Gerardo Cardona
email: gerardo.cardona@mines-paristech.fr

Alain Sarlette

Pierre Rouchon

Continuous-time Quantum Error Correction with Noise-assisted Quantum Feedback

Keywords: Quantum control, quantum error correction, stochastic systems, Lyapunov method

We address the standard quantum error correction using the three-qubit bit-flip code, yet in continuous-time. This entails rendering a target manifold of quantum states globally attractive. Previous feedback designs could feature spurious equilibria, or resort to discrete kicks pushing the system away from these equilibria to ensure global asymptotic stability. We present a new approach that consists of introducing controls driven by Brownian motions. Unlike the previous methods, the resulting closed-loop dynamics can be shown to stabilize the target manifold exponentially. We further present a reduced-order filter formulation with classical probabilities. The exponential property is important to quantify the protection induced by the closed-loop error-correction dynamics against disturbances. We study numerically the performance of this control law and of the reduced filter.

INTRODUCTION

Developing methods to protect quantum information in the presence of disturbances is essential to improve existing quantum technologies [START_REF] Matthew D Reed | Realization of three-qubit quantum error correction with superconducting circuits[END_REF], [START_REF] Ofek | Extending the lifetime of a quantum bit with error correction in superconducting circuits[END_REF]). Quantum error correction (QEC) codes, encode a logical state into multiple physical states. Similarly to classical error correction, this redundancy allows to protect quantum information from disturbances by stabilizing a submanifold of steady states, which represent the nominal logical states [START_REF] Daniel | Quantum error correction[END_REF], Nielsen and Chuang [2002]. As long as a disturbance does not drive the system out of the basin of attraction of the original nominal state, the logical information remains unperturbed. To stabilize the nominal submanifold in a quantum system, a syndrome diagnosis stage performs quantum non-destructive (QND) measurements extracting information about code disturbances without perturbing the encoded data. Based on this information, a recovery feedback action restores the corrupted state. QEC is most often presented as discretetime operations towards digital quantum computing, see e.g. [START_REF] Michael | Quantum computation and quantum information[END_REF]. Not only the design of the underlying control layer, but also the proposal of analog quantum technologies, like solving optimization problems by quantum annealing, motivate a study of QEC in continuous-time, among them reservoir engineering and measurement-based feedback.

Reservoir engineering couples the target system to a dissipative ancillary quantum system, such that the entropy introduced by errors on the target system is evacuated through the dissipation of the ancillary one. Reservoir engineering for autonomous QEC has been investigated in [START_REF] Murch | Cavity-assisted quantum bath engineering[END_REF], [START_REF] Cohen | Dissipationinduced continuous quantum error correction for superconducting circuits[END_REF], [START_REF] Guillaud | Repetition catqubits: fault-tolerant quantum computation with highly reduced overhead[END_REF]. An advantage of this approach is that there is no need for external control logic. However, the challenge is to implement the specific ancillary system and coupling within experimental constraints.

Experimental progress on performing high-fidelity quantum measurements now allows to consider measurementbased feedback in continuous-time. In the context of QEC, this has been addressed in [START_REF] Ahn | Continuous quantum error correction via quantum feedback control[END_REF][START_REF] Ahn | Quantum error correction for continuously detected errors[END_REF], [START_REF] Sarovar | Practical scheme for error control using feedback[END_REF], [START_REF] Mabuchi | Continuous quantum error correction as classical hybrid control[END_REF], essentially as proposals illustrated by simulation. The short dynamical timescales of experimental setups is a main difficulty towards implementing complex feedback laws. Furthermore, data acquisition and processing leads to latencies in the feedback loop. This motivates the development of efficiently computable control techniques that are robust against unmodeled dynamics.

In this paper we establish analytical results about the convergence rate of QEC systems towards the nominal submanifold, a prerequisite for analytically quantifying the protection of quantum information. To obtain exponential convergence in a compact space, it is necessary to suppress any spurious unstable equilibria that might remain in the closed-loop dynamics. As we noted in [START_REF] Cardona | Exponential stochastic stabilization of a two-level quantum system via strict lyapunov control[END_REF], this problem is greatly simplified by considering stochastic processes to drive the controls (see also [START_REF] Zhang | Locally optimal measurement-based quantum feedback with application to multi-qubit entanglement generation[END_REF] for feedback laws with similar stochastic terms). Therefore in the present paper, in the context of QEC, we propose a noise-assisted quantum feedback, acting with Brownian noise whose gain is adjusted in real-time. We show via standard stochastic Lyapunov arguments that this new approach renders the target subspace, containing the nominal encoding of quantum information, globally exponentially stable thanks to feedback from syndrome measurements. Furthermore, our strategy allows to work with a reduced state estimator: while other feedback schemes require to keep track of quantum coherences, the proposed feedback scheme allows for the implementation of a reduced filter that only tracks the populations on the various joint eigenspaces of the measurement operators.

The paper is organized as follows. Section 2 presents the dynamical model of the three-qubit bit-flip code, which is the most basic model in QEC. In section 3 we introduce our approach to feedback using noise and we prove exponential stabilization of the target manifold of the three-qubit bitflip code. It presents as well the reduced order filter that follows from the feedback scheme. Section 4 examines the performance of this feedback and reduced filter to protect quantum information from bit-flip errors. Remark 1. (Stochastic Calculus): We will consider concrete instances of Itō stochastic differential equations (SDEs) on R n of the form dx = µ(x)dt + σ(x)dW, (1) where W is a standard Brownian motion on R k , and µ, σ are regular functions of x with image in R n and R n×k respectively, satisfying the usual conditions for existence and uniqueness of solutions ( [Khasminskii, 2011, Chapter 3]) on S, a compact and positively invariant subset of R n .

We will use results on stochastic stability [START_REF] Khasminskii | Stochastic stability of differential equations[END_REF]). Consider (1) with µ(x) = σ(x) = 0 for x ∈ S 0 ⊂ S, thus S 0 is a compact set of equilibria. Let V (x), a nonnegative real-valued twice continuously differentiable function with respect to every x ∈ S \ S 0 . Its Markov generator associated with (1) is

AV = i µ i ∂ ∂x i V + 1 2 i,j σ i σ j ∂ 2 ∂x i x j V, (2)
and E[V (x t )] = V (x 0 ) + E t 0 AV (x s )ds . Theorem 2. [START_REF] Khasminskii | Stochastic stability of differential equations[END_REF]). If there exists r > 0 such that AV (x) ≤ -rV (x), ∀x ∈ S \ S 0 , then V (x t ) is a supermartingale on S with exponential decay:

E[V (x t )] ≤ V (x 0 ) exp(-r t) .
If V is a meaningful way to quantify the distance to a target set { x : V (x) = 0 } ⊇ S 0 , then this theorem establishes an exponential convergence result in the sense of expectation of V . Analysis in the rest of this paper consists in defining a function V and constructing controls that ensure exponential convergence in the above sense.

CONTINUOUS-TIME DYNAMICS OF THE THREE-QUBIT BIT-FLIP CODE

The general model for a quantum system subject to several measurement channels (see, e.g., [START_REF] Barchielli | Quantum trajectories and measurements in continuous time: the diffusive case[END_REF]) is an Itō stochastic differential equation of the type

dρ t = k D L k (ρ)dt + √ η k M L k (ρ)dW k , (3)
dY k = √ η k Tr (L k + L † k )ρ dt + dW k .
We have used the standard super-operator notation

D L (ρ) = LρL † -1 2 (L † Lρ + ρL † L) , M L (ρ) = Lρ + ρL † -Tr ρ(L + L † ) ρ
, where L † denotes the complex conjugate transpose of L. The state ρ belongs to the set of density matrices S = {ρ ∈ C n×n : ρ = ρ † , ρ positive semidefinite , Tr (ρ) = 1} on the Hilbert space of the system H C n×n ; the {W k } are independent standard Brownian motions and the {dY k } correspond to the measurement processes of each measurement channel. The η k ∈ [0, 1] express the corresponding measurement efficiencies, i.e. the ratio of the corresponding channel linking the system to the outside world which is effectively captured by the measurement device; channels k with η k = 0 represent pure loss channels.

The simplest way to model the feedback stage consists in applying an infinitesimal unitary operation to the openloop evolution, ρ t+dt = U t (ρ t + dρ t )U † t , where U t = exp(-i j H j u t,j dt) with H j hermitian operators denoting the control Hamiltonians that can be applied, and each u t,j dt a real control input. The fact that u t,j dt may contain stochastic processes requires to treat this feedback action with care, we will come back to this in the next section.

Dynamics of the three-qubit bit-flip code

The three-qubit bit-flip code corresponds to a Hilbert space H = (C 2 ) ⊗3 C 8 , where ⊗ denotes tensor product (Kronecker product, in matrix representation). We denote I n the identity operator on C n and we write X k , Y k and Z k the local Pauli operators acting on qubit k, e.g. X 2 = I 2 ⊗ σ x ⊗ I 2 . We denote {|0 , |1 } the usual basis states, i.e. the -1 and +1 eigenstates of the σ z operator on each individual qubit [START_REF] Michael | Quantum computation and quantum information[END_REF]).

The encoding on this 3-qubit system is meant to counter bit-flip errors, which map a ±1 eigenstate of Z k to the ∓1 eigenstate for each k = 1, 2, 3. The nominal encoding for a logical information 0 (resp. 1) is on the state |000 (resp. |111 ). A single bit-flip on e.g. the first qubit brings this to X 1 |000 = |100 (resp. |011 ), which by majority vote can be brought back to the nominal encoding.

In the continuous-time model (3), bit-flip errors occurring with a probability γ k dt 1 during a time interval [t, t + dt] are modeled by disturbance channels, with L k+3 = √ γ k X k and η k+3 = 0, k = 1, 2, 3. The measurements needed to implement "majority vote" corrections, so-called syndromes, continuously compare the σ z value of pairs of qubits. The associated measurements correspond in (3) to

L k = √ Γ k S k for k = 1, 2, 3, with S 1 = Z 2 Z 3 , S 2 = Z 1 Z 3 , S 3 = Z 1 Z 2 and Γ k representing the measurement strength.
This yields the following open-loop model:

dρ = 3 k=1 Γ k D S k (ρ)dt+ η k Γ k M S k (ρ)dW k + 3 s=1 γsD Xs (ρ)dt. (4)
We further define the operators:

Π C = 1 4 I 8 + 3 k=1 S k , Π j := X j Π C X j , j ∈ {1, 2, 3}, (5)
corresponding to orthogonal projectors onto the eigenspaces of the measurement syndromes. Π C projects onto the nominal code C := span(|000 , |111 ) (+1 eigenspace of all the S k ), whereas Π j projects onto the subspace where qubit j is flipped with respect to the two others. For each k ∈ {C, 1, 2, 3}, we write p t,k := Tr (Π k ρ t ) ≥ 0 the so-called population of subspace k, i.e. the probability that a projective measurement of the syndromes would give the output corresponding to subspace k. By the law of total probabilities, k∈{C,1,2,3} p t,k = 1 for all t.

Behavior under measurement only

We have the following behavior in absence of feedback actions and disturbances. Lemma 3. Consider (4) with γ s = 0 for s ∈ 1, 2, 3.

(i) For each k ∈ {C, 1, 2, 3}, the subspace population p t,k is a martingale i.e. E(p t,k |p 0,k ) = p 0,k for all t ≥ 0. (ii) For a given ρ 0 , if there exists k ∈ {C, 1, 2, 3} such that p 0, k = 1 and p 0,k = 0 for all k = k, then ρ 0 is a steady state of ( 4). (iii) The Lyapunov function

V (ρ) = k∈{C,1,2,3} k =k √ p k p k decreases exponentially as E[V (ρ t )] ≤ e -rt V (ρ 0 ) for all t ≥ 0, with rate r = 4 min k∈{1,2,3} η k Γ k .
In this sense the system exponentially approaches the set of invariant states described in point (ii).

Proof. The first two statements are easily verified, we prove the last one. The variables ξ j = √ p j , j ∈ {1, 2, 3, C} satisfy the following SDE's:

dξ C = -2ξ C k∈{1,2,3} η k Γ k (1 -ξ 2 C -ξ 2 k ) 2 dt + 2ξ C k∈{1,2,3} η k Γ k (1 -ξ 2 C -ξ 2 k ) dW k , dξ j =C = -2ξ j η j Γ j (1 -ξ 2 C -ξ 2 j ) 2 + k∈{1,2,3}\j η k Γ k (ξ 2 C + ξ 2 k ) 2 dt + 2ξ j η j Γ j (1 -ξ 2 C -ξ 2 j ) dW j - k∈{1,2,3}\j η k Γ k (ξ 2 C + ξ 2 k ) dW k , while V = k∈{C,1,2,3} k =k ξ k ξ k . Noting that 2(1-ξ 2 C - ξ 2 k ) and 2(ξ 2 C + ξ 2 k ) just correspond to 1 ± Tr (ρS k ), we only have to keep track of ± signs to compute AV = -2 k∈{C,1,2,3} j∈{C,1,2,,3}\k ξ j ξ k l∈{1,2,3} j,k,l η l Γ l
where, for each pair (j, k), the selector j,k,l ∈ {0, 1} equals 1 for two l values, namely C,k,l = k,C,l = 1 if l = k ∈ {1, 2, 3} and j,k,j = j,k,k = 1 for j, k ∈ {1, 2, 3}. This readily leads to AV ≤ -4 min k∈{1,2,3} (η k Γ k ) V . We conclude by Theorem 2 and noting that V = 0 necessarily corresponds to a state as described in point (ii).

The above Lyapunov function describes the convergence of the state towards Tr (Πkρ) = 1, for a random subspace k ∈ {C, 1, 2, , 3} chosen with probability p 0, k. We now address how to render a particular subspace globally attractive, namely the one associated to Π C and nominal codewords.

ERROR CORRECTION VIA NOISE-ASSISTED FEEDBACK STABILIZATION

Controller design

Error correction requires to design a control law satisfying two properties:

• Drive any initial state ρ 0 towards a state with support only on the nominal codespace C = span{|000 , |111 }. This comes down to making Tr (Π C ρ t ) converge to 1. • For Tr (Π C ρ 0 ) = 1 and in the presence of disturbances γ s = 0, minimize the distance between ρ t and ρ 0 for all t ≥ 0.

We now directly address the first point, the second one will be discussed in the sequel.

As mentioned in the introduction, this problem has already been considered before, yet without proof of exponential convergence. Towards establishing such proof, we introduce a key novelty into the feedback signal: we drive it by a stochastic process. Indeed, noise can be as efficient as a deterministic action to exponentially destabilize a spurious equilibrium where k = C; in turn, using noise simplifies the study of the average dynamics, both in the analysis via Theorem 2 and towards implementing a quantum filter to estimate ρ. We thus introduce noise-assisted quantum feedback, where the control input consists of pure noise with state-dependent gain; i.e. we take u j dt = σ j (ρ)dB j , with B j (t) a Brownian motion independent of any W k (t).

As control Hamiltonians we take H j = X j , thus rotating back the bit-flip actions. The closed-loop dynamics in Itō sense then writes:

dρ = 3 k=1 Γ k D S k (ρ)dt + η k Γ k M S k (ρ)dW k + 3 s=1 γsD Xs (ρ)dt + 3 j=1 -iσ j (ρ)[X j , ρ]dB j + σ j (ρ) 2 D X j (ρ)dt . (6)
The last term can be viewed as "encouraging" a bit-flip with a rate depending on the value of σ j and thus on ρ. The remaining task is to design the gains σ j . For this many options will work -its only essential role is to "shake" the state when it is close to Tr (Π C ρ) = 0, since the open loop already ensures stochastic convergence to either Tr (Π C ρ) = 0 or Tr (Π C ρ) = 1. The following hysteresisbased control law, illustrated by Fig. 1, depends only on the p t,k and should not be too hard to implement. Select real parameters α j and β j such that 1 2 < β j < α j < 1 for j ∈ {1, 2, 3}, and take a constant c > 0.

(1) If p j ≥ α j then take σ j = 6cηj Γj 2αj -1 ;

(2) If p j ≤ β j then take σ j = 0;

(3) In the hysteresis region, i.e. for values of p j ∈]β j , α j [: keep the previous value of σ j .

Closed-loop exponential convergence

We propose the closed-loop Lyapunov function: Theorem 4. Consider ( 6) with all γ s = 0 and feedback gains (σ j ) as specified just before section 3.2. Then

V (ρ) = V 1 (ρ) + V 2 (ρ) + V 3 (ρ) (7) with V k (ρ) = √ p k + p 1 + p 2 + p 3 for k = 1, 2, 3.
E[V (ρ t )] ≤ V (ρ 0 )e -rt
, ∀t ≥ 0, with the exponential convergence rate estimated as:

r = min j∈{1,2,3} η j Γ j min c , 4 3 √ 2 min (s,x 1 ,x 2 ,x 3 )∈K g(s, x 1 , x 2 , x 3 )
where g(s, x 1 , x 2 , x 3 ) is given in (9) below and

K = (s, x 1 , x 2 , x 3 ) ∈ [0, 1] 4 x 1 + x 2 + x 3 = 1; sx j ≤ α j
Proof. By design of the hysteresis, well-posedness of the solution then follows from standard arguments on the construction of solutions of SDE's. The proof then consists in showing that V (ρ t ) on S is an exponential supermartingale satisfying A(V ) ≤ -rV . Towards this we partition the state-space into Q := ∪ 3 j=1 ρ ∈ S | p j ≥ α j and S \ Q on which we compute the Markov generator A(V ) separately.

Let us write from (6) the expression of AV (ρ) = E dV t | ρ t = ρ /dt for any value of the control gain vector σ. We exploit here the following formula based on Itō rules and valid for any non-negative operator F :

d Tr (F ρ) = Tr (F dρ) 2 Tr (F ρ) - (Tr (F dρ)) 2 4 Tr (F ρ) Tr (F ρ)
.

We detail below the computations when η j ≡ η and Γ j ≡ Γ (the formulas in the general case are slightly more complicated). With

F 1 = 2Π 1 + Π 2 + Π 3 and V 1 (ρ) = √ f 1 = Tr (F 1 ρ), we get AV 1 (ρ) = 2σ 2 1 1-f 1 +σ 2 2 1-2(p 1 +p2) +σ 2 3 1-2(p 1 +p 3 ) 2 √ f 1 -4ηΓ (p 2 +p 3 )(1-f 1 ) 2 + p 1 +(p 1 +p 3 )(1-f 1 ) 2 + p 1 +(p 1 +p 2 )(1-f 1 ) 2 f 1 √ f 1 - σ 2 1 Tr 2 ([X 1 ,ρ]F 1 )+σ 2 2 Tr 2 ([X 2 ,ρ]F 1 )+σ 2 3 Tr 2 ([X 3 ,ρ]F 1 ) 4f 1 √ f 1 . Since √ f 1 ≥ 1 3 √ 2 V , we have AV 1 (ρ) ≤ 2σ 2 1 1-f 1 +σ 2 2 1-2(p 1 +p2) +σ 2 3 1-2(p 1 +p 3 ) 2 √ f 1 - 4ηΓV (p 2 +p 3 )(1-f 1 ) 2 + p 1 +(p 1 +p 3 )(1-f 1 ) 2 + p 1 +(p 1 +p 2 )(1-f 1 ) 2 3 √ 2f 2 1 .
Via circular permutation and summation, we get

AV (ρ) ≤ 3 j=1 σ 2 j (ρ)g j (ρ) -4ηΓ 3 √ 2 g(ρ)V (ρ) (8)
where g j (ρ) =

1-fj √ fj + 1-2(pj +p j ) 2 √ f j + 1-2(pj +p j ) 2 √ f j with {j, j , j } = {1, 2, 3} and g(ρ) = (p2+p3)(1-f1) 2 + p1+(p1+p3)(1-f1) 2 + p1+(p1+p2)(1-f1) 2 (2p1+p2+p3) 2 + (p3+p1)(1-f2) 2 + p2+(p2+p1)(1-f2) 2 + p2+(p2+p3)(1-f2) 2 (2p2+p3+p1) 2 + (p1+p2)(1-f3) 2 + p3+(p3+p2)(1-f3) 2 + p3+(p3+p1)(1-f3) 2 (2p3+p1+p2) 2
.

When ρ ∈ Q, we have p j ≥ α j > 1/2 for a unique j ∈ {1, 2, 3}, since p 1 + p 2 + p 3 ≤ 1. Assume first that

p 1 ≥ α 1 , thus σ 1 = 6cηΓ 2α1-1 and σ 2 (ρ) = σ 3 (ρ) = 0. Since g(ρ) ≥ 0, inequality (8) implies AV ≤ 6cηΓ 2α1-1 1-f1 √ f1 + 1-2(p1+p2) 2 √ f2 + 1-2(p1+p3) 2 √ f3 . Since f 1 ≥ 2α 1 , 1 -2p 1 ≤ 0, f 1 ≤ 2 and V ≤ 3 √ 2 we get AV ≤ 6cηΓ 2α1-1 1-2α1 √ f1 = -6cηΓ V √ f1 V ≤ -cηΓV.
We get a similar inequality when p 2 ≥ α 2 or p 3 ≥ α 3 . Thus ∀ρ ∈ Q, AV (ρ) ≤ -cηΓV (ρ). Consider now ρ ∈ S \ Q. Then, p j < α j for all j. Since σ j (ρ) = 0 when p j ≤ 1/2 we have σ 2 j (ρ)g j (ρ) ≤ 0. From ( 8), we have AV (ρ) ≤ -4ηΓ 3 √ 2 g(ρ)V (ρ). Let us prove that g(ρ) ≥ r for any ρ ∈ S \ Q. With s = p 1 + p 2 + p 3 and x j = p j /s, g can be seen as a function of (s, x 1 , x 2 , x 3 ),

g(ρ) = g(s, x 1 , x 2 , x 3 ) (x 2 +x 3 )(1-f 1 ) 2 + x 1 +(x 1 +x 3 )(1-f 1 ) 2 + x 1 +(x 1 +x 2 )(1-f 1 ) 2 (1+x 1 ) 2 + (x 3 +x 1 )(1-f 2 ) 2 + x 2 +(x 2 +x 1 )(1-f 2 ) 2 + x 2 +(x 2 +x 3 )(1-f 2 ) 2 (1+x 2 ) 2 + (x 1 +x 2 )(1-f 3 ) 2 + x 3 +(x 3 +x 2 )(1-f 3 ) 2 + x 3 +(x 3 +x 1 )(1-f 3 ) 2 (1+x 3 ) 2 (9)
with f j = 1 -s -sx j . Here (s, x 1 , x 2 , x 3 ) belongs to the compact set s ∈ [0, 1], x j ≥ 0, j x j = 1 and sx j ≤ α j for all j. On this compact set, g is a smooth function. Moreover it is strictly positive since g = 0 implies that s = 1 and x j = 1 for some j ∈ {1, 2, 3} which would not satisfy sx j ≤ α j . This means that min ρ∈S\Q g(ρ) > 0.

Taking all things together, we have proved that AV (ρ) ≤ -rV (ρ) always holds. We conclude with Theorem 2.

For a heuristic estimate of r, take s = α j with x j = 1 for some j to get r ∼ min j∈{1,2,3} η j Γ j min c, 8 √ 2 (1 -ᾱ) 2 , with ᾱ = max j∈{1,2,3} α j . Typically one would take c = 1 and α 1 = α 2 = α 3 = α close to 1. When η j Γ j are all equal, such a rough estimate simplifies to r = 4 √ 2(1 -α) 2 ηΓ .

Reduced quantum filter

Towards implementing the control law we have to reconstruct in real-time the quantum state estimate ρ via a quantum filter. For (6), this filter reads:

dρ = 3 k=1 Γ k D S k (ρ)dt + 3 s=1 γ s D Xs (ρ)dt + 3 k=1 η k Γ k M S k (ρ) dY k -2 η k Γ k Tr (S k ρ) dt + 3 j=1 -iσ j (ρ)[X j , ρ]dB j + σ j (ρ) 2 D Xj (ρ)dt . ( 10
)
where dY k = 2 √ η k Γ k Tr (S k ρ) dt + dW k is the measurement outcome of syndrome S k , and the random dB j applied to the system are accessible too a posteriori.

Instead, we can replace the state ρ t in the feedback law, by ρ t corresponding to the Bayesian estimate of ρ t knowing its initial condition ρ 0 and the syndrome measurements dY k between 0 and the current time t > 0, but not the dB j . Then ρ t obeys to the SME:

d ρ = 3 k=1 Γ k D S k ( ρ)dt + 3 j=1 (γ s + σ 2 j ( ρ))D Xj ( ρ)dt + 3 k=1 η k Γ k M S k ( ρ) dY k -2 η k Γ k Tr (S k ρ) dt (11)
where 6) where σ j (ρ) is replaced by σ j ( ρ). Denote pj = Tr (Π j ρ) and ŝk = Tr (S k ρ). Then we have

dY k = 2 √ η k Γ k Tr (S k ρ) dt + dW k with ρ governed by (
dŝ 1 = -2(γ 2 + σ 2 2 + γ 3 + σ 3 3 )ŝ 1 dt + 2 η 1 Γ 1 (1 -ŝ2 1 ) dY 1 -2 η 1 Γ 1 ŝ1 dt + 2 η 2 Γ 2 (ŝ 3 -ŝ1 ŝ2 ) dY 2 -2 η 2 Γ 2 ŝ2 dt + 2 η 3 Γ 3 (ŝ 2 -ŝ1 ŝ3 ) dY 3 -2 η 3 Γ 3 ŝ3 dt (12)
with p1 = (1 + ŝ1 -ŝ2 -ŝ3 )/4. The formulas for dŝ 2,3 and p2,3 are obtained via circular permutation in {1, 2, 3}. Since the feedback law depends only on the populations pj , it can be implemented with the exact quantum filter reduced to (ŝ 1 , ŝ2 , ŝ3 ) ∈ R 3 . Contrarily to the full quantum filter (10), here the syndrome dynamics ŝk are independent of any coherences among the different subspaces and we get a closed system on classical probabilities, driven by the measurement signals.

ON THE PROTECTION OF QUANTUM INFORMATION

It is well-known in control theory that exponential stability gives an indication of robustness against unmodeled dynamics. In the present case, this concerns the first control goal, namely stabilization of ρ t close to the nominal subspace C in the presence of bit-flip errors γ s = 0. About the second control goal, namely keeping the dynamics on C close to zero such that logical information remains protected, the analysis of the previous section is less telling.

We can illustrate both control goals by simulation. As in [START_REF] Ahn | Continuous quantum error correction via quantum feedback control[END_REF] we set as initial condition ρ 0 = |000 000| and simulate 1000 closed-loop trajectories under the feedback law of section 3.1. We compare the average evolution of this encoded qubit with a single physical qubit subject to a σ x decoherence of the same strength, since this is the situation that the bit-flip code is meant to improve. Parameter values and simulation results are shown on Figure 2 where we consider that the quantum filter perfectly follows (6). Figure 3 corresponds to a more realistic situation where the same feedback law relies on the reduced order quantum filter (12) corrupted by errors and feedback latency: we observe a small change of performance but still a clear improvement compared to a single qubit. Tr((;+X 1 ; X 1 +X 2 ; X 2 +X 3 ; X 3 ); 0 ) Fig. 2. Ideal situation where the feedback of subsection 3.1 is based on ρ governed by ( 6). Solid red: mean overlap of the state with the code space. Solid black: mean fidelity of the logical qubit versus ρ 0 . Solid blue: mean correctable fidelity under active quantum feedback. For the three solid curves, the initial state is chosen as ρ 0 = |000 000| and closed-loop simulation parameters based on (6) are Γ j = 1, γ j = 1/64, η j = 0.8, and for the feedback law β j = 0.6, α j = 0.95, c = 3/2. Dashed line, for comparison: mean fidelity towards |0 0| for a single physical qubit without measurement nor control and subject to bitflip disturbances with γ = 1/64.

Regarding the first control goal, we observe that the controller indeed confines the mean evolution to a small neighborhood of C, for all times, as expected from our analysis. Regarding the second criterion, the distance between ρ t and ρ 0 cannot be confined to a small value for all times. Indeed, majority vote can decrease the rate of information corruption but not totally suppress it; as corrupted information is irremediably lost, ρ t progressively converges towards an equal distribution of logical 0 and logical 1. However, for the protected 3-qubit code, this information loss is much slower than for the single qubit; this indicates that the 3-qubit code with our feedback law indeed improves on its components.

In our feedback design, making α j closer to 1/2 would improve the convergence rate estimate in Theorem 4; however, this also has a negative effect on the logical information, since it means that we turn on the noisy drives more often. Analytically computing the optimal tradeoff is the subject of ongoing work. Similary, making c larger would accelerate the recovery action but increase the level of noise, and we want to keep the induced motion slower than the measurement timescale. Simulations (not reproduced here) clearly show that intermediate values of the control parameters deliver better overall results.

CONCLUSION

We have approached continuous-time quantum error correction in the same spirit as [START_REF] Ahn | Continuous quantum error correction via quantum feedback control[END_REF], and showed how introducing Brownian motion to drive control fields yields exponential stabilization of the nominal codeword manifold. The main idea relies on the fact that the SDE in open loop stochastically converges to one of a few steadystate situations, but on the average does not move closer to any particular one. It is then sufficient to activate noise only when the state is close to a bad equilibrium, in order to induce globale convergence to the target ones. This general idea can be extended to other systems with this property, and in particular to more advanced errorcorrecting schemes. In the same line, while we have proposed particular controls with hysteresis, proving a similar property with smoother control gains should not be too different. The convergence rate obtained is dependent on our choice of Lyapunov function and on the values of α j ; from parallel investigation it seems possible to get a closedloop convergence rate arbitrarily close to the measurement rate.

However, unlike in classical control problems, the key performance indicator is not how fast we approach the target manifold. Instead, what matters is how well, in presence of disturbances, we preserve the encoded information. Towards this goal, we should refrain from disturbing the system with feedback actions; accordingly, we have noticed that taking α j closer to 1 can improve the code-word fidelity, despite leading to a slower convergence rate estimate. A theoretical analysis of information protection capabilities is the subject of ongoing work.

Fig. 1 .

 1 Fig. 1. for α j ≡ α and β j ≡ β, the 6 active feedback zones in the simplex (p 1 , p 2 , p 3 ) p 1 , p 2 , p 3 ≥ 0, p 1 +p 2 +p 3 ≤ 1 .

Fig. 3 .

 3 Fig. 3. Simulation similar to Fig. 2 for a more realistic case where feedback is based on the reduced order filter (12) and includes modeling/measurement errors and feedback latency. Marked with subscript * , the parameter values used in (12) are as follows: γ * = 0.8γ, Γ * = 0.9Γ, η * = 0.9η; constant measurement bias according to dY * ,1 = dY 1 +

The authors would like to thank K. Birgitta Whaley and Leigh S. Martin for discussions on continuous-time QEC.


Pierre Arnoux
email: pierre@pierrearnoux.fr

Jorge Soto-Andrade
email: sotoandrade@uchile.cl

From concrete to abstract and back: Metaphor and Representation

Keywords: Metaphor, Representation, Enaction, Concrete, Abstract

We are interested in exploring from an enactivist perspective the role of metaphor and representation in the emergence of the abstract from the concrete and also in sense making of abstract mathematical notions. After introducing our theoretical framework, we present and comment on some illustrative examples, explored with several cohorts of learners, which include both prospective mathematics secondary teachers and first year university students studying science and the humanities.

Introduction

We claim that representation and metaphor (between which we distinguish, see below) play a key role in the teaching and learning of mathematics. More precisely, in this paper we focus on the ways in which metaphorising and representing can foster the emergence of abstract concepts from concrete a-didactic situations [START_REF] Brousseau | Théorie des situations didactiques[END_REF] and backwards, making (concrete) sense of abstract concepts. We argue, moreover, that taking advantage of such concrete situations can afford insights and motivations for classical abstract mathematical notions, which are friendlier to the learners than the traditional approach where such notions are quite often "parachuted from Olympus". Indeed, many mathematical concepts emerge as procedural computations, which are not supposed to take a meaning by themselves. Every time a new representation, or metaphor, is found for such a "procedural" concept, this results in a deeper understanding, and quite often in new questions and results. A very famous example is that of complex numbers, invented as "impossible" numbers to solve cubic equations, whose now classical geometric representation via the complex plane was only found two centuries later. Other examples, like the metaphor of "arithmetic geometry," are now at the base of important mathematical research programs. We argue that it is important to help students develop such representations explicitly, rather than treating them as trivial side-products of teaching. It is however a big challenge to figure out under which conditions representing and metaphorising foster mathematical thinking processes as hypothesised above.

We intend here to pursue our recent research on metaphorising, representing and enacting [START_REF] Arnoux | Using mental imagery processes for teaching and research in mathematics and computer science[END_REF]Diaz-Rojas & Soto-Andrade, 2015;[START_REF] Soto-Andrade | Metaphors and cognitive styles in the teaching-learning of mathematics[END_REF], 2014[START_REF] Soto-Andrade | Enactive metaphorising in the learning of mathematics[END_REF], by presenting and commenting some non-obvious examples of enactive metaphors and representations of mathematical concepts and their use in a-didactic situations (Brousseau,1998). The contextual background of our examples involves several cohorts of learners in Chile and France, mainly prospective secondary mathematics teachers.

1. Which sorts of metaphorising and representing can foster the emergence of abstract notions from concrete situations and convey a better grasp of them for the students? 2. ow can teachers support, in a friendly manner, learners in developing their representational and metaphorical competences?

Theoretical background

Metaphorising and Representing in cognitive science and mathematics education

Metaphorising and Representing nowadays play undoubtedly a key role in cognitive science and mathematics education. See [START_REF] Goldin | Mathematical Representations[END_REF] and Soto-Andrade (2014) for related comprehensive surveys. Notice that we prefer to focus in on metaphorising, a verb denoting a process, rather than on metaphor, a noun denoting an object. Indeed, what is a metaphor for someone may not be a metaphor for someone else, and for yet another person it could be a representation instead (see below). What we observe is actually the process of metaphorising carried out by someone, a cognitive subject. Regarding metaphorising ("looking at something and seeing something else", in metaphorical terms), widespread agreement has arisen in cognitive science that our ordinary conceptual system, in terms of which we both think and act, is fundamentally metaphorical in nature [START_REF] Gibbs | The Cambridge Handbook of Metaphor and Thought[END_REF][START_REF] Johnson | Metaphors we live by[END_REF]. In mathematics education metaphor typically appears not just as a rhetorical device, but as a powerful cognitive tool, that helps us in grasping or constructing new concepts, as well as in solving problems in an efficient way and user-friendly way [START_REF] English | Mathematical reasoning: Analogies, metaphors, and images[END_REF][START_REF] Lakoff | Where Mathematics comes from[END_REF][START_REF] Sfard | Metaphors in education[END_REF][START_REF] Soto-Andrade | Metaphors and cognitive styles in the teaching-learning of mathematics[END_REF], 2014).

Although in the literature the same object is sometimes described either as a representation or as a metaphor, here we draw a distinction: we re-present something given beforehand, usually to explain concepts already constructed, but we metaphorise to try to fathom something unknown or to construct a concept. Recall that [START_REF] Lakoff | Where Mathematics comes from[END_REF] highlight the intensive use we make of conceptual metaphors that appearmetaphoricallysuch as inference-preserving mappings (arrows) "going upwards" from a rather concrete 'source domain' into a more abstract 'target domain', enabling us to fathom the latter in terms of the former. Then representations naturally appear as arrows going the other way around, downwards from the more abstract domain to the more concrete one (Soto-Andrade, 2014). Indeed, our approach to the learning of mathematics emphasises the poietic (from the Greek poiesis = creation, production) role of metaphorising, which brings concepts into existence. For instance, we bring the concept of probability into existence when, while studying a symmetric random walk on the integers, we look at the walker (a frog jumping on a row of stones in a pond, say) and we see it splitting into two equal halves that go right and left instead of being equally likely to jump right or left (Diaz-Rojas & Soto-Andrade, 2015). This 'metaphoric sleight of hand' which turns a random process into a deterministic one, allows us to reduce probabilistic calculations to deterministic ones, where we just need to keep track of the walker's splitting into pieces: The probability of finding the walker at a given location after n jumps is just the portion of the walker landing there after n splittings.

In the same vein, imagine that one is trying to figure a struggle between two producers A and B for a consumer market, who each monthas a consequence of intensive marketing strategiesentice consumers of the other brand to change their choice, say 20% of consumers of A going to B but only 10% conversely. If one is familiar with jumping frogs, one could metaphorise the evolution of the market as the random walk of a frog between two rocks, tagged A and B, with corresponding transition probabilities. However, someone who is more familiar with market struggles would rather metaphorise the frog's random walk as a market evolution, to benefit of his/her economic intuition.

It can be argued that we often introduce new concepts via metaphor, but giving a new meaning (and often, just meaning) to a concept that has already been taught involves a representation. We may have internal representations [START_REF] Goldin | Representations and the psychology of mathematics education: Parts I and II (special issues)[END_REF][START_REF] Goldin | Mathematical Representations[END_REF] which are operationally equivalent to metaphors, as the ways a cognitive subject has of figuring out concepts unfamiliar or still opaque.

Methodology and experimental background

Our methodology relies mainly on qualitative approaches like participant observation techniques and ethnographic methods [START_REF] Brewer | Ethnographic and Qualitative Research in Education[END_REF][START_REF] Spradley | Participant observation[END_REF].

Regarding our experimental background, several cohorts of students have participated in preliminary tests at the University of Aix-Marseille, France: 60 first year university students majoring in science and humanities in a mathematics course in 2015-2018 (usually organized in 10 groups of 6) and 40 fifth year teacher students in 2 mathematics courses, of 20 students each, in 2016-2018 (usually organized in 5 groups of 4 each).

At the University of Chile, in Santiago, three cohorts of prospective secondary school mathematics and physics teachers (45 students each, on the average), in a one-semester yearly course in elementary number theory, have been involved in our teaching and learning following a metaphoric and enactivist approach, from 2016 to 2018. Working most of the time in random groups of 3 to 4, (defined by blind picking of coloured Lego cubes from a bag) they were observed and monitored by the teacher and an assistant as participant observer or ethnographer [START_REF] Spradley | Participant observation[END_REF][START_REF] Brewer | Ethnographic and Qualitative Research in Education[END_REF]).

Since we were especially interested in evaluating the impact of our approach on the student's engagement and problem solving and problem posing abilities, we observed mainly: their level of participation and horizontal (peer) interaction, the emergence of "research questions", i.e. questions they ask themselves, to be tackled by themselves (not questions addressed to the teacher to ask for a clarification or explanation) and (idiosyncratic) metaphors, arising spontaneously or under prompting, accompanying gestural language of learners and teacher, expression and explicit acknowledgement of affective reactions. Snapshots of their written products in problem solving activities were taken and processed in a worksheet, also for evaluation purposes, and some videos of their enacting moments were recorded.

Illustrative examples and case studies

We present and discuss here a couple of paradigmatic examples, regarding concrete ways to introduce and motivate in a friendly way Pythagorean triples as well as arithmetical congruences. In fact, we could describe them under the same roof as arithmetic in a discrete "modular universe": a 2D pixeled, grid-like one (the lattice ) in the first case, and a 1D cyclic one (the polygon formed by the integers modulo m) in the second case. For other examples, related to computer science (finite automata) and probability, see [START_REF] Arnoux | Using mental imagery processes for teaching and research in mathematics and computer science[END_REF], Diaz-Rojas and Soto-Andrade (2015) and Soto-Andrade (2018).

Example 1: Pythagorean triples, with Lego bricks

This activity has been carried out at the University of Aix-Marseille, in several contexts, with the cohorts described above: first year science and humanities students and fifth year teacher students (last year of teacher initial formation in mathematics).

It has been tested informally in the last two years, using qualitative monitoring by an assistant, and posterior evaluation by the participants; we are setting up now an interdisciplinary team (mathematicians, didacticians and cognitive scientists) to study it more in depth in the framework of a methodology unit next year.

We give to groups of 4 to 6 science and humanities students or teacher students one 16  16 horizontal plaque, two elementary 1  1 bricks, and one 1  16 brick. The question is: for which positions on the plaque can the small bricks be connected by the long one? See Figure 1, showing a 8  8 plaque, two 1  1 bricks in a good position, and a 1  8 brick superposed on them; it is of course cumbersome to explain this in writing, but the real model leaves no room for hesitation. By construction, the plaque is endowed with a discrete grid (a metaphor for with the canonical metric, points being the centres of the small circles and 1  1 bricks playing the role of dots); no explanation of this is needed. But some time is needed to understand that the long brick can only be set on the pair of small bricks if the distance between these two bricks is an integer multiple of the unit distance: after half an hour, most groups discover the concept of Pythagorean triples as implied by this situation (they all know the Pythagorean Theorem, but not Pythagorean triples). This exercise is very rich. It works well, because all the distances are quadratic numbers, hence, if they are not integers, they are quite far from an integer: at small distance, there are only exact solutions, no "near-solutions". It is also very engaging for students, because it is concrete. It evolves naturally into finding the classical Babylonian solutions (it may be pertinent at that point to present the famous Plimpton 322 tablet). One can then ask to find non-trivial rational points on the unit circle; students usually start by trial and error, and come first to the conjecture that there are no such points. It takes some time to make the relation with the previous question; this can be continued with a geometric parametrisation by the rational lines through the point (-1,0), which makes the link with the quadratic equation, and, via elementary geometry, with the formula for sine and cosine as a function of the tangent of the half-angle: these formulas appear as a reformulation of the Babylonian formula. Most students seem surprised to see that there can be links between very different domains of mathematics. This can be taken to much more elaborate questions, like the number of triples within a bounded distance, and in fact to difficult research questions. One remark is that, in our experience, it is not important whether or not the students have already studied these notions: in the case of prospective teachers, they are always surprised by the exercises, and do not immediately link them to notions they know quite well, but in an abstract way.

Example 2: Congruence mod m and dynamical systems

We report on some developments of our metaphoric enactive approach to arithmetic congruences, with third year prospective maths and physics secondary teachers taking a one semester course in elementary number theory, in 2016, 2017 and 2018, at the University of Chile. Since congruences mod m are unavoidable in this course, we wanted to motivate them or make their study friendlier. To this end, following a radical enactivist approach [START_REF] Proulx | From problem solving to problem posing, and from strategies to laying down a path in solving: taking Varela's ideas to Mathematics Education Research[END_REF][START_REF] Soto-Andrade | Enactive metaphorising in the learning of mathematics[END_REF], we just proposed a situational seed first and let the action emerge freely.

From our theoretical perspective, taking into account the previous mathematical training of our teacher students, at secondary school and also in most courses of their initial formation, we could predict a "metaphoric deficit" in their understanding of arithmetical congruences. Eventually they will be able to recite the definition and calculate, but without having a favourite metaphor (or representation) for them, and also with no appreciation of their usefulness. Notice that if they metaphorise the integers mod m as a finite "shadow" of the integers, they might easily have the idea that a necessary condition for a property to hold for the integers is that it holds for its projection onto their arithmetical shadows, something much easier to investigate. So, if a Diophantine equation is solvable (in the integers) its shadow should be solvable in the shadow integers mod m. In this way, they get necessary conditions for a Diophantine equation to be solvable, which allows them to prove unsolvability in several cases. Indeed, we found that although our prospective teachers know by heart the definition of congruence mod m in the integers, when we ask them how they imagine, metaphorise or visualise congruence mod m or which are their internal representations for it in the sense of [START_REF] Goldin | Mathematical Representations[END_REF], they are at a loss. After a while, some of them think of kangaroos, rabbits, frogs, jumping on the integer line, or they begin to paint the integers in different colours (five colours for congruence mod 5). Slowly the metaphor emerges in several groups, of winding the integer number line on a polygon, e. g. a pentagon for congruence mod 5. They naturally carry out a spiral-like winding on the plane to begin with. If nobody suggests a different way, we ask them what they do when they want to wind up a long garden hose. They realise then that it is smarter to wind it in 3D, cylinder like. From there they come to visualise congruence mod 5, say, as a helical winding of the integer line above a regular pentagon. They metaphorise then this congruence relation as a covering space! Typically, however, this helix does not remind them of the intuitive construction of the imaginary exponential that they were exposed to in Calculus 1, where the real line was wound around or above the unit circle. Our enactivist theoretical perspective emphasises processes and dynamics more than (the more traditional) objects and static structures. So setting the integers mod m as a stage, for instance m = 12 (or even m = 6 or 7, for simpler examples of different nature), we prompt the students to wonder what interesting phenomena may arise in these universes with just 6, 7 or 12 sites, where they can add and multiply.

Among other ideas, they can look at the transformation M 2 given by multiplying by 2, for example, which when iterated launches a dynamical system in the integers mod 12. The students try then to study the generated dynamics, something more natural for those who have developed a systemic perspective on phenomena, more often biology and physics students or humanistic students than mathematics students. Working for an hour, in random groups of 3 to 4 (defined by blind picking of coloured Lego cubes from a bag), a class of 30 to 40 students dutifully iterates M 2 and tries to represent the phenomenon. They explore then M 3 and so on. See Figure 2, which shows some drawings by the students, for m = 7 and m = 12. More generally, they investigate the dynamics spawned by the multiplication by an integer k in the integers mod m, discovering its variegated (forward) orbit structure. They metaphorise idiosyncratically these orbits as forward trajectories and they see fixed points, sinks, attractors, 2cycles, 3-cycles, etc., and become able to figure out the "fate" of different integers mod m. They begin to conjecture on the number of orbits, or the number of n-cycles for a given n. Level of engagement and participation is high, as well as their horizontal interaction, where several ideas and approaches meet. Some of them want to find an arithmetical explanation of this dynamic geometric behaviour. Others do not. But we suggest everyone to look at this geometric phenomenon with arithmetical eyes and vice versa. Most of them draw pictures, but others just write down numbers and tables, and compare. We prompt them to exert some hermeneutical effort [START_REF] Isoda | Mathematical thinking[END_REF], so that all try to understand other's viewpoints. In this way, some predict the existence of an "ubiquitous orbit" for some k for prime m. The students are quite excited and motivated by this challenge when working in small random groups, where horizontal interaction is fostered. Quite often some groups want absolutely to show the teacher and assistant their progress so far. Each group summarises its findings in a report on a sheet of their notebooks, which is photographed by the teacher and assistant. Samples of their work are shown in Figure 2. These reports are evaluated, and a huge worksheet is set up with the corresponding grades. At the end of the course each student gets a grade in group work which is only taken into account if it is higher than the average the grades in the usual compulsory traditional tests taken by the student during the course. Interestingly, in our courses the percentage of (the non-compulsory) student attendance is much higher than the average (80 to 85% against 50%).

Overall, we see that students previously exposed to traditional teaching show a severe metaphorical and dynamical deficit, remaining confined in the arithmetic-algebraic realm. However, progressively, most of them turn out to be able to open up to geometric and dynamical insights, even though initially they remain tied to the purely arithmetical approach that they are more familiar with. At the end of the work session consensus arises on the importance of being able to "change register" and move seamlessly between the arithmetic-algebraic realm and the geometricvisual one. Most students seem surprised to discover, in this way, links between domains of mathematics usually perceived as being far apart, like arithmetic and dynamical systems.

Discussion and open ends

We have shown how Pythagorean triples can be concretely embodied and enactively explored and also how a dynamical systems approach can make the arithmetic modulo m more lively, visual and motivating.

Our examples show that concrete embodiment of abstract mathematical notions and properties, before or instead of their "abstract parachuting", may bear a dramatic impact on the level of participation and engagement of the students in otherwise unappealing tasks. Such impact is also seen on their mathematical performance and their ability to put forth their own research questions, like investigating the "fate" of various numbers mod m or wondering about an arithmetical explanation of the dynamical phenomena observed in the universe of integers mod m.

Our didactic activities are not staircases to climb sequentially but a rather highly intertwined space to explore, where moving flexibly among several representations or metaphors is a must for a meaningful and fruitful mathematical experience.

Our examples also show that representations and metaphors appear sometimes deeply intertwined in a circular relationship, so that it may be somewhat artificial to try to separate them. If we focus in on metaphorising and representing it may very well happen that in a class, some students are metaphorising while others are representing. Also, quite often the teacher will be striving to representin a friendly way for the studentsa notion that she is familiar with (maybe in a rather abstract way though), while students will be metaphorising the same notion, which they hardly fathom. This is visible in our preliminary example of the frog's random walk: the teacher will be eventually representing the probability of finding the frog in a given stone after m jumps by the portion of the frog landing there, while the students will be metaphorically just constructing this probability as the frog's portion landing in this stone.

The same holds for Pythagorean triples and arithmetical congruences. Some arithmetically minded students may realise for the first time that Pythagorean triples may be embodied as an integer fitting phenomenon on a grid and that congruence modulo 5 may be represented by winding the integer number line over a regular pentagon. Other students may discover arithmetical congruences, which they ignored or felt to be an esoteric notion, with the help of this enactive geometric winding. Notice here the emerging connection between arithmetical congruences and geometric congruences.

A recurring observation in all our examples, is the surprise that arises among the learners when they realise hitherto hidden connections between realms of mathematics which seemed to lie far apart. Furthermore, it appears that usual problem solving, as found in the literature, tends to neglect the important role of metaphorisation and representation, as a learner's first reaction when tackling a problem that looks opaque to him or her. Not only because this may allow him or her to solve an otherwise unyielding problem but also because it may allow him or her to "see" a solution, turning a hitherto blind calculation into pellucid insight.

The various metaphors we have presented here are only a sample; there are many others, in analysis, probability, computer science, etc. Invariably, when these metaphors are presented or emerge, they attract the attention of students, and make for a motivating and meaningful classroom experience. They may be however hard to find, so a systematic "catalogue" of them would be most commendable.

Figure 1 :

 1 Figure 1: Linking brick for Pythagorean triples

Figure 2 :

 2 Figure 2: Orbit structure for M k (k = 2, 3, 4, 5) in the integers mod 7 and mod 12

Acknowledgment

Funding from PIA-CONICYT Basal Funds for Centres of Excellence Project BF0003 and DAAD Project 573 35022 D (Uni. Bielefeld -U. of Chile) is gratefully acknowledged.


Arash Farnam

Alain Sarlette

About strong string stability of a vehicle chain with time-headway control

This paper deals with the problem of string stability in a chain of acceleration-controlled vehicles, i.e. how input disturbances affect the distributed system for very long chains. There exist variants of string stability, like avoiding that a local disturbance gets amplified along the chain, or more strongly ensuring that the output vector's p-norm remains bounded for any bounded vector of input disturbances independently of the string length. They are all impossible to achieve with any linear controller if the vehicles only use relative information of few vehicles in front. Previous work has shown that adding absolute velocity into the controller, allows to at least avoid amplification of a local disturbance. In this paper, we consider the stronger definitions of string stability, under this same relaxation of using absolute velocity in the controller. We prove that the influence from input vector to output vector cannot be bounded independently of chain length in the most popular 2-norm sense, with any bounded stabilizing linear controller; while a proportional derivative (PD) controller can guarantee it in the practically relevant ∞-norm sense. Moreover, we identify the disturbance acting on the leader as the main issue for string stability.

I. INTRODUCTION

Platooning of vehicles is a method for increasing the capacity of roads and reducing fuel consumption, see e.g. automated highway systems [START_REF] Chu | Decentralized control of high-speed vehicular strings[END_REF]. Platoon coordination control must prevent the vehicles from too sudden acceleration/braking or even colliding, while maintaining short intervehicle distances for better performance. The most fundamental platoon is the vehicle chain, where all vehicles are aligned after each other. During the recent years, numerous works have considered different control strategies to stabilize each vehicle at a desired distance from its predecessor and follower in such chains [START_REF] Chu | Decentralized control of high-speed vehicular strings[END_REF], [START_REF] Sheikholeslam | Longitudinal control of a platoon of vehicles[END_REF], [START_REF] Levine | On the optimal error regulation of a string of moving vehicles[END_REF], [START_REF] Rogge | Vehicle Platoons Through Ring Coupling[END_REF]. The maybe unexpected challenge is the early observation that some bad behavior cannot be overcome with any linear controller that feeds back relative distances between the vehicles.

More precisely, string instability is a situation where the spacing error between consecutive vehicles grows unbounded when the number of vehicles increases to infinity, and string stability is the situation where this is avoided. (We intentionally keep the definition of "spacing error" loose at this point as there are several versions, to be detailed below.) This Arash Farnam is with the ID Lab, Department of Electronics and Information Systems (ELIS), Faculty of Engineering and Architecture, Ghent University; Technologiepark Zwijnaarde 914, 9052 Zwijnaarde(Ghent), Belgium arash.farnam@ugent.be., Alain Sarlette. is with ELIS, Ghent University, Belgium; and with the QUANTIC lab, INRIA Paris; 2 rue Simone Iff, 75012 Paris, France. alain.sarlette@inria.fr concept has spurred a lot of discussion and research since its definition in [START_REF] Swaroop | String stability of interconnected systems[END_REF], [START_REF] Swaroop | String stability of interconnected systems: an application to platooning in automated highway systems[END_REF]. Basically, it is known since [START_REF] Swaroop | String stability of interconnected systems[END_REF], [START_REF] Swaroop | String stability of interconnected systems: an application to platooning in automated highway systems[END_REF] that string stability cannot be achieved in a homogeneous chain of interconnected second-order integrators (e.g. accelerationcontrolled vehicles), with any controller that is linear and whose local control actions are determined from the relative distance to a few directly preceding vehicles. This has attracted attention as a prototypical, unavoidable shortcoming of linear systems [START_REF] Sheikholeslam | Longitudinal control of a platoon of vehicles[END_REF], [START_REF] Levine | On the optimal error regulation of a string of moving vehicles[END_REF]. When each vehicle only reacts to its immediate predecessor, a straightforward proof of string instability follows from the Bode integral theorem [START_REF] Looze | Tradeoffs and limitations in feedback systems[END_REF]. Indeed, the transfer function from error on vehicle i -1 to error on the following vehicle i, takes the form of a complementary sensitivity function, which unavoidably amplifies some frequencies of the disturbance [START_REF] Swaroop | String stability of interconnected systems: an application to platooning in automated highway systems[END_REF].

To investigate in more detail this problem and options to solve it, a distinction among several string stability notions was made. A weaker version on which researchers have concentrated first [START_REF] Klinge | Time headway requirements for string stability of homogenous linear unidirectionally connected systems[END_REF], [START_REF] Knorn | Passivitybased control for multi-vehicle systems subject to string constraints[END_REF], [START_REF] Öncü | Cooperative Adaptive Cruise Control: network-aware analysis of string stability[END_REF], [START_REF] Ploeg | Lp string stability of cascaded systems: application to vehicle platooning[END_REF], [START_REF] Ploeg | Graceful degradation of Cooperative Adaptive Cruise Control[END_REF], [START_REF] Milanés | Modeling cooperative and autonomous adaptive cruise control dynamic responses using experimental data[END_REF], and which we here label L 2 string stability (see Section II), is to avoid that a single, local L 2 -bounded disturbance signal would have unbounded effects far off in the chain. The Bode integral argument indeed shows that particular local disturbances would grow unbounded, unavoidably, with the simpler controllers. Solving this problem is a prerequisite for stronger versions, here labeled (L 2 , p ) string stability, where one requests that a bounded vector disturbance signals, as the chain becomes infinitely long, induces a bounded vector of inter-vehicle distance errors in the same norm. For instance, (L 2 , ∞ ) string stability would request that if all the vehicles are subject to an L 2 -bounded input disturbance, then each inter-vehicle distance error should remain L 2 -bounded. This appears to be the most "practical" formulation, at least when sticking to the traditional L 2 norm for signals in time. An (L ∞ , ∞ ) version, corresponding to BIBO string stability, would arguably be even more significant in practice; but this would depart too much from the existing literature, and we leave it for future research in order to provide a significant novel point with L 2 signal norm. The (L 2 , p ) string stability is the version considered originally by [START_REF] Swaroop | String stability of interconnected systems: an application to platooning in automated highway systems[END_REF], just with input disturbances replaced by initial conditions. In standard work [START_REF] Barooah | Error amplification and disturbance propagation in vehicle strings with decentralized linear control[END_REF], [START_REF] Seiler | Disturbance propagation in vehicle strings[END_REF], [START_REF] Stüdli | From vehicular platoons to general networked systems: String stability and related concepts[END_REF], the (L 2 , 2 ) version has in fact been the most popular proxy, for the benefit of easier analysis; it is called "general L 2 " string stability in the review paper [START_REF] Stüdli | From vehicular platoons to general networked systems: String stability and related concepts[END_REF].

Several lines of work have shown that L 2 string stability can be solved by adding a sufficiently strong feedback term proportional to absolute velocity -thus slightly enlarging the setting compared to purely relative information. The absolute velocity feedback can be obtained from a natural drag force [START_REF] Rogge | Vehicle Platoons Through Ring Coupling[END_REF], although this would be less in line with developing ever more fuel efficient transportation means. In a somewhat subtler way, the absolute velocity term can also be obtained from a so-called time headway policy, where the desired distance from a vehicle to its predecessor increases with the vehicle's velocity [START_REF] Klinge | Time headway requirements for string stability of homogenous linear unidirectionally connected systems[END_REF], [START_REF] Knorn | Passivitybased control for multi-vehicle systems subject to string constraints[END_REF]. This has the advantage to not favor slow vehicles, but it makes the effective distance velocity-dependent, where absolute velocity of the chain is a priori uncontrolled; so the interfacing with velocity control would have to be carefully investigated. It also remains to be seen exactly how (accurately) the absolute velocity is obtained in practice. While this absolute velocity solution has gathered serious attention as solving L 2 string stability [START_REF] Klinge | Time headway requirements for string stability of homogenous linear unidirectionally connected systems[END_REF], [START_REF] Knorn | Passivitybased control for multi-vehicle systems subject to string constraints[END_REF], [START_REF] Öncü | Cooperative Adaptive Cruise Control: network-aware analysis of string stability[END_REF], [START_REF] Ploeg | Lp string stability of cascaded systems: application to vehicle platooning[END_REF], [START_REF] Ploeg | Graceful degradation of Cooperative Adaptive Cruise Control[END_REF], [START_REF] Milanés | Modeling cooperative and autonomous adaptive cruise control dynamic responses using experimental data[END_REF], it appears that no result so far has established its power for the stronger yet practically important (L 2 , p ) versions. Those have only been investigated with even more information, e.g. controllers relying on absolute position and/or on non-deteriorated knowledge, throughout the chain, of the leader's velocity profile [START_REF] Besselink | String Stability and a delay-based spacing policy for vehicle platoons subject to disturbances[END_REF].

Therefore, the possibilities for time-headway to satisfy (L 2 , p ) string stability have remained, somewhat surprisingly, open to date. Establishing these results is precisely the purpose of the present paper. We have both a positive result -characterizing a PD controller which satisfies the "practical" (L 2 , ∞ ) string stability as requested in [START_REF] Besselink | String Stability and a delay-based spacing policy for vehicle platoons subject to disturbances[END_REF]; and a negative result -suggesting why these results were missing, namely because the more standard (L 2 , 2 ) string stability notion cannot be satisfied by any controller that has bounded DC gain. We furthermore track the (L 2 , 2 ) problem down to the effect of the leading vehicle only. This specificity of the string stability issue with time headway might come as a little surprising, and motivate further research towards circumventing string instability in (L k , p ) sense: (i) the actual relevance of different norm choices; (ii) other contexts than vehicles, featuring possibly noise models with a "perfectly clean" leader; and (iii) the careful use of unbounded controllers, like a PID, in presence of other noise sources and contingencies. Note that under undirected coupling, controllers with integrator terms were proven to be unstable [START_REF] Barooah | Error amplification and disturbance propagation in vehicle strings with decentralized linear control[END_REF]Thm.1.1]. The concrete contribution of this paper is thus to establish a simple way to satisfy the strong and practical (L 2 , ∞ ) string stability; and to clarify that if one truly wants the (L 2 , 2 ) version, then a more careful analysis using unbounded controllers will be necessary.

The paper is organized as follows. Section II formalizes the problem setting and different definitions of string stability. In section III we prove the impossibility result for (L 2 , 2 ) string stability with bounded controllers, while in section IV we show how (L 2 , ∞ ) string satbility is achieved with the same controllers. In section V an example illustrates our positive result.

II. PROBLEM SETTING A. Model of Vehicle Chain

Consider N + 1 vehicles, whose position along the road at time t we denote by x(t) = (x 0 (t), x 1 (t), x 2 (t), ..., x N (t)) ∈ R N +1 , with index 0 denoting the leader. The focus of this work lies on the relative position of consecutive vehicles, while their absolute value remains free. More precisely, we assume that the control objective is to stabilize the subspace {x ∈ R N +1 : x i = x i-1 -r for i = 1, 2, ..., N } for some given desired inter-vehicle distance r > 0. Note that r > 0 implies that vehicle i is behind vehicle i -1 when they move with positive velocity. The configuration error vector thus writes e(t) = (e 1 (t), e 2 (t), ..., e N (t)) with

e i = x i-1 -x i -r , i = 1, 2, ..., N . (1)
The value of x 0 can then be independently guided as e.g. a trajectory tracking command. The N vehicles are modeled as isolated pure double-integrators with u i and d i as acceleration control input and disturbance input, respectively:

ẍi (t) = u i + d i , i = 0, 1, 2, ..., N . (2)
To stabilize e 1 = e 2 = ... = e N = 0, each vehicle i adapts u i as a function of observed information about its neighboring vehicles. We here consider a unidirectional nearest-neighbor chain, in line with much of the literature, where the feedback controller u i can depend on the relative information of one vehicle in front e.g. their relative position x i-1 -x i and relative velocity ẋi-1 -ẋi . In addition, we allow u i to depend on the absolute velocity ẋi of the corresponding vehicle like in [START_REF] Klinge | Time headway requirements for string stability of homogenous linear unidirectionally connected systems[END_REF], [START_REF] Knorn | Passivitybased control for multi-vehicle systems subject to string constraints[END_REF], [START_REF] Ploeg | Lp string stability of cascaded systems: application to vehicle platooning[END_REF] and other papers, but unlike e.g. [START_REF] Öncü | Cooperative Adaptive Cruise Control: network-aware analysis of string stability[END_REF] we do not allow vehicle-to-vehicle communication. We suppose that the leading vehicle is a virtual one and assume u 0 = 0 as customary.

B. Different Definitions of String Stability

There are several variants of string stability in the literature, as mentioned in the introduction. In the following, we consider three of them: so called L 2 norm, (L 2 , 2 ) norm, and (L 2 , ∞ ) norm string stability. The L 2 norm [START_REF] Klinge | Time headway requirements for string stability of homogenous linear unidirectionally connected systems[END_REF], [START_REF] Knorn | Passivitybased control for multi-vehicle systems subject to string constraints[END_REF], [START_REF] Öncü | Cooperative Adaptive Cruise Control: network-aware analysis of string stability[END_REF], [START_REF] Ploeg | Lp string stability of cascaded systems: application to vehicle platooning[END_REF], [START_REF] Ploeg | Graceful degradation of Cooperative Adaptive Cruise Control[END_REF], [START_REF] Milanés | Modeling cooperative and autonomous adaptive cruise control dynamic responses using experimental data[END_REF] and (L 2 , 2 ) norm [START_REF] Barooah | Error amplification and disturbance propagation in vehicle strings with decentralized linear control[END_REF], [START_REF] Seiler | Disturbance propagation in vehicle strings[END_REF], [START_REF] Stüdli | From vehicular platoons to general networked systems: String stability and related concepts[END_REF] versions are most standard in the literature. The last one is closer to realistic concerns, but it seems that it was only recently examined in [START_REF] Besselink | String Stability and a delay-based spacing policy for vehicle platoons subject to disturbances[END_REF], while [START_REF] Stüdli | From vehicular platoons to general networked systems: String stability and related concepts[END_REF] also mentions versions with L 2 replaced by L p regarding the norm over time. The L 2 norm of a time-dependent scalar signal is denoted

x i (•) = +∞ -∞ |x i (t)| 2 dt.
For a time-dependent vector e.g. x(t) , a lower index will indicate the discrete norm used on the vehicle index dimension: the

(L 2 , 2 ) norm is x(•) 2 = N i=0 +∞ -∞ |x i (t)| 2 dt and the (L 2 , ∞ ) norm is x(•) ∞ = max i ( x i (•) ). Definition ((L 2 , 2 ) String Stability): The vehicle chain is (L 2 , 2 )
string stable if, with the closed-loop dynamics, for every > 0 there exists δ > 0 such that: d(•) 2 < δ implies e(•) 2 < , uniformly for all N = 1, 2, ... .

Definition (L 2 String Stability):

The vehicle chain is L 2 string stable if, with the closed-loop dynamics, for every > 0 there exists δ > 0 such that: d(•) 2 < δ implies e i (•) < , uniformly for all N = 1, 2, ... and for all i ∈ {1, 2, ..., N }.

Definition ((L 2 , ∞ ) String Stability): The vehicle chain is (L 2 , ∞ )
string stable if, with the closed-loop dynamics, for every > 0 there exists δ > 0 such that: d j (•) < δ for all j ∈ {0, 1, ..., N } (equivalently d(•) ∞ < δ) implies e i (•) < for all i (equivalently e(•) ∞ < ), uniformly for all N = 1, 2, ... . Remark 1 (admissible disturbances): These variants of string stability sometimes restrict the structure of the disturbance vector, e.g. assuming d j = 0 for all j > 0 to model disturbance on the leader only [START_REF] Yamamoto | Bounded disturbance amplification for mass chains with passive interconnection[END_REF], or the opposite. Disturbances on the leader are indeed special, both practically since this is the "active" boundary of the chain, and for analysis since the controller on the leading vehicle is different; we will see that some results can indeed differ. A most notable point is that standard string stability studies assume disturbances on the inputs and/or initial conditions, but not on the measurements.

In a nutshell, the focus of string stability is that the configuration error must be bounded uniformly in N . The weaker notion is L 2 string stability, as it bounds the sum of disturbance inputs but requests a bounded effect just independently for each e i . L 2 string stability is a necessary condition but not sufficient to guarantee the stronger versions: (L 2 , 2 ) string stability, where the sum-of-squares of the e i must be bounded too; and (L 2 , ∞ ) string stability, where the e i are considered individually but also the input disturbances d i need to be bounded only individually. We would argue that the latter is closest to a realistic physical situation -a still more physical property might be the full BIBO version where also the L ∞ norm is taken over time, but this would depart too much from the well-established literature to be our focus here. A priori, the (L 2 , 2 ) and (L 2 , ∞ ) string stability are not in a definite relation with respect to each other.

Several papers have considered the impossibility of L 2 string stability under relative information feedback, and proved how alternative settings using e.g. absolute velocity feedback [START_REF] Rogge | Vehicle Platoons Through Ring Coupling[END_REF], [START_REF] Klinge | Time headway requirements for string stability of homogenous linear unidirectionally connected systems[END_REF], [START_REF] Öncü | Cooperative Adaptive Cruise Control: network-aware analysis of string stability[END_REF], [START_REF] Ploeg | Lp string stability of cascaded systems: application to vehicle platooning[END_REF], [START_REF] Ploeg | Graceful degradation of Cooperative Adaptive Cruise Control[END_REF], [START_REF] Milanés | Modeling cooperative and autonomous adaptive cruise control dynamic responses using experimental data[END_REF] do allow to achieve L 2 string stability with appropriate tuning. It has also been proved that it is possible to achieve L 2 norm string stability based on relative measurements only, using symmetric bidirectional controllers [START_REF] Yamamoto | Bounded disturbance amplification for mass chains with passive interconnection[END_REF] in which vehicles react not only to their predecessor but also symmetrically to their follower, but only provided disturbances are assumed to act exclusively on the leader. However, regarding (L 2 , 2 ) string stability, as well as the less-studied (L 2 , ∞ ) string stability, the situation is more negative. In the symmetric bidirectional control setting of [START_REF] Yamamoto | Bounded disturbance amplification for mass chains with passive interconnection[END_REF], it has been proved that (L 2 , 2 ) norm string stability cannot be achieved using any linear symmetric bidirectional controllers, see [START_REF] Barooah | Error amplification and disturbance propagation in vehicle strings with decentralized linear control[END_REF], [START_REF] Seiler | Disturbance propagation in vehicle strings[END_REF]. For the controllers with absolute velocity feedback, satisfying the stronger versions of string stability has remained open so far, and we precisely set to answer this point.

C. Considered control situations

The first observations of string instability (see e.g. [START_REF] Swaroop | String stability of interconnected systems: an application to platooning in automated highway systems[END_REF]) were made when each control input u i is reacting just to the relative distance e i with the vehicle in front, and trying to stabilize a constant inter-vehicle distance r. A relatively simple proof allows to check that string stability in L 2 sense, and thus a fortiori the stronger variants, is impossible in this case, with any linear controller that avoids pole cancellation. We recall the proof here as some related computations will be used below. A common assumption for the whole paper is that the controller should not be based on perfect pole cancellation, i.e. K(0) = 0. Let u i (s) = K(s)e i (s). From (1),( 2), the closed-loop equation for the e i writes

e i = T (s) e i-1 + 1 s 2 + K(s) (d i-1 -d i ) , with
T (s) = K(s) s 2 + K(s) = R(s) 1 + R(s)
where R(s) = K(s)/s 2 . This takes the form of a complementary sensitivity function. Now assume for example that there is only disturbance on the leading vehicle, so

e i = T (s) i-1 1 s 2 + K(s) d 0 .
To guarantee L 2 string stability, with N unboundedly large, it is then necessary in particular that |T (jω)| ≤ 1 at all frequencies ω. One concludes that this is impossible for a stable system, from the statement of Bode's Complementary Sensitivity integral [START_REF] Looze | Tradeoffs and limitations in feedback systems[END_REF] which we recall below.

Proposition 1: Assume that the loop transfer function R(s) of a system has (at least) a double pole at s = 0. If the associated feedback system is stable, then the complementary sensitivity function T (s) = R(s) 1+R(s) must satisfy:

∞ 0 ln | T (jω) | dω/ω 2 = π k 1 q (T ) k ≥ 0 ,
where {q So, no linear controller of the type u i (s) = K(s)e i (s) can achieve string stability.

For that reason, researchers have investigated controllers that depend not only on the relative distance between the vehicles x i-1 -x i but also on the absolute velocity ẋi of the vehicle itself. Regarding implementation, this can appear from velocity damping [START_REF] Rogge | Vehicle Platoons Through Ring Coupling[END_REF] or from a so-called timeheadway policy [START_REF] Klinge | Time headway requirements for string stability of homogenous linear unidirectionally connected systems[END_REF] where desired inter-vehicle distance r would depend on ẋi . In the latter case, the form e i = x i-1 -x i -h ẋi -r 0 has been proposed with constant r 0 > 0 and time-headway parameter h > 0, as illustrated on Fig. 1. The linear controller u i (s) = K(s)e i (s), in presence of timeheadway spacing policy, implies the closed-loop equation 3) is < 1 at all frequencies ω = 0, and its H ∞ norm equals T (0) = 1, if and only if the following condition holds:

e i = K(s) s 2 + (1 + hs)K(s) e i-1 (3)
+ 1 s 2 + (1 + hs)K(s) (d i-1 -(1 + hs)d i ) ,
h > max ω K R (jω) (2 -ω 2 K R (jω)) + ωK J (jω) (4) where K R (jω) = 1 2 ( 1 K(jω) + 1 K(jω) * ), K J (jω) = 1 2j ( 1 K(jω) -1 K(jω) * )
, and the maximization runs over all ω for which the argument of the square root is positive.

Proof: We just write 1/|T (jω)| 2 = | -ω 2 /K(jω) + (1 +
hjω)| 2 > 1 and we group real and imaginary parts to finally isolate h.

For particular controllers one can get easier criteria. E.g. for a PD controller K(s) = bs + a, it is not hard to see that if a > 2b 2 then the right hand side in (4) is decreasing with ω, and one gets the simple condition h > 2/a. Note that the system with time headway is not subject to the Bode Integral, because we have T (s) = R(s) 1+R(s) with R(s) = K(s)/(s 2 + hsK(s)) having a single pole at s = 0.

III. IMPOSSIBILITY OF (L 2 , 2 ) STRING STABILITY USING BOUNDED LINEAR CONTROLLERS WITH TIME-HEADWAY

We now consider the impossibility of achieving (L 2 , 2 ) string stability using any bounded stabilizing controller K(s), in particular any controller satisfying |K(0)| < ∞, even in presence of time headway. (As we just recalled, without time headway i.e. for h = 0, it is already impossible to just achieve L 2 string stability.) We do this in two steps to identify that the main culprit is the disturbance on the leading vehicle: in essence, we can avoid that it gets amplified, but we cannot damp it fast enough along the chain with any bounded linear controller. In the literature, disturbances are mostly expected either everywhere, or exclusively on the leader like in [START_REF] Yamamoto | Bounded disturbance amplification for mass chains with passive interconnection[END_REF]. However, from an academic research point of view, it might be useful to know that only disturbances acting on the leader are causing the problem. The restriction to bounded controllers reflects the existing literature and avoids discussing other, unmodeled issues that might arise when e.g. an integral term is present. In any case, the general conclusion may explain why a result about more than L 2 string stability was still missing regarding controllers with time-headway.

A. No disturbance on leader, d 0 = 0 While d 0 = 0 is not a practical situation, we treat it first to show, by linearity, that all problems essentially arise from d 0 . We will show indeed that for d 0 = 0, one can achieve (L 2 , 2 ) string stability using PD controllers with time headway.

Theorem 1: There exists a pair (K(s), h) , where h ≥ 0 is a sufficiently large constant time-headway satisfying Proposition 2 and K(s) = bs + a is a stabilizing PD controller, such that the system (3) is (L 2 , 2 ) string stable provided d 0 = 0.

Proof:

The key point is to recognize that two effects of d i tend to compensate each other in e m with m > i. Indeed, we rewrite ( 3) as

e 1 = -L(s)d 1 e i = -L(s)d i + i m=2 T (s) i-m P (s)d m-1 , with P (s) = s 2 (s 2 +(1+hs)K(s)) 2 and L(s) = 1+hs s 2 +(1+hs)K(s)
and T (s) defined as in Prop.2. We will assume that we place ourselves in the conditions of Prop.2, satisfying the related condition for a PD controller; it is not hard to check that the system is always stable with such PD controller. We next rewrite the dynamics in matrix form: e(s) = -L(s)A + P (s)B(s) d(s)

with the N × (N + 1) matrices A =          0 1 0 . . . 0 0 0 1 . . . 0 0 0 0 . . . 0 . . . . . . . . . . . . . . . 0 0 0 . . . 1          , B(s) =          0 0 0 . . . 0 0 1 0 . . . 0 0 T (s) 1 . . . 0 . . . . . . . . . . . . . . . 0 T (s) N -2 T (s) N -3 . . . 0          .
We first use the triangle inequality to bound

e(s) 2 ≤ |L(s)| A 2 + |P (s)| B 2 ) d(s) 2
with the induced matrix norms, i.e.

D 2 = λ max (D * D)
with * the complex conjugate transpose. The proof now comes down to proving a bounded norm, independent of N and s = jω, for the coefficient in front of d 2 .

For the first term, since A * A = diag(0, 1, 1, 1, ..., 1), we immediately have |L(s)| A 2 = |L(s)|, and the latter can be bounded independently of s = jω for a stable system.

For the second term, we obtain that the element (m, n) of the matrix B * B equals

T (s) m-n N -m j=0 |T (s)| 2j
for m, n ∈ {2, 3, ..., N }, m ≥ n, symmetrically for n > m, and zero for the remaining terms. The Gerschgorin circle theorem thus says that all the eigenvalues of B * B(jω) are comprised in the circles of respective center and radius

c (m) = N -m j=0 |T (jω)| 2j , r (m) =   N -m j=0 |T (jω)| 2j     N n=2,n =m |T | |m-n|   .
With a PD controller satisfying Proposition 2, we have |T (jω)| < 1 for all ω > 0 and we can bound each sum by the result of an infinite geometric series. This yields

|P (jω)| 2 B(jω) 2 2 ≤ |P (jω)| 2 max m (c (m) + r (m) ) ≤ 1 1 -|T (jω)| 2 • 2 1 -|T (jω)| • |P (jω)| 2 = |L(jω)| 2 1 -|T (jω)| 2 • 2|R(jω)| 2 1 -|T (jω)| where R(s) = s 2 / (1+hs)
s 2 +(1+hs)K(s) . Every factor in this expression is bounded at large frequencies, so there just remains to investigate the limit at ω = 0. For the first factor we have

|L(jω)| 2 1 -|T (jω)| 2 = 1 |-ω 2 + (1 + jhω)K(jω)| 2 -|K(jω)| 2 1 a 2 • 1 h 2 ω 2
for ω close to zero. For the second factor, we have |R(s)| ω 2 a , while 2 is of order ω 4 /ω 3 and converges to zero for low ω. This gives a uniform bound on |P (jω)| 2 B(jω) 2 2 at all frequencies and thus concludes the proof.

B. Disturbance concentrated on d 0

The chain's reaction to disturbances on the leader is slightly different, and we now show that this precludes the achievement of (L 2 , 2 ) string stability with any linear controller of bounded DC gain.

Theorem 2: There exists no pair (K(s), h) , with h ≥ 0 a constant time-headway and K(s) a stabilizing controller with K(0) finite, which would guarantee (L 2 , 2 ) norm string stability of system (3) when d 0 = 0.

Proof: We consider only a disturbance input d 0 that affects the leading vehicle, which leads to

e 1 = 1 s 2 + (1 + hs)K(s) d 0 (5)
e i = T (s) i-1 1 s 2 + (1 + hs)K(s) d 0 , i ∈ {2, 3, ..., N } with T (s) defined as in Prop. 2. Then N i=1 |e i (s)| 2 = N -1 i=0 | T (s) | 2i • |d 0 (s)| 2 2 |s 2 + (1 + hs)K(s)| 2 .
Take some β > 0 and define α > 0 such that

|s 2 + (1 + hs)K(s)| 2 | s=jω < α for all ω ∈ (-β, β
). Now select any ∈ (0, β), and take an input disturbance concentrated at low frequencies such that

- |d 0 (jω)| 2 dω ≥ 1 2 +∞ -∞ |d 0 (jω)| 2 dω . Then e(.) 2 2 ≥ - N -1 i=0 |T (s)| 2i |d 0 (s)| 2 |s 2 + (1 + hs)K(s)| 2 s=jω dω ≥ d 0 (.) 2 2 2 1 α min ω∈(-, ) N -1 i=0 | T (jω) | 2i . (6)
Since T (0) = 1, for any given K(s) and h and any δ > 0, there will always exist an such that min ω∈(-, ) |T (jω)| 2 > 1 -δ. As δ can tend towards 0 and N towards infinity, the geometric sum in the second line of (6) then cannot be bounded independently of N .

Theorem 2 implies the impossibility to achieve (L 2 , 2 ) string stability in any cases where disturbances are expected at least on the leading vehicle.As we mentioned earlier, this might explain why results in the literature are restricted to L 2 string stability, because the next-most popular setting would indeed be (L 2 , l 2 ) with bounded controllers K(s). Luckily, there are two possible workarounds for this negative result. A first one is to allow K(s) with unbounded DC gain, like a PID controller; we indeed show in [START_REF] Farnam | Towards impossibility and possibility results for string stability of platoon of vehicles[END_REF] that a PID can be tuned to satisfy the (L 2 , l 2 ) definition of string stability with respect to input disturbances. However, the unbounded DC gain might require to investigate other effects more carefully, as unmodeled measurement noises or saturation effects could seriously deteriorate the situation. Another solution is to recognize that (L 2 , l ∞ ) string stability might be a satisfactory achievement in practice. Indeed, for the latter case, we have the positive result that we present next.

IV. SATISFYING (L 2 , ∞ ) STRING STABILITY WITH PD CONTROLLER We now turn to the positive part of the results, repeating a similar analysis to show how one does guarantee string stability in the "practical" (L 2 , ∞ ) sense using a PD controller with time headway.

Theorem 3: There exists a pair (K(s), h) , where h ≥ 0 is a sufficiently large constant time-headway satisfying Proposition 2 and K(s) = bs + a is a stabilizing PD controller, such that the system (3) is (L 2 , ∞ ) string stable.

Proof: The stability of the system is easy to achieve with any positive tuning of the PD controllers as they lead to second-order polynomials in the denominators, also with time headway.

For string stability, consider the worst case where there are disturbance inputs satisfying d i = δ on all the vehicles i ∈ {0, 1, ..., N }. Following the notations of Thm.1 and Thm.2, we have

e i ≤ max s=jω i m=2 |T i-m P | + |T i-1 L 0 | + |L| δ (7)
where L 0 (s) = For the remaining term, we have

i m=2 |T i-m (jω)P (jω)| ≤ 1 1 -|T (jω)| • |P (jω)| (8) = 1 | -ω 2 + (1 + hjω)K(jω) | -| K(jω) | • | -ω 2 | | -ω 2 + (1 + hjω)K(jω) | .
We first check its behavior at low frequencies. By Taylor expansion we find

1 |-ω 2 +(1+jωh)K(jω)|-|K(jω)| 1 ωa and | -ω 2 -ω 2 +(1+hjω)K(jω) | ω 2
a . For ω = 0 thus, (8) converges to 0. At low frequencies ω > 0, the deviation from 0 in the right-hand side of ( 8) is independent of i and of N , and this provides a bound independent of i and N for the left-hand side. For any given controller satisfying Proposition 2, it is thus straightforward to identify some ω 0 > 0 such that i m=2 |T i-m (jω)P (jω)| < 1/a for instance, for all ω ∈ (-ω 0 , ω 0 ). There remains to prove that the same term remains bounded independently, of i and N , for all ω > ω 0 . With the proposed PD controller, for any ω 0 > 0, there exists α < 1 such that | T (jω) |≤ α for all ω > ω 0 ; this is checked for instance by ensuring a monotone decreasing Bode amplitude diagram, as we will draw in the example below. Then we have, for all ω > ω 0 , a uniform bound on To illustrate Thm.3 about (L 2 , ∞ ) string stability, we simulate this system for a string length N = 150, applying pseudo-random L 2 -norm bounded disturbances on all the vehicles; we just need to drop the last vehicles to get the behavior for shorter strings, thanks to the unidirectionality of the coupling. Figure 3 shows the L 2 norms of errors e i as a function of i = 1, 2, ..., 150. The error appears saturated, illustrating (L 2 , ∞ ) string stability, as expected from Thm.3. Repeating the simulation with other disturbance inputs, similarly normalized, we have always found a bound of about e i < 6 for all i. Computing the (L 2 , 2 ) string stability criterion with the same controller and same disturbance input, we must appropriately rescale d as a function of N to keep d 2 bounded. The corresponding criterion, shown on on Fig. 4, appears to saturate and suggests string stability. However, this is due to the fact that our pseudorandom disturbance will give less and less weight to the leader disturbance d 0 as N increases. Indeed, from Thms.1-2, we have seen that it is d 0 alone which causes the string instability, and this at low frequencies.

To illustrate the failure of (L 2 , 2 ) string stability, as established by Thm.2, we must specifically apply a disturbance concentrated on the leader and at low frequencies. Fig. 5 shows the same string stability criteria as on Figs. 3 and4, but with the only disturbance d 0 = sin(ωt) and three cases of different frequency ω. As expected from the theorems, the (L 2 , ∞ ) criterion, always below 1, keeps decreasing along the chain since |T (jω)| < 1 for all ω > 0. The (L 2 , 2 ) criterion however grows unbounded as N goes up and ω goes down, as the damping along the chain becomes weaker. Applying the same disturbance to another vehicle, e.g. d 2 = sin(ωt), we have observed that, as predicted by Thm.1, the (L 2 , 2 ) criterion does remain bounded (figure not shown due to space constraints). e i as a function of i when L 2 -norm bounded disturbances are applied on all the vehicles. The (L 2 , ∞) string stability criterion for chain length N is given by e(.) ∞ = max i≤N ( e i (.) ); it remains bounded e(.) ∞ < 6 for any N . for unidirectional vehicle chains. In this paper we have shown both theoretically and in simulations how two stronger definitions of string stability behave when using such controllers. From a more practical perspective, we have proved that a simple PD feedback with time-headway, does solve the (L 2 , ∞ ) version of string stability. Furthermore, we have pinned down a specific context for the impossibility to achieve (L 2 , 2 ) string stability: the use of bounded controllers, as favored in the existing literature (e.g. PD control); in combination with the dominating presence of low-frequency disturbance on the leader, whose controller is subject to a boundary condition. This illustrates how subtle differences in definition can change the conclusions on string stability. Thus proxies on norms and error models are unfortunately not a great option and a careful study of the practically relevant criterion appears necessary for each application. It seems particularly relevant for future work to examine explicitly the BIBO-type version, namely (L ∞ , ∞ ) string stability, which has gathered less attention in the literature than its L 2 proxies. Pushing this practical concern further, one could also seek to establish a precise quantitative tradeoff between absolute-velocity feedback h, control parameters in K, chain length N , and admissible local disturbances δ/ , thus allowing to tune the controller to precision requirements δ for any finite N . This could enable to lower the value of h, gain some leeway towards attaining a better tradeoff with other control performance criteria, and possibly achieve stronger norms like (L 2 , 2 ) bounds in this N -dependent-tradeoff sense.

  This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office.

k

  } are the zeros of R(s) in the open right half plane. In particular, if |T (jω)| < 1 at some frequencies, then necessarily |T (jω)| > 1 at other frequencies.

Fig. 1 .

 1 Fig. 1. Feedback controller using time-headway policy

for i = 2 , 3 , 1 s 2 +

 2312 ..., N , and e 1 = (1+hs)K(s) (d 0 -(1+hs)d 1 ). This controller was motivated by the following result [10]. Proposition 2: The norm at s = jω of transfer function T (s) = K(s) s 2 +(1+hs)K(s) in (

1 1 -

 1 |T (jω)| has a leading term of order 1/ω at low frequencies. Thus in fact |P (jω)| 2 B(jω) 2

1 s 2 + 2 (s 2 +

 1222 (1+hs)K(s) , while T (s) = K(s) L 0 (s), L(s) = 1+hs s 2 +(1+hs)K(s) and P (s) = s (1+hs)K) 2 . By satisfying Proposition 2, we know that | T (jω) |< 1 for all ω > 0, and then with the PD controller both |L 0 (s)| = |T (s)|/|a + bs| and |L(s)| = |T (s)| |1+hs| |a+bs| are bounded uniformly for all s = jω. The last two terms in (7) are thus bounded independently of i and of N .

1 1 -

 1 Fig.2we show the Bode diagram of transfer function T (s) and we can see that T (s) is monotonically decreasing, as needed in the proof of Theorem 3.To illustrate Thm.3 about (L 2 , ∞ ) string stability, we simulate this system for a string length N = 150, applying pseudo-random L 2 -norm bounded disturbances on all the vehicles; we just need to drop the last vehicles to get the behavior for shorter strings, thanks to the unidirectionality of the coupling. Figure3shows the L 2 norms of errors e i as a function of i = 1, 2, ..., 150. The error appears saturated, illustrating (L 2 , ∞ ) string stability, as expected from Thm.3. Repeating the simulation with other disturbance inputs, similarly normalized, we have always found a bound of about e i < 6 for all i. Computing the (L 2 , 2 ) string stability criterion with the same controller and same disturbance input, we must appropriately rescale d as a function of N to keep d 2 bounded. The corresponding criterion, shown on on Fig.4, appears to saturate and suggests string stability. However, this is due to the fact that our pseudorandom disturbance will give less and less weight to the leader disturbance d 0 as N increases. Indeed, from Thms.1-2, we have seen that it is d 0 alone which causes the string instability, and this at low frequencies.To illustrate the failure of (L 2 , 2 ) string stability, as established by Thm.2, we must specifically apply a disturbance concentrated on the leader and at low frequencies. Fig.5shows the same string stability criteria as on Figs.3 and 4, but with the only disturbance d 0 = sin(ωt) and three cases of different frequency ω. As expected from the theorems, the (L 2 , ∞ ) criterion, always below 1, keeps decreasing along the chain since |T (jω)| < 1 for all ω > 0. The (L 2 , 2 ) criterion however grows unbounded as N goes up and ω goes down, as the damping along the chain becomes weaker. Applying the same disturbance to another vehicle, e.g. d 2 = sin(ωt), we have observed that, as predicted by Thm.1, the (L 2 , 2 ) criterion does remain bounded (figure not shown due to space constraints).

Fig. 2 .

 2 Fig. 2. Bode magnitude diagram of T (s) for the example of Section V. Note that we have avoided log-scales because it would squeeze the most important features, namely where ||T (jω)|| 1.

2 Fig. 4 . 2 Fig. 5 .

 2425 Fig.4. Error (L 2 , 2 )-norm e(.) 2 as a function of N when the same disturbances are applied as on Fig.3, up to rescaling such that d(.) 2 < 1. This curve appears to saturate at large N .