0 0   N k,si - na j=1     T i,j,k ⊗ Q aj 0 diag (T i,j,k ) ⊗ s aj 0 0 0 0 0 diag (T i,j,k ) ⊗ s aj 0 trace(T i,j,k r aj ) 0 0 0 0 0     - n d g=1     0 0 0 0 0 Θ i,g,k ⊗ Q dg diag (Θ i,g,k ) ⊗ s dg 0 0 diag (Θ i,g,k ) ⊗ s dg trace(Θ i,g,k r dg ) 0 0 0 0 0     +     0 0 0 0 0 0 0 0 0 0 0 V si A k x 0 0 0 V si A k x 0 -Σ -1 si     ≥ 0 (13)
where N k,si := N k 0 0 0 r Qs i ×r Qs i .

(ii) There exists an admissible initial state x 0 ∈ S p such that system (1) is safe over the time-window 0, k f for all abnormal inputs of S a and disturbances of S d :

∀k ∈ 0, k f , ∀u a k ∈ R m , ∀d k ∈ R w , (6)
-(9) hold. Proof: By the Schur Lemma [START_REF] Jönsson | A lecture on the S-procedure[END_REF], ( 12) is equivalent to (9), i.e. x 0 ∈ S p . In addition, by Lemma 3 coupled with the application of the Schur Lemma, (13) implies that the system is safe at each time step k. As this is true for all k ∈ 0, k f , the system is safe over the whole interval.

Similarly to Lemma 3, condition (ii) is a feasibility problem with LMI constraints, relying on the S-procedure.

Remark 3: The factorization of symmetric matrices Q p h and Q si may be obtained using an eigenvalue decomposition. For instance:

∃U p h ∈ R l×l , ∃Σ p h ∈ R r Qp h ×r Qp h , Q p h = U p h Σ p h 0 0 0 U p h = U p h I 0 Σ p h I 0 U p h
The matrix V p h can then be defined as V p h := I 0 U p h .

B. System critical-time computation

To compute the critical-time and an associate x 0 , Algorithm 2 is proposed. Based on Lemma 4, k f is incremented until there is no more solution to the LMI feasibility problem. The resulting under-estimate k * f of the critical-time k c , ensures that the system with initial state x * 0 is safe until k * f . Theorem 2: Let (Σ) be defined by [START_REF] Dibaji | A Systems and Control Perspective of CPS Security[END_REF] and consider the sets S p , S s , S a and S d as in Lemma 4. Denote k * f and x * 0 the solutions returned by Algorithm 2. Then the system (Σ) with admissible initial state x * 0 ∈ S p is safe over the time-window

I(k * f ) = 0, k * f
and for all abnormal inputs of S a and disturbances of S d . Proof: This follows by application of Lemma 4.

Algorithm 2: System critical-time computation In Lemma 4, the sets were assumed to be ellipsoidal, potentially partially rank-degenerated, that is described by quadratic constraints such as in Definition 1 with Q ≤ 0. One may notice that halfspace sets, i.e. when Q = 0, may be straightly considered by simply removing appropriate lines and columns in [START_REF] Scherer | Stability Analysis by Dynamic Dissipation Inequalities: on merging Frequency-Domain Techniques with Time-Domain Conditions[END_REF] or [START_REF] Iwasaki | Generalized KYP Lemma: Unified Frequency Domain Inequalities with Design Applications[END_REF]. For instance, if Q p h = 0, (12) is replaced by the following LMI

1 k f ← 0 2 x 0 k f ← [ ] /* empty set /* 3 f ← f alse 4 while not(f ) do 5 k f ← k f + 1 6 x 0 ← x 0 k f 7 x 0 k f ← Find x 0 ∈ R n subject to: ∀k ∈ 0, k f ,
r p h + 2s p h (Cx 0 ) ≥ 0 VI. ILLUSTRATION
In this section, a numerical example is provided to illustrate the main results established in the previous sections. For the sake of clarity, no disturbance d is considered. Moreover, it is assumed that the performance constraint leads to y 0 ∈ [y min , y max ], where y min = 0.363 and y max = 3.27, for all k ∈ Z the system is safe if

x k (1) ∈ [x 1min , x 1max ] and x k (2) ∈ [x 2min , x 2max ]
, where x 1min = x 2min = 0.10, x 1max = 6.90 and x 2max = 1.72, while the actuator limitations allow only inputs u a ∈ [u min , u max ], where u min = 0 and u max = 6. The performance set S p and the actuator-limitations set S a are then defined using ( 2), ( 4) and Lemma 1, while the safety set S s is defined by (3) and

Q s1 s s1 s s1 r s1 =   -1.00 0 3.50 0 0 0 3.50 0 -0.690   Q s2 s s2 s s2 r s2 =   0 0 0 0 -1.00 0.912 0 0.912 -0.172  
For illustration purposes, the system is simulated starting from the initial points x 01 = 1.4536 0.3629 and x 02 = 5.8143 1.4517 under either a DoS attack, i.e. u a = 0, or an attack by upper saturation, i.e. u a = u max , that occurs from k a = 10. The evolution of x( 2) is plotted in Fig. 1.

Applying Theorem 2, Algorithm 2 is solved using the Robust Control Toolbox of MATLAB. The maximal timeindex is found k * f = 38 with x * 0 = 1.9035 0.9946 . To provide a comparative insight of this result, a DoS attack and an attack by upper saturation are applied to the system for different initial points and the related critical-time index k c is computed for each attack. For the sake of illustration, these initial points are arbitrarily chosen such that x {λ} 0 = λ • x 0 , where λ ∈ R. The simulation results plotted in Fig. 2 highlight the tradeoff arising when both attacks are considered. The index k * f suggests to be an adequate tradeoff to ensure the maximal time for defense mechanisms. In this paper, a novel criterion, namely critical time, was investigated to characterize the degree of resilience of controlled CPS subject to actuator anomalies, including attacks or faults. This criterion has been defined as the maximum time-horizon for which a system is ensured to be safe after the occurrence of an anomaly. Using the framework of quadratic constraints, sufficient conditions have been established to compute the critical time of discrete-time LTI systems through iterative LMI-based algorithms. This work paves the way for further investigations. This includes extension to more general attack models, for instance replay or covert attacks, and system dynamics as nonlinear or multi-agent systems, the use of other convex sets such as zonotopes [START_REF] Combastel | A Distributed Kalman Filter with symbolic Zonotopes and Unique Symbols Provider for Robust State Estimation in CPS[END_REF] and the integration of communication and networking models. This is a topic of our current research. APPENDIX A. Proof of Lemma 3 Proof: First, denote I := 0, k f -1 and z I := u a I d I . Using [START_REF] Megretski | System Analysis via Integral Quadratic Constraints[END_REF], rewrite σ si (x k f ) and define ρ si (z I ) as follows

σ si (x k f ) = z I 1 Q si s si s si r si z I 1 :=ρs i (z I )
where

Q si := P k f G k f Q si P k f G k f s si := P k f G k f Q si A k f x 0 + s si r si := x 0 A k f Q si A k f x 0 + 2s si A k f x 0 + r si

8 ∃ 9 ∃

 89 T i,j,k ∈ D k ≥ 0, i = 1, . . . , n s , j = 1, . . . , n a , Θ i,g,k ∈ D k ≥ 0, i = 1, . . . , n s , g = 1, . . . , n d , such that (12), (13) hold 10 f ← isempty(x 0 k f ) /* true if no solution /* 11 k f ← k f -1 12 return k f , x 0 C. Lemma 4 with halfspace sets

  Consider a system given by (1) with:

Fig. 1 .

 1 Fig. 1. Illustration of DoS and Upper saturation attacks on x 2 starting from two different initial points. Grey parts represent unsafe regions.

Fig. 2 .

 2 Fig. 2. Illustration critical time index kc for different initial state x 0 and trade-off between DoS attack and attack by saturation

1 Q 1 ρ 1 Q 1 whereQFinally

 1111 Moreover, define ρ aj ,k (z I ) and ρ dg,k (z I ) as ∀k ∈ I,ρ aj ,k (z I ) := z I aj ,k s aj ,k s aj ,k r aj ,k z I dg,k (z I ) := z I dg,k s dg,k s dg,k r dg,k z I aj ,k := E k ⊗ Q aj 0 0 0 w•k f Q dg,k := 0 m•k f 0 0 E k ⊗ Q dg s aj ,k := diag (E k ) ⊗ s aj 0 m×(w•k f ) r aj ,k := r aj s dg,k := 0 w×(m•k f ) diag (E k ) ⊗ s dg r dg,k := r dg E k := This implies ρ aj ,k (z I ) = σ aj (u a k ), ρ dg,k (z I ) = σ dg (d k ). Thus, condition (ii) is equivalent to find if, ∀i = 1, . . . , n s , ρ si (z I ) ≥ 0 holds for all z I ∈ R (m+w)•k f such that, ∀k ∈ I, ∀j = 1, . . . , n a , ∀g = 1, . . . , n d , ρ aj ,k (z I ) ≥ 0 and ρ dg,k (z I ) ≥ 0.Applying the S-procedure, this is implied by:∃{τ i,j,k ≥ 0}, ∃{θ i,g,k ≥ 0}, such that ∀z I ∈ R (m+w)•k f , ρ si (z I )j,k ρ aj ,k (z I ) g,k ρ dg,k (z I ) ≥ 0 that is equivalent to Q si s si s si r si -na j=1 k f -1 k=0 τ i,j,k Q aj ,k s aj ,k s aj ,k r aj ,k n d g=1 k f -1 k=0 θ i,g,k Q dg,k s dg,k s dg,k r dg,k ≥ 0 j,k Q aj ,k s aj ,k s aj ,k r aj ,k =   T i,j ⊗ Q aj 0 diag (T i,j) ⊗ s aj 0 0 0 diag (T i,j ) ⊗ s aj 0 trace(T i,j r aj )Similar reasoning for Θ i,g leads to condition (i).

This study has been carried out with financial support from the French National Research Agency (ANR) in the framework of the Investments for the Future Programme IdEx Bordeaux-SysNum (ANR-10-IDEX-03-02). The financial support from the "Conseil Régional de la Nouvelle-Aquitaine" is also gratefully acknowledged.


Agnès Van Zanten

Claire Maxwell

Elite education and the State in France: durable ties and new challenges

Introduction

This article examines the two-way relationship between state and elite education by drawing on Max [START_REF] Weber | The Profession and Vocation of Politics[END_REF] 1994) conceptions of the state, bureaucracy and social status groups, and those studies that have extended his work. Weber, influenced by the observations made about the dynamics of the Prussian but also of the French state, emphasised the strong relationship between the emergence of modern state institutions and the development of bureaucracy. Already underway during the Ancien Régime, the training and recruitment of specialised agents to manage matters of the state was strongly reinforced in France by the Revolutionaries and later by Bonaparte in order to do away with patrimonial forms of rule and to establish a new type of state [START_REF] Chazel | Éléments pour une Reconsidération de la Conception Wébérienne de la Bureaucratie[END_REF]. According to Weber, bureaucracy played a key role in this process of new state formation because it provided a strong basis for the two dimensions of government: authority, the constitution and maintenance of a legitimate social order; and power, the capacity to act effectively to solve practical problems [START_REF] Duran | La Bureaucratie a-t-Elle un Avenir? [Does Bureaucracy Have a Future[END_REF].

According to Weber, bureaucracy developed thanks to an alliance between political authorities and administrative agents keen to extend their field of power, which in turn encouraged close interaction between these two groups, and led to the constitution of the state as an institution, understood as separate from the rest of society. However, another central purpose of the bureaucratic consolidation of the state, according to Weber, was the promotion of capitalism. Far from conceiving state and economic development as two independent processes, he emphasised the cooperation of the bourgeoisie in the creation and extension of modern state institutions. These institutions were critical to members of this group to establish and sustain those economic and cultural activities that were the basis of their power and that allowed them to occupy a dominant position in society after the Revolution [START_REF] Chazel | Communauté Politique. Etat et Droit dans la Sociologie Wéberienne: Grandeur et Limites de l'Entreprise [Political Community, State and Law in Weberian Sociology: Possibilities and Limits[END_REF].

Although Weber's analysis is anchored in a specific historical context and did not directly focus on the role of education, his work inspires two central questions that need to be considered when studying the relationship between the state and education. The first concerns the specific relationship that is created between the state and those educational institutions whose purpose is to train future state agents. Drawing on Weber's conception of bureaucracy, this question requires that we explore how educational institutions support and extend the authority and power of government through working to promote and persevere with the legitimacy of the state and the effectiveness of its political institutions. Similarly, it requires a close examination of how the state promotes the necessity and validity of the elite education institutions themselves.

The second central question that arises from a Weberian-inspired study of the relationship between the state and elite education institutions is the need to analyse whether and how the dominant social classesthat is, according to Bourdieu's (1984) model of the social space, those groups who have managed to secure high volumes of the valued capitals in a given society and at a given periodwork to reproduce or improve their social positions within and outside the state through their control over elite education institutions. Here, it is useful to introduce a further concept of Weber'ssocial closureto examine the processes through which social collectivities seek to restrict access to positions of power to a limited set of people, largely by closing off opportunities to 'outsiders'. As the modern state has an increased capacity to control the allocation of resources, it plays a crucial role in mediating the struggle between different collectivities for these positions of power.

These two research questions facilitate the examination of some of the broader questions with which the field of elite studies still needs to concern itself: a coherent definition of elites, understanding whether and how they engage in processes of social closure to limit the 'circle of eligibles' (Parkin 1974, 44), and the role of education in shaping and sustaining elites [START_REF] Maxwell | Elites: Some Questions for a New Research Agenda[END_REF]. We focus on the case of France in this paper because it fits with our areas of expertise and because the state has had a much stronger involvement in the creation, support and control of elite education institutions, when compared with many other countries, such as England, the USA and others. Using France in the first instance also allows us to examine the utility of Weber's theorisations for extending our understandings of the mechanisms that shape and sustain elite education systems, and will, we hope, therefore inspire further work drawing on a similar framework in other national contexts.

In France, an examination of the relationship between the state and the elite education institutions and how they mutually reinforced each other's legitimacy became a focus for political scientists and policy and organisational sociologists during the 1970s. These academics focused on how particular educational tracks shaped the political and administrative systems, and who took up senior positions within these bureaucratic structures [START_REF] Birnbaum | Les Sommets de l'État. Essai sur l'Élite du Pouvoir en France [The Heights of the State[END_REF][START_REF] Birnbaum | La Classe Dirigeante Française. Dissociation, Interpénétration, Intégration [The French Ruling Class. Dissociation, Interpenetration, Integration[END_REF][START_REF] Suleiman | Elites in French Society. The Politics of Survival[END_REF]. The second key research question identified abovehow processes of social closure are used by dominant social classeswas a key concern engaged with by Pierre Bourdieu in the last decades of the twentieth century. The State Nobility [START_REF] Bourdieu | The State Nobility[END_REF] was an ambitious study that examined both questionsfocusing on the state's capacity to control the elite education sector objectively (through the development of specific organisations and institutional mechanisms) and subjectively (the inculcation of social structures in the mind; Ball 2010) [START_REF] Bourdieu | Rethinking the State. Genesis and Structure of the Bureaucratic Field[END_REF]. The State Nobility therefore begins to trace the complex interpenetration of political, social and institutional priorities that are advanced through the elite education system in France [START_REF] Bourdieu | The State Nobility[END_REF][START_REF] Van Zanten | Bourdieu as Education Policy Analyst and Expert: A Rich but Ambiguous Legacy[END_REF].

In this paper we seek to present an analysis of the relationship between the state and elite education that explores the two central research questions which Weber's work encourages us to examine. Drawing on these classical French studies, especially the work of Bourdieu, but also integrating relevant international research literature, we examine both continuities and change in relations, and processes that embed these, between the state, the elite education system and the dominant social classes.

In the first section that follows, we take a historical perspective to examine the durable links between French political and elite educational institutions. We examine the content of the 'charters' [START_REF] Kamens | Colleges and Elite Formation: The Case of Prestigious American Colleges[END_REF][START_REF] Meyer | The Charter: Conditions of Diffuse Socialization in Schools[END_REF]) of these institutionstheir licence and mandate about what and how to teach, and the kinds of individuals they aim to produce. We also consider the crucial role of links between these elite institutions and the state in preparing their graduates for power [START_REF] Cookson | Preparing for Power. America's Elite Boarding Schools[END_REF]. We then explore how their curricula and institutional 'habituses' [START_REF] Reay | Always Knowing' and 'Never Being Sure': Familial and Institutional Habituses and Higher Education Choice[END_REF] are shaped by changing distributions of power among different status groups in the broader society and the kinds of educational strategies they pursue [START_REF] Karabel | Status-Group Struggle, Organizational Interests, and the Limits of Institutional Autonomy. The Transformation of Harvard, Yale, and Princeton, 1918-1940[END_REF][START_REF] Khan | Privilege: The Making of an Adolescent Elite at St. Paul's School[END_REF]).

In the second section that follows, we focus on new challenges to elite education. We first show why and how meritocracy, an institutional 'myth' [START_REF] Meyer | Institutionalized Organisations. Formal Structure as Myth and Ceremony[END_REF] promoted by elite educational institutions and by the state in their common effort to sustain their legitimacy and political order [START_REF] Duran | La Bureaucratie a-t-Elle un Avenir? [Does Bureaucracy Have a Future[END_REF], but also by the dominant groups seeking to defend their privileged status, is being challenged. In the following subsection, we assess the impact of processes of internationalisation within higher education on elite tracks. We examine to what extent these processes are 'dis-embedding' the French national elite system and replacing national principles and procedures with global norms, as has been the case in some other contexts (van Zanten 2015a). We also consider how this might be affecting the relationships between the state, elite institutions and social groups [START_REF] Wagner | Les Classes Sociales dans la Mondialisation [Social Classes and Globalisation[END_REF][START_REF] Brown | The Global Auction. The Broken Promises of Education, Jobs and Rewards[END_REF].

Durable but complex ties

The creation of an elite system for and by the state The state has exerted a profound influence on elite schooling in France. It fostered the creation of special elite higher education institutions and has strictly controlled the preparation for, and conditions imposed around, admissions into these elite schools; the kinds of curricula they teach; and the specific positions occupied by graduates from these elite higher education institutions in key public administration, political and other stateconnected organisations.

This influence dates back to the last quarter of the eighteenth century when royalty initiated the creation of institutions that would train agents qualified to lead the army, civil engineering projects and agricultural development [START_REF] Green | Education and State Formation: Europe, East Asia and the USA[END_REF]. This trend was reinforced rather than halted by the political upheavals of the eighteenth, nineteenth and twentieth centuries as the revolutionaries, Napoleon, the state representatives of the Third Republic and even the provisional government established after the fall of the Vichy regime in 1944who created the last major public institution of this type, l'Ecole Nationale d'Administration (ENA), in 1945all actively supported the need to train professionals that would both increase the state's capacity to control and rule and help re-affirm their authority. The creation of these 'special training schools', which came progressively to be referred to as the grandes écoles [START_REF] Belhoste | La Préparation aux Grandes Écoles Scientifiques au Xixe Siècle: Établissements Publics et Institutions Privées [The Development of Scientific Grandes Ecoles in the 19th Century: Public Establishments and Private Institutions[END_REF], was also related to generalised political mistrust towards universities. The universities were considered by representatives of all types of political regimes in France as overly oriented towards scholarly pursuits and either too influenced by the Church or too autonomous to produce the kinds of competent military and civil servants that would be loyal to the state.

Elite educational institutions were therefore used from the outset as a means of sustaining the efficacy and legitimacy of the state. Central to this process was the introduction of selection by merit through competitive examinationsthe concoursfor a specific number of places, which were annually determined by the state. The concours had a clear practical functionproviding the state with highly skilled professionals according to its technical needsbut also a strong normative component. It was instituted by the French revolutionaries to end the favouritism and nepotism linked to birth and social status associated with the Ancien Regime and to give priority to personal qualities assessed through what were perceived to be neutral procedures. First implemented by the most famous grande école -Polytechniquethe concours progressively became the symbol of a new 'Republican elitism' and was integrated into the recruitment processes of almost all grandes écoles [START_REF] Belhoste | L'anatomie d'un Concours. L'organisation de L'examen d'admission à L'ecole Polytechnique de la Revolution à nos Jours [The Anatomy of a Concours. The Organisation of the Admission Exam at the Ecole Polytechnique From the French Revolution to Today[END_REF].

The establishment of the competitive concours had a significant effect on the relationship between the grandes écoles and the secondary schooling sector because of the need to prepare students to take it. From 1802, training was being given to take the written and oral examinations required to gain entry into the grandes écoles, located in secondary schools. Later on, these classes préparatoires became a provision that students took for two years after their baccalauréat (end of secondary school examinations) and became officially viewed as a form of higher education [START_REF] Belhoste | La Préparation aux Grandes Écoles Scientifiques au Xixe Siècle: Établissements Publics et Institutions Privées [The Development of Scientific Grandes Ecoles in the 19th Century: Public Establishments and Private Institutions[END_REF], although largely still taught in secondary school buildings. This is still the case today.

The curriculum provided within these elite education tracks has been strongly influenced by both their relationship to the state and the division of work between the grandes écoles and the preparatory classes. The curricular 'charter' [START_REF] Meyer | The Charter: Conditions of Diffuse Socialization in Schools[END_REF] of the public grandes écoles has thus been driven by their initial purpose, which is to train students to become civil servants within the administrative and technical grand corps of the state. Such 'channeling' has also demanded that students be socialised into particular kinds of understandings of the purpose and value of the state, supported by a strong and expert bureaucracy [START_REF] Birnbaum | Les Sommets de l'État. Essai sur l'Élite du Pouvoir en France [The Heights of the State[END_REF][START_REF] Birnbaum | La Classe Dirigeante Française. Dissociation, Interpénétration, Intégration [The French Ruling Class. Dissociation, Interpenetration, Integration[END_REF]. Such an inculcation of values is further embedded through the meritocratic nature of the recruitment processes into the grandes écoles, given the small size of these institutions, and the strong probability that graduates will secure a prestigious position among the political and administrative elite, all of which foster a sense of solidarity among its students [START_REF] Suleiman | Elites in French Society. The Politics of Survival[END_REF].

As found by colleagues in the USA of students attending private elite boarding schools [START_REF] Gaztambide-Fernández | The Best of the Best: Becoming Elite at an American Boarding School[END_REF][START_REF] Khan | Privilege: The Making of an Adolescent Elite at St. Paul's School[END_REF], students come to understand their education and future positions they will hold within and outside political institutions as merited.

Knowledge within the grandes écoles is also understood differentlynot as necessarily valued in and of itself, as in universities, but seen as a technical and power resource [START_REF] Belhoste | La Formation d'une Technocratie[END_REF]. Furthermore, while the curriculum focuses on the kinds of knowledges needed for the particular field students are being prepared for, these institutions also emphasise the development of a range of generalist skills that, they believe, will help these future professionals take on a variety of roles within and beyond their sphere of technical expertise (Mendras and Suleiman 1995;[START_REF] Thoenig | L'Ere des Technocrats [The Era of Technocrats[END_REF]. Such an emphasis on generalist skills becomes an elite marker for these graduates, as it does for bureaucractic elites in other countries [START_REF] Mangset | Contextually-bound Authoritative Knowledge: A Comparative Study of British, French and Norwegian Administrative Elites' Merit and Skills[END_REF].

A specific focus on the curricula of those grandes écoles preparing students for high-status administrative positions can be found in two relatively recent studies examining how students are trained at ENA, and are prepared for the concours to enter ENA at Sciences Po. [START_REF] Eymeri | La Fabrique des Enarques[END_REF] describes 'a second-hand culture', where knowledge is informed by textbooks rather than the analysis of original work, and classes are taught by state agents rather than university professors. The main form of pedagogy is delivery of knowledge through exercises that encourage students to focus on facts rather than on theoretical understandings of issues, and where they are required to adopt unambiguous positions rather than embrace the complexity of the social, financial and political nature of the problems they are discussing. [START_REF] Oger | Le Façonnage des Elites de la République[END_REF], meanwhile, analysed the content of the written and oral examinations of the ENA concours and found that both conformity and originality were in fact rewarded. Yet [START_REF] Oger | Le Façonnage des Elites de la République[END_REF] has also suggested that the kinds of intellectual and social qualities being examined for are associated with being members of the upper classes of French society, an argument also made by [START_REF] Bourdieu | Les Catégories de l'entendement Professoral [The Categories of Professorial Understanding[END_REF] in their analyses of teachers' perceptions of students' performance within preparatory classes.

The curriculum of the grandes écoles must also be understood in relation to the division of labour that occurs between them and the preparatory classes. While ENA and Sciences Po perhaps present a different case, because in the past Sciences Po has played a preparatory role for entry into ENA and has prided itself on offering a kind of pedagogical counter-model of preparatory classes [START_REF] Garrigou | Les Elites contre la République[END_REF], the preparatory classes leading to the other grandes écoles (those specialising in engineering, management, sciences and humanities) provide a highly demanding and ambitious programme of study. The teaching approach, provided by secondary school teachers, is still very close to that of secondary schools but, because of the strong focus on success at the concours, teachers behave more as coaches. Students are expected to work hard under conditions of urgency, even 'organized panic' [START_REF] Bourdieu | The State Nobility[END_REF], which is seen to aid them in ensuring they will succeed at the concours, but also as mirroring the working environment they will encounter when they take up their administrative and political elite positions. These teachers, who see themselves as 'at the service of elites' [START_REF] Rauscher | Les Professeurs des Classes Préparatoires aux Grandes Écoles: Une Élite au Service des Élites[END_REF], are nevertheless careful to create an environment in which their students' talents are encouraged to emerge and to provide them each with personalised attention [START_REF] Darmon | Classes Préparatoires. La Fabrique d'une Jeunesse Dominante[END_REF]van Zanten 2015a).

Elite tracks were therefore initially created by the state to fulfil its need for a highly skilled workforce of bureaucrats and public servants who would stabilise French society and support its growth in particular ways. The legitimacy of the creation of elite tracks has been strongly supported by the socalled meritocratic concours, which seeks to recruit 'the best of the best' (Gaztambide-Fernández 2009) and create a closed elite system based on exclusive interactions between the classes préparatoires and the grandes écoles. Studies have also highlighted the complementary curricular character of these two elite educational segments. In preparatory classes, students acquire a general disposition, learning to cope quickly and effectively with external demands; while in the grandes écoles they continue to acquire general skills alongside more specialised ones and prepare more directly for positions of power.

Private elite institutions and the public/private divide among class fractions

It is, however, important to underline that the state has never had a total monopoly on elite education, nor has it been able to completely regulate the struggles that have emerged between different social groups in relation to elite educational tracks. Private education is particularly important when considering gender and elite schooling. Access to the Napoleonic state lycées created in 1802 was solely for boys. Similar public institutions for girls were only created in 1880. The purpose of the introduction of state provision for girls was, however, largely driven by a desire to limit the ideological influence of Catholic institutions on future female citizens, rather than seeking to promote women's work [START_REF] Offen | The Second Sex and the Baccalauréat in Republican France, 1880-1924[END_REF]). Thus, these schools for girls had a different curriculum and were not preparing them for the baccalauréat, or entry into higher education [START_REF] Mayeur | L'enseignement Secondaire des Jeunes Filles sous la Troisième République [The Secondary Education of Young Girls Under The Third Republic[END_REF].

However, even after the introduction of publically-funded secondary girls' schools, the majority of families who made up the aristocracy and bourgeoisie continued to send their daughters to Catholic private schools, especially those families who were practising Catholics and whose power and prestige were linked to economic resources (industrialists, bankers, members of the highly-paid professions) as they sought out an environment where girls could be educated among students from similar social milieus and inculcated into a 'domestic' ideology (de Saint-Martin 1990).

The situation was not the same for boys, who might attend Catholic private institutions but were also likely to be sent to prestigious state lycées in order to prepare them for the elite professions. However, although Catholic schools were chosen by the upper classes mainly for ideological and social reasons, over the course of the first half of the twentieth century these schools began to focus more on providing 'excellence' in their academic provision. This move was driven by a series of changes in the law: the 1905 law that enforced the separation between church and state and banned religious teachers from such schools; the introduction of coeducational, state-funded secondary schooling in 1924; and the 1959 Debré law that provided public funding to private schools [START_REF] Peretz | La Création de l'enseignement Secondaire Libre de Jeunes Filles à Paris (1905-1920) [The Creation of Private Secondary Schools for Young Girls in Paris (1905-1920)[END_REF]. These processes contributed to the growth and take-up of preparatory classes in private Catholic lycées.

Work by van Zanten (2015a) nevertheless shows that some significant differences remain between prestigious public and private 'prépas', as these classes are called in colloquial language. The selection process, as well as the formal curriculum of these preparatory classes (dictated by the content of the concours), are the same. However, private preparatory classes are allowed to ask for more personal details from applicants, as well as a 'letter of motivation'. Moreover, within private provision, the focus on socialisation is stronger, as these classes seek to develop the 'whole person' through providing a 'total curriculum', including compulsory cultural, social and physical extracurricular activities [START_REF] Maxwell | Becoming Accomplished: Concerted Cultivation among Privately Educated Young Women[END_REF]. Significantly, there is also a greater emphasis placed on the development of group solidarity between students, and on identifying with the specific institution being attended instead of with the state, as is the case in public preparatory classes.

With regard to the grandes écoles, today at least one-third of them are private institutions. Private tertiary providers started emerging after 1870 as a result of economic developments in France. Special institutions were created that were linked to business sectors. Despite important initial differences between these and the state-funded grandes écoles, those wishing to become 'top institutions' developed a process of 'mimetic isomorphism' [START_REF] Dimaggio | The Iron Cage Revisited: Institutional Isomorphism and Collective Rationality in Organizational Fields[END_REF] with their public counterparts, to gain prestige and increase their influence. One of the main strategies in this respect was the adoption of the concours as the main form of selection, thereby benefitting from the 'nobility' bestowed upon public institutions through this meritocratic contest [START_REF] Bourdieu | The State Nobility[END_REF]. Additionally, recruitment was focused on students who had completed their two years of preparatory classes, thereby raising the academic qualifications of their entrants significantly [START_REF] Languille | L'Essec, de l'Ecole Catholique des Fils à Papa à la Grande Ecole de Gestion [The Essec -Grande Ecole de Commerce. Moving from the Catholic School attended by your Father to the Big Management School[END_REF]. A further, important development has been that the most prestigious of these private tertiary education providers have acquired a special status through securing funding from regional chambers of commerce.

However, the selection processes, curricula and types of socialisation provided through the grandes écoles do still vary in important ways [START_REF] Allouch | L'Ouverture Sociale comme Configuration[END_REF]). These dimensions are not only shaped by their public or private status, or the specialisation for different fields of work, but, significantly by the particular social groups [START_REF] Karabel | Status-Group Struggle, Organizational Interests, and the Limits of Institutional Autonomy. The Transformation of Harvard, Yale, and Princeton, 1918-1940[END_REF] they are most likely to recruit. This has important implications for the different institutional 'charters' [START_REF] Kamens | Colleges and Elite Formation: The Case of Prestigious American Colleges[END_REF][START_REF] Meyer | The Charter: Conditions of Diffuse Socialization in Schools[END_REF]) and institutional habituses [START_REF] Reay | Always Knowing' and 'Never Being Sure': Familial and Institutional Habituses and Higher Education Choice[END_REF]) of these tertiary providers. As [START_REF] Bourdieu | The State Nobility[END_REF] found, at the Ecole Normale Supérieure (ENS), for instance, which trains mostly scientists and higher education professors and whose students are most likely to come from the intellectual fractions of the upper class, scholarly knowledge and intellectual pursuits have always been prioritised. On the other hand, managerial schools that attract a significant number of students from the economic fractions of the upper class have, from the onset, promoted 'soft skills' and held an instrumental view of knowledge [START_REF] Lazuech | L'exception Française[END_REF]. In these institutions, extracurricular activitiesincluding election campaigns for student associations, sport events and various social and networking activitiesare foregrounded [START_REF] Abraham | Du Souci Scolaire au Sérieux Managérial, ou Comment Devenir un 'Hec' [From Academic Concern to Managerial Seriousness. Or How to Become a 'Hec[END_REF]). This process is deemed necessary so the 'purist' [START_REF] Brown | The Mismanagement of Talent. Employability and Jobs in the Knowledge Economy[END_REF] students recruited from the preparatory classes via the very selective concours learn the necessary skills to succeed as future leaders in the private sector. These differences can already be observed in the curricula of various preparatory classes. Thus, literary and scientific classes are much more intellectually demanding, while those preparing for management schools have a greater focus on other forms of socialisation [START_REF] Darmon | Classes Préparatoires. La Fabrique d'une Jeunesse Dominante[END_REF].

Differences between graduates from different institutions and tracks, however, tend to fade once they enter the job market because of the varied career trajectories pursued by many. The grandes écoles and the grands corps have for a long time tolerated, for instance, that a proportion of civil servants will initially work in the private sector immediately following their studies, in the anticipation that they will eventually return to serve the state. An alternative trajectory is for graduates to enter public administration upon completion of their grandes écoles training, but later move into private industry. This movement between careers in the public and private sectors, with an increasing focus on pursuing careers outside public administration, grew during the second half of the twentieth century. Private firms appreciate having employees who have gained particular kinds of informational and social capital through their time in public administration, seeing these as useful during negotiations with state representatives in a state-dominated economy. Similarly, the state values civil servants who have insider knowledge of the private sector.

Studies have shown that a significant proportion of CEOs of the 40 largest French private firms are graduates from public grandes écoles (27% from Polytechnique and 23% from ENA in the 1990s; 22% from Polytechnique, 19% from Sciences Po, 16% from ENA in 2007) alongside graduates from the major private management school, l'Ecole des Hautes Etudes Commerciales [START_REF] Bauer | Radiographie des Grands Patrons Français. Les Conditions d'accès au Pouvoir[END_REF][START_REF] Dudouet | Les Dirigeants Français du Cac 40: Entre Elitisme Scolaire et Passage par l'État [The Leaders of the Biggest 40 French Firms. Academically Elite with Experience of Working in the State[END_REF]. Similarly, among the 200 most important French companies, 47% in the 1980s and 28% in 2007 had among their directors people who had had a previous career in the public sector [START_REF] Bauer | Les 200[END_REF][START_REF] Dudouet | Les Grands Patrons et L'État en France [CEOs and The State in France[END_REF]. Birnbaum et al.'s (1978) analyses of three editions of Who is Who in France? demonstrated that already in the 1970s a strong interpenetration between political and economic elites was occurring due to their common social background and academic trajectories through the grandes écoles, but also because they shared residential and leisure spaces, as well as membership of the same clubs and associations.

In this section we have highlighted how the various elite tracks have been sustained over time with a growing mixed economy of providersthe private and Church sectors. Although elite tracks variously serve different upper-class fractions and have been shown to engage in different socialisation processes, the effect nevertheless seems to be that the elite nature of the education provided is securedthrough a focus on academic excellence (in a small group of state institutions, as well as some private, mostly Catholic, lycées and preparatory classes), selection based on merit via the concours, and an acceptance by the state and private industry that graduates from all the grandes écoles bring a set of skills, expertise and experience that will be a welcome resource in both sectors.

New challenges

The erosion of meritocratic legitimacy and new efforts to recreate it In France, the existence of separate higher education tracks leading to elite jobs has been historically justified on the basis of the meritocratic nature of their admission procedures. Merit is defined as a combination of talent and effort understood as individual characteristics that are independent of birth and family resources [START_REF] François | L'École des Chances. Qu'est-ce qu'une École Juste?[END_REF]). Yet existing data show that the proportion of working-class students who have been able to pursue the French elite tracks has always been quite small [START_REF] Baudelot | Les Normaliens: Origines Sociales et Recrutement Social des Normaliens 1914-1992 [The Normaliens: Social Origins. The Social Recruitment of Normaliens, 1914-1992[END_REF]. The illusion of equal access across social classes has nevertheless been maintained for most of the twentieth century.

The hold of such a myth has been possible, first, because of the abstract definition of meritocracy as equality of opportunity, not as equality of results. Second, most classes préparatoires are located in public secondary schools, so they are seen as accessible to students from all social classes. In practice, this has meant that these tracks have been opened to those social groups with high cultural capital but relatively low economic resources, such as the children of secondary and primary school teachers. Research has found that during the greatest period of the democratisation of elite tracks , students from these social groups were almost as successful in securing places in preparatory classes and the grandes écoles as students with upper-class backgrounds [START_REF] Albouy | Les Inégalités Sociales d'accès aux Grandes Ecoles [Social Inequalities in Access to Grandes Ecoles[END_REF]Euriat and Thelot 1995). Additionally, studies have identified that many of these middle-class students' grandparents had been farmers or manual workers, giving credence to the idea that the French elite education system made it possible to move from the bottom to the top of the social ladder within two generations [START_REF] Veltz | Faut-Il Sauver les Grandes Écoles? [should Grandes Ecoles Be Saved?[END_REF].

However, despite the relative democratisation that occurred during the 'Glorious Thirty' period )a period of significant economic growth in France, which also introduced the comprehensivisation of the education systemthe last decades of the twentieth century heralded a return to stronger processes of social closure around elite education tracks. First, concerns about the possible negative effects of the successive waves of expansion of French universities in the 1960s and 1980s led to a growing proportion of upper-class parents, who themselves had been university educated, opting for a more selective, elite track for their own children's education. Second, as the provision of preparatory classes was expanded and diversified during the 1990s, partly as a result of increasing demand from middle-class parents, upper-class families increasingly adopted strategies to secure their own child's success in gaining access to the 'prépas' that were seen as both the most intellectually demanding and the most socially selective through activities such as intensely supervising their education at home, providing them with private tuition and extracurricular activities, and carefully choosing curricular options and schools [START_REF] Van Zanten | Choisir son École. Stratégies Parentales et Médiations Locales [Choosing Schools. Parental Strategies and Local Mediations[END_REF](van Zanten , 2015b)).

Third, the expansion of the secondary school system coupled with its 'segregative democratisation' from 1960 to 1990 [START_REF] Merle | Le Concept de Démocratisation de l'Institution Scolaire: Une Typologie et sa Mise à l'Epreuve [The Concept of Democratisation of Educational Institutions. A Typology and Its Empirical Verification[END_REF] has meant that recruitment to classes préparatoires has been increasingly influenced by the subjective assessments made by selection panel members. Research has found that members of these panels are more likely to mistrust the academic value attached to students applying from socially and ethnically diverse lycées as they tend to consider that teachers working in these schools are 'overindulgent' in their assessments and cover curricular subjects in less depth. Conversely, students with similar grades but who come from lycées with good reputations, usually located in Paris or in other urban areas where upper-class and middle-class families reside, are advantaged in the selection process (van Zanten 2015a).

The meritocratic basis of elite education is also challenged when considering the under-representation of female students in some of the tracks, especially the scientific tracks and grandes écoles specialising in engineering. However, the reasons for this are different to the factors that affect accessibility for lower-class backgrounds. First, it is not that young women do less well than young men in terms of attainment (in fact, the opposite is truealthough gender differences are more acute among lower-class than upper-class groups), but that they are less likely than their male counterparts to choose the scientific track at the lycée level and therefore cannot apply to the scientific preparatory classes that would channel them into the most prestigious engineering schools and to Polytechnique (where only 13% of students starting in 2013 were women).

Second, women are often aware of their reduced chances of occupying top positions within the civil service. The administrative and political fields in France have been strongly monopolised by men who hold a male-centred vision of political leadership and authority, and who have been found to be more likely to nominate other men to key positions [START_REF] Milewski | L'Inégalité entre les Femmes et les Hommes dans la Haute Fonction Publique: Du Constat aux Moyens d'y Remédier [The Inequality Between Women and Men in Senior Civic Service. From Evidence to Possible Solutions[END_REF][START_REF] Sineau | Profession Femme Politique. Sexe et Pouvoir sous la Cinquième République [Female Politicians[END_REF]. Such processes of nepotism have been reinforced by the 'incestuous links' between certain grandes écoles (that only became coeducational in the 1970s) and the grands corps [START_REF] Suleiman | Elites in French Society. The Politics of Survival[END_REF]. These factors explain to a large extent why female students, despite the fact that they have outnumbered male students at Sciences Po and other political science institutes for several years, were much less likely until recently to take the ENA concours (and also less likely to pass) [START_REF] Eymeri | La Fabrique des Enarques[END_REF]. Finally, even if female students are over-represented in the grandes écoles specialising in management, their distribution between the different sub-fields of management is very unequal, revealing again processes of self-selection according to the prestige, gender image and male composition of the different professions they feed into [START_REF] Buscatto | Le 'Plafond de Verre' Dans Tous ses Eclats. La Féminisation des Professions Supérieures au Xxe Siècle [The 'Glass Ceiling' in All Its Manifestations. The Feminization of the Professions During the 20th Century[END_REF].

These facts, in particular those concerning inequalities in access to elite tracks for lower-class students, began to receive considerable attention after Sciences Po, in 2001, launched a new 'equality of opportunity' programme through a series of partnerships with lycées in disadvantaged areas (initially with seven in 2001, growing to 100 in 2014). This initiative gave way to heated debates not only within the institution itself, especially by the upperclass students who criticised its 'demagogic' character as students from the disadvantaged lycées were not required to take the competitive entrance examination as per this new initiative, but also within the media, political circles, and between the Director of Sciences Po and those of other grandes écoles.

Meanwhile, private-sector firms, especially those with international markets, created new 'diversity' programmes similar to those of their American or European competitors [START_REF] Bereni | Faire de la Diversité une Richesse pour L'entreprise': La Transformation d'une Contrainte Juridique en Catégorie Managériale ['Turning Diversity into an Asset For Business'. The Transformation of a Legal Constraint into a Managerial Category[END_REF], and pushed elite educational institutions for more symbolic action around increasing diversity. President Sarkozy's right-wing government also sought to promote limited forms of affirmative action in politics and the administration as part of his effort to distinguish himself from the old Right and well as from the Left [START_REF] Simon | Comment la Lutte contre les Discriminations est Passé à Droite [How the Fight Against Discrimination was Recuperated by the Right[END_REF]. The grandes écoles responded with a number of widening participation programmes, led by the ESSECa prestigious management school. During the same period, Polytechnique and the most prestigious engineering schools developed other initiatives aimed at increasing the number of women entering their institutions, driven by similar market and political pressures (these included the law in 2000 encouraging parity in political bodies, as well as a more recent 2013 law enforcing quotas for women among new high-level civil servants).

The initiatives aimed at increasing the participation of lower-class students or women entering elite tracks have nevertheless been very modest in scope. Most grandes écoles have only offered additional tutoring for students from disadvantaged lycées when preparing for the concours. While Sciences Po made the concours non-mandatory for these students, other grandes écoles would not contemplate such a move or agree to change the content or form of the examination in ways that might advantage these lower-class students. Furthermore, selection panel members of the public preparatory classes were very reluctant to consider taking into account students' social background as they believed strongly in the neutrality and symbolic legitimating power of their existing procedures.

Nevertheless, despite these limited changes, elite educational institutions have managed to deflect potential criticism by emphasising that the problem of social and gender inequality lies with society more generally, including within the family and the rest of the school system. These institutions have claimed that by developing small interventions to tackle the issue (sometimes with funding received from private firms, who thereby also meet their corporate social responsibility), and through the official integration of these initiatives into a 'comprehensive' government policy around equal opportunity that is promoted by intensive media coverage, they are doing their bit. In this way they have succeeded in maintaining their legitimacy while also being publicly perceived as important policy actors committed to social justice. This arguably further embeds the material and symbolic resources of these elite institutions and demonstrates the ways elite institutions have worked in partnership with the state to at least discursively support the call for improved equality of opportunity, while also demonstrating their autonomy to do so in ways that suits them (van Zanten 2010).

Twists and turns in the march towards internationalisation

Pressures towards internationalisation constitute a second significant challenge to the relationship between the state and elite education institutions. The French governmentwho was from the onset directly involved in the framing of the Bologna declaration on higher education, designed to ensure comparability in the standards and quality of higher education qualifications in European countries [START_REF] Ravinet | La Coordination Européenne 'à la Bolognaise' Réflexions sur l'instrumentation de l'espace Européen d'enseignement Supérieur [European Coordination 'A la Bolognaise'. Reflections on the Creation of a European Space for Higher Education[END_REF])has enforced its main components. These include: the establishment of three different cycles in higher education studies (three to four years for a bachelor's degree, one to two years for a master's degree and three to four years for a PhD); and the introduction of the European Credit Transfer and Accumulation System. State-funded higher education institutions must implement these new norms, and private institutions have also been strongly encouraged to adopt them.

While elite institutions have engaged to some extent with these changes, they have also worked hard to preserve their institutional interests in the process. For the Ministry of Higher Education, the most rational and effective way to adopt the Bologna Process would have been to create a unified higher education system in which elite tracks would have become part of a reformed university system. Raised and muted at other points in time, such a solution has been largely dismissed by fractions of the administrative and political elites and completely rejected by the elite institutions themselves, fearing that such a change would signal the end of their prestigious status and their autonomy.

Grandes écoles have instead re-organised their curricula in such a way that their students can now gain a master's upon graduating (either awarded by the grande école itself, or via partnerships with selected universities). Meanwhile, preparatory classes have obtained European Credit Transfer and Accumulation System credits for their students, who after completing two years of these specialist classes can now, by enrolling on a few additional university courses, gain a bachelor's degree, or even go straight onto a master's programme if they have completed three years of preparatory classes.

Furthermore, both the state and elite education institutions are under considerable pressure to secure a top position in internationally-recognised ranking schemes. Yet for elite institutions, the highly specialised nature of their provision makes their integration in these international comparison exercises challenging. A central dimension in these rankings and in the definition of 'world-class excellence' [START_REF] Hazelhorn | Rankings and the Reshaping of Higher Education[END_REF]) is research output. However, for the grandes écolesdue to their links to the state, their professional orientation and their separation from universitiesresearch has, until recently, not featured as a central aspect of their activity [START_REF] Cytermann | Universités et Grandes Ecoles[END_REF]. 'Excellence' has been and still is, to a large extent, understood as being about the academic (and social) qualities of their students, emphasised through the rigorous meritocratic selection processes insisted upon, and demonstrated in the prestigious nature of the careers of their graduates in public and private organisations. Second, the kinds of indicators of research output that are used in rankings (Nobel prizes and Field medals as well as publications in highimpact journals and number of citations) are highly dependent on institutional size. Given the small size of most grandes écoles (Polytechnique recruits 400 students each year, the ENS and the most prestigious engineering schools 100, and ENA only 80), this makes their ability to compete on a world stage around research output extremely difficult.

In the context of considerable fragmentation of the French higher education field with pressures to compete at an international level, in 2007 the government promoted the creation of new networks of higher education institutions called Pôles de recherche et d'enseignement supérieur (PRES) with a view to increasing the concentration of research and teaching activities into one single entity in different regions and big cities [START_REF] Aust | Napoléon Renversé? Institutionnalisation des Pôles de Recherche et d'enseignement Supérieur et Réforme du Système Académique Français [Napoleon Overthrown? Institutionalization of Research and Higher Education Networks and Reform of the French Academic System[END_REF]. While several of the 26 PRES that were created included universities and grandes écoles, the interaction between the two types of institutions has remained very limited. Grandes écoles applied for PRES status both because the price of abstention in terms of their relationship to the state would have been high, but also because higher education institutions integrated into PRES could benefit from additional financial resources that are channelled into these networks.

A more ambitious approach led by the state from 2013 has been higher education institutions forming 'communities of universities', aimed at forging stronger associations and promoting opportunities for mergers. The most important of these communities to date is the proposed Campus Paris-Saclay, which is due to start in 2015. It will bring together 11 grandes écoles, including Polytechnique, and two universities, as well as seven research centres in a cluster situated at the southwest of the Parisian region. Campus Paris-Saclay will produce joint research publications, as well as offering a PhD and master's programme for French and international students. Although it is too early to anticipate the outcomes of this new development, it is likely that such institutional frameworks will allow grandes écoles when they are in the majority, as in this case, to control most of the major decision-making processes. Meanwhile, from a macro perspective, it can be argued that the formation of these different types of networks have accentuated rather than reduced hierarchies within the French higher education system, with prestigious grandes écoles and universities collaborating and the less elite institutions left to associate with each one other or remain isolated.

Another major aspect of becoming a world-class institution in the global race for income and prestige is the ability to attract 'excellent' students from abroad, especially those coming from emerging countries that have not as yet established competitive national higher education systems of their own. French grandes écoles have two major challenges in such a race.

The first challenge is that the main language of instruction is French. The French state has not strongly encouraged a move towards the mainstreaming of English in higher education. This can be partly understood by the fact that, traditionally, foreign students have come from France's former colonies or from countries where one of the dominant languages was French, as well as from French-speaking families living abroad. Although these groups of students have, for the most part, not been understood as 'excellent' and most do not have an upper-class background, the government has welcomed them as part of its efforts to defend the global presence of the French language and its culture.

While most grandes écoles are conscious of the need to attract new generations of foreign academically-able students, recruitment practices have been very diverse across the sector. The desire to internationalise their student body and academic staff cohorts has depended on each institution's relationship to the state, on the type of disciplines taught and whether graduates are being prepared for a national or international job market, as well as the kinds of resources the grandes écoles have available. In this respect there is a great divide between state and private grandes écoles, with private institutions having more funding and more impetus to internationalise [START_REF] Darchy-Koechlin | Les Élites Étrangères des Grandes Écoles face au Modèle Français de l'Excellence [Foreign Elites in Grandes Ecoles: Engaging with the French Model of Excellence[END_REF].

Alongside the challenges posed by French medium instruction and a largely French-speaking academic teaching staff, the long-standing commitment to the concours for recruitment into the grandes écoles makes the recruitment of international students extremely difficult. In the most famous state grandes écoles -ENS, Polytechnique, ENAthe concours is open only to national students who are considered civil servants in training and receive a salary from government. Foreign students, meanwhile, are admitted following a different procedure. In other grandes écoles, foreign students could, in theory, take the competitive examination. Yet because most foreign students will not have completed the initial preparatory classes beforehand, they would be significantly disadvantaged during the admissions process. Therefore, most grandes écoles have created specific admission systems for them. The institution of alternative admissions procedures for foreign students has created tensions between the French and non-French students, especially in the smaller state grandes écoles. This links closely to the founding principle of the elite education systema commitment to meritocracydefined here as securing admission through the concours, the best method for measuring excellence (Darchy-Koechlin and Draelants 2010).

In fact, in a context of internationalisation, new tensions have emerged between the three members of the triad sustaining elite education in France. On the one hand, among state officials there is a desire to attract foreign students without compromising France's position as a former colonial power or the importance of French as a global cultural resource. On the other hand, elite institutions are keen to attract the most able students rather than pursue these political goals, but without radically transforming their recruitment procedures. Finally, domestic studentsthe majority of whom come from the dominant social classesactively defend the meritocratic legitimacy of the concours. Wishing to secure the national advantages that stem from an elite education system for them, they tend to see foreign students as illegitimate competitors for elite status.

A third focus related to the internationalisation agenda is the promotion of international mobility of French students. This again is rendered difficult by the specific features of French elite education. Within the preparatory classes, the curriculum is dominated by conservative French pedagogical traditions, evident, for instance, in the practice of teachers awarding their students very low grades for their work in an attempt to prepare them for the way they will be evaluated against others at the concours. While the most prestigious of these classes have now taken some action, such as providing an explanation in English of their grading system, such a system is still likely to complicate acceptance onto foreign university courses for the few but growing number of students opting for this option, many of whom are taking preparatory classes in literature and humanities for which there are very limited opportunities for further study in the grandes écoles.

Most of the grandes écoles are working to promote the possibility of studying abroad or provide internship opportunities through reciprocal partnerships with prestigious foreign universities. Yet the latter arrangements have been difficult to sustain as the state grandes écoles are free of charge, while many of the foreign elite institutions are not. Furthermore, although French upper-class students from elite institutions are open to these experiences, most of them intend to remain living and working in France and indeed display a strong allegiance to the French educational model and to the French nation-state [START_REF] Power | Self, Career and Nationhood: The Contrasting Aspirations of British and French Elite Graduates[END_REF].

Attempts to internationalise the recruitment basis of the elite French higher education system have thus been limited in its successespecially for grandes écoles traditionally established to produce the grands corps of the state bureaucracy. Here the state, the institutions themselves and the social classes who dominate in such institutions are all hesitant to facilitate the recruitment of foreign students because this will imply significant changes in targeting, admission processes, curricula and the medium of instruction that threaten geopolitical strategies, institutional prestige and social domination and legitimacy. Similarly, efforts to increase the international mobility of students from elite institutions between France and other prestigious institutions have been stymied by the organisation of the French elite tertiary system, which is quite different to most other countries, the pedagogical and curriculum content of many French elite education programmes, and the main actors involved -the state, the institutions and the upper-class students and familieswho display a general ambivalence for transnational mobility.

Conclusion

Inspired by Weber's theorisation of the importance of state bureaucracies in shaping modern nation-states, we have considered, in this paper, how elite education tracks were established and have been maintained by the state in France. The state saw these elite tracks as necessary to train high-status civil servants who could manage its affairs effectively and stabilise the political environment. The state therefore created and has actively sustained a segregated higher education system. Many young people planning to work for the private sector also follow these elite tracks. Thus, given that most dominant agents in the public and private sectors have been educated through the grandes écoles system, it is not surprising that the elite education sector has received considerable support to sustain its separation from the universities.

Elite education in France is seen as a public good and is negotiated between the state and the dominant social classes. Despite upper-class fractions competing for elite status in French society throughout history, there has remained strong support for elite education tracks among them as this is one of the key mechanisms for excluding other groups from vying for elite positions [START_REF] Bourdieu | The State Nobility[END_REF][START_REF] Van Zanten | La Competition entre Fractions des Classes Moyennes Supérieures et la Mobilisation des Capitaux autour des Choix Scolaires [Competition Between Upper-Middle Class Fractions and the Mobilization of Capitals around School Choice[END_REF]. Furthermore, the elite institutions themselves are actors within the sector, actively defending their own interests, usually through forming strategic alliances with the other two sets of actors within the triad. It is important to understand this particular political and social context, which has shaped the development and legitimacy of elite education institutions in France, when seeking to make sense of how the elite education sector has responded to, and will most probably continue to respond to, challenges to its legitimacysuch as those recently experienced, including pressures to widen participation and internationalise higher education.

The French elite education system is strongly constricted in its ability to significantly innovate by its history and the strong three-way relationship that exists between the state, elite education tracks and dominant social classes. While efforts to widen participation and internationalise have made visible the existence of different perspectives between groups of senior civil servants and CEOs, elite institutions and the upper classes, as well as between fractions of dominant social groups, our analysis suggests that overall the triad dynamic tends to reproduce particular normative understandings and institutional procedures [START_REF] North | Institutions, Institutional Change and Economic Performance[END_REF][START_REF] Pierson | Path Dependence, Increasing Returns, and the Study of Politics[END_REF], although the process of reproduction is not as seamless as it has been in the past. The actors in this triad have worked hard to maintain the 'meritocratic consensus' governing elite education tracks, which had acted as an effective closure mechanism for the state, elite institutions and the upper classes. However, the legitimacy conferred onto elite education institutions, dominant social groups and those who take up key positions across the private and public sectors through this consensus has become the object of regular and growing contestation and scrutiny in the last decades. Furthermore, an ongoing commitment to the concours as a way of recruiting students onto elite tracks and the kinds of curricula found in the grandes écoles can be understood as an obstacle to French influence on global dynamics.

The specific features of the French elite educational system, and their preservation in identical or renewed forms in current times, raise the question of the degree to which these observations about relations between the state, elite education tracks and dominant social classes can be applied elsewhere. This broader question will require scholars to consider two issues. First, we need to appreciate differences in the composition and purpose of the state in different countries. In France, the state has been argued to be 'strong' because it relies on a large bureaucracy. Furthermore, through maintaining a relatively strict separation between public and private domains, the state seeks to actively create a set of universal 'French' norms and practices with which it expects all its citizens to comply, ignoring its increasingly diverse population [START_REF] Birnbaum | Défense de l'État 'Fort': Réflexions sur la Place du Religieux France -Etats Unis [On Defense of the 'Strong' State. Reflections on the Importance of Religion in France and The United States][END_REF]. Analysts in other countries may want to consider the 'styles of statism' [START_REF] Baldwin | Beyond Weak and Strong: Rethinking the State in Comparative Policy History[END_REF] different ways in which associations between the state and the society it governs are structured (King and Le Galès 2011)before seeking to consider how the state shapes and legitimates elite education tracks. Second, scholars of elite education should begin to compare the interests of, values espoused by and strategies employed by elite educational institutions and dominant social classes in relation to elite education tracks across various national contexts.

We suggest that the theoretical model used here to interpret institutional developments and change in the French context may support such a crossnational analyses. This approach has firstly emphasised that the 'charters' of elite institutions are strongly shaped by the broader relations governing societies [START_REF] Dimaggio | The Iron Cage Revisited: Institutional Isomorphism and Collective Rationality in Organizational Fields[END_REF][START_REF] Meyer | The Charter: Conditions of Diffuse Socialization in Schools[END_REF]. Second, we have demonstrated that even when there is considerable national and international pressure for change, elite institutions are able to, in different ways, defend their interests through the significant economic, cultural, social and symbolic resources they have accumulated over time and due to the strong support they receive from the state and the dominant classes [START_REF] Karabel | Status-Group Struggle, Organizational Interests, and the Limits of Institutional Autonomy. The Transformation of Harvard, Yale, and Princeton, 1918-1940[END_REF].

While our work has taken the nation-state as the specific space being examined, increasingly education, and particularly elite education, must also be understood within a broader global context. Research by Jane Kenway and colleagues, for instance, understands the elite schools in their study as shaped by global forces such as mobility of upper-class and middle-class families, desires for transnational identities, the impact of colonial legacies, international ventures promoting cooperation between schools, and so forth (McCarthy and Kenway 2014), and examines how these shape elite education charters. Yet, alongside the necessary focus on the global, most elite education institutions are still bounded in different ways by their relationships to the national state. This requires scholars of elite education to understand the ways in which the state and dominant social groups influence the context in which elite institutions have the potential to change or remain the same when negotiating their positions within the maelstrom of ongoing national and global influences.