Daniele Astolfi
email: daniele.astolfi@univ-lyon1.fr

Swann Marx
email: swann.marx@ls2n.fr

Nathan Van De Wouw

Repetitive Control Design Based on Forwarding for Nonlinear Minimum-Phase Systems

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

Introduction

Repetitive Control (RC) techniques for continuous-time systems have been developed at the end of the 80's for tracking and/or rejecting periodic signals of known period T , see [START_REF] Hara | Repetitive control system: A new type servo system for periodic exogenous signals[END_REF]. Supported by the well-known "internal model principle" (see, e.g., [START_REF] Francis | The internal model principle of control theory[END_REF]) stating that a robust controller needs to incorporate a copy of the model generating the references/disturbances, the main idea of RC relies on the use of a delay as universal generator of periodic signals. Such a delay (of time T ) is then embedded in the feedback regulator in order to achieve asymptotic regulation of the desired output. As a consequence, one of the main difficulties in RC is to guarantee the stability of a closed-loop system incorporating a delay, i.e., being an infinite-dimensional system. The aforementioned technical issues and the fact that, from a practical point of view, the implementation of infinite-dimensional control solutions is considered as challenging, widely justified the research of finite-dimensional versions of RC schemes. Indeed, after the seminal work of [START_REF] Tomizuka | Discrete-time domain analysis and synthesis of repetitive controllers[END_REF], a large part of the RC-community approached the problem in the discrete-time domain, see for instance, [START_REF] Longman | On the theory and design of linear repetitive control systems[END_REF][START_REF] Escobar | Filters with linear-phase properties for repetitive feedback[END_REF][START_REF] Owens | Discrete time linear optimal multi-periodic repetitive control: a benchmark tracking solution[END_REF][START_REF] Zhang | Multirate repetitive control for pwm dc/ac converters[END_REF]. This partial change of paradigm made also a step towards computational techniques such as it-erative learning control and run-to-run control, see, e.g., [START_REF] Longman | Iterative learning control and repetitive control for engineering practice[END_REF][START_REF] Wang | Survey on iterative learning control, repetitive control, and run-to-run control[END_REF][START_REF] Wang | Predictive iterative learning control with experimental validation[END_REF].

Despite the large progress in practical applications (see, among them, [START_REF] Kasac | Passive finitedimensional repetitive control of robot manipulators[END_REF][START_REF] Kurniawan | A survey on robust repetitive control and applications[END_REF][START_REF] Blanken | Multivariable repetitive control: Decentralized designs with application to continuous media flow printing[END_REF][START_REF] Mattavelli | Repetitive-based control for selective harmonic compensation in active power filters[END_REF]), when considering the problem of a RC-design in the continuous-time domain, many questions are still open. In particular, it is not clear whether an exact (i.e., infinite-dimensional) RC-scheme can be used (from a theoretical point of view) to achieve asymptotic convergence of the desired regulated output. In fact, one of the main limitations of existing continuous-time schemes is that only "boundary information" (namely, the values of the delayed signal at instants t and t + T ) is used for control purposes. Such a constraint, however, strongly restricts the class of systems to which a RC-scheme can be applied, that is nonlinear systems which are strictly input passive (in other words, with a direct feedthrough term). We refer to [START_REF] Hara | Repetitive control system: A new type servo system for periodic exogenous signals[END_REF] for a proof for linear systems where it is shown that exponential stability of a (continuous-time) linear system incorporating a pure delay in the RC-scheme can be achieved only for systems having zero-relative degree between the input and the regulated output; alternatively, see [START_REF] Califano | Stability analysis of nonlinear repetitive control schemes[END_REF][START_REF] Califano | A stability analysis based on dissipativity of linear and nonlinear repetitive control[END_REF] for a proof in the context of nonlinear systems based on dissipativity operators.

Somehow, to our point of view, the observation in [1, Proposition 2] limited the developments of infinitedimensional continuous-time research in RC, since mostly finite-dimensional approximations have been proposed after this seminal work. Among them, we recall the use of low-pass filters in [START_REF] Omata | Nonlinear repetitive control with application to trajectory control of manipulators[END_REF][START_REF] Weiss | Repetitive control of mimo systems using h ∞ design[END_REF], and the use of an approximated harmonic representation of the delay via Fourier series, see, e.g., [START_REF] Mattavelli | Repetitive-based control for selective harmonic compensation in active power filters[END_REF][START_REF] Ghosh | Nonlinear repetitive control[END_REF]. Note that equivalent designs involving the use of such a harmonic representation have been also studied in output regulation literature, see, e.g., [START_REF] Paunonen | Internal model theory for distributed parameter systems[END_REF][START_REF] Paunonen | Robust controllers for regular linear systems with infinite-dimensional exosystems[END_REF][START_REF] Astolfi | Approximate regulation for nonlinear systems in presence of periodic disturbances[END_REF][START_REF] Astolfi | Francis-wonham nonlinear viewpoint in output regulation of minimum phase systems[END_REF][START_REF] Serrani | Semi-global nonlinear output regulation with adaptive internal model[END_REF] and references therein.

The objective of this work is therefore to study the problem of designing an exact RC-scheme (that is, an infinitedimensional one) for nonlinear systems with global convergence properties. To the best of the authors' knowledge, this problem is still open. We focus in particular on systems which are minimum-phase, with a global, well defined1 arbitrary relative degree and globally Lipschitz nonlinearities and we propose a new approach for the RCfeedback design. For this, we follow the idea of using an "in-domain feedback" (that is, all the values of the delayed signals between t and t + T ) instead of a pure "boundary feedback" (that is, only the values at t and t + T ). Although such an idea is not new in discretetime RC-schemes (see, e.g., [START_REF] Tomizuka | Discrete-time domain analysis and synthesis of repetitive controllers[END_REF][START_REF] Owens | Discrete time linear optimal multi-periodic repetitive control: a benchmark tracking solution[END_REF][START_REF] Wang | Predictive iterative learning control with experimental validation[END_REF]), to the best of the authors' knowledge, it has never been investigated in the literature of continuous-time RC-schemes. In order to construct the feedback, differently from classical approaches, we represent the delay as a transport equation2 , resulting in an overall system which is in the so-called "feedforward-form" [START_REF] Mazenc | Adding integrations, saturated controls, and stabilization for feedforward systems[END_REF]. Then, inspired by the "forwarding approach" (see [START_REF] Astolfi | Francis-wonham nonlinear viewpoint in output regulation of minimum phase systems[END_REF][START_REF] Mazenc | Adding integrations, saturated controls, and stabilization for feedforward systems[END_REF] and references therein) we construct a new (partial) state feedback law, and we prove that the proposed dynamical regulator satisfies the internal model property, namely it can reproduce any periodic trajectory. We also show that the proposed control law can be equivalently written in the pure time-domain without the use of the transport equation. The latter, however, is essential for the overall analysis: indeed, given to the infinite-dimensional properties of the RC-scheme, well-posedness, regularity and stability of the solutions of the resulting closed-loop system are rigorously addressed. Global asymptotic stability of the regulated output to the desired periodic reference is finally proved.

With respect to existing literature of continuous-time RC-schemes, our result is completely new as it certificates the existence with a constructive design of an infinitedimensional RC-design achieving exact asymptotic stability, for nonlinear systems with arbitrarily relative degree. The design of the proposed forwarding-control law is inspired by our recent preliminary work [START_REF] Marx | Forwarding design for stabilization of acoupled transport equation/ode with acone-bounded input nonlinearity[END_REF] in the context of stabilization of a partial differential equation equation coupled with a linear scalar ordinary differential equation. The extension to the proposed nonlinear paradigm is however non-trivial.

This work is organized as follows. First, in Section 2 we detail the problem statement. The main result concerning the design of a RC-scheme for minimum-phase nonlinear systems with unitary relative degree is presented in Section 3. Next, using the results in Section 3 as a stepping stone, in Section 4, we address the case of systems having a relative degree larger than one. Illustrative examples are provided in Section 5. Conclusions are derived in Sec-tion 6. Proofs are postponed to the Appendix.

Notation. We denote with R the space of real numbers and C the space of complex numbers. Set R + := [0, ∞). For a function w : (t, x) ∈ R + ×[0, 1] → w(t, x) ∈ R, the notation w t (respectively, w x ) denotes the partial derivative of w with respect to the variable t (respectively, with respect to the variable x). We keep the notation for the weak and the strong definition of partial derivatives. When a function w depends only on the variable of the time t (respectively, space x), we denote its derivative by ẇ (respectively, w ). Let L 2 (0, 1) be the Hilbert space of real-valued squareintegrable functions over the interval (0, 1). We denote f, g L 2 = 1 0 f g the inner product between f, g ∈ L 2 (0, 1) and f L 2 the induced norm. Let H 1 (0, 1) ⊂ L 2 (0, 1) be the Hilbert space of real-valued absolutely continuous functions over [0, 1] with square-integrable derivative. We denote f, g H 1 = f, g L 2 + f , g L 2 the inner product between f, g ∈ H 1 (0, 1) and f H 1 the induced norm.

Problem Statement for Nonlinear Repetitive Control

Repetitive Control Scheme Representation

Any periodic signal of period T can be generated by the time-delay system represented in Figure 1a and described by the following transfer function (where s ∈ C represents the Laplace variable)

R(s) = exp(-T s) 1 -exp(-T s) , (1)
see for instance [START_REF] Hara | Repetitive control system: A new type servo system for periodic exogenous signals[END_REF]. Hence, in order to solve the problem of reference tracking of disturbance rejection of periodic signals, it is reasonable to include such a delay model in the regulator, as expected from the internal model principle stated in [START_REF] Francis | The internal model principle of control theory[END_REF]. Regulators following this paradigm are usually denoted in literature as Repetitive Control (RC), see [START_REF] Hara | Repetitive control system: A new type servo system for periodic exogenous signals[END_REF].

In this work, we represent the delay exp(-T s) of the transfer function [START_REF] Hara | Repetitive control system: A new type servo system for periodic exogenous signals[END_REF], by means of the (alternative but equivalent) following transport equation representation:

η t (t, x) = -1 T η x (t, x) ∀ (t, x) ∈ R + × [0, 1], η(t, 0) = p(t) ∀ t ∈ R + , η(0, x) = 0 ∀ x ∈ [0, 1], (2)
where η(t, x) is a state variable taking values in R for each (t, x) ∈ R + × [0, 1], and p is the source of the delay equation. It is possible to see that the output η(t, 1) of system (2) is equal to the delayed version of the source input p(t) at any t ≥ T . Indeed, recall that, as shown in [START_REF] Bastin | Stability and boundary stabilization of 1-d hyperbolic systems[END_REF]Chapter 2.1], the general solution to (2) is given by

η(t + (x -x )T, x) = η(t, x ) (3)
for all t ∈ R + and for all 0 ≤ x ≤ x ≤ 1. Therefore, by using (3) with the boundary conditions in (2), we obtain As a consequence, the input-output relation of a standard RC-scheme (1) represented in Figure 1a, which can be computed as

η(t + T, 1) = η(t, 0) = p(t
y(t) = L -1 exp(-sT )L p(t) , p(t) = e(t) + y(t),
where we denoted with L the Laplace transform and by L -1 its inverse, can be equivalently described, by using the delay representation given by (2), as

η t (t, x) = -1 T η x (t, x) ∀ (t, x) ∈ R + × [0, 1], η(t, 0) = p(t) ∀ t ∈ R + , η(0, x) = 0 ∀ x ∈ [0, 1], p(t) = e(t) + y(t) ∀ t ∈ R + , y(t) = η(t, 1) ∀ t ∈ R + , (4)
as depicted in Figure 1b. Note that, from a conceptual point of view, the representation (4) can be interpreted as the continuous-time and continuous-space version of the discrete RC-scheme proposed in [START_REF] Tomizuka | Discrete-time domain analysis and synthesis of repetitive controllers[END_REF], in which the transport is represented, in the discrete-time domain, as a chain of N delays discrete operators z -1 . In particular, compare Figure 1b to [START_REF] Tomizuka | Discrete-time domain analysis and synthesis of repetitive controllers[END_REF]Fig. 2].

The objective of this work is therefore to develop a new RC-feedback design using the representation (4). Note that, differently from classical schemes, we will allow us to use not only the information of the delay at the boundaries η(t, 0) and η(t, 1), but also the full-state information of η(t, x) at all x ∈ [0, 1]. Such a choice, which is, to the best of authors' knowledge, not exploited in other works concerning continuous-time RC-schemes, gives us a larger degree of freedom in the feedback design, allowing to achieve asymptotic stability of the overall closed-loop system for a system of arbitrarily large relative degree.

Implementation aspects related to the discretization of the proposed representation will not be discussed in this brief article and will be addressed in future works.

Problem Statement

In this work, we consider the problem of designing a RCscheme for minimum-phase nonlinear systems that can be written, possibly after a change of coordinates, in the form:

ż = f (t, z, e) ė = q(t, z, e) + u , (5)
where (z, e) ∈ R r × R is the system state, with the zdynamics being the so-called zero-dynamics, e ∈ R is the regulated output, and u ∈ R is the control input. The more general case of higher relative degree will be discussed in Section 4, since, by using a standard change of coordinates, the higher relative degree case can be reduced to the unitary one (see, for instance, [START_REF] Serrani | Semi-global nonlinear output regulation with adaptive internal model[END_REF]).

The following assumptions on the functions f, q in (5) are adopted throughout this paper.

Assumption 1. The functions f, q : R × R n × R are globally Lipschitz, C 2 in all their arguments and periodic with respect to the first argument. In particular, there exists

f , q > 0 such that |f (t, z, e) -f (t, ẑ, ê)| ≤ f |z -ẑ| + f |e -ê|, ( 6
)
|q(t, z, e) -q(t, ẑ, ê)| ≤ q |z -ẑ| + q |e -ê|, (7)
for all t ∈ R + , z, ẑ ∈ R n and e, ê ∈ R, and there exists T > 0 such that

f (t + T, z, e) = f (t, z, e), q(t + T, z, e) = q(t, z, e), (8)
for all t ∈ R + and (z, e) ∈ R n × R.
Assumption 2. The zero-dynamics ż = f (t, z, 0) admits a unique C 2 T -periodic bounded solution z(t) which is globally uniformly stable3 . In particular, there exists a positive definite function V : R × R n → R + and class K ∞ functions 4 α, ᾱ and real numbers α, γ > 0 satisfying

α(|z -z(t)|) ≤ V (t, z -z(t)) ≤ ᾱ(|z -z(t)|) (9)
for all (t, z) ∈ R + × R n and

∇V (t, z -z(t)), f (t, z, e) -f (t, z(t), 0) ≤ -α|z -z(t)| 2 + γ|e| 2 (10) for all (t, z, e) ∈ R + × R n × R.
Remark. (About Assumptions 1 and 2) Assumption 1 states that all functions characterizing the vector field of (5) are globally Lipschitz, while Assumption 2 states that the unperturbed zero-dynamics of system (5) possesses an attractive steady-state which is globally asymptotically stable and that the z-dynamics with nonzero e(t) is input-to-state stable with respect to e. Note that Assumption 2 is rather standard in the context of global stabilization (in which z(t) = 0 and q(t, z(t), 0) = 0 for all t ≥ 0), see, for instance, [START_REF] Isidori | Nonlinear control systems[END_REF]Chapter 9.2]. We conjecture it is possible however to relax Assumptions 1 and 2 by requiring only locally Lipschitz functions and by restricting the attention to a semi-global framework in which solutions starts from a known compact set, see, for instance, [START_REF] Isidori | Nonlinear control systems[END_REF]Chapter 9.3] or [START_REF] Serrani | Semi-global nonlinear output regulation with adaptive internal model[END_REF]. Since, in this work, we deal with infinite-dimensional systems, we restrict the analysis to the global framework, while keeping in mind the fact that all the analysis could be done also under milder conditions, although with possible mathematical developments that would unnecessarily over-complicate the exposition of the developments of this work.

Depending on the considered framework, system (5) may represent a scenario in which the output e represents the error between a desired output and a given periodic reference (i.e., output tracking), or a scenario in which the function q represents the composition of the coupling of the (z, e)-dynamics with a periodic perturbation affecting the e-dynamics (i.e., disturbance rejection). Hence, without loss of generality, the objective of this work is to design a dynamical control feedback for system [START_REF] Escobar | Filters with linear-phase properties for repetitive feedback[END_REF] such that the trajectories of the overall closed-loop system are bounded (forward in time) and moreover the state-variable e is regulated asymptotically to zero with a global domain of attraction, namely

lim t→∞ e(t) = 0, ∀ (z(0), e(0)) ∈ R n × R, (11)
possibly in a robust manner with respect to f and q, since we desire to use as little information as possible about f, q in the control feedback design. For instance, we would like to use only the knowledge of the constants α, γ, q and the period T in Assumptions 1 and 2. Note that a simple high-gain feedback u = -κe, with κ > 0, cannot in general ensure [START_REF] Kasac | Passive finitedimensional repetitive control of robot manipulators[END_REF] since the origin of the zero-dynamics is not an equilibrium and the function q(t, z, 0) is in general not equal to zero.

Forwarding-Based RC-Design for Unitary Relative Degree Systems

Consider system [START_REF] Escobar | Filters with linear-phase properties for repetitive feedback[END_REF]. Assumption 1 states that all functions are T -periodic and so is the attractive steady-state z of the zero-dynamics of system (5) established by Assumption 2. A direct consequence is that in order to guarantee invariance of the solution (z(t), e(t)) = (z(t), 0) for the total dynamics of (5), the control law needs to be able to generate the signal u(t) = -q(t, z(t), 0) which is T -periodic. Such observation justifies therefore the development of a RC-scheme in order to achieve the control objective [START_REF] Kasac | Passive finitedimensional repetitive control of robot manipulators[END_REF] for system [START_REF] Escobar | Filters with linear-phase properties for repetitive feedback[END_REF]. To this end, we first equip system (5) with the internal model (1) and, as a second step, we design a new partial state-feedback law, which is a function of e and η(•, x), to stabilize the extended system (1), [START_REF] Escobar | Filters with linear-phase properties for repetitive feedback[END_REF]. Inspired by the forwarding-stabilization technique developed in [START_REF] Marx | Forwarding design for stabilization of acoupled transport equation/ode with acone-bounded input nonlinearity[END_REF], we propose the following RCforwarding-based scheme, consisting of the internal model unit

η t (t, x) = -1 T η x (t, x) ∀ (t, x) ∈ R + × [0, 1], (12a) η(t, 0) =η(t, 1) + e(t) ∀ t ∈ R + , (12b)
η(0, x) =η 0 (x) ∀ x ∈ [0, 1], (12c)
with initial conditions η 0 ∈ L 2 (0, 1), and the stabilizing feedback law

u(t) = -κe(t) + v(t) , (13a)
v(t) = µ 1 0 (η(t, x) -M (x)e(t))M (x)dx , (13b)
where κ, µ > 0 are design parameters and the function

M : [0, 1] → R in (13b
) is defined as the solution of the following two-point boundary value problem:

M (x) = κT M (x), M (0) = M (1) + 1. ( 14
)
Note that the solution to ( 14) is given by

M (x) = exp(κT x) 1 -exp(κT ) , x ∈ [0, 1]. ( 15
)
The main motivations justifying the RC-design ( 12)-( 14) can be summarized as follows:

• The regulator [START_REF] Kurniawan | A survey on robust repetitive control and applications[END_REF] with input e = 0 and output v given by (13b) can be seen as a "universal generator" of C 2 T -periodic signals, hence it satisfies the internal model property, see forthcoming Lemma 1.

In particular, the objective of the internal model unit is therefore to (automatically) generate the correct steady-state input u(t) ≡ -q(t, z(t), 0), with z given by Assumption 2.

• The feedback law ( 13) is composed of two terms. The first term in (13a) is -κe, which is a proportional (high-gain) feedback term needed to stabilize the (z, e)-dynamics, which may be unstable in openloop due to the term q. The second term in (13a), v, as defined in (13b), is needed to stabilize the ηdynamics and is based on forwarding techniques (see [START_REF] Astolfi | Francis-wonham nonlinear viewpoint in output regulation of minimum phase systems[END_REF] or [START_REF] Marx | Forwarding design for stabilization of acoupled transport equation/ode with acone-bounded input nonlinearity[END_REF]). In particular, suppose to ignore the term q (which is dominated by κe) in the e-dynamics and consider the system ė = -κe. Then, we can consider the e-dynamics to be fast and η-dynamics as slow relative to that fast time scale. As consequence, the main strategy consists in designing a feedback law, that stabilizes the η-dynamics on an invariant manifold that depends on e, which is designed to be linear (since we ignored the nonlinear term q) according to M (x)e(t), with M defined in [START_REF] Mattavelli | Repetitive-based control for selective harmonic compensation in active power filters[END_REF]. Indeed, the stabilizing action v contains a term in η -M e for this purpose. If this term converges to zero, then v converges to zero and, since e converges to zero when v = 0, one can hope that both η and z converges to zero if M is defined in an appropriate way.

Following previous design ideas, we now prove the internal model property of the regulator ( 12), (13b).

Lemma 1. Let M be the solution to [START_REF] Mattavelli | Repetitive-based control for selective harmonic compensation in active power filters[END_REF]. For any C 2 Tperiodic function ψ : R + → R and any µ > 0, there exists η0 ∈ L 2 (0, 1), such that the solution to the system

ηt (t, x) = -1 T ηx (t, x), ∀ (t, x) ∈ R + × [0, 1], η(t, 0) = η(t, 1) ∀ t ∈ R + , η(0, x) = η0 (x) ∀ x ∈ [0, 1], v(t) = µ 1 0 η(t, x)M (x)dx ∀ t ∈ R + , ( 16
) satisfies v(t) ≡ ψ(t) for all t ∈ R + .

Proof.

First of all, note that, due to the boundary condition η(t, 0) = η(t, 1), it follows from (3) that the solution to ( 16) is periodic, namely η(t + T, x) = η(t, x) for all (t, x) ∈ R + × [0, 1]. Now, let us define the function Ψ : R + → R as Ψ(t) := ψ(t) -κψ(t). The function Ψ is C 1 and T -periodic due to the properties of ψ. Now let η0 (x) = T µ Ψ((1 -x)T ) for all x ∈ [0, 1]. Periodicity of Ψ implies η0 (0) = η0 (1) and C 1 of Ψ implies in particular η0 ∈ H 1 (0, 1). Furthermore, by using (3) and the above initial condition, the solution to η(t, x) to ( 16) satisfies

η(t, x) = η(0, x -t T ) = T µ Ψ(t + (1 -x)T ),
for all x ∈ [0, 1] and all 0 ≤ t ≤ T x. Periodicity of η allows to extend the previous relation to all times, namely,

η(t, x) = T µ Ψ(t + (1 -x)T ) ∀(t, x) ∈ R + × [0, 1].
Therefore, by using previous relations, the definition of M in [START_REF] Califano | Stability analysis of nonlinear repetitive control schemes[END_REF], the expression for v as in [START_REF] Califano | A stability analysis based on dissipativity of linear and nonlinear repetitive control[END_REF], and periodicity of ψ, we obtain

v(t) =T 1 0 Ψ(t + (1 -x)T ) exp(κT x) 1 -exp(κT ) dx = T 0 Ψ(t + T -s) exp(κs) 1 -exp(κT ) ds = T 0 [ ψ(t -s) -κψ(t -s)] exp(κs) 1 -exp(κT ) ds = -1 1 -exp(κT ) ψ(t -s) exp(κs) T 0 = ψ(t)
for all t ∈ [0, T ], where in the last equality periodicity of ψ is again used. Finally, since η is H 1 (0, 1) and T -periodic, we conclude that the previous relation holds for all t ∈ R + , concluding the proof.

Clearly, Lemma 1 shows that the property that [START_REF] Kurniawan | A survey on robust repetitive control and applications[END_REF] with input e = 0 and output v given by (13b) is a "universal generator" of C 2 T -periodic signals, for an appropriated choice of the initial conditions η 0 . To the aim of regulation, however, such initial conditions η 0 do not need to be known, as they will be automatically "learned" over the time due to the stability properties of the overall closedloop system trajectories.

We can now state the main result of this work, concerning the properties of the closed-loop system (5), ( 12), [START_REF] Blanken | Multivariable repetitive control: Decentralized designs with application to continuous media flow printing[END_REF]. The proof is postponed to the Appendix. Theorem 1. Suppose Assumptions 1, 2 hold. Then, there exists κ ≥ 1 (which depends on the constants q , α, γ defined in Assumptions 1, 2) such that, for any κ ≥ κ and any µ > 0, the following statements hold.

(i) For any initial condition (z 0 , e 0 , η 0 ) ∈ R n × R × L 2 (0, 1), the closed-loop system (5), ( 12), ( 13), ( 14) admits a unique solution (z, e, η)

∈ C 0 ([0, ∞); R n × R × L 2 (0, 1)
), satisfying (z(0), e(0), η(0, x)) = (z 0 , e 0 , η 0 ), which is bounded for all t ≥ 0, namely

|z(t)| + |e(t)| + η(t, •) L 2 ≤ δ, ∀ t ∈ R + (17)
for some δ > 0.

(ii) Let the set A be defined, for all t ∈ R + , as

A(t) := (z, e, η) ∈ R n × R × L 2 (0, 1) : |z(t) -z(t)| = 0, |e(t)| = 0, η(t) -η(t) L 2 = 0 , ( 18
)
where z is defined in Assumption 2 and η is the solution of system (16) for the particular choice ψ(t) = -q(t, z(t), 0). Then, the set A is globally asymptotically stable 5 for any solution of the closed-loop system (5), ( 12), ( 13), ( 14) starting from R n × R × L 2 (0, 1).

In summary, Theorem 1 states that the regulator ( 12)-( 14) solves the repetitive control problem for system (5) defined in Section 2, namely the solutions of the overall closed-loop system are bounded (forward in time) and [START_REF] Kasac | Passive finitedimensional repetitive control of robot manipulators[END_REF] holds. Remark. (Robustness properties) We care to stress that the regulation objective (11) is obtained robustly with respect to model uncertainties on the functions f, q, in the sense that we do not know the explicit knowledge of them. Indeed, for the design of the feedback law (12), ( 13), we only need to know the period T and the constants α, γ, q of Assumptions 1, 2 (that will determine the value of κ of the statement of Theorem 1, see (A.8) in the proof ), and not the functions f, q itself. As a consequence, the result of Theorem 1 holds robustly for all such functions satisfying these assumptions.

Remark. (Alternative Time-Domain Representation)

In view of Theorem 1, we can select, without loss of generality, the initial condition η(0, x) = 0 for the regulator [START_REF] Kurniawan | A survey on robust repetitive control and applications[END_REF]. With such a choice, the solution to (12) is given by η(t, 0) = e(t) + e(t -T ) for all t ≥ T . Hence, by using the explicit solution (3) of a transport equation, it is readily seen that η(t, x) = e(t -xT, 0) + e(t -xT -T ). As a consequence, by operating a change of coordinates x → s := T x, and using the definition of M in (15), the resulting repetitive-control based regulator (12)-( 13) can be alternatively written, in the pure time-domain (i.e., without the use of the transport-equation), as

u(t) = -κ + µ 2κT exp(2κT ) -1 (exp(κT ) -1) 2 e(t) + v(t) v(t) = µ T T 0 exp(κs) 1 -exp(κT ) e(t -s) + e(t -s -T ) ds,
where e(t) ≡ 0 for all t < 0. The transport-equation representation (2) is therefore not needed for the feedbackimplementation, although it is instrumental for obtaining the above feedback law and essential to establish the stability results of Theorem 1.

RC-Design for Higher Relative Degree Systems via Partial State-Feedback

In this section, we present a RC-approach for systems with relative degree higher than one, that is, for minimumphase systems of arbitrary relative degree r of the form ż = f (t, z, ξ 1 ), ξi = ξ i+1 , i = 1, . . . , r -1 ξr = q(t, z, ξ 1 , . . . , ξ r ) + u, e = ξ 1 , [START_REF] Ghosh | Nonlinear repetitive control[END_REF] where (z , ξ ) ∈ R n × R r is the system state, with the z ∈ R n -dynamics being the zero-dynamics and ξ := (ξ 1 , . . . , ξ r ) ∈ R r representing the derivatives of the output e that we aim to regulate to zero. We suppose that the functions f, q in (19) satisfy the following assumption. Assumption 3. The functions f, q in (19) are globally Lipschitz, C 2 in their arguments and periodic with respect to the first argument. Moreover, the function f satisfies 6 Assumption 2.

Based on the design proposed in Section 3, in order to solve the control objective [START_REF] Kasac | Passive finitedimensional repetitive control of robot manipulators[END_REF] for system [START_REF] Ghosh | Nonlinear repetitive control[END_REF], we propose the following regulator:

η t (t, x) = -1 T η x (t, x) η(t, 0) = η(t, 1) + θ(t) η(0, x) = η 0 (x) u(t) = -κθ(t) + µ 1 0 (η(t, x) -M (x)θ(t))M (x)dx ( 20
)
6 where in inequality [START_REF] Wang | Predictive iterative learning control with experimental validation[END_REF], the variable e has to be replaced by ξ 1 .

with M defined as in ( 14) and θ as

θ := ξ r + r-1 i=1 g r-i a i ξ i , (21)
with g > 0 and a i > 0, i = 1, . . . , r -1, parameters to be defined. We now can present the following result on the properties of the resulting closed-loop system [START_REF] Ghosh | Nonlinear repetitive control[END_REF], [START_REF] Paunonen | Internal model theory for distributed parameter systems[END_REF].

Theorem 2. Suppose Assumption 3 holds. Let a i > 0, i = 1, . . . , r -1, be selected such that the polynomial p 0 (λ

) := λ r-1 + a r-1 λ r-2 + • • • + a 2 λ + a 1 is Hurwitz.
Then, there exists g ≥ 1 and, for any g > g , there exists κ ≥ 1 (which depends on the constants q , α, γ defined in Assumptions 1, 2, and the choice of g and a 1 , . . . , a r-1 ) such that, for any κ ≥ κ and any µ > 0, the following statements hold.

(i) For any initial condition (z 0 , ξ 0 , η 0 ) ∈ R n × R r × L 2 (0, 1), the closed-loop system (19), ( 20), ( 21), admits a unique solution (z, ξ, η) ∈ C 0 ([0, ∞); R n ×R r × L 2 (0, 1)), satisfying (z(0), ξ(0), η(0, x)) = (z 0 , ξ 0 , η 0 ), which is bounded for all t ≥ 0, namely

|z(t)| + |ξ(t)| + η(t, •) L 2 ≤ δ, ∀ t ∈ R +
for some δ > 0.

(ii) Let the set A be defined, for all t ∈ R + , as

A(t) := (z, ξ, η) ∈ R n × R r × L 2 (0, 1) : |z(t) -z(t)| = 0, |ξ(t)| = 0, η(t) -η(t) L 2 = 0 ,
where z is defined in Assumption 2 and η is the solution of system (16) for the particular choice ψ(t) = -q(t, z(t), 0). Then, the set A is globally asymptotically stable for any solution of the closed-loop system (19), ( 20), [START_REF] Paunonen | Robust controllers for regular linear systems with infinite-dimensional exosystems[END_REF], starting from R n × R r × L 2 (0, 1).

Proof. Following [24, Section V-A], the main idea of the proof consists in transforming system (19) into a system of unitary relative degree by means of a linear change of coordinates, and then to re-apply the results of Theorem 1.

In particular, with the definition of θ given in [START_REF] Paunonen | Robust controllers for regular linear systems with infinite-dimensional exosystems[END_REF] in mind, by applying the following change of coordinates [START_REF] Ghosh | Nonlinear repetitive control[END_REF], we obtain the transformed dynamics ż = f (t, z, g 1-r Cε) ε = gAε + gBθ θ = q 0 (t, ε, θ) + q 1 (ε, θ) + u e = g 1-r Cε, [START_REF] Astolfi | Approximate regulation for nonlinear systems in presence of periodic disturbances[END_REF] in which ε :

ξ i → ε i := g r-i ξ i , i = 1, . . . , r -1, ξ r → θ := ξ r + r-1 i=1 g r-i a i ξ i , to system
= (ε 1 , . . . , ε r-1 ) ∈ R r-1 , B := 0 r-2,1 1 , C := 1 0 1×r-2 , A := 0 r-2,1 I r-2 -a 1 -a 2 • • • -a r-1 , q 0 (t, ε, θ) := q(t, z, g 1-r ε 1 , . . . , g -1 ε r-1 , θ - r-1 i=1 a i ε i ), q 1 (ε, θ) := g r-2 i=1 a i ε i+1 + ga r-1 (θ - r-1 i=1 a i ε i ).
Now recall that the subsystem z is input-to-state stable with respect to ξ 1 and therefore to g 1-r Cε. Indeed, under Assumption 3 (and therefore by Assumption 2), inequality [START_REF] Wang | Predictive iterative learning control with experimental validation[END_REF], in which e is substituted with ξ 1 , holds. Furthermore, the matrix A is Hurwitz by construction since it is the companion form of the Hurwitz polynomial p(λ). As a consequence, for g large enough, system [START_REF] Astolfi | Approximate regulation for nonlinear systems in presence of periodic disturbances[END_REF] with output θ and zero-dynamics (z, ε) is still minimum-phase. In other words, the subsystem (z, ε) is input-to-state stable with respect to θ. This, in particular, can be established by means of the Lyapunov function 9), [START_REF] Wang | Predictive iterative learning control with experimental validation[END_REF], and P solution to the Lyapunov equation P A + A P = -2I. Indeed, by using (10), we can compute the time-derivative of W along the solutions of the closed-loop system [START_REF] Astolfi | Approximate regulation for nonlinear systems in presence of periodic disturbances[END_REF], [START_REF] Paunonen | Internal model theory for distributed parameter systems[END_REF], denoted as Ẇ (t), as

W (t, z, ξ) = V (t, z) + ε P ε with V satisfying (
Ẇ (t) ≤ -α|z -z(t)| 2 + γg 1-r |Cε| -2g|ε| 2 + 2gε P Bθ ≤ -α|z -z(t)| 2 -(g -γ)|ε| 2 + g|P | 2 θ 2
for all t ∈ R + , showing the desired input-to-state stability properties for g > g , g = max{1, γ}. Also, we deduce that the equilibrium of the zero-dynamics of ( 22) is given by (z, ε) = (z, 0). Furthermore, it follows by linearity of the coordinate transformation that if q is a C 2 T -periodic globally Lipschitz function, then so are the functions q 0 , q 1 defined in [START_REF] Astolfi | Approximate regulation for nonlinear systems in presence of periodic disturbances[END_REF].

We conclude therefore that the system [START_REF] Astolfi | Approximate regulation for nonlinear systems in presence of periodic disturbances[END_REF] satisfies the assumptions of Theorem 1 when considered as a unitaryrelative degree system with (z, ε) zero-dynamics, that is, a system of the form [START_REF] Escobar | Filters with linear-phase properties for repetitive feedback[END_REF]. We can therefore apply Theorem 1 to show that solutions of the closed-loop system [START_REF] Astolfi | Approximate regulation for nonlinear systems in presence of periodic disturbances[END_REF], [START_REF] Paunonen | Internal model theory for distributed parameter systems[END_REF] are bounded (forward in time) and converges asymptotically to the set

(z, ε, θ, η) ∈ R n × R r-1 × R × L 2 (0, 1) : |z -z(t)| = 0, |ε(t)| = 0, |θ(t)| = 0, η(t) -η(t) L 2 = 0
for κ large enough and any µ > 0. Note that the value of κ depends on the choice of g which has to be selected first. By construction, the set {(ε, θ) ∈ R r-1 × R : ε = 0, θ = 0}, corresponds to the set {ξ = 0}, concluding the proof.

Some Illustrative Examples

The literature on control problem for minimum-phase systems is vast and and hence so is the number of potential applications to which the results in this paper could be applied. In this section, we concisely discuss three applications of the proposed RC-design technique developed in Sections 3 and 4.

Tracking control for frictional systems with unknown friction characteristics.

Friction is an ubiquitous phenomenon in mechanical systems that is difficult to model and often requires to be compensated, for instance, in order to achieve required positioning performance in high-tech motion systems. Consider a two-dimensional (normalized) mechanical system of the form θ = F ( θ) + u where position ϑ and speed θ are measurable, u ∈ R is the control force and F is an unknown (smooth enough) globally Lipschitz function characterizing the friction. Suppose that we desire the controller to ensure ϑ to asymptotically track any arbitrary periodic reference ϑ ref (t) with known bounded first and second-order derivatives. By selecting the error coordinates

ξ 1 := ϑ -ϑ ref (t), ξ 2 := θ -θref (t),
we obtain a system in the form [START_REF] Ghosh | Nonlinear repetitive control[END_REF], with n = 2, in which there is no zero-dynamics z and the function q is given by

q(t, ξ) := F (ξ 2 + θref (t)) -θref (t),
which is T -periodic and globally Lipschitz. As the system has no-zero dynamics, the minimum-phaseness prerequisite is trivially verified. Assumption 3 is therefore satisfied. Hence, we can employ the RC-design of Section 4 to address the desired tracking problem. Note that for the design of the the regulator [START_REF] Paunonen | Internal model theory for distributed parameter systems[END_REF], only the knowledge of a (possible conservative) approximation of the Lipschitz constant of the friction characteristic F is required, which is beneficial given the typical uncertainty in friction models. As a consequence Theorem 2 can be used to show that a controller of the form ( 12), ( 13), ( 14) solves the periodic tracking problem at hand.

Neural Network Function Approximation

As discussed in [START_REF] Ortega | Nonlinear pi control of uncertain systems: an alternative to parameter adaptation[END_REF], many papers in neural network control have been devoted to the solution of the tracking problem for the n-dimensional system χi = χ i+1 , i = 1, . . . , n -1,

χn = n i=1 a i χ i + N i=1 δ i 1 + α i exp(-β i χ i ) + u
where N is a known positive integer and all the parameters a i , i = 1, . . . , n, and α i , β i , δ i > 0, i = 1, . . . , N , are unknown. In particular, we can apply the design procedure of Section 4 to design a state-feedback law ensuring χ 1 to asymptotically track a desired periodic smooth reference χ ref (t). By setting

ξ i := χ 1 -χ (i-1) ref (t), i = 1, . . . , n,
the system is transformed into the form [START_REF] Ghosh | Nonlinear repetitive control[END_REF] in which there is no zero-dynamics and the function q is given by

q(t, ξ) := -χ (n) ref (t) + n i=1 a i (ξ i + χ (i-1) ref (t)) + N i=1 δ i 1 + α i exp(-β i (ξ i + χ (i-1) ref (t))
, which is T -periodic and globally Lipschitz. In order to design the control law [START_REF] Paunonen | Internal model theory for distributed parameter systems[END_REF], we need to suppose that the parameters δ i , a i , satisfy

n i=1 |a i | + N i=1 |δ i β i | ≤ c
for some known c > 0. Indeed, in this case, an overapproximation of the Lipschitz constant of the function q is given by q = c. As the system has no-zero dynamics, the minimum-phaseness prerequisite is trivially verified. Hence Assumption 3 is satisfied and we can use apply Theorem 2 to show that a controller of the form ( 12), ( 13), ( 14) solves the periodic tracking problem at hand.

Simulation Example

As an illustrative simulation example, consider the following system ż = -z 3 + cos(2πt) -0.5 + e ė = 0.5 + 2 arctan(z)e + u

verifying Assumptions 1 and 2 with T = 1 on any given compact set, and select the controller ( 12), ( 13), ( 14) with κ = 2 and µ = 10. Figure 2 (resp., Figure 3) shows the evolution of the trajectories of (z, e) (resp., η(t, •) L 2 ), for the initial conditions (z 0 , e 0 ) = (1, -1) and η 0 : x → sin(x).

Simulations have been performed using the method of characteristics (see, e.g., [START_REF] Evans | Partial differential equations[END_REF]Section 2.1]) to compute the exact solution of ( 12) at any time with a spatial discretization interval of 10 -2 , while the ODEs have been discretized using the Euler method with time step 10 -2 . Figure 2 shows the asymptotic convergence of e to zero with a slow residual dynamics typical of forwarding approaches.

Conclusions

In this work, we addressed the problem of repetitive control for nonlinear systems. Differently from standard approaches, we represent here the delay as a transport equation and we propose a new forwarding-based (partial) state feedback that uses not only the boundary information of the transport equation representing the delay, but all its state. This allows to establish asymptotic stability of a RC-scheme for nonlinear minimum-phase systems with arbitrarily relative degree and constant high-frequency gain. Implementation issues related to the (time and/or space) discretization of the regulator [START_REF] Paunonen | Internal model theory for distributed parameter systems[END_REF] have not been discussed in this manuscript, since providing a numerical scheme for spatio-temporal discretization would need a careful study of its convergence, and this is out of the (theoretical) scope of this brief article. Nevertheless, a proper discretization of the proposed infinite-dimensional scheme surely deserves of further research, since different approaches can be employed, such as the use of low-pass filters [START_REF] Hara | Repetitive control system: A new type servo system for periodic exogenous signals[END_REF][START_REF] Omata | Nonlinear repetitive control with application to trajectory control of manipulators[END_REF][START_REF] Weiss | Repetitive control of mimo systems using h ∞ design[END_REF], Fourier approximations [START_REF] Ghosh | Nonlinear repetitive control[END_REF] [START_REF] Astolfi | Approximate regulation for nonlinear systems in presence of periodic disturbances[END_REF], [START_REF] Paunonen | Internal model theory for distributed parameter systems[END_REF] or discrete-delay representations [START_REF] Tomizuka | Discrete-time domain analysis and synthesis of repetitive controllers[END_REF][START_REF] Owens | Discrete time linear optimal multi-periodic repetitive control: a benchmark tracking solution[END_REF][START_REF] Wang | Predictive iterative learning control with experimental validation[END_REF]. It is worth stressing, that the proposed approach also opens the doors for new results in RC-design for discrete-time nonlinear systems, possibly combining recent results in forwarding stabilization [START_REF] Mattioni | Forwarding stabilization in discrete time[END_REF].

From the theoretical point of view, future works include the extension of the proposed technique to pure output-feedback designs for higher relative degree systems; the study of RC-schemes for non-minimum phase systems and/or multi-variable systems; the use of adaptive schemes that possibly estimate online the period T ; and the extension of the proposed technique to the context of cooperative output control problems for multi-agent systems, see, e.g., [START_REF] Wang | Global cooperative control framework for multiagent systems subject to actuator saturation with industrial applications[END_REF][START_REF] Casadei | Multipattern output consensus in networks of heterogeneous nonlinear agents with uncertain leader: A nonlinear regression approach[END_REF].

Let ψ(t) := -q(t, z(t), 0), with z given by Assumption 2. In view of Assumptions 1 and 2, the function ψ is C 2 and T -periodic. Hence, consider the following change of coordinates 

 z(t) e(t) η(t)   →   ζ(t) e(t) φ(t)   :=   z -z(t) e(t) η(t, x) -η(t, x) -M (x)e(t)  
where η is the solution of ( 16) satisfying v(t) ≡ ψ(t) = -q(t, z(t), 0) for all t ≥ 0, and z is defined by Assumption 2. By using the following identity

- 1 T [η x (t, x) -ηx (t, x)] + κM (x)e(t) = - 1 T φ x (t, x)
obtained from the definition of M in ( 14), we obtain

ζ(t) = F (t, ζ, e) ė(t) = ∆(t, ζ, e) -κe(t) + µ 1 0 φ(t, x)M (x)dx φ t (t, x) = -1 T φ x (t, x) -M (x)∆(t, ζ, e) -µM (x) 1 0 φ(t, x)M (x)dx φ(t, 0) = φ(t, 1) φ(0, x) = φ 0 (x) (A.1
) where φ 0 (x) := η 0 (x) -η0 (x) -M (x)e(0) ∈ H 1 (0, 1), and the functions F, ∆, defined as for all (t, ζ, e) ∈ R + × R n × R. Note that, by construction, F (t, 0, 0) = 0 and ∆(t, 0, 0) = 0 for all t ∈ R + . Hence, it can be verified that the origin (ζ, e, φ) = (0, 0, 0) in the R n × R × L 2 (0, 1)-topology is an equilibrium point of (A.1). Furthermore, by recalling the definitions of ζ and φ, we have the existence of class K ∞ functions β and β satisfying

F (t, ζ, e) := f (t, ζ + z(t), e) -f (t, z(t), 0), (A.2) ∆(t, ζ, e) := q(t, ζ + z(t), e) -q(t, z(t), 0), (A.
β(|z -z(t)| + |e| + η -η L 2 ) ≤ |ζ| + |e| + φ L 2 ≤ β(|z -z(t)| + |e| + η -η L 2 ). (A.6)
Therefore, showing that the origin (ζ, e, φ(•, x)) = 0 of (A.1) is stable, coincides, in the original coordinates (z, e, η), in showing that the set A defined in ( 18) is stable. In other words, the items (i) and (ii) of Theorem 1 will be equivalently proved in the coordinates (A.1). Using the Duhamel formula (i.e., variation of constants formula), one can write the solution to (A.1) with an integral formula which exists thanks to a Banach fixed point theorem, as it has been done in [START_REF] Chitour | L p -asymptotic stability analysis of a 1d wave equation with a nonlinear damping[END_REF]. Thus, one can prove that solutions of the system (A.1) exist and are unique for some small interval of time, uniformly in the initial time t 0 and in the initial conditions (ζ 0 , e 0 , φ 0 ). This is stated in the next lemma. We omit the proof for space reasons.

Lemma 2 (Well-posedness of (A.1)). For any t 0 > 0 and any initial conditions (ζ 0 , e 0 , φ 0 ) ∈ R n × R × L 2 (0, 1) (resp. R n × R × H 1 (0, 1) with the compatibility condition φ 0 (0) = φ 0 (1)), there exists a positive constant τ > 0 sufficiently small such that there exists a unique solution (ζ, e, φ) ∈ C 0 ([t 0 , t 0 + τ ]; R n × R × L 2 (0, 1)) (resp. C 1 ([t 0 , t 0 + τ ]; R n × R × L 2 (0, 1)) ∩ C 0 ([t 0 , t 0 + τ ]; R n × R × H 1 (0, 1))) to system (A. for some given U > 0. By definition the set Ω U is a bounded subset of R n × R × L 2 (0, 1). Hence, consider any initial condition (ζ 0 , e 0 , φ 0 ) ∈ Ω U . In view of Lemma 2, there exists τ > 0 such that [t 0 , t 0 + τ ) is the maximal interval of existence of the corresponding solutions(ζ, e, φ) to system (A.1). Note that since all the bounds and constants selected in the proof are uniform in t 0 (in particular, they do not depend on it), without loss of generality, in the reset of the proof we simply set t 0 = 0.

We compactly denote by U (t) the expression for U along such solutions. By using [START_REF] Wang | Predictive iterative learning control with experimental validation[END_REF] Note that, by using the fact that φ(t, 0) = φ(t, 1), we obtain 1 0 φ(t, x)φ x (t, x)dx = 0.

Then, by using Young's inequality 7 and Lipschitz properties of ∆ in (A.5), we have the following bounds: 2e∆(t, ζ, e) ≤ 2 q |e| (|e| + |ζ|) ≤ 3 q |e| 2 + q |ζ| 2

Figure 1 :

 1 Figure 1: Equivalent representations of an internal-model designs in RC-schemes.

Figure 2 :

 2 Figure 2: Closed-loop system (23), (12), (13): trajectories of z(t) (dotted red line) and e(t) (blue line) for t ∈ [0, 30].

Figure 3 :

 3 Figure 3: Closed-loop system (23), (12), (13): evolution of η(t, •) L 2 for t ∈ [0, 30].

  3) satisfy, in view of Assumption 1, the following inequalities: |F (t, ζ, e)| ≤ f |ζ| + f |e| , (A.4) |∆(t, ζ, e)| ≤ q |ζ| + q |e |, (A.5)

  [START_REF] Hara | Repetitive control system: A new type servo system for periodic exogenous signals[END_REF]. Now consider the Lyapunov functionalU (t, ζ, e, φ) := cV (t, ζ) + e 2 + µ 1 0 φ(t, x) 2 dx,with c > 0 to be selected and V given by Assumption 2. In view of Assumption 2, the function U satisfies, for all(t, ζ, e, φ) ∈ R + × R n × R × L 2 (0, 1), α U (|ζ| + |e| + |φ| L 2 (0,1) ) ≤ U (t, ζ, e, w) ≤ ᾱU (|ζ| + |e| + |φ| L 2 (0,1) ), (A.7)for some class K ∞ functions α U , ᾱU . Define the setΩ U := (t, ζ, e, φ) ∈ R + × R n × R × L 2 (0,1) : U (t, ζ, e, φ) ≤ U ,

  , the derivative of U satisfiesU (t) ≤ -cα|ζ| 2 + cγ|e| 2 + 2e∆(t, ζ, e) -2κ|e| 2

In the sense of[START_REF] Isidori | Nonlinear control systems[END_REF] Chapter 4].

The use of a transport-equation representation for the delay has been recently used in[START_REF] Califano | Stability analysis of nonlinear repetitive control schemes[END_REF][START_REF] Califano | A stability analysis based on dissipativity of linear and nonlinear repetitive control[END_REF] but for analysis purposes only.

See[START_REF] Khalil | Nonlinear systems[END_REF] Definition

4.4].[START_REF] Longman | On the theory and design of linear repetitive control systems[END_REF] A function α : R + → R + is said to be of class K if α is continuous, increasing, and α(0) = 0. If moreover lim x→+∞ α(x) = +∞, we say that α is of class K∞.

See, for instance,[START_REF] Yoshizawa | Stability theory by liapunov's second method[END_REF] Chapter 4, §15] for the definition of set stability. Note that, because of the boundary condition η(t, 0) = η(t, 1) of the transport equation, exponential stability of the closed-loop system cannot be guaranteed in view of the presence of an infinitenumber of poles on the imaginary axis and the use of a bounded (in the sense of[START_REF] Tucsnak | Observation and control for operator semigroups[END_REF] Page 24]) control operator. See also[START_REF] Paunonen | Internal model theory for distributed parameter systems[END_REF][START_REF] Paunonen | Robust controllers for regular linear systems with infinite-dimensional exosystems[END_REF] for other examples of such phenomenon.

In particular, 2ab ≤ νa 2 + ν -1 b 2 for any ν > 0.

$ This research was partially supported by the French Grant ANR ODISSE (ANR-19-CE48-0004-01) and was also conducted in the framework of the regional programme "Atlanstic 2020, Research, Education and Innovation in Pays de la Loire", supported by the French Region Pays de la Loire and the European Regional Development Fund.

Appendix A. Proof of Theorem 1

The proof of Theorem 1 is organized as follows. First, we modify system [START_REF] Escobar | Filters with linear-phase properties for repetitive feedback[END_REF], [START_REF] Kurniawan | A survey on robust repetitive control and applications[END_REF], [START_REF] Blanken | Multivariable repetitive control: Decentralized designs with application to continuous media flow printing[END_REF] by means of some changes of coordinates. Existence and uniqueness of solutions is discussed first. Finally, a Lyapunov-based analysis is presented, allowing to establish boundedness of solutions and convergence to the desired stead-state. and

Hence, combining all the bounds, we obtain

Therefore, by selecting

for any arbitrarily small > 0, the previous inequality gives, for any κ ≥ κ ,

(A.9) The second inequality in (A.9) implies also U (t) ≤ U (0) for all t ∈ [0, τ ). This shows that the solution (ζ, e, φ) remain in Ω U for all (positive) times (and any t 0 ). So, from its definition, τ is infinite, namely the solution exists for all t ∈ R + , and in particular satisfies

for all t ≥ R + , where we used (A.7). Finally, the inequality (A.6) can be obtained by combining inequality (A.10) with (A.6) and the following bounds

Note that in view of Assumption 2 and Lemma 1, z and η0 are bounded. This concludes the proof of the item (i) of Theorem 1. Next, in order to show item (ii), we follow the main arguments used in proof of [27, Theorem 2], which are adapted to the analysis of system (A.1). In particular, by using the first inequality of (A.9), we also obtain ( 1 0 φ(t, x)M (x)dx) 2 dt < +∞. Such inequalities, combined with the bound (A.10) and the fact that all functions are globally Lipschitz, implies that also the functions t → e(t), t → ζ(t), and t → 1 0 φ(t, x)M (x)dx are also globally Lipschitz and therefore uniformly continuous. Hence, by applying Barbalat's lemma, we obtain lim t→+∞ |ζ(t)| = 0, lim t→+∞ | 1 0 φ(t, x)M (x)dx| = 0 and lim t→+∞ |e(t)| = 0. Finally, for the last part of the proof, one can follow the same steps of the proof of [START_REF] Marx | Forwarding design for stabilization of acoupled transport equation/ode with acone-bounded input nonlinearity[END_REF]Theorem 2], where it can be shown, by using precompactness of solutions and LaSalle's invariance principle arguments for infinite-dimensional systems (see [START_REF] Slemrod | Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control[END_REF]Theorem 3.1]), that (ζ, e, φ) converge asymptotically to (0, 0, 0) in the R × L 2 (0, 1)-topology. Standard density argument (see, e.g., [START_REF] Marx | Cone-bounded feedback laws for m-dissipative operators on Hilbert spaces[END_REF]Lemma 1]) allows also to conclude that such a result holds for initial condition (ζ 0 , e 0 , φ 0 ) in R×L 2 (0, 1). Therefore, the proof concludes by noting that the origin of (A.1) coincides, in the z, e, η coordinates, with (z, e, η) = (z, 0, η), that is the set A defined in [START_REF] Weiss | Repetitive control of mimo systems using h ∞ design[END_REF].


Pauline Loxq

Nathalie Debono

Süleyman Gülcemal

Jean-Claude Daran

Eric Manoury

Rinaldo Poli

Bekir Çetinkaya

Agnès Labande

Palladium(II) complexes with planar chiral ferrocenyl phosphane/(benz)imidazol-2-ylidene ligands

We describe here the first examples of planar chiral ferrocenyl phosphane/benzimidazol-2-ylidene ligands and their coordination chemistry with palladium(II). All ligand precursors, namely enantiopure ferrocenyl phosphane/(benz)imidazolium salts, and all enantiopure palladium complexes have been fully characterised by 1 H, 31 P and 13 C NMR, mass spectrometry and X-ray diffraction methods for seven examples. The potential of these very bulky bidentate ligands in catalysis was evaluated and compared to their imidazol-2-ylidene analogues. The influence of sterics was shown to be non-negligible as the bulkiest ligand gave the lowest activities in the asymmetric Suzuki-Miyaura reaction.

Introduction

The synthesis and coordination chemistry of functionalised Nheterocyclic carbene (NHC) ligands and their applications in catalysis has attracted the attention of several groups, [START_REF] Lee | [END_REF][2][3] including ours. [4][5][6][7] These strong -donor ligands proved very efficient in catalysis, even surpassing phosphanes in many cases. 8 One advantage of using N-heterocyclic carbenes is the high stability conferred to the metal complexes, thus avoiding the use of excess ligand or catalyst deactivation. Fine tuning the stereoelectronic properties of the N-heterocyclic carbene may allow us finding useful trends in the search of the best catalyst, particularly in terms of enantioselectivity. Recently, we described the synthesis of 1,1 'disubstituted ferrocenyl phosphane/benzimidazol-2-ylidene ligands for applications in rhodium-catalysed hydrosilylation of carbonyl groups. It was shown in this case that the replacement of the imidazol-2-ylidene moiety by the more bulky, electron-rich benzimidazol-2-ylidene bearing two methyl groups on the aromatic ring enhanced the catalytic activity dramatically. 6 It is now commonly admitted that strongly electron-donating, bulky ligands favour oxidative addition on palladium and help stabilise the intermediate species in C-C coupling reactions such as for instance the Suzuki-Miyaura reaction. [START_REF]General reviews on Suzuki-Miyaura cross-coupling reaction: N. Miyaura in Metal-Catalyzed Cross-Coupling Reactions[END_REF][10][11] Furthermore, we have recently turned our attention to the asymmetric version of this reaction, obtaining encouraging results with the use of palladium complexes containing enantiomerically pure phosphane/imidazol-2-ylidene ligands (Figure 1). 7 We therefore turned our efforts towards the synthesis of palladium complexes containing novel chiral 1,2-disubstituted ferrocenyl phosphane/(benz)imidazol-2ylidene ligands in enantiomerically pure form and describe here their synthesis. Their catalytic properties in the asymmetric Suzuki-Miyaura reaction [11][12][13][14][15] have been evaluated and compared to those of the above mentioned phosphane/imidazol-2-ylidene complexes. 7 Finally, the activity of a phosphane/imidazol-2-ylidene ligand in the asymmetric 1,2-addition of phenylboronic acid to 4-nitrobenzaldehyde has been evaluated. [16][17][18][19] Scheme 1 Palladium(II) complexes described in our previous work. 7 50

Results and Discussion

Synthesis of (benz)imidazolium salts

With the intention of varying the stereoelectronic properties of the ferrocenyl phosphane-NHC ligands in a rational way, we decided to focus on the NHC moiety without changing the 55 diphenylphosphano group. The σ-donating properties of benzimidazol-2-ylidene ligands are situated between those of unsaturated imidazol-2-ylidenes and of saturated imidazolin-2ylidenes, which has been demonstrated in several examples of catalytic applications. 20,21 Apart from the electronic properties, 60 their structure should bring more bulkiness to the bifunctional ligands. In order to probe the effect of bulky substituents on both 00-00 activity and enantioselectivity, we have also changed the nature of the N-substituent.

2 | Journal Name, [year], [vol],
The synthesis of the phosphane/imidazolium salts with R = Me (3a) has already been described, both in racemic 5 and enantiomerically pure form. 7 The synthesis of other racemic and enantiomerically pure imidazolium and benzimidazolium salts was carried out following these established protocols. 5 Racemic alcohol rac-1 was used mainly to develop and optimise the synthesis of various salts and palladium complexes, but only enantiopure (S)-palladium complexes were used in catalytic reactions since we expected them to exhibit similar levels of activity compared to the previously described neutral palladium complex [PdCl2(Im Me )] (Scheme 1). 7 Imidazolium salt 2b was obtained in moderate yield from ferrocenyl alcohol 1 and N- [(2,4,6-trimethylphenyl)methyl] imidazole (Scheme 2). The synthesis of benzimidazolium salts 2c,d, however, proved more difficult. Benzimidazoles are poorer nucleophiles compared to imidazoles, thus giving greater amounts of side products and 20 lower yields of benzimidazolium salts. Their purification was also less straightforward: the excess of N-substituted benzimidazoles that are not consumed during the reaction cannot be removed with aqueous acidic washings due to their lower hydrophilicity and have a tendency to trail on silica when the same eluting conditions as for the imidazolium salts are applied. Eventually, their purification was successful by eluting the N-substituted benzimidazole excess with pure diethyl ether and then the benzimidazolium salts with a DCM/acetone mixture on a silicagel chromatography column. The 1 H NMR spectra of (benz)imidazolium salts 2b-d show the expected signal above δ 8.5, characteristic of the acidic C2 proton. AB systems were observed for the diastereotopic protons of the methylene bridge situated between the ferrocenyl and (benz)imidazolyl moieties for all salts, as well as for protons of the methylene bridge linking the (benz)imidazolyl and mesityl groups 35 (salts 2b and 2d), which is consistent with the introduction of planar chirality at ferrocene and with the generation of rather rigid structures due to steric crowding. The imidazolium and benzimidazolium salts exhibited different reactivity during the diphenylthiophosphano group deprotection by 55 Raney nickel. Indeed, whereas the desulfurisation of the thiophosphane/imidazolium salt 2b to give 3b went to completion at room temperature in 16 h, longer reaction times and moderate heating (40°C) were necessary to achieve complete desulfurization of the thiophosphane/benzimidazolium salts 2c,d to give 3c,d ( 31 P 60 NMR monitoring). This is a first confirmation that steric crowding is more important in the benzimidazolium family, with possible implications in catalysis. As expected, upon deprotection of the diphenylthiophosphano group, the 31 P NMR signal of (benz)imidazolium salts shifted from δ ca. 40.5 for 2 to ca. -27 for 3.

Synthesis of palladium(II) complexes

The synthesis and X-ray structural characterisation of complex 4a, obtained by deprotonation of 3a both in racemic and enantiopure forms, has been previously published. ‡,7 We have now extended this procedure to the synthesis of the analogous complexes with the NHC ligands obtained from 3b, 3c and 3d. All phosphane/(benz)imidazolium salts were deprotonated with tBuONa in the presence of the palladium precursor PdCl2(PhCN)2 to give neutral N-heterocyclic carbene/phosphane palladium complexes 4b-d in moderate yields. All products are air-stable and could be purified by column chromatography on silicagel. The absence of the acidic proton characteristic of (benz)imidazolium salts in the 1 H NMR spectra indicated carbene formation. The 31 P NMR spectra of all complexes showed a major species with a signal between δ 3.86 and 4.56, along with ca. 2% of a second species around δ 15. This minor species was obtained whatever the complex and mass spectrometry analysis of the mixture by electrospray revealed a small amount of palladium dimer along with the expected monomeric structure. The carbenic carbon in 13 C NMR was found at δ 156.0 for complex 4b bearing an imidazol-2ylidene moiety and went downfield for complexes 4c,d possessing the benzimidazol-2-ylidene group with, respectively, δ 166.70 and 166.74. The same trend has been observed for other neutral palladium(II) NHC complexes in the literature. 21,22 The structures of all new complexes were established by X-ray crystallography and compared with that of 4a. 7 The bond lengths and angles are all within the expected range for neutral palladium complexes with a slightly distorted square-planar geometry (Figures 2 and3, Table 1). Pd-C(NHC) (1.973(2) Å, 4b, to 1.992(6) Å, 4d) and Pd-P (2.2444(8) Å, 4a, to 2.2546(19) Å, 4d) bond lengths compare well with those in other previously published neutral palladium complexes bearing chelating NHC/phosphane ligands 21,22 As observed previously for complex 4a, [START_REF]General reviews on Suzuki-Miyaura cross-coupling reaction: N. Miyaura in Metal-Catalyzed Cross-Coupling Reactions[END_REF] the Pd-Cl bonds trans to the NHC are all slightly but significantly shorter than the Pd-Cl bonds trans to the diphenylphosphano groups, which is surprising since the more donating NHC should exert a greater trans influence than the 45 phosphane, but this was also observed elsewhere. 21,22 Similarly to the (benz)imidazolium salts, the C-C bond of the heteroaromatic unit is slightly longer in the case of benzimidazol- 2-ylidenes (1.372(5) Å for 4c and 1.391(8) Å for 4d) than in the case of imidazol- 2-ylidenes (1.350(6) Å for 4a and 1.345(4) Å for 4b), 50 which could denote some delocalisation of the double bond due to the presence of the fused benzene ring. No other significant differences in bond lengths are notable between the four structures. However, the C(NHC)-Pd-P bite angle is significantly narrower in complex 4a with 84.95( 9)° and increases with the steric bulkiness 55 of the ligand to reach 91.94 (19)° in complex 4d.

Asymmetric Suzuki-Miyaura reaction

Palladium(II) complexes 4b-d were evaluated in the asymmetric version of the Suzuki-Miyaura cross-coupling reaction between naphthyl bromides and naphthylboronic acid. We previously demonstrated that complex 4a is active for the coupling of 10 substituted naphthyl bromides with naphthylboronic acid at low catalyst loadings and gives, as expected, similar levels of enantioselectivities with the (R) and (S) enantiomers. 7 Therefore, the other complexes were tested directly under the same conditions with the enantiomerically pure ligands possessing the (S) configuration. As expected, all complexes are active for the coupling of various naphthyl bromides with a catalytic charge of 0.5 mol% at 40°C. All reactions were stopped after 24 h to maximise conversions and allow comparison of the results.

The first observation to be made is that steric hindrance in the 20 catalyst indeed plays a role since for each reaction the product yield decreases as the steric crowding on the complex increases. Whereas 4a gives excellent yields of the expected coupling product for most substrates, 4b gives the products in only moderate yields and the unexpectedly low yield for the ethoxy substrate cannot be rationally explained. Complexes 4c and 4d bearing the more bulky benzimidazol-2ylidene backbone also give moderate yields of the expected 40 coupling products. Although the yields were low with all complexes except for 4a, it seems that steric bulk could be slightly beneficial for the asymmetric induction in some cases, since slightly better e.e.s were obtained with 4b (up to 46%), bearing a CH2Mes substituent in place of a methyl group. Again, 45

unexpectedly low e.e.s were obtained with complex 4c, which cannot only be explained in terms of steric congestion since complex 4d, which possesses the bulkiest ligand, gives better e.e.s than complex 4c. Increasing the catalytic charge from 0.5 mol% to 2 mol% did not improve the level of enantioselectivity or the yield 50 of coupling product (entry 16). The naphthyl bromide bearing a phosphonate substituent at the 2-position has been widely used in asymmetric Suzuki-Miyaura reactions and usually gives better e.e.s than other substrates, as it seems that a chelating functionality on one coupling partner is beneficial to achieve good 55 enantioselectivities. 11,14 However, in our case none of the expected coupling product was obtained whatever the complex used, only a mixture of naphthyl bromide and the corresponding dehalogenated product. Hydrodehalogenation has already been observed in the conditions of the Suzuki-Miyaura coupling 23 and this could indicate that the oxidative addition takes place but the following step, i.e. the transmetallation of the boronic acid, does not occur. One of the reasons could be the bulkiness of our ligands, but chelating diphosphanes have also been reported not to give conclusive results with this substrate. 13 Therefore, the chelating nature of our ligands, which have two strongly coordinating groups, could be detrimental to the reaction in certain cases.

Asymmetric 1,2-addition of phenylboronic acid to 4nitrobenzaldehyde 10

The addition products such as diarylmethanols, are very important intermediates in the synthesis of biologically active compounds. 24 This reaction has been well studied with rhodium complexes, with the first example of enantioselective 1,2-addition of boronic acids to aldehydes described by Miyaura in 1998. 25 Diarylmethanols have thus been obtained with very good enantioselectivities (up to 94% ee), 26 but the use of cheaper palladium complexes in the asymmetric version of this reaction is much less common. [17][18][19] Indeed, only two recent examples show poor to moderate enantioselectivities, using a planar chiral ferrocenyl phosphine (up 20 to 11% ee) 18 or axially chiral NHC ligands (up to 65% ee). 19 We thus turned to the 1,2-addition of PhB(OH)2 to 4nitrobenzaldehyde, hoping to improve the previously measured enantioselectivities. As the synthesis of 4a is more straightforward than that of 4b-d, we therefore carried out preliminary tests with enantiopure (R)-4a only. The reaction was first conducted in THF using KOH as base and 5 mol% of palladium complex (entry 1).

Scheme 4 1,2-Addition of phenylboronic acid to 4-nitrobenzaldehyde. The selectivity with THF/KOH was not satisfactory as a nonnegligible amount of 4-nitrobenzyl alcohol from the Canizzarro reaction, as well as other minor by-products, were observed in the 1 H NMR spectrum of the crude mixture. Changing the solvent/base couple to dioxane/CsF was no better: although the 1 H NMR spectrum of the crude mixture showed almost complete consumption of the starting aldehyde, several unidentified byproducts were also observed (entries 2,3). Only the toluene/K2CO3 couple gave interesting results in terms of selectivity and yield (entries 4,5). Unfortunately, the levels of enantioselectivity measured for complex (R)-4a were poor, with up to 13% e.e. in THF (entry 1), and are comparable to those obtained by Suzuma et al. 18 However, this is a preliminary work and a more thorough 50 study of the reaction conditions with this and other chiral palladium complexes will be carried out in order to improve the efficiency of this challenging reaction.

Conclusions

We have described in this paper the synthesis of a new family of 55 chiral Pd complexes, containing enantiomerically pure phosphane/(benz)imidazol-2-ylidene ligands. A first application of these complexes to the asymmetric Suzuki-Miyaura reaction has revealed that the bulkiness of ferrocene conjugated with that of benzimidazol-2-ylidene seems detrimental to the catalytic activity.

60

The influence of the NHC structure on enantioselectivity is not entirely predictable and the best e.e. of 46% was obtained with 4b, only a 4% improvement compared to the previously investigated complex 4a, and to the detriment of the yield. Preliminary tests in the catalysed 1,2-addition of phenylboronic acid to 4-65 nitrobenzaldehyde with complex 4a showed moderate activities and low enantioselectivities. Future work will aim at designing other types of chiral functionalised N-heterocyclic carbenes in order to find the best compromise between activity and enantioselectivity.

70

Experimental

General considerations

All manipulations were performed under an inert atmosphere of dry argon by using vacuum line and Schlenk tube techniques. Solvents for syntheses were dried and degassed by standard methods before use. 1D-and 2D-NMR spectra were recorded on Bruker AV500 and Bruker AV300 spectrometers. 1 H and 13 C chemicals shifts (δ) are given in ppm (the residual peak of deuterated solvent was used as reference). 31 P chemicals shifts are 85 reported in ppm. Peaks are labeled as singlet (s), doublet (d), triplet (t), multiplet (m) and broad (br). The proton and carbon assignment were performed by COSY, NOESY, HSQC, 1 H- 13 C HMBC, 1 H-31 P HMBC experiments. MS spectra were performed by the mass spectrometry service of the Paul Sabatier University. Optical 90 rotations were measured in a 1 dm cell with a Perkin-Elmer 241 photopolarimeter. GC chromatograms were recorded on a HP 4890A GC equipped with a DB-5 MS UI capillary column and the products were identified by comparison with authentic samples. HPLC chromatograms were recorded on a Shimadzu LC-95 2010A(HT) equipped with a Chiralcel OJ column. Commercial chemicals were from Acros, Aldrich, Alfa Aesar, or Fluka and used as received. 1,2-Ferrocenyl alcohol 1, 27 imidazolium salts 2a,3a 5 and palladium complex 4a 7 were prepared according to previously reported procedures. Abbreviations: CpA: cyclopentadienyl ring 00-00 with PPh2 or PPh2(=S) substituents; CpB: unsubstituted cyclopentadienyl ring; Fc: ferrocenyl; Im: imidazolyl; BIm: benzimidazolyl; Mes: mesityl. C q : quaternary carbon.

6 | Journal Name, [year], [vol],

Syntheses FcPPh2(=S)CH2Im +CH2Mes BF4 -2b

To a solution of ferrocenyl alcohol 1 (100 mg, 0.23 mmol) in dry CH2Cl2 (10 mL) was quickly added HBF4 (54%wt in Et2O, 50 µL, 0.35 mmol), immediately followed by N-2,4,6 trimethylbenzylimidazole (110 mg, 0.55 mmol). After 5 minutes, the mixture was washed with 2M aq. HCl, water, sat. aq. NaHCO3 and water, dried (MgSO4), filtered and concentrated in vacuo. The residue was purified by precipitation with methyl tert-butyl ether from a CH2Cl2 solution to give a yellow solid (80 mg, 50% yield). [START_REF] Lee | [END_REF] H NMR (300 MHz, CDCl3, 298K) δ 8.61 (1H, s, NCHN), 7. 78-7.71 (2H, m, CH PPh2), 7. 60-7.47 (4H, m, CH PPh2), 7.40 (1H, m, CH PPh2), 7.30 (3H, m, CH PPh2) [α] CH2Cl2).

FcPPh2(=S)CH2BIm +CH3 BF4 -2c

To a solution of ferrocenyl alcohol 1 (418 mg, 0.97 mmol) in dry CH2Cl2 (20 mL) was quickly added HBF4 (54%wt in Et2O, 330 µL, 2.42 mmol), immediately followed by N-methyl 40 benzimidazole (466 mg, 2.91 mmol). After 5 minutes, the mixture was washed with 2M aq. HCl, water, sat. aq. NaHCO3 and water, dried (MgSO4), filtered and concentrated in vacuo. The residue was purified by precipitation with a mixture of petroleum ether and methyl tert-butyl ether (80/20) from a CH2Cl2 solution to give an orange solid (193 mg, 27% yield). X-ray quality crystals of rac-2c were obtained by slow diffusion of diethyl-ether from a dichloromethane solution. [START_REF] Lee | [END_REF] H NMR (300 MHz, CDCl3, 298K) δ 8.79 (1H, s, NCHN), 7. 73-7.69 (3H, m, CH PPh2+BIm), 7.55-7.47 (4H, m, CH PPh2), 7.21-7.15 (3H, m, CH PPh2), 6.95-6.93 (3H, m, 50 CH PPh2+BIm), 6.57 (1H, AB syst., JHH=14 Hz, Fc-CH2), 5.23 (1H, br. s, CpA), 5.21 (1H, AB syst., JHH=14 Hz, Fc-CH2), 4.58 (1H, br. s, CpA), 4.48 (5H, s, CpB), 3.78 (1H, br. s, CpA), 3.64 (3H, s, NCH3), 2.37 (3H, s, CH3 BIm), 2.16 (3H, s, CH3 BIm). 13 C{ 1 H} NMR (75.5 MHz, CDCl3, 298K) δ 140.47 (NCHN), 55 136.95+136.98 (C q BIm), 133.08 (d, JCP=80 Hz, C q PPh2), 131.99 (d, JCP=86 Hz, C q PPh2), 131.85 (d, JCP=2 Hz, CH PPh2), 131.77 (d, JCP=12 Hz, CH PPh2), 131.25 (d, JCP=11 Hz, CH PPh2), 130.94 (d, JCP=2 Hz, CH PPh2), 129.78+129.24 (C q BIm), 128.34 (d, JCP=13 Hz, CH PPh2), 127.61 (d, JCP=12 Hz, CH PPh2) CH2Cl2).

FcPPh2(=S)CH2BIm +CH2Mes BF4 -2d

To a solution of ferrocenyl alcohol 1 (474 mg, 1.09 mmol) in dry CH2Cl2 (20 mL) was quickly added HBF4 (54%wt in Et2O, 380 70 µL(2.75 mmol, 2.5 equiv), immediately followed by N-2,4,6 trimethylbenzyl benzimidazole (800 mg, 2.88 mmol, 2.7 equiv). After 5 minutes, the mixture was washed with 2M aq. HCl, water, sat. aq. NaHCO3 and water, dried (MgSO4), filtered and concentrated in vacuo. The residue was purified by precipitation 75 with a mixture of petroleum ether and methyl tert-butyl ether (80/20) from a CH2Cl2 solution, followed by column chromatography on silica gel (eluents: diethyl ether, then CH2Cl2, then CH2Cl2/Acetone 90/10) to give an orange solid (481 mg, 57% yield). X-ray quality crystals of rac-2d were obtained by slow 80 diffusion of diethyl ether into a dichloromethane solution. [START_REF] Lee | [END_REF] H NMR (500 MHz, CDCl3, 301K) δ 8.68 (1H, s, NCHN), 7. 72-7.68 (3H, m, CH PPh2+BIm), 7.58-7.55 (1H, m, CH PPh2), 7.51-7.47 (2H, m, CH PPh2), 7.22-7.18 (1H, m, CH PPh2), 7.07-7.00 (3H, m, CH PPh2+BIm), 6.94 (2H, s, CH Mes), 6. 90-6.85 (2H, m, CH PPh2) General procedure for the synthesis of (benz)imidazolium salts FcPPh2CH2Im/BIm +R BF4 -3b-d: Raney nickel (50 to 100 equivalents) was washed with degassed methanol (3 X 10 mL), degassed diethyl ether (3 X 10 mL) and, finally degassed acetonitrile (3 X 10 mL). A solution of (benz)imidazolium salt 2b-110 d in acetonitrile (20 mL) was added and the mixture was heated at 40°C for 20h. After cooling to room temperature, the supernatant was filtered through a short path of Celite ® and the Raney nickel was rinsed with acetonitrile. The filtrate was concentrated and the solid residue was dried in vacuo.

FcPPh2CH2Im +CH2Mes BF4 -3b: yellow solid (220 mg, 88% yield). [START_REF] Lee | [END_REF] H NMR (300 MHz, CDCl3, 298 K) δ 8.73 (1H, s, NCHN), (S)-3c: [α] CH2Cl2). FcPPh2CH2BIm +CH2Mes BF4 -3d: yellow solid (500 mg, 82% yield). [START_REF] Lee | [END_REF] H NMR (500 MHz, CDCl3, 301 K) δ 8. 90 (1H, s, NCHN), 7.50-7.47 (3H, m, CH BIm+PPh2), 7.44-7.38 (3H, m, CH PPh2), 6.98 (1H, t, JHH=7.5 Hz, CH PPh2), 6.94 (2H, s, CH Mes), 6.81 CH3 Mes), 20.92 (CH3 BIm), 20.37 (p-CH3 Mes), 20.26 (CH3 BIm). 31 [20][21][22][23][24][25][26][27][28][29][30][31][32]CH2Cl2).

General procedure for Suzuki-Miyaura coupling reactions: in a typical run, a mixture of naphthyl halide (0.29 mmol, 1 equiv), naphthylboronic acid (0.34 mmol, 1.2 equiv), potassium carbonate (0.69 mmol, 2.4 equiv) and catalyst (0.5 mol%) in 2 mL of toluene (+ 50 μL of CH2Cl2 for complete solubilisation) was stirred for 24h at 40°C. The solution was allowed to cool to room temperature, washed with water and the aqueous phase was extracted with diethyl ether. The organic phase was dried over MgSO4, filtered and concentrated in vacuo. The residue was purified by flash chromatography on silica gel (hexane or hexane/ethyl acetate 9/1).

  40

Fig. 1

 1 Fig. 1 ORTEP representations of benzimidazolium salts rac-2c (top) and rac-2d (bottom). Ellipsoids are shown at the 50% level. All hydrogens are omitted for clarity. Selected bonds (Å), rac-2c: C(23)-C(24) = 1.388(8); rac-2d: C(23)-C(24) = 1.390(3).The structure of the intermediates 2c and 2d (Figure1) was 45

Scheme 3

 3 Scheme 3 Synthesis of neutral NHC-phosphane palladium(II) complexes.

Fig. 2

 2 Fig. 2 ORTEP representation of complex (S)-4b. Ellipsoids are shown at the 50% level. All hydrogens are omitted for clarity.

Fig. 3

 3 Fig.3 ORTEP representations of complexes (S)-4c (top) and rac-4d (bottom). Ellipsoids are shown at the 50% level. All hydrogens are omitted for clarity.

Scheme 4

 4 Scheme 4 Asymmetric Suzuki-Miyaura reaction between 2-substituted αnaphthyl bromides and α-naphthylboronic acid.

a

  Reagents and conditions: naphthyl bromide (1.0 equiv), boronic acid (1.2 equiv), Pd cat. (0.5 mol%), K 2 CO 3 (2.4 equiv), toluene, 24h. b Determined by integration of characteristic 1 H NMR signals in the crude mixture. c Isolated yield. d Determined by HPLC with a Chiracel-OJ 35 column. e The commercial 1-bromo-2-methylnaphthalene contains ca. 6% of 2-methylnaphthalene. f reference 7. g 2 mol% of Pd catalyst were used.

a

  Reagents and conditions: 4-Nitroenzaldehyde (1 equiv), phenylboronic acid (2 equiv), KOH (3 equiv), THF (5 mL), 24h. b Reagents and conditions: 4-Nitroenzaldehyde (1 equiv), phenylboronic acid (1.5 equiv), 35 base (2 equiv), solvent (1 mL). c Isolated yield. d Determined by HPLC with a Chiracel-AD-H column and compared with published results.18,19

90 CH3C

 90 Mes), 139.51 (NCHN), 137.94 (C q o-CH3C Mes), 137.23+137.20 (C q CH3C BIm), 134.01 (d, JCP=86 Hz, C q PPh2), 132.00(d, JCP=11 Hz, CH PPh2), 131.99 (d, JCP=87 Hz, C q PPh2), 131.92(d, JCP=3 Hz, CH PPh2), 131.20 (d, JCP=3 Hz, CH PPh2), 130.83 (d, JCP=11 Hz, CH PPh2), 130.07 (CH Mes), 95 129.72+129.60 (C q BIm), 128.66 (d, JCP=13 Hz, CH PPh2), 127.75 (d, JCP=12 Hz, CH PPh2), 124.95 (C q NCH2C Mes), 114.79+112.33 (CH BIm), 83.05 (d, JCP=12 Hz, C q CpA), 77.28 (CH CpA, seen by HSQC{31P}), 76.12 (d, JCP=12 Hz, CH CpA), 73.91 (d, JCP=94 Hz, C q CpA), 71.59 (CH CpB), 71.17 (d, JCP=10 100 Hz, CH CpA), 46.72 (CH2-Fc), 45.77 (CH2-Mes), 21.10 (p-CH3 Mes), 20.73+20.69 (CH3 BIm), 20.03 (o-CH3 Mes). 31 P{ 1 H} NMR (202.5 MHz, CDCl3, 301K) δ 40.88. MS (ESI) m/z 693 [M + ] (100). HRMS (ES + ) calcd for C42H42N2PSFe 693.2156; found 693.2164. (S)-2d: [α] CH2Cl2).

  105

  P{ 1 H} NMR (202.5 MHz, CDCl3, 298 K) δ 3.86. MS (DCI-CH4) m/z 801 [M -Cl] (100), 767 [M -2Cl] (58). HRMS (Cl+) calcd for C42H41N2PFePdCl 801.1080; found 801.1118. (S)-4d: [α]

Table 1

 1 Selected

	Distance (Å)	4a	4b	4c	4d
	Pd-C(NHC)	1.991(3)	1.973(2)	1.975(4)	1.992(6)
	Pd-P

distances (Å) and angles (°) for complexes 4b-d and 60 comparison with those of 4a.

7

2.2444(8) 2.2449(5) 2.2475(9) 2.2546(19) Pd-Cl(trans-NHC) 2.3397(8) 2.3414(6) 2.3043(12) 2.3395(16) Pd-Cl(trans-P) 2.3614(9) 2.3798(5) 2.3347(10) 2.3725(18) C=C (NHC) 1.350(6) 1.345(4) 1.372(5) 1.391(8) Angle (°) N-C(NHC)-N 106.4(3) 106.1(2) 106.4(3) 107.3(5) C(NHC)-Pd-P 84.95(9) 88.90(6) 89.94(10) 91.94(19) N-C(CH 2 )-C(Cp) 112.2(3) 109.22(19) 111.1(3) 110.7(5) N-C(CH 2 )-C(Mes) -115.3(2) -114.7(5)

Table 2

 2 Catalytic performances of complexes 4a-d in the reaction of

	30
	Scheme 4.
	Entry	R	Pd cat.	Conv (%) b	Yield (%) c	e.e. (%) d
	1	Me e	( S)-4a	n.d.	88	40 (S) f
	2		(R)-4a	95	88	42 (R) f
	3		(S)-4b	63	58	46 (S)
	4		(S)-4c	60	48	9 (S)
	5		(S)-4d	62	47	37 (S)
	6	OMe	(S)-4a	n.d.	95	35 (R) f
	7		(R)-4a	n.d.	93	33 (S) f
	8		(S)-4b	88	65	31 (R)
	9		(S)-4c	67	54	17 (S)
	10		(S)-4d	67	57	22 (R)
	11	OEt	(S)-4a	n.d.	95	23 (R) f
	12		(R)-4a	n.d.	92	24 (S) f
	13		(S)-4b	22	22	26 (R)
	14		(S)-4c	80	65	13 (S)
	15		(S)-4d	79	62	23 (R)
	16		(S)-4d	75	54	28 (R) g

Table 3

 3 Catalytic

	Entry Yield (%) a t (h) Solvent Base 24 THF KOH 22	13 (R)
	2 b	24	dioxane	CsF	22	3 (R)
	3 b	4	dioxane	CsF	32	7 (R)
	4 b 5 b	24 4	toluene toluene	K 2 CO 3 K 2 CO 3	69 45	12 (R) 11 (R)

performances of complex (R)-4a in the reaction of Scheme 5. c E.e. (%) d 1

  Mes).13 C{ 1 H} NMR (75.5 MHz, CDCl3, 298K) δ 139.88 (C q Mes, C-CH2), 137.93 (2 C q Mes, C-o-CH3), 134.91 (NCHN), 134.66 (d, JCP=86 Hz, C q PPh2 ), 132.08 (d, JCP=56 Hz, C q PPh2), 131.90(d, JCP=11 Hz, CH PPh2), 131.70 (d, JCP=3 Hz, CH PPh2),

		, 7.15 (1H, s, CH Mes), 6.88 (2H,
		s, NCH=CH), 6.51 (1H, s, CH Mes), 6.40 (1H, AB syst., JHH=14
		Hz, CH2-Fc), 5.22 (1H, AB syst., JHH=14 Hz, CH2-Fc), 5.21 (1H,
		br. s, CpA), 5.14 (1H, AB syst., JHH=14 Hz, CH2-Mes), 4.91 (1H,
	20	AB syst., JHH=14 Hz, CH2-Mes), 4.53 (1H, br. s, CpA), 4.35 (5H,
		s, CpB), 3.86 (1H, br s., CpA), 2.28 (3H, s, p-CH3 Mes), 2.13 (6H,
		s, o-CH3 JCP=9 Hz, CH CpA), 71.23 (5 CH CpB), 47.91 (CH2-Fc or CH2-
		Mes), 47.40 (CH2-Fc or CH2-Mes), 21.04 (p-CH3 Mes), 19.61 (o-
		CH3 Mes). 31 P{ 1 H} NMR (121 MHz, CDCl3, 298K) δ 40.65. MS
		(ESI) m/z 615 [M + ] (100), 415 [M + -C13H16N2] (8). HRMS (ES + )
	35	calcd for C36H36N2PSFe 615.1687; found 615.1688. (S)-2b:

  , 60 114.22+111.58 (CH BIm), 82.37 (d, JCP=13 Hz, C q CpA), 77.24 (CH CpA), 76.38 (d, JCP=10 Hz, CH CpA), 75.27 (d, JCP=91 Hz, C q CpA), 71.39 (5 CH CpB), 71.18 (d, JCP=10 Hz, CH CpA), 45.60 (CH2-Fc), 33.37 (NCH3), 20.64+20.52 (CH3 BIm). 31 P{ 1 H} NMR (121.5 MHz, CDCl3, 298K) δ 40.34. MS (ESI) m/z 575 [M + ] (100),

	65
	415 [M + -C10H12N2] (18). HRMS (ES + ) calcd for C33H32N2PSFe
	575.1374; found 575.1367. (S)-2c: [α]

  , 856.48 (1H, AB syst., Fc-CH2), 5.51 (1H, AB syst., Fc-CH2), 5.40 (1H, br, CpA), 5.32 (1H, AB syst., CH2-Mes), 4.98 (1H, AB syst., CH2-Mes), 4.57 (1H, br s, CpA), 4.42 (5H, s, CpB), 3.81 (1H, s, CpA), 2.34-2.33 (9H, m, CH3 BIm+p-CH3Mes), 2.17 (6H, s, o-CH3 Mes).13 C NMR{ 1 H} (125.8 MHz, CDCl3, 301K) δ 139.97 (C q p-

Journal Name, [year],[vol],00-00 | 5

financial support of this work.

Acknowledgements

We thank the CNRS, the Ministère de l'Enseignement Supérieur et de la Recherche (doctoral grant to P.L.), the Institut Universitaire de France (grant to S.G.) and the Agence Nationale de la Recherche (ANR-07-JCJC-0041, postdoctoral grant to N.D.) for

Enantiomeric excesses were determined by HPLC with a Chiralcel OJ column (eluent: Hexane/iPrOH 9/1, rate: 1 mL/min, T=30°C). Retention time for 2-methylbinaphthyl: 6.42 min (R isomer) -9.93 min (S isomer). Retention time for 2-methoxybinaphthyl: 9.22 min (S isomer) -15.66 min (R isomer). Retention time for 2-50 ethoxybinaphthyl: 7.66 min (S isomer) -11.90 min (R isomer). General procedure for 1,2-addition reactions: A mixture of 4nitrobenzaldehyde (0.2 mmol, 1 equiv), phenylboronic acid (0.4 mmol, 2 equiv (THF) or 0.3 mmol, 1.5 equiv (other solvents)), base (0.6 mmol, 3 equiv (THF) or 0.4 mmol 2 equiv (other solvents)) 55 and catalyst (5 mol%) in a solvent (THF: 5 mL, other solvents: 1 mL) was stirred for 24h at 65°C. After cooling to room temperature, water was added and the aqueous phase was extracted with tBuOMe. The organic phases were washed with brine, dried (MgSO4), filtered and concentrated in vacuo. After analysis by 1 H 60 NMR (d6-acetone), the residue was purified by flash chromatography on silica gel (petroleum ether/EtOAc 8/1 then 4/1) to give pure (4-nitrophenyl)(phenyl)methanol as a yellow solid. It was analysed by chiral HPLC using a Chiracel AD-H column, (hexane/2-propanol = 80:20, flow rate = 0.5 mL/min, wavelength 65 = 214 nm, 13.9 min (R isomer) and 16.4 min (S isomer). X-ray structural analyses. Single crystal of each compound was mounted under inert perfluoropolyether at the tip of glass fiber and cooled in the cryostream of either an Oxford-Diffraction XCALIBUR CCD 70 diffractometer for rac-2c and (S)-4c, a Bruker APEXII diffractometer for rac-2d and (S)-4b or an Agilent Technologies GEMINI EOS diffractometer for rac-4d. The structures were solved by direct methods (SIR97 28 ) and refined by least-squares procedures on F 2 using SHELXL-97. 29 All 75 H atoms attached to carbon were introduced in calculation in idealised positions and treated as riding models. In four of the five compounds, rac-2d, (S)-4b, (S)-4c and rac-4d, it was difficult to locate some disordered solvents, and the SQUEEZE option from PLATON 30 has been used. The absolute configuration for 80 compounds (S)-4b and (S)-4c has been determined by refining the Flack's parameter. 31 The drawing of the molecules was realised with the help of ORTEP32. 32 Crystallographic data (excluding structure factors) have been deposited with the Cambridge Crystallographic Data Centre as 85 supplementary publication no. CCDC 940117-940121. Copies of the data can be obtained free of charge on application to the Director, CCDC, 12 Union Road, Cambridge CB2 1EZ, UK (fax: (+44) 1223-336-033; e-mail: deposit@ccdc.cam.ac.uk).

Notes and references