Jawhar Ghommam
email: jawhar.ghommam@gmail.com

Ahmed Chemori

Adaptive RBFNN Finite-Time Control of Normal Forms for Underactuated Mechanical Systems

Keywords: Adding Power Integrator, Underactuated Mechanical Systems, Model Uncertainties, RBFNN, Lyapunov Method

This paper presents a constructive design of a continuous finite time controller for a class of mechanical systems known as underactuated systems that satisfy the symmetry properties . An adaptive radial basis function neural networks (RBFNN) finite time control scheme is proposed to stabilize the underactuated system at a given equilibrium, regardless of the various uncertainties and disturbances that the system contains. First, a coordinate transformation is introduced to decouple the control input so that an n-th order underactuated system can be represented into a special cascade form. Next an adaptive robust finite-time controller is derived from the adding a power integrator (API) technique and the RBFNN to approximate the nonlinear unknown dynamics in the new space, whose bounds are supposedly unknown. The stability and finite-time convergence of the closed-loop system are established by using Lyapunov theory. To show the effectiveness of the proposed method, simulations are carried out on the rotary inverted pendulum, a typical example of an underactuated mechanical system .

Introduction

Underactuated systems [START_REF] Choukchou-Braham | Analysis and Control of Underacuated Mechanical Systems[END_REF][START_REF] Fantoni | Non-linear control for underactuated mechanical systems[END_REF][START_REF] Jadlovska | A complex overview of modeling and control of the rotary single inverted pendulum system[END_REF][START_REF] Jadlovska | Modeling and optimal control of underactuated mechanical systems a survey[END_REF][START_REF] De Luca | Stabilization of underactuated planar 2r manipulator[END_REF][START_REF] Spong | The swing up control problem for the acrobot[END_REF] are mechanical systems with less control inputs than degrees of freedom, consequently they have at least one unactuated degree of freedom. The control synthesis of such mechanical systems represents a challenging control problem and can be considered much more difficult than the synthesis of fully-actuated systems in [START_REF] Bortoff | Pseudolinearization of the acrobot using spline functions[END_REF][START_REF] Iba | Controlling the inverted pendulum by means of a nested saturation function[END_REF][START_REF] She | Global stabilization of 2-dof underactuated mechanical systems-an equivalent-input-disturbance approach[END_REF][START_REF] Spong | Energy based control of a class of underactuated mechanical system[END_REF]. In this context, our main focus of this paper will be on a class of underactuated systems enjoying symmetry properties [START_REF] Olfati-Saber | Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles[END_REF], which, based on an appropriate global transformation, will be reduced to normal form, so that it can easily controlled using standard techniques like reminiscent backstepping technique.

In practice, most underactuated mechanical systems are uncertain and multi-variable in character. It is important then to investigate effective robust and adaptive control techniques for such kind of systems . The authors in [START_REF] Carlos | Stabilization of the cart pole system: by sliding mode control[END_REF] proposed a robust controller based on a combination of a PD controller and a twisting-like algorithm to stabilize the damped cart pole system. In this approach however, uncertain model parameters have not been considered. To deal with model uncertainties, a time-scale approach along with the Lyapunov design have been proposed in [START_REF] Ravichandran | Robust stabilization of a class of underactuated mechanical systems using time scaling and lyapunov redesign[END_REF]. The authors in [START_REF] Park | Swing-up and stabilization control of inverted pendulum systems via coupled sliding-mode control method[END_REF] used a sliding mode technique for the cart pole system to stabilize the system in presence of disturbances. Model uncertainties however have not been tackled in this work. In [START_REF] Adhikary | Integral backstepping sliding mode control for underactuated systems: Swingup and stabilization of the cart-pendulum system[END_REF][START_REF] Chen | Adaptive control of rotary inverted pendulum system with time-varying uncertainties[END_REF], disturbances and model uncertainties have been fully considered in a technique that involves a backstepping procedure combined with sliding mode, applied to the inverted pendulum system after the system was converted into a normal form. Integral sliding mode control was also applied in [START_REF] Ming | Adaptive slidingmode control for two-wheeled inverted pendulum vehicle based on zero-dynamics theory[END_REF][START_REF] Xu | Design and implementation of integral sliding mode control on an underactuated two-wheeled mobile robot[END_REF] to deal with uncertainties in the two-wheeled mobile inverted pendulum. The control law being designed is based on the linearized system dynamics which resulted in a small region of attraction for the equilibrium. The problem becomes more challenging whenever non-parametric uncertainties and unknown time-varying disturbances are considered in the design of effective and robust control law for the underactuated system.

An important research issue which has attracted the attention of the control community is the finite-time convergence and finite time stabilization [START_REF] Amato | Finite-time control of linear systems subject to parametric uncertainties and disturbances[END_REF][START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF][START_REF] Haimo | Finite-time controllers[END_REF][START_REF] Hong | Finite-time stabilization and stabilizability of a class of controllable systems[END_REF][START_REF] Hong | Stabilization of uncertain chained form systems within finite settling time[END_REF][START_REF] Huang | Global finite-time stabilization of a class of uncertain nonlinear systems[END_REF], that not only ensures faster convergence rate, but also guarantees better disturbance rejection and greater robustness to uncertainty. Rigorous analysis for finitetime stability has been established in [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF][START_REF] Hong | On an output feedback finite-time stabilization problem[END_REF][START_REF] Moulay | Finite time stability conditions for non autonomous continuous systems[END_REF]. Several interesting design schemes for finite stabilization of certain class of nonlinear systems using state-feedback and output-feedback can be found in [START_REF] Li | Global finite-time stabilisation by output feedback for a class of uncertain nonlinear systems[END_REF][START_REF] Qian | Global finite-time stabilization by output feedback for planar systems without observable linearization[END_REF], just to quote a few, and references therein. Although interesting broad results have been obtained by using finite-time stabilizing controllers in the literature, they may not be effective in dealing simultaneously with unknowns and time-variations in the systems. In the aforementioned works these unknowns are either constants or timevarying only, that is, not state-dependent. Different attempts using adaptive design schemes have been conducted in [START_REF] Hong | Finite-time stabilization of nonlinear systems with parametric and dynamic uncertainties[END_REF][START_REF] Hong | Adaptive finitetime control of nonlinear systems with parametric uncertainty[END_REF] to resolve the problem encountered in the presence of unknowns and time-variation in the system. However, the adaptive scheme has been applied to smooth nonlinear systems of the p-normal form [START_REF] Cheng | On normal form of nonlinear systems[END_REF], However, to the authors' best knowledge, there is no result on adaptive finite time control scheme for underactuated system with symmetry [START_REF] Olfati-Saber | Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles[END_REF] embodying time varying uncertainties with unknown bounds. Therefore, it is of interest to develop a concise method that is capable of globally finite-time stabilizing the underactuated mechanical system of class I (i.e., enjoying the symmetry properties of mechanical systems) with more involved unknowns and time-variations.

In this paper, an adaptive finite-time control strategy is investigated for a class of underactuated mechanical systems with tree structures by using radial basis neural networks approximation functions. The purpose of this paper is two-fold: to generalise the Olfati-Saber's transformation [START_REF] Olfati-Saber | Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles[END_REF] to n-th order underactuated systems into a special upper-triangular form with mismatched time-varying uncertainties, and to use the adding a power integrator (API) technique in combination of the radial basis neural networks approximation functions as a basic design tool for finite stabilization of the underactuated mechanical system.

The main contribution of this paper is twofold:

1. A finite time controller is constructed based on the adding a power integrator technique [START_REF] Lin | Adding one power integrator: A tool for global stabilization of high order lower-triangular systems[END_REF][START_REF] Lin | Adaptive regulation of highorder lower-triangular systems: Adding a power integrator technique[END_REF] and backstepping method. The key feature of this technique is that it only requires knowledge of the upper bounds of nonlinearities involved in the system dynamics, therefore simple recursive domination design can be obtained for an explicit construction of a smooth state feedback controller. The explicit form of the finite time controller appears to be more concise compared with the finite time controller proposed in [START_REF] Chen | Adaptive control of rotary inverted pendulum system with time-varying uncertainties[END_REF][START_REF] Huang | Adaptive Control of Underactuated Mechanical Systems[END_REF][START_REF] Lee | Finite time control of nonlinear underactuated systems using terminal sliding surface[END_REF][START_REF] Yang | Robust finite time controller design for second order underactuated mechanical systems[END_REF] for underactuated mechanical systems. Furthermore, by virtue of the adding a power integrator, finite time stabilization of the underactuated system can be obtained as opposed to [START_REF] Chen | Adaptive control of rotary inverted pendulum system with time-varying uncertainties[END_REF][START_REF] Huang | Adaptive Control of Underactuated Mechanical Systems[END_REF] where only global uniform ultimate boundedness is achieved. Moreover the inherent chattering problem due to discontinuity of the sliding mode approaches is straightforwardly circumvented.

2. The proposed scheme does not need any prior information about the bound of mismatched uncertainties. The tuning rule incorporated to estimate the mismatched uncertainty using the RBFNNN borrows much from the adaptive tracking [START_REF] Krstic | Nonlinear and Adaptive Control Design[END_REF] approach including the notion of tuning functions. The proposed scheme can guarantee that the state of the closed loop system converges to the origin in finitetime and the parameter estimations are bounded.

The reminder of this paper is organized as follows: Section 2 discusses the control problem formulation of a class of of mechanical underactuated systems along with the coordinate transformation that transforms this model into an upper-triangular form representation. Section 3 propose a finite-time stabilization control strategy for the n-th order underactuated system. In Section 4, simulation results with an application to the rotary inverted pendulum system are presented and discussed. Finally, Section 5 draws a conclusion for this paper.

2 Problem formulation and preliminaries

Problem formulation

Consider an underactuated mechanical system described by [START_REF] Reyhanoglu | Dynamics and control of a class of underactuated mechanical systems[END_REF] M 1 (q)q u + M 2 (q)q a + N 1 (q, q) = 0

(1)

M ⊤ 2 (q)q u + M 3 (q)q a + N 2 (q, q) = B(q)u (2)
Where the first equation (1) represents the unactuated dynamics while the second equation denotes the actuated dynamics, with q = [q ⊤ u , q ⊤ a ] ⊤ is the generalized coordinates such that q a ∈ R n is the unactuated degrees of freedom, and q u ∈ R m represents the actuated degrees of freedom. The inertia matrix of the underactuated system is defined by

M = M 1 (q) M 2 (q) M ⊤ 2 (q) M 3 (q) (3)
which is positive symmetric and positive definite for all q ∈ R n+m . B(q) is an invertible matrix and u is the generalized control forces produced by the m actuators. N i (q), i = 1, 2 are the Coriolis, centrifugal and gravitational force vectors. Control of underactuated system such as the system presented in (1)-( 2), most often resorts to the wellknown collocated partial feedback linearization proposed by [START_REF] Spong | Underactuated mechanical systems[END_REF], where the objective is to transform the dynamics of the underactuated system into a reduced-order system with cascaded nonlinear system in strict feedback form with zero-dynamics. However, with this approach there is an inherent requirement that the model parameters of the mechanical system must be perfectly known to be able to design controller for the transformed normal form.

To address this observation, the control design in this paper avoids the collocated feedback linearization entirely. Before deriving the control scheme, the equations of motion of the underactuated system are converted to a reduced form through simple manipulation of ( 1)-(2) as in [START_REF] Spong | Underactuated mechanical systems[END_REF] qa = J 1 (q)u + R 1 (q, q) (4) qu = J 2 (q)u + R 2 (q, q)

where

R 1 (q, q) = (M 3 -M ⊤ 2 M -1 1 M 2 ) -1 [M ⊤ 2 M -1 1 N 1 (q, q) -N 2 (q, q)] J 1 (q) = (M 3 -M ⊤ 2 M -1 1 M 2 ) -1 B(q) R 2 (q, q) = N 1 -M -1 1 M 2 R 1 (q, q) J 2 (q) = -M -1 1 M 2 J 1 (q)
The dynamic equations ( 4)-( 5) define the coupled dynamics of the underactuated system (1)- [START_REF] Amato | Finite-time control of linear systems subject to parametric uncertainties and disturbances[END_REF], where it can be seen that the control input appears both in the actuated subsystem and the unactuated subsystem of the original system, which significantly increases complexity of the control design for underactuated systems. To resolve this issue, the author in [START_REF] Olfati-Saber | Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles[END_REF], provided a suitable global change of coordinates that decouples the actuated from the unactuated dynamics with respect to the control input and allows to transform the system into a cascaded normal form.

Without loss of generality, an n-th order underactuated system can be described with a chain structure as follows ẍi = φ i (x 1 , . . . , x i+1 )u + γ i (x 1 , . . . , x i+1 , ẋ1 , . . . , ẋi ),

ẍn = φ n (x)u + γ n (x, ẋ) i = 1, . . . , n -1 (6)
where x = [x 1 , . . . , x n ] ⊤ ∈ R n are the state variables of the underactuated system and u ∈ R is the control input, γ i (x, ẋ) = 0 , φ i (x) are nonlinear functions for all i = 1, . . . n. We will use the technique presented in [START_REF] Olfati-Saber | Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles[END_REF] to transform the system (6) to a system of uppertriangular form. As such, we introduce the following coordinate changes

z i = x i - xi+2 0 φ i+1 (s) φ i+3 (s) ds i = 1, 3, 5 . . . , n -3 z i = x i - φ i+1 (x) φ i+2 (x) i = 2, 4, 6, . . . , n -2 (7) z n-1 = x n-1 z n = x n
Remark 1 Note that the integral term in [START_REF] Cheng | On normal form of nonlinear systems[END_REF] has no real physical meaning except that the mathematical expression will help to globally transform the system (6) into normal form, where the actuated dynamics are decoupled from the unactuated ones. More about this transformation can be found in details in [START_REF] Olfati-Saber | Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles[END_REF].

Hence, the system dynamics in a new Z-space coordinate can be obtained. Taking the time derivative of ( 7), we have

żi = z i+1 + d i (z), 1 ≤ i ≤ n -1 żn = φ n (z)u + d n (z) (8)
y = h(z)
where d i (z) are defined as follows:

d 1 (z) = φ 2 (z) φ 4 (z) z 4 - d dt z3 0 φ 3 (s) φ 4 (s) ds d 2 (z) = -z 3 + γ 3 (z) - φ 3 (z) φ 4 (z) γ 4 - d dt φ 3 (z) φ 4 (z) z 4 . . . ( 9
)
d n-3 = φ n-2 (z) φ n (z) z n - d dt zn-1 0 φ n-2 (s) φ n (s) ds d n-2 = -z n-1 + γ n-2 (z) - φ n-2 (z) φ n (z) γ n - d dt φ n-2 (z) φ n (z) z n d n-1 = 0 d n = φ n (z)
To control the normal form (8), we will consider the following assumptions :

Assumption 1 All modeling terms in (1)-( 2), including M 1 , M 2 , M 3 , N 1 and N 2 are considered unknown functions of time.

Assumption 2 The uncertain control coefficient φ n (z) is bounded as φ min ≤ φ n (z) ≤ φ max . Furthermore, it is convenient to assume that φ n (z) = 0.
Remark 2 Note that the terms d i (z) ∈ R, i = 1, 2, . . . , n in the transformed system (8) are regarded as matched and unmatched uncertainties. Also by construction, since d i (z), i = 1, . . . n are functions of the states it will not be practical to assume that their variation bounds are known. In addition, it cannot be assumed to be linearly parameterizable into a multiplication of a known regressor and a vector of unknown constant parameters, this is mainly due to their complex expressions in [START_REF] Fantoni | Non-linear control for underactuated mechanical systems[END_REF].

Remark 3 In many robotic systems such as those classified in [START_REF] Olfati-Saber | Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles[END_REF] as of Class-I underactuated systems, the sign of the control coefficient φ n (.) is independent of the change of coordinates and is usually known and is nonzero.

In this paper, we consider a class of underactuated systems of the form [START_REF] Chen | Adaptive control of rotary inverted pendulum system with time-varying uncertainties[END_REF], which can be transformed without collocated feedback linearization and through the global change of coordinates (8) into the cascaded form [START_REF] Choukchou-Braham | Analysis and Control of Underacuated Mechanical Systems[END_REF]. Here, denote z = [z 1 , z 2 , . . . , z n ] ⊤ ∈ Υ , where Υ is a compact set of R n , u ∈ R is the control input and d i (z) ∈ R, i = 1, . . . , n -1 are unmatched uncertainties while d n is the matched uncertainty. Under Assumptions 2.1 and 2.2, the idea is to design the controller u such that the system state z tracks in finite time the desired trajectory

z * d = [z * 1d , z * 2d , . . . , z * nd ] ⊤ ∈ Υ d where Υ d is a compact set of Υ .
This means that for any finite initial condition z i (0), i = 1, . . . n, there exists a finite time T , called the settling time of system ( 8) such that lim

t→T e i = lim t→T z i -z * id = 0, i = 1, . . . , n (10)

Finite time stability

In what follows, we review some basic concepts of finitetime stability, its definition and some useful lemmas.

We denote the solution of the normal form system (8) by z(t, z 0 ), where z 0 is the initial state.

Definition 1 [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF] The equilibrium point z = 0 of ( 8) is finite-time convergent if there is an open neighborhood ℘ ⊂ R n of the origin and a function T : ℘ -{0} → (t 0 , +∞), such that ∀z 0 ∈ ℘, the solution z(t) of the system (8) exists and unique for all t ∈ [0, T (z 0 , t 0 )), such that lim t→T z(t) = 0 and z(t) = 0 ∀t > T . The instant time T is called the settling time. The equilibrium z = 0 of the system (8) is finite-time stable if it is Lyapunov stable and finite convergent. In particular if ℘ = R n , the equilibrium is said to be global finite-time stable (GFTS).

Lemma 1 [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF]: Consider the non-Lipschitz continuous autonomous system ż = f (z), f (0) ∈ R. Assume there are C 1 function V (z) defined on a neighborhood ℑ ⊂ R of the origin, and real numbers c > 0 and 0 < α < 1 such that

1. V (z) is positive definite on ℑ 2. V (z) + cV (z) α ≤ 0, ∀z ∈ ℑ
Then the origin z is locally finite-time stable. If ℑ = R and V (z) is radially unbounded, then the origin z = 0 is globally finite time stable. Moreover, it can be verified that the settling time being dependent on the initial state

z(0) = z 0 satisfies T z (z 0 ) ≤ V (z0) 1-α c(1-α)
for all z 0 in some open neighborhood of the origin.

Lemma 2 [START_REF] Qian | A continuous feedback approach to global strong stabilization of nonlinear systems[END_REF] For any x i ∈ R, i = 1, . . . , n and 0 < p ≤ 1, the following inequality holds

(|x 1 |+. . .+|x n |) p ≤ |x 1 | p + . . . + |x n | p ≤ n 1-p (|x 1 | + . . . + |x n |) p . when 0 < p = p1
p2 ≤ 1, where p 1 and p 2 are positive odd integers then the following holds |x p -y p | ≤ 2 1-p |x -y| p , for any x, y ∈ R.

Lemma 3 [START_REF] Qian | A continuous feedback approach to global strong stabilization of nonlinear systems[END_REF] For any x ∈ R, y ∈ R, c > 0, d > 0 and γ(x, y) > 0 a real-valued function, the following holds:

|x| c |y| d ≤ (cγ(x, y)/(c + d))|x| c+d + (dγ(x, y) -c/d /(c + d))|y| c+d .

RBF Neural Network Approximation

Consider a function f (z) : R m → R. Suppose that f (z) is unknown smooth nonlinear function and it can be approximated over a compact set Ω ⊆ R m with the following RBFNN:

f (z) = W * ⊤ ̟(z) + δ f (z), ∀z ∈ Ω (11)
where the node number of the NN is l. More nodes mean more accurate approximation. W * ∈ R l represents the optimal weight vector, which is defined by

W * = arg min W sup z∈Ω f (z) -W ⊤ ̟(z) (12)
where W represents the estimate of

W * , ̟(z) = [̟ 1 ( z- θ 1 ), ̟ 2 ( z -θ 2 ), . . . , ̟ l ( z -θ l )] ⊤ : Ω → R l rep-
resents the regressor vector, with ̟ i (.) being an RBF and θ i (i = 1, . . . , l) are distinct points in the state space (also called centers). The elements of the regressor vector are chosen to be as the Gaussian function:

̟ i ( z-θ i ) = exp -(z -θ i ) ⊤ (z -θ i ) σ 2 , i = 1, . . . , l (13)
where θ = [θ 1 , θ 2 , . . . , θ l ] ⊤ is the center vector of the Gaussian basis function, and σ is the spread of the Gaussian basis function. δ f (z) is the approximation error that is bounded over Ω, such that |δ f (z)| ≤ δf , where δf is an unknown constant.

Remark 4 Note that in RBF neural network approximation, the center parameters θ i and the spread σ are generally chosen according to the scope of the input value. If the parameter values are badly chosen, the Gaussian function will not be effectively mapped and the RBF network will be invalid. In this paper, In order to reduce computational burden, we deliberately set θ i and σ in the range of the input of the RBF and therefore we select the center θ i and the spread σ values as constants.

Main results

The purpose of this section is to devise a control law that stabilizes the system (8) in finite time. Given the fact that the Z-system (8) contains unmatched uncertainty, traditional adaptive techniques like the backstepping technique [START_REF] Khalil | Nonlinear Systems[END_REF] and the multiple-surface sliding [START_REF] Qian | Hierarchical sliding mode control for a class of simo under-actuated systems[END_REF] cannot be used, because the variation bounds of d i , i = 1, . . . , n are not assumed to be known. To solve the problem of finite-time stabilization with unmatched uncertainties, we propose a novel robust finite-time control scheme whereby a continuous recursive finite-time stabilizing control law is derived from the adding power integrator (API) technique [START_REF] Qian | A continuous feedback approach to global strong stabilization of nonlinear systems[END_REF] and the RBFNN technique.

In the following, to construct the finite-time controller we first define a set of virtual controllers z * 1d , . . . , z * nd and new variables e 1 , . . . , e n as follows

z * 1d = 0, e 1 = z 1 q 1 1 -z * 1 q 1 1d z * 2d = -β 1 e q2 1 , e 2 = z 1 q 2 2 -z * 1 q 2 2d . . . (14)
z * (n)d = -β n-1 e qn n , e n = z 1 qn n -z * 1 qn nd
and choose the following controller and the adaptive law:

u = -sign(φ n (.))ξ(z)e qn+1 ˙ Θ = Ξ(z, e n ) ( 15
)
where the parameters 1 = q 1 > q 2 > . . . > q m :=

4n+3-2m 4n+1

with m = 2, . . . , n, the gain parameters β i are positive constant to be determined during the control design. z = [z ⊤ , Θ ⊤ ] ⊤ , Θ is the parameter estimation that will be defined later, ξ(z) and Ξ(z n , e n ) are smooth nonnegative functions to be also determined later.

The design procedure consists of n steps, throughout the n -1 steps, the virtual controllers z * (m-1)d will be designed. Upon the completion of step n, the fast control term ξ(z) as well as the update law in (15) will be determined.

Step 1: From ( 8) and the change of variables in ( 14), one has ė1 = z 2 + d 1 (z). For the first step of the induction, we choose a Lyapunov function as

V 1 (z) = 1 1 + S e 1+S 1 ( 16
)
where S = (4n -1)/(4n + 1). The time derivative of [START_REF] Huang | Adaptive Control of Underactuated Mechanical Systems[END_REF] gives

V1 (z) = e S 1 ė1 = e S 1 z * 2 + e S 1 (z 2 -z * 2d ) + e S 1 d 1 (z)
By introducing the virtual control z * 2d = -β 1 e S 1 , where β 1 is a design parameter, V1 (z) rewrites:

V1 (z) = -β 1 e 2S 1 + e S 1 (z 2 -z * 2d ) + e S 1 d 1 (z) (17)
In order to proceed with the second step of the design procedure, one needs to reveal the error variable e 2 into (17). This can be done by upper-bounding the second term of the righthand side of ( 17) by an expression that involves the error variable e 2 . Using Lemma 2 and 3, the second term of ( 17) can be upper-bounded as follows

e S 1 (z 2 -z * 2d ) ≤ 2 1-S |e 2 | S |e 1 | S ≤ 1 2 |e 1 | 2S +C 2 |e 2 | 2S (18)
where C 2 is a positive constant. In light of ( 18), the time derivative of V 1 (z), can be written as follows:

V1 (z) ≤ -β 1 e 2S 1 + 1 2 |e 1 | 2S + C 2 |e 2 | 2S + e S 1 d 1 (z) ( 19
)
Step 2: By examining the dynamics of e 2 , one has

ė2 = 1 q2 (z 3 + d 2 (z))z 1 q 2 -1 2 + S q2 β 1 ė1 e S-1 1 z 1 q 2 -1 2d
. Clearly from this dynamic equation, it becomes hard to design a virtual controller z * 3d to stabilize e 2 using standard backstepping technique (i.e, by constructing quadratic Lyapunov function). According to [START_REF] Huang | Global finite-time stabilization of a class of uncertain nonlinear systems[END_REF] and [START_REF] Qian | A continuous feedback approach to global strong stabilization of nonlinear systems[END_REF] , a C 1 , positive definite and proper Lyapunov candidate function is constructed as

V 2 (z) = V 1 (z) + W 2 (z 1 , z 2 ) (20)
with

W 2 (z 1 , z 2 ) = z2 z * 2d (χ 1 q 2 -z * 1 q 2 2d ) 1+S-q2 dχ (21)
being differential, positive definite and proper function, then similar to the proof in [START_REF] Lin | Adaptive regulation of highorder lower-triangular systems: Adding a power integrator technique[END_REF][START_REF] Wang | Adaptive finite time coordinated consensus for high-order multiagent systems: adjustable fraction power feedback approach[END_REF], it is easy to show that the Lyapunov function candidate V 2 (z) is positive definite, proper and satisfies

V 2 (z) ≤ max{ 1 1+S , 2}(e 1+S 1 + e 1+S 2
). Note also from [START_REF] Lin | Adaptive regulation of highorder lower-triangular systems: Adding a power integrator technique[END_REF], that the function W 2 (z 1 , z 2 ) has the following property ∂W 2 (.)

∂z 2 = e 1+S-q2 2 ∂W 2 (.) ∂z 1 = -(1 + S -q 2 ) ∂z * 1 q 2 2d ∂z 1 z2 z * 2d (χ 1 q 2 -z * 1 q 2 2d ) S-q2 dχ = - ∂z * 1 q 2 2d ∂z 1 (z 2 -z * 2d )
Hence, the derivative of W 2 (.) can be computed as follows

Ẇ2 (.) = e 2 ż2 -((z 2 -z * 2d ) ∂z * 1 q 2 2d ∂z 1 ż1 ) = e 2 (z 3 -z * 3d ) -(z 2 -z * 2d ) ∂z * 1 q 2 2d ∂z 1 ż1 + e 2 z * 3 +e 2 d 2 (z) (22)
In order to proceed further, the second and third terms on the right side of ( 22) should be estimated. Using the fact that

(z 2 -z * 2d ) ∂z * 1 q 2 2d ∂z 1 ż1 ≤ 2 1-q2 e q2 2 ∂z * 1 q 2 2d ∂z 1 z 2 +2 1-q2 e q2 2 ∂z * 1 q 2 2d ∂z 1 d 1 (z)
Let the first lumped unmatched uncertainty be

τ 1 = e 2 d 2 (z) + 2 1-q2 β 1 q 2 1 e q2 2 d 1 (z), it is then straightforward to obtain Ẇ2 ≤ e 2 (z 3 -z * 3d ) + e 2 z * 3d + 2 1-q2 β 1 q 2 e q2 2 |z 2 | + τ 1 (23)
According to Lemma 2, the third term of the right hand side of ( 23) can be bounded such that

|z 2 | ≤ |e 2 + z * 1 q 2 2d | q2 ≤ |e 2 | q2 + β 1 |e 1 | q2 which implies that |e| q2 2 2 1-q2 β 1 q 2 |z 2 | ≤ 2 1-q2 β 1 q 2 |e 2 | 2q2 + 2 1-q2 β 1+ 1 q 2 |e 2 | q2 ×|e 1 | q2 ≤ 2 2(1-q2)-1 β 2(1+ 1 q 2 ) |e 2 | 2q2 + 1 2 |e 1 | 2q2 +2 1-q2 β 1 q 2 |e 2 | 2q2 = 1 2 |e 1 | 2q2 + C2 |e 2 | 2q2 (24)
where C2 = 2 2(1-q2) β 2(1+ 1 q 2

) + 2 1-q2 β 1 q 2 . Combining ( 19), ( 23) and ( 24) yields the following V2 (.) ≤ -(

β 1 -1)|e 1 | 2S + (C 1 + C2 )|e 2 | 2S +e 2 z * 3d + e 2 (z 3 -z * 3d ) + τ 1 (25)
At this stage, it is worth to note that q 3 = q 2 -2 4n+1 , which implies that q 3 = 1 + 2q 2 = 1 + 2S. Therefore by selecting β 1 > n -1 -κ, where κ > 0 is a design parameter, and the virtual control

z * 3d = -β 2 e q3 2 with β 2 ≥ n -2 + κ + C 2 + C2 , we obtain V2 ≤ -(n-2+κ)|e 1 | 2S -(n-2+κ)|e 2 | 2S +e 2 (z 3 -z * 3d )+τ 1 (26)
From [START_REF] Lin | Adaptive regulation of highorder lower-triangular systems: Adding a power integrator technique[END_REF], it is clear that in the k-th step the time derivative of V k should also be upper bounded by two negative terms, a crossing product and the lumped uncertainties. This observation will be shown by inductive steps as follows.

Step k (k := 3 . . . n -1): We proceed to the derivation of the virtual control by using an inductive argument. Suppose at step k-1, there exists a C 1 Lyapunov candidate function V k-1 (z 1 , . . . , z k-1 ), which is positive definite, proper that verifies

V k-1 (.) ≤ max{ 1 1 + S , 2} k-1 m=1 e 1+S m ( 27
)
and a set of virtual controllers and coordinate transformations defined by

z * 1d = 0, e 1 = z 1 q 1 1 -z * 1 q 1 1d z * 2d = -β 1 e q2 1 , e 2 = z 1 q 2 2 -z * 1 q 2 2d . . . ( 28
) z * (k)d = -β k-1 e q k k , e k = z 1 q k k -z * 1 q k kd
where β 1 > 0, . . . , β k-1 > 0 are positive design parameters such that,

Vk-1 (.) ≤ -(n -k + 1 + κ) k-1 m=1 e 2S m + e 1+S-q k-1 k-1 ×(z k -z * kd ) + k-2 m=1 τ m (29)
Now let us claim that ( 27) and ( 29) are true at step k.

To prove this claim, consider the following Lyapunov candidate function

V k (z 1 , . . . , z k ) = V k-1 (.) + W k (z 1 , . . . , z k ) (30)
with

W k (.) = z k z * dk (χ 1 q k -z * 1 q k dk ) 1+S-q k dχ (31)
From the previous step and according to [START_REF] Lin | Adaptive regulation of highorder lower-triangular systems: Adding a power integrator technique[END_REF][START_REF] Wang | Adaptive finite time coordinated consensus for high-order multiagent systems: adjustable fraction power feedback approach[END_REF] , it can be observed that W k (.) has the following properties:

∂W k (.) ∂z k = e 1+S-q k (32) ∂W k (.) ∂z m = -(1 + S -q k ) ∂z * 1 q k dk ∂z m z k z * dk (χ 1 q k -z 1 q k dk ) S-q k dχ (33)
also, it is easy to show that V k (.) is C 1 , proper and positive definite, which verifies

V k (.) =≤ max{ 1 1 + S , 2} k m=1 e 1+S m
The time derivative of V k (.) satisifies

Vk (.) ≤ -(n -k + 1 + κ) k-1 m=1 e 2S m + e 1+S-q k-1 k-1 (z k -z * dk ) +e 1+S-q k k z k+1 + e 1+S-q k k d k (z) + k-1 m=2 ∂W k (.) ∂z m żm + k-1 m=1 τ m ≤ -(n -k + 1 + κ) k-1 m=1 e 2S m + e 1+S-q k-1 k-1 (z k -z * dk ) +e 1+S-q k k z k+1 + k-1 m=2 ∂W k (.) ∂z m z m+1 (34)
+ e 1+S-q k k d k (z) + k-1 m=2 ∂W k (.) ∂z m d m (z) + k-1 m=1 τ m
If the k-th lumped unmatched uncertainty is defined as

τ k = e 1+S-q k k d k (z) + k-1 m=2 ∂W k (.) ∂zm d m (z), then (35) rewrites: Vk (.) ≤ -(n -k + 1 + κ) k-1 m=1 e 2S m + e 1+S-q k-1 k-1 (z k -z * dk ) +e 1+S-q k k z k+1 + k-1 m=2 ∂W k (.) ∂z m z m+1 + k m=1 τ m (35)
Next we bound the second and the fourth term of the right hand side of [START_REF] Qian | A continuous feedback approach to global strong stabilization of nonlinear systems[END_REF]. First according to Lemma 3, it holds that

|e 1+S-q k-1 k-1 (z k -z * dk )| ≤ 2 1+S-q k-1 |e k-1 | 1+S-q k-1 |e k | q k ≤ |e k-1 | 2S 2 + C k |e 2S k | (36)
with C k a positive constant. As for the fourth term, it is easy to obtain the following:

k-1 m=2 ∂W k (.) ∂z m z m+1 ≤ (1 + S -q k )2 1-q k |e k | S × k-1 m=2 ∂z * dk ∂z m z m+1 (37)
To further bound the fourth term in [START_REF] Qian | A continuous feedback approach to global strong stabilization of nonlinear systems[END_REF], we need to conduct the analysis by inductive argument and assume that at step k -1, the following holds

k-2 m=2 ∂z * dk ∂z m z m+1 ≤ k-1 m=1 γ (k-1)m e S m ( 38
)
where γ (k-1)m ≥ 0, then show the inequality also holds for k. Therefore, we have

k-1 m=2 ∂z * dk ∂z m z m+1 ≤ -β 1 q k k-1 k-1 m=2 ∂e k-1 ∂z m z m+1 (39) ≤ β 1 q k k-1 z 1 q k -1 k-1 q k-1 z k + k-2 m=2 ∂z * 1 q k -1 k-1 ∂z m z m+1
In light of the definition of the tracking error, it is worth recalling that e m = z

|z m | ≤ |e m + z * 1 qm m | qm ≤ |e m | qm + β m-1 |e m-1 | qm (40)
Applying [START_REF] She | Global stabilization of 2-dof underactuated mechanical systems-an equivalent-input-disturbance approach[END_REF] to the inequality [START_REF] She | Global stabilization of 2-dof underactuated mechanical systems-an equivalent-input-disturbance approach[END_REF] and using the assumption of the inequality (38) yields:

k-1 m=2 ∂z * dk ∂z m z m+1 ≤ β 1 q k k-1 1 q k-1 |e k-1 | 1-q k-1 + β 1 q k-1 -1 k-2 ×e 1-q k-1 k-2 |e k | q k + β k-1 |e k-1 | q k + k-2 m=2 ∂z * dk ∂z m z m+1 ≤ k m=1 γ km |e m | S (41)
with γ km being a positive constant. Therefore,

k-1 m=2 ∂W k (.) ∂z m z m+1 ≤ (1 + S -q k )2 1-q k |e k | S × k m=1 γ km |e m | S ≤ 1 2 k-1 m=1 e 2S m + Ck |e k | 2S (42)
with Ck ≥ 0 being a positive constant. Substituting [START_REF] Qian | Global finite-time stabilization by output feedback for planar systems without observable linearization[END_REF] and ( 42) into (35) leads to

Vk (.) ≤ -(n -k + 1 + κ) k-1 m=1 e 2S m + (C k + Ck )|e k | 2S + 1 2 k-1 m=1 e 2S m + |e k-1 | 2 + e 1+S-q k k z * k+1 + k m=1 τ k + e 1+S-q k k (z k+1 -z d(k+1) ) * ) ≤ -(n -k + κ) k-1 m=1 e 2S m + (C k + Ck )|e k | 2S +e 1+S-q k k (z k+1 -z * d(k+1) ) + e 1+S-q k k z * k+1 + k m=1 τ k (43)
By introducing the virtual control z * (k+1

)d = -β k e q k+1 k with β k selected such that β k ≥ n-k +κ+C k + Ck > 0, we get Vk (.) ≤ -(n -k + κ) k m=1 e 2S m + e 1+S-q k k (z k+1 -z * (k+1)d ) + k m=1 τ k (44)
This completes the inductive proof.

step n: This is the final stage of the design where the real control input appears in the dynamics. For this step, according to the inductive previous steps, the n-th part of the Lyapunov candidate function can be constructed accordingly

V n (z 1 , . . . , z n ) = V n-1 (.) + W n (z 1 , . . . , z n ) ( 45
)
where

W n (.) = zn z * nd (χ 1 qn -z 1 qn n ) 1+S-qn dχ (46)
Then it is obvious to conclude that V n (.) is C 1 , positive definite and satisfies

V n (.) ≤ max{ 1 1 + S , 2} n m=1 e 1+S m
From the above inductive argument, one can conclude that

Vn ≤ -κ n-1 m=1 e 2S m + e 1+S-qn n żn + (C n + Cn )e 2S n + n-1 m=1 τ m ≤ -κ n-1 m=1 e 2S m + e 1+S-qn n (φ n u + d n (z)) +(C n + Cn )e 2S n + n-1 m=1 τ m ≤ -κ n-1 m=1 e 2S m + φ n e 1+S-qn n u + (C n + Cn )e 2S n +e qn+1 n e 1+S-qn-qn+1 n d n (z) + e -qn+1 n n-1 m=1 τ m ( 47
)
where C n > 0 and Cn > 0 are positive constants. Denote by G the total lumped matched and unmatched uncertainty defined as

G = e 1+S-qn-qn+1 n d n (z) + e -qn+1 n n-1 m=1 τ m
Since G is unknown continuous function, it cannot be directly compensated by the design of the control input u. Based on the RBFNN approximation, the function G can be modeled by RBFNN on the compact set Υ as

G = W ⊤ ̟(z) + δ G (z), ∀z ∈ Υ (48)
where W ∈ R M is the weight vector, M > 1 is the NN node number and δ G (z) ∈ R is the approximation error satisfying |δ G (z)| ≤ δG . Substituting (48) into (47), we get

Vn ≤ -κ n-1 m=1 e 2S m + φ n e 1+S-qn n u + (C n + Cn )e 2S n +e qn+1 n (W ⊤ ̟(z) + δ G (z)) ≤ -κ n-1 m=1 e 2S m + φ n e 1+S-qn n u + (C n + Cn )e 2S n +[W ⊤ , δG ] ̟(z) 1 e qn+1 n ( 49
) By denoting Θ = [W ⊤ , δG ] ⊤ ∈ R M+1 and ̟(z) = [̟(z), 1] ⊤ ∈ R M+1 , we have Vn ≤ -κ n-1 m=1 e 2S m + φ n e 1+S-qn n u + (C n + Cn )e 2S n +Θ ⊤ ̟(z)e qn+1 n ≤ -κ n-1 m=1 e 2S m + sign(φ n )φ min e 1+S-qn n u +(C n + Cn )e 2S n + Θ ⊤ ̟(z)e qn+1 n (50)
Obviously, Θ is an unknown parameter and ̟(z) is a known vector function. The actual control law can therefore be designed as follows

u = - sign(φ n ) φ min e qn+1 (n̺ + Θ ⊤ ̟(z)) = -sign(φ n )ξ(z)e qn+1 (51)
where ξ(z) = φ -1 min (n̺ + Θ ⊤ ̟(z)), ̺ > 0 is a designed parameter such that ξ(z) > 0 and Θ is the estimation of the unknown parameter Θ. Substituting (51) into (50), it gives

Vn ≤ -κ n-1 m=1 e 2S m + (C n + Cn )e 2S n -n̺e 2S n +e qn+1 n Θ ⊤ ̟(z) (52)
where Θ = Θ -Θ is the estimate error of the matched and unmatched uncertainty. It is easy to see that ˙ Θ = -˙ Θ. Then, to be able to design the parameter update law for Θ, we will consider the Lyapunov function candidate

V T = V n (.) + 1 2 Θ ⊤ Γ -1 Θ ( 53
)
where Γ is a positive definite diagonal matrix. Taking the time derivative of (53) along the solutions of (52), results in

VT ≤ -κ n-1 m=1 e 2S m + (C n + Cn )e 2S n -n̺e 2S n +e qn+1 n Θ ⊤ ̟(z) -Θ ⊤ Γ -1 ˙ Θ ≤ -κ n-1 m=1 e 2S m + (C n + Cn )e 2S n -n̺e 2S n -Θ ⊤ (Γ -1 ˙ Θ -e qn+1 n ̟(z)) (54)
From (54), one can choose the adaptive law as follows:

˙ Θ = Γ ̟(z)e qn+1 n := Ξ(z, e n ) ( 55
)
which results in

VT ≤ -κ n-1 m=1 e 2S m -(n̺ -C n -Cn )e 2S n ( 56
)
We are now ready to announce the main result of this paper.

Theorem 1 Consider the n-th order underactuated system in the X-space represented by [START_REF] Chen | Adaptive control of rotary inverted pendulum system with time-varying uncertainties[END_REF], through the coordinate transformation [START_REF] Choukchou-Braham | Analysis and Control of Underacuated Mechanical Systems[END_REF], with Assumptions 2.1-2.2 and under the finite time controller (51) and the adaptive law (55), the closed-loop of the underactuated system is finite time stable in the sense of Definition 2.1 and the parameter estimation Θ is bounded.

Proof The proof is conducted similarly to the analysis presented in [START_REF] Hong | Finite-time stabilization of nonlinear systems with parametric and dynamic uncertainties[END_REF]. First to ensure (56) is negative definite, one can select ̺ such that such that n̺-C n -Cn > κ, then (56) rewrites

VT ≤ -κ n m=1 e 2S m (57)
Obviously, from (57) it can be inferred that e i , i = 1, . . . , n and Θ are bounded, so does Θ because Θ is a constant vector. Moreover, it can be seen from ( 55) that Θ does not change its sign because ˙ Θ is nonnegative. Therefore, without loss of generality, we assume

Θ 1 ∈ [0, C],
where C is a constant depending on initial values of e i (0) and Θ(0).

Take the Lyapunov function

Ṽ (e, Θ) = V T - 1 2 Θ ⊤ Γ -1 Θ := V n (.)
which is positive definite for all e 1 , e 2 , . . . , e n and for any fixed Θ.

Noting that V n (.) ≤ max{

V ≤ VT + Θ ⊤ Γ -1 ˙ Θ ≤ -κ n m=1 e 2S m + e qn+1 Θ ⊤ ̟(z) ≤ - κ 2 n m=1 e 2S m - κ 2 e 2S n + e 2S Θ 1 ζ 0 (59) ≤ - κ 2 n m=1 e 2S m - κ 2 e 2S n 1 - 2(C + Θ) κ ζ 0 ≤ - κ 2 max{ 1 1+S , 2} Ṽ 2S 1+S - κ 2 e 2S n 1 - 2(C + Θ) κ ζ 0
where Θ 1 ≤ Θ, with Θ is an unknown bound. Define a continuous function V (e n , z) = 2(C+ Θ) κ ζ 0 which satisfies V (0, z) = 0. It is easy to show then that for a given Θ 1 ∈ [0, C], there exists a constant ǫ > 0 such that for any e ∈ Υ = (e, Θ) : Ṽ (e, Θ) ≤ ǫ , V < 1 and therefore e 2S n (1 -V ) > 0 because e 2S n is positive. Therefore once (e, Θ) ∈ Υ , it will never escape.

From the above analysis two cases for finite time convergence are considered.

If the initial conditions are (e(0), Θ(0)) ∈ Υ , it is straightforward to show that V ≤ -

κ 2 max{ 1 1+S ,2} Ṽ 2S 1+S ,
since α := 2S 1+S < 1 then Ṽ finite time convergent according Lemma 2.1. Therefore e = [e 1 , e 2 , . . . , e n ] ⊤ becomes 0 within time T such that

T ≤ 2 max{ 1 1 + S , 2} Ṽ (e(0), Θ(0)) 1-α κ(1 -α)
If in the seconde case, the initial conditions (e(0), Θ(0)) / ∈ Υ , then the first thing to do is to estimate the maximum reaching time of (e(t), Θ(t)) to Υ . Before the state enters Υ , we have Ṽ (e, Θ) > ǫ, therefore Therefore, (e, Θ) will enter Υ within T 2 :

V T (e(0), Θ(0)) ≥ V T (e(0), Θ(0)) -V T (e(T 2 ), Θ 2 (T 2 )) ≥ T2 0 κ n m=1 e 2S m ds ≥ T2 0 κ 2 max{ 1 1+S , 2} Ṽ 2S 1+S ds ≥ κ 2 max{ 1 1+S , 2}
T 2 ≤ 2 max{ 1 1+S , 2}V T (e(0), Θ(0)) + Θ(0) ⊤ Γ Θ(0) κ * ǫ 2S 1+S
after the reaching time T 2 , the state will be in Υ . It will however take

T 1 ≤ 2 max{ 1 1 + S , 2} ǫ 1-α κ(1 -α)
to attain the origin, thus e = [e 1 , . . . , e n ] ⊤ = 0 within T ≤ T 1 + T 2 , therefore the closed-loop of the underactuated system is finite time convergent. This completes the proof.

Simulation results

In this section, the effectiveness and the robustness of the proposed adaptive finite time controller augmented by the RBFNN is evaluated through two simulation cases on a rotary inverted pendulum (see Fig. 1), whose system dynamics are described by the following state space representation with ∆ = (J + ml 2 )(I + ml 2 sin(x 3 ) 2 ) + mJL 2 + m 2 l 2 L 2 sin(x 3 ) 2 , m and J are respectively the mass and the moment of inertia of the pendulum, l is the distance of the center of mass of the pendulum to its end point, I is the moment of inertia of the arm and F d is the external disturbance acting on the pendulum. For convenience, the numerical values of the main physical parameters of the rotary inverted pendulum are provided in Table 1. Interested readers can refer to the literature [START_REF] Huang | Adaptive Control of Underactuated Mechanical Systems[END_REF] for more details. The initial conditions are chosen as x = [-π 4 , 0, π 3 , 0] ⊤ , the desired state is x d = [0, 0, 0, 0] ⊤ . The control gains are chosen to be κ = 10, β 1 = 13, β 2 = 12, β 3 = 11, β 4 = 20 and Γ ii = 500. Different simulation scenarios have been considered. The first simulation scenario is the control of uncertain cart-pole system without external disturbances. The second one considers the case of an uncertain cart-pole system with the presence of external disturbances.

ẋ1 = x 2 ẋ2 = f 2 (x) + b 2 (x)u ẋ3 = x 4 (

Scenario 1: Uncertain rotary inverted pendulum system without external disturbances

In this scenario, we assume that the system dynamics is completely unknown. The obtained simulation results are shown in Figures 2 to 3. It is clear in Figure 2 how the arm and the pendulum angles converge in finite time to zero despite the uncertainties present in the system dynamics. The control effort being deployed is shown in Figure 3(a). Figure 3(b) shows the convergence to zero of the state variables in the Z-space. Clearly the algorithm performs well in the presence of model parameters' uncertainties.

Scenario 2: Uncertain rotary inverted pendulum system with external disturbances

In this simulation scenario, on top of the uncertainties that the system dynamics contains, a periodic perturbation d(t) = sin(2t) is added as an external excitation to the rotary inverted pendulum in order to test the robustness of the proposed control approach. The obtained simulation results are depicted in Figure 4 and Figure 5. From Figure 4, it can be noticed that despite the existing uncertainties and the considered external disturbance, the controller is able to restore the system to its desired values. The control effort after the application of the disturbances is shown Figure 5(e). Clearly the control input remains with reasonable amplitude even with the external perturbation. The convergence of the state variables in Z-space are shown in Figure 5(a)-(d). From these figures it can be seen that our control design is robust to tolerate significant variation of the system parameters as well as the external disturbances. Fig. 3 Evolution of the control effort and Z-space state versus time.

Conclusion

In this paper, the problem of finite time stabilization was addressed for the control of a class of uncertain un- deractuated mechanical systems. By integrating a fractional power feedback control method with an adaptive RBFNNs scheme, the uncertainties in the system can be effectively handled and finite time stabilization is achieved. Future work is to extend the current design technique to tracking control problem of underactuated systems operating under modelling uncertainties and stochastic perturbations with fault-tolerant control. Effective strategies that could handle all these challenges can be found in [START_REF] Li | Adaptive fault-tolerant stochastic shape control with application to particle distribution control[END_REF] and [START_REF] Shen | A separated approach to control of markov jump nonlinear systems with general transition probabilities[END_REF]. However, the solutions presented so far in these works, can only ensure asymptotic stability. Our endeavor will be to introduce finitetime convergence to stochastic system model.

1 qm m -z * 1 qm

 11 md and z * m = -β m-1 e qm m-1 , it can then be inferred that z * 1 qm m = -β 1 qm m-1 e m-1 and therefore the following inequality holds

Fig. 1

 1 Fig.1Schematic view of the rotary inverted pendulum[START_REF] Olfati-Saber | Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles[END_REF] .

  60) ẋ4 = f 4 (x) + b 4 (x)u where, we designated by [θ, θ, φ, φ] ⊤ = x ⊤ = [x 1 , x 2 , x 3 , x 4 ] ⊤ with θ is the angular displacement of the arm and φ is the angular displacement of the pendulum and f 2 , f 4 , b 2 and b 4 are defined in (61):

f 2 (

 2 x) = (J + ml 2 )mlx 4 [Lx 4 sin(x3) -lx 2 sin(2x 3 )] ∆ -m 2 l 2 L cos(x 3 )[g sin(x 3 ) + 0.5lx 2 2 sin(2x 3 )] ∆ -mlL cos(x 3 ) ∆ F d b 2 (x) = J + ml 2 ∆ f 4 (x) = ml sin(x 3 )[(I + mL 2 l sin(x 3 ) 2 )g -mlL 2 x 2 3 cos(x 3 )] ∆ -ml 2 x 2 sin(2x 3 )x 2 [mlLx 4 cos(x 3 ) + 0.5x 2 (I + mL 2 + l 2 sin(x 3 ) 2 )] ∆ + (I + mL 2 + ml 2 sin(x 3 ) 2 )

Fig. 2

 2 Fig.2System response arm and pendulum angle versus time.

Fig. 4

 4 Fig.4System response arm and pendulum angle versus time.

Fig. 5

 5 Fig. 5 Evolution of the control effort and Z-space state versus time.

Table 1

 1 Physical parameters of the rotary inverted pendulum

	Parameter	Nominal Value (unit)	Added Uncertainty
	g	9.8ms -2	0
	m	5.38 × 10 -2 kg	0.0263 + 0.01 sin 40t
	I	1.75 × 10 -2 kg.m 2	0.01
	J	1.98 × 10 -4 kg.m 2	0.01
	l	0.113m	0.07 + 0.01 sin 2t
	L	0.215m	0.07 + 0.01 sin 2t


Stefano Olla
email: olla@ceremade.dauphine.fr

Ceremade

Fourier and the science of today/Fourier et la sciences d'aujourd'hui Role of conserved quantities in Fourier's law for diffusive mechanical systems

Keywords: Diffusive Transport, Linear Response, Hydrodynamic Limit, Non-equilibrium Stationary States, Weak Coupling limit

Energy transport can be influenced by the presence of other conserved quantities. We consider here diffusive systems where energy and the other conserved quantities evolve macroscopically on the same diffusive space-time scale. In these situations the Fourier law depends also from the gradient of the other conserved quantities. The rotor chain is a classical example of such systems, where energy and angular momentum are conserved. We review here some recent mathematical results about diffusive transport of energy and other conserved quantities, in particular for systems where the bulk Hamiltonian dynamics is perturbed by conservative stochastic terms. The presence of the stochastic dynamics allows to define the transport coefficients (thermal conductivity) and in some cases to prove the local equilibrium and the linear response argument necessary to obtain the diffusive equations governing the macroscopic evolution of the conserved quantities. Temperature profiles and other conserved quantities profiles in the non-equilibrium stationary states can be then understood from the non-stationary diffusive behaviour. We also review some results and open problems on the two step approach (by weak coupling or kinetic limits) to the heat equation, starting from mechanical models with only energy conserved.

Introduction

Fourier's law claims that the local energy current is proportional to the local gradient of temperature and the ratio of these quantities, which is a function of the local temperature, is called the thermal conductivity. However in many realistic systems energy is not the only conserved quantity and the interplay between extra conserved quantities and energy has a deep impact on the thermal properties of the system, for example generating uphill diffusion.

Dynamics that have more conserved quantities, either than energy, often present different time scales for the macroscopic evolution of these. In the present review we are interested in systems where conserved quantities evolve macroscopically in the same diffusive time scale, and their macroscopic evolution is governed by a system of coupled diffusive equations. One example is given by the chain of coupled rotors, whose dynamics conserves the energy and the angular momentum. In Sections 2 and 3 we show that, as a consequence of a linear response argument and certain symmetries in the corresponding Onsager matrix, the macroscopic evolution of these two quantities follows a diffusive system of PDE (14) or equivalently [START_REF] Iacobucci | Stationary non-equilibrium states in rotors models[END_REF]. In terms of the evolution of the temperature profile, the usual heat diffusion can be counterbalanced by a local increase of temperature due to the gradient of the momentum profile (see [START_REF] Iubini | Boundary induced instabilities in coupled oscillators[END_REF]). It seems that there is some universality about equations [START_REF] Iubini | Boundary induced instabilities in coupled oscillators[END_REF], on the role of the gradients of the other conserved quantities. We give some examples where the deterministic Hamiltonian dynamics is perturbed by a stochastic terms conservative of energy and other quantities. In some of these stochastic models the hydrodynamic limit can be proven mathematically.

This macroscopic coupled evolution generates interesting stationary profiles of temperatures in the nonequilibrium stationary states with thermal and mechanical forces acting on the boundaries of the system, as explained in section 4.

In the last section we will report on some results concerning the two step approach for the Fourier law, for dynamics that conserves only energy.

Linear response and Onsager matrix

For simplicity let us consider first a spatially homogeneous dynamics on a lattice Z such that there are two conserved quantities: momentum and energy. A typical and commonly studied example is given by the rotor chain, where the configurations are given by {q i (t) ∈ S 1 , p i (t) ∈ R, i ∈ Z}, where S 1 is the unit circle. Nearestneighbor rotators interact through a periodic potential V (q i+1 -q i ). For example V (r) = 1 -cos(2πr), but the considerations below are valid for more general interactions. The Hamiltonian dynamics is given by qi (t) = p i (t), ṗi (t) = V (q i+1 (t) -q i (t)) -V (q i (t) -q i-1 (t)).

In the following we will denote r i = q i -q i-1 . There are two locally conserved (or balanced) quantities: the momentum p i and the total energy e i = p 2 i 2 + V (r i ). In fact we have d dt p i (t) = j p i-1,i (t) -j p i,i+1 (t), j p i,i+1 (t) = -V (r i+1 (t)), d dt e i (t) = j e i-1,i (t) -j e i,i+1 (t), j e i,i+1 (t) = -p i (t)V (r i+1 (t)),

where j α i,i+1 (t), α = p, e, are the corresponding instantaneous currents. Notice that r i is not a balanced quantity.

One of the main conditions needed in order to have a macroscopic autonomous diffusive evolution of these conserved quantities is that there are no other translation invariant conserved quantities for the dynamics of the infinite system. This ergodic property is very hard to prove for the deterministic dynamics. We may consider also some stochastic perturbation of the dynamics (1) that have the same conserved quantities. The purpose of such perturbations is in fact to guarantee that there are no other balanced quantities [START_REF] Fritz | Stationary states of random Hamiltonian systems[END_REF]. One example is to add a random flip of the signs of r i : for each particle i, at exponential times with rate γ, r i changes to -r i , independently from the other particles j = i. This operation does not change the momentum.

A precise way to state this ergodic property of the infinite dynamics is the characterization of the stationary and translation invariant probability measures, i.e. that the Gibbs measures

dν β,p = i∈Z e -βei+βppi Z β,βp dp i dr i , β > 0, p ∈ R, (3)
are the only stationary and translation invariant probability measures for the dynamics (within a certain class of regular distributions). We will denote by < • > β,p the expectation with respect to ν β,p , as well as the expectation of a function of the path of the dynamics under ν β,p . These Gibbs measures are called equilibrium and they have the following time-reversible property: if {(r(t), p(t)) = (r i (t), p i (t), i ∈ Z) , t ∈ [-t 0 , t 0 ]} are stationary distributed with marginal ν β,p , then {r i (t) = r i (-t), pi (t) = -p i (-t)} follow the same dynamics, but with marginal ν β,-p , i.e. for any function of the path F (r, p) we have

< F (r(•), p(•)) > β,p =< F (r(•), p(•)) > β,-p .
In the specific case of the rotators, there is also a rotational symmetry of the equilibrium dynamics, i.e. under the equilibrium (β, p), the distribution of the path of ri (t) = r i (t), pi (t) = p i (t) -p is the same as the one at equilibrium (β, 0), i.e. with zero average velocity.

One way to define the transport diffusion coefficients is through linear response. Notice that for any equilibrium state, we have that < j p i,i+1 > β,p = 0 =< j e i,i+1 > β,p for any values of β > 0 and p ∈ R, consequently we do not expect any ballistic evolution in these systems. We have to understand how these expectations behave if we impose a gradient of temperature or of momentum, at the first order for small gradients. Given 1 , 2 , consider the inhomogeneous Gibbs measure

dµ 1 , 2 = i∈Z e -(β+ 1i)ei+(βp+ 2i)pi Z β+ 1 i,βp+ 2i dp i dr i . (4)
Starting at time t = 0 with this inhomogeneous measure we expect that, at the first order in ( 1 , 2 ),

< j p 0,1 (t) > µ 1 , 2 = K p,p t 2 + K p,β t 1 + o( 1 , 2 ) < j e 0,1 (t) > µ 1 , 2 = K β,p t 2 + K β,β t 1 + o( 1 , 2 ), (5)
and then we are interested in the limit as t → ∞ for the coefficients K u,v t . Defining e = e i β,p , a straightforward development at the first order gives

K p,p t = i i j p 0,1 (t)(p i (0) -p) β,p , K p,β t = - i i j p 0,1 (t)(e i (0) -e) β,p , K β,p t = i i j e 0,1 (t)(p i (0) -p) β,p , K β,β t = - i i j e 0,1 (t)(e i (0) -e) β,p , (6)
assuming that the sums in (6) converge. By using the symmetries of the dynamics (rotational, time stationarity, time reversibility), and recalling that j p 0,1 (0)p i (0) β,0 = 0, we have

K p,p t (β, p) = i i j p 0,1 (t)p i (0) β,0 = - i i j p 0,1 (0)p i (t) β,0 = - t 0 ds i i j p 0,1 (0) j p i-1,i (s) -j p i,i+1 (s) β,0 = - t 0 ds i j p 0,1 (0)j p i,i+1 (s) β,0 . (7)
So we define the limit as t → ∞, assuming that it exists, as

K p,p (β, p) = - ∞ 0 ds i j p 0,1 (0)j p i,i+1 (s) β,0 = K p,p (β, 0) := K p,p (β). (8)
Notice that K p,p is only a function of β and does not depend on p. This is a consequence of the rotational symmetry of the dynamics. We define similarly K p,β (β, p), K β,p (β, p), K β,β (β, p). Similar calculations give, recalling that j p 0,1 (0)e i (0) β,p = 0 for any p and β,

K p,β t = - i i j p 0,1 (t)(e i (0) -e) β,p = - i i j p 0,1 (0)(e i (t) -e) β,-p = - t 0 ds i j p 0,1 (0)j e i,i+1 (s) β,-p = - t 0 ds i j p 0,1 (0)p i (s)j p i+1,i+2 (s) β,-p = t 0 ds i j p 0,1 (s)p i (0)j p i+1,i+2 (0) β,p = -pK pp t + t 0 ds i j p 0,1 (s)(p i (0) -p)j p i+1,i+2 (0) β,p = -pK pp t + t 0 ds i j p 0,1 (s)p i (0)j p i+1,i+2 (0) β,0 (9)
The second term on the right hand side is equal to 0, since t 0 ds i j p 0,1 (s) is symmetric by time reversal s → t -s while p i (0)j p i+1,i+2 (0) is antisymmetric. This implies that K p,β = -pK pp . A similar argument gives K β,p (β, p) = K p,β (β, -p) = pK pp (β).

(10) Similarly,

K β,β t = - i i j e 0,1 (t)(e i (0) -e) β,p = i i j e 0,1 (0)(e i (t) -e i (0)) β,-p = t 0 ds i j e 0,1 (0)j e i,i+1 (s) β,-p = t 0 ds i j e 0,1 (0)j e i,i+1 (s) β,p , (11)
Using again the symmetries and recalling that j e i,i+1 = p i j p i+1,i+2 we obtain also that

K β,β (β, p) = K β,β (β, 0) -p 2 K p,p (β). ( 12
)
Finally thanks to the symmetries of the system, all the coefficients can be calculated from K p,p (β) and K β,β (β) := K β,β (β, 0), computed at zero average velocity. Relations [START_REF] Das | Nonlinear Fluctuating Hydrodynamics for the Classical XXZ Spin Chain[END_REF] and [START_REF] Dolgopyat | Energy transfer in a fast-slow Hamiltonian system[END_REF] were already noted in [START_REF] Iubini | Coupled transport in rotor models[END_REF]. One of the main mathematical problems in dealing with the deterministic infinite dynamics, is in proving that the limits defining K p,p (β) and K β,β (β) exist and are finite. If this is an open problem for the deterministic dynamics, stochastic perturbations can help. In fact adding a random independent flip of the r i 's one can prove that K p,p (β) and K β,β (β) are well defined and finite, by adapting the argument used in [START_REF] Bernardin | Transport Properties of a Chain of Anharmonic Oscillators with Random Flip of Velocities[END_REF].

The diffusive macroscopic equations

Macroscopic diffusive equations for the rotors model

The linear response analysis of the previous section gives a heuristic argument for the macroscopic equations governing the evolution in the diffusive space-time scale. In order to state the macroscopic equations, we need some thermodynamic functions. The internal energy (or thermal energy) as a function of β is given by u(β) = -∂ β log Z β,0 , where Z β,0 is the partition function appearing in [START_REF] Bernardin | Green-Kubo Formula for Weakly Coupled Systems with Noise[END_REF]. The temperature is given by T = β -1 , and the heat capacity is defined as c v (T ) = du dT = β 2 Var β,0 (e i ). The thermodynamic entropy is S(u) = inf β>0 {βu + log Z β,0 }, and β(u) = S (u) provides the inverse function of u(β).

The linear response argument (5) suggests the following macroscopic equations for the evolution of the profiles p(t, x), e(t, x) of the conserved quantities:

∂ t p = -∂ x K p,p (β)∂ x (βp) + K p,β (β, p)∂ x β ∂ t e = -∂ x K β,p (β, p)∂ x (βp) + K β,β (β, p)∂ x β , (13)
with the profiles β(t, x) := β(u(t, x)), and u(t, x) = e(t, x) -1 2 p 2 (t, x). By using the relations ( 10) and ( 12), the only coefficient involved are K p,p (β) and K β,β (β) = K β,β (β, 0). Expressing the equations in terms of the temperature profile T (t, x) = β -1 (t, y), (13) are equivalent to

∂ t p = ∂ x (D p (T )∂ x p) ∂ t e = ∂ x D p (T )∂ x p 2 2 + κ(T )∂ x T , (14)
where

D p (T ) := -T -1 K p,p (T -1 ) momentum diffusivity, κ(T ) := T -2 K β,β (T -1 ) thermal conductivity. ( 15
)
Alternatively, rewriting (14) as closed equations in p(t, x) and e(t, x):

∂ t p = ∂ x (D p ∂ x p) ∂ t e = ∂ x (D p -D e )∂ x p 2 2 + D e ∂ x e , (16)
with the energy diffusivity defined by D e = κ(T ) cv(T ) . It is more interesting to close the equations in the temperature profile T (t, x) obtaining

∂ t p = ∂ x (D p (T )∂ x p) c v (T )∂ t T = ∂ x (κ(T )∂ x T ) + D p (T )(∂ x p) 2 . ( 17
)
There are two remarkable consequences from the equations ( 17):

-The gradient in the temperature does not contribute to the diffusion of the momentum, but the momentum diffusivity depends only on the temperature. -The gradient of the momentum increases locally the temperature. The time evolution of the temperature is composed by the usual heat diffusion term ∂ x (κ(T )∂ x T ) plus an increase due to inhomogeneities in the momentum distribution. The total energy e(t, x) is the sum of the mechanical energy p 2 (t, x)/2 and an internal energy u(t, x). Notice that ∂ t u = c v (T )∂ t T , i.e. the momentum diffusion decreases the mechanical energy, that dissipates into internal energy, and the term D p (T )(∂ x p) 2 is the rate of dissipation of the mechanical energy in internal energy. We can also read this in the increase of the total entropy of the system:

d dt S(u(t, x))dx = D p (T ) T (∂ x p) 2 + κ(T ) T 2 (∂ x T ) 2 dx. (18)

Mathematical problems

We have obtained the equations ( 17) heuristically from the linear response argument and some symmetries of the dynamics. In fact they are quite general, the details of the microscopic dynamics are contained in the macroscopic transport coefficients D p (T ) and κ(T ). A rigorous mathematical statement would be given by a hydrodynamic limit: consider the empirical profile distributions

p (t, x) = i p i ( -2 t)δ i (x), ê (t, x) = i e i ( -2 t)δ i (x). (19)
These are random variables valued on distributions on R. We would like to prove that, as → 0, they converge in probability to the solution p(t, x), e(t, x) of ( 16). We have already mentioned that for the deterministic dynamics, even the existence of D p (T ) and κ(T ) is an open problem. Adding stochastic conservative terms to the dynamics can help to prove the existence of the transport coefficient [START_REF] Bernardin | Transport Properties of a Chain of Anharmonic Oscillators with Random Flip of Velocities[END_REF], but still the proof of the hydrodynamic limit is a difficult task. The main problem is the following. In the hydrodynamic limit, on the microscopic scale, gradients have order , but one has to look at a time scale -2 . In the linear response argument we have first made an expansion in the first order of the gradients (cf. ( 5)) because of a large space scale, and subsequently we took the large time limit. In the hydrodynamic limit we have to take the limit in space and time together, with the diffusive scaling. It should be mentioned here that this possible mathematical statement about the diffusive limit does not explain the superdiffusive behavior that the rotor chain may have at low temperatures in certain space-time scales. This is due to a kind of metastable low temperature states, where rotors have mostly an oscillatory behaviour, like an FPU chain of anharmonic springs, and the r i behave like an almost conserved quantity (see discussion and simulations in [START_REF] Spohn | Fluctuating hydrodynamics for a chain of nonlinearly coupled rotators[END_REF][START_REF] Suman | Role of conserved quantities in normal heat transport in one dimenison[END_REF] and for a related model in [START_REF] Das | Nonlinear Fluctuating Hydrodynamics for the Classical XXZ Spin Chain[END_REF]).

A technique to approach this hydrodynamic limit problem in stochastic dynamics was developped by Varadhan [START_REF] Varadhan | Nonlinear diffusion limit for a system with nearest neighbor interactions II, Asymptotic problems in probability theory: stochastic models and diffusions on fractals[END_REF] (see also chapter 7 in [START_REF] Kipnis | Scaling Limits of Interacting Particle Systems[END_REF]). It consists in decomposing, by approximations, the microscopic currents in a gradient term (dissipation) plus a term in the range of the generator L of the dynamics (fluctuation). In the present context this would mean

j p 0,1 ∼ -D p (T )(p 1 -p 0 ) + LF p , j e 0,1 ∼ -D e (T )(e 1 -e 0 ) -p(D p (T ) -D e (T ))(p 1 -p 0 ) + LF e , ( 20
)
where the approximation symbol means that there exist sequences of local functions F p , F e in the domain of the generator L such that the space-time variance of the difference, calculated for the dynamics in the equilibrium with average momentum p and temperature T , vanishes. Such fluctuation-dissipation decompositions allow to substitute, locally, currents with gradients and eventually close the equations after the hydrodynamic limit. In [START_REF] Varadhan | Nonlinear diffusion limit for a system with nearest neighbor interactions II, Asymptotic problems in probability theory: stochastic models and diffusions on fractals[END_REF] Varadhan used first this approach to prove the hydrodynamic limit for a non-gradient reversible stochastic Ginzburg-Landau dynamics (one conserved quantity). Reversibility here is intended for a dynamics that has a self-adjoint generator with respect to the stationary Gibbs measures. Then Quastel, in his PhD thesis [START_REF] Quastel | Diffusion of colors in the simple exclusion process[END_REF], proved by this technique the diffusion of colors in the symmetric simple exclusion process (still reversible but with two conserved quantities). There have been many other results extending this method to non-reversible dynamics. In the context of a chain of anharmonic oscillators with a stochastic perturbation conserving only the energy, the fluctuation-dissipation decomposition has been proved in [START_REF] Olla | Macroscopic energy diffusion for a chain of anharmonic oscillators[END_REF]. One of the limitations of this non-gradient approach is that it requires that the generator of the dynamics L has a very 'large' symmetric part S, whose finite dimensional version has a spectral gap uniformly bounded with respect to the dimension; furthermore L should satisfy a spectral sector condition with respect to S. This requires to consider only stochastic perturbations that act on both positions and velocities (S locally elliptic).

Macroscopic diffusive equations for the harmonic chain with bulk noise

In the case when κ and D e are constant independent of the temperature T , there are examples of dynamics where the hydrodynamic limit can be proven rigorously. This is the case of a chain of harmonic oscillators whose Hamiltonian dynamics is perturbed by a random sign change of the interparticle distance. The dynamics is defined as in [START_REF] Bálint | On the limiting Markov process of energy exchanges in a rarely interacting ball-piston gas[END_REF], but now r i ∈ R and V (r) = r 2 2 . Furthermore any particle i has an independent Poisson process with intensity γ, when it rings r i changes sign. Momentum and energy are conserved and it can be proven that the empirical distribution defined by [START_REF] Komorowski | Diffusive propagation of energy in a non-acoustic chain[END_REF] converges to the solution of ( 16), with explicit κ and D e depending only on γ. Of course these coefficients diverge for γ → 0 as the deterministic harmonic chain does not have a diffusive behaviour.

A rigorous mathematical proof of such hydrodynamic limit for a different dynamics of springs, with random perturbations that do not conserve momentum but conserves r i can be found in [START_REF] Bernardin | Hydrodynamics for a system of harmonic oscillators perturbed by a conservative noise[END_REF][START_REF] Komorowski | Macroscopic evolution of mechanical and thermal energy in a harmonic chain with random flip of velocities[END_REF]. In [START_REF] Komorowski | Macroscopic evolution of mechanical and thermal energy in a harmonic chain with random flip of velocities[END_REF] is considered a harmonic chain, but with a random flip of the velocities sign. In this case the two conserved quantities are the volume stretch r i (with currents j r i,i+1 = -p i ) and the energy. In [START_REF] Komorowski | Macroscopic evolution of mechanical and thermal energy in a harmonic chain with random flip of velocities[END_REF] it is proven that the empirical distributions r (t, x)

= i r i ( -2 t)δ i (x), T (t, x) = i 1 2 p 2 i ( -2 t)δ i (x). (21)
converge to the solution r(t, x), T (t, x) of

∂ t r = 1 2γ ∂ 2 x r ∂ t T = 1 2γ (∂ x r) 2 + 1 4γ ∂ 2 x T, (22)
that have the same structure as [START_REF] Iubini | Boundary induced instabilities in coupled oscillators[END_REF]. Similar results with a different stochastic perturbation are contained in [START_REF] Bernardin | Hydrodynamics for a system of harmonic oscillators perturbed by a conservative noise[END_REF]. In these models, where the transport coefficients are constant, the fluctuation-dissipation relations [START_REF] Kipnis | Scaling Limits of Interacting Particle Systems[END_REF] are exact for explicit local functions F . Still, the equations ( 22) are non linear and the proof in [START_REF] Komorowski | Macroscopic evolution of mechanical and thermal energy in a harmonic chain with random flip of velocities[END_REF] demands the application of Wigner distributions techniques in order to control the separate evolution of the thermal and mechanical energy.

For the anharmonic chain with velocity flip, the corresponding macroscopic equations are given by [START_REF] Olla | Microscopic derivation of an adiabatic thermodynamic transformation[END_REF]29]

∂ t r = 1 2γ ∂ 2 x τ (r, T ) c v (r, T )∂ t T = ∂ x (κ(r, T )∂ x T ) + 1 2γ (∂ x τ (r, T )) 2 , (23)
where τ (r, T ) is the thermodynamic tension at temperature T and volume r, c v (r, T ) the corresponding heat capacity. The thermal conductivity κ(r, T ) may depend on r in a non explicit way, as we do not have the same symmetries as in the rotor model. Notice that in this case, thanks to the noise in the dynamics, the current of the volume strain is j r 0,1 = -p 1 and it has an explicit fluctuation-dissipation decomposition of the (20) type:

j r 0,1 = -p 1 = 1 2γ Lp 1 - 1 2γ (V (r 1 ) -V (r 0 )) , (24)
that explains the first of the equations (23).

Dynamics with 3 conserved quantities

There are dynamics with 3 conserved quantities that evolve macroscopically in the diffusive space-time scaling. In [START_REF] Komorowski | Diffusive propagation of energy in a non-acoustic chain[END_REF] we study a harmonic chain where the potential energy does not depend on the volume strain or the tension of the system, but on its bending or curvature. This implies that we add springs between next nearest neighbor particles with negative potential, such that the total potential energy is given by

1 2 i (q i+1 -q i-1 -2q i ) 2 . ( 25
)
To the Hamitonian dynamics we add a random exchange, with intensity γ, of velocities between nearest neighbor particles. It results that there are three conserved quantities:

k i = q i+1 -q i-1 -2q i , p i , e i = p 2 i 2 + k 2 i 2 (26)
We call k i the curvature or bending. In [START_REF] Komorowski | Diffusive propagation of energy in a non-acoustic chain[END_REF] we prove that the corresponding empirical distributions converge to k(t, x), p(t, x), e(t, x) solutions of the diffusive system

∂ t k = -∂ 2 x p ∂ t p = ∂ 2 x k + γ∂ 2 x p ∂ t T = 1 γ ∂ 2 x T + γ (∂ x p) 2 , (27)
where the temperature profile is defined as T (t, x) = e(t, x) -p 2 (t,x)

2

. We can see this as a diffusive perturbation of the Bernoulli's beam wave equation ∂ 2 t k = -∂ 4 x k. Notice the similar structure as in equations ( 17) and [START_REF] Komorowski | Macroscopic evolution of mechanical and thermal energy in a harmonic chain with random flip of velocities[END_REF], with a heating term γ (∂ x p) 2 .

It is an open question if similar macroscopic equations will hold for a non linear dynamics with potential V (k i ), without any stochastic term. Numerical dynamical simulations have been inconclusive about this possible diffusive behaviour.

Non-equilibrium stationary states

Stationary temperature profiles and up-hill diffusion in rotor model

In this section we review some results concerning the non-equilibrium stationary states for the rotors model contained in [START_REF] Iacobucci | Stationary non-equilibrium states in rotors models[END_REF], more details can be found there. Let us consider the finite dynamics on N + 1-rotors as in [START_REF] Bálint | On the limiting Markov process of energy exchanges in a rarely interacting ball-piston gas[END_REF], where we add boundary forces or heath baths such that gradients are imposed on the conserved quantities in the corresponding stationary state. In order to establish a gradient in the temperature profile, we apply at the boundary two Langevin heat baths at different temperatures T L , T R , while constant forces τ L , τ R act respectively on the first rotor on the LHS and last rotor on the RHS. The equations of motion read as:

ṙi (t) = p i (t) -p i-1 (t), 1, . . . , N, ṗi (t) = V (r i+1 (t)) -V (r i (t)), i = 2, . . . , N -1, dp 0 (t) = (τ L + V (r 1 (t)) -γp 0 (t)) dt + 2γT L dw L (t), dp N (t) = (τ R -V (r N (t)) -γp N (t)) dt + 2γT R dw R (t), (28)
where w L (t), w R (t) are two independent Wiener processes.

We expect here that an hydrodynamic limit holds for the empirical profile distribution [START_REF] Kipnis | Scaling Limits of Interacting Particle Systems[END_REF] scaling with = N -1 , with the equation ( 17) in x ∈ [0, 1] provided with the boundary conditions

p(t, 0) = τ L γ , T (t, 0) = T L , p(t, 1) = τ R γ , T (t, 1) = T R .
As t → ∞ the system, at fixed size N , should approach a stationary state µ N ss depending in principle on T L , T R , τ L , τ R and γ. This probability distribution is called non-equilibrium stationary state (NESS) and only in the case T L = T R = β -1 and τ L = τ R = τ it coincides with the equilibrium measure ν β,p with p = γ -1 τ .

From the mathematical side, the study of the NESS and its asymptotic properties as N → ∞ is harder than the non stationary behavior. This is because in the NESS the time scales are hidden. For the rotors model, even the existence of the NESS is an open problem, and only recently there have been some progress for N = 2 and 3 [START_REF] Cuneo | On the relaxation rate of short chains of rotors interacting with Langevin thermostats[END_REF][START_REF] Cuneo | Non-equilibrium steady state and subgeometric ergodicity for a chain of three coupled rotors[END_REF].

Assuming the existence of the NESS, by stationarity the expectation of the currents of the conserved quantities have to be homogeneous along the chain, i.e. denoting with < • > ss the expectation with respect to µ N ss , we have

J p N :=< j p i,i+1 > ss , J e N :=< j e i,i+1 > ss , i = 1, . . . , N -1, (29)
are constant in i. Taking into account also the boundary currents we have

J p N = -τ R + γ < p N > ss = τ L -γ < p 0 > ss , J e N = γ T L -< p 2 0 > ss -τ L < p 0 > ss = γ < p 2 N > ss -T R + τ R < p N > ss . (30)
The diffusive behaviour implies that J p N , J e N ∼ O 1 N , and we expect that lim

N →∞ N J p N = J p = -D p (T ss (x))∂ x p ss (x) lim N →∞ N J e N = J e = -D p (T ss (x))∂ x p ss (x) 2 2 -κ(T ss (x))∂ x T ss (x), (31)
where p ss (x), T ss (x) are the stationary solutions of equations ( 14) with boundary conditions

p ss (0) = τ L γ , T ss (0) = T L , p ss (1) = τ R γ , T ss (1) = T R .
Notice that the energy current is the sum of the heat current J Q (x) = -κ(T ss (x))∂ x T ss (x), and the

mechanical energy current -D p (T ss )∂ x p 2 ss 2
. They can be of opposite signs, giving rise to the phenomenon of uphill diffusion (J e of the same sign as the gradient of temperature, cf. [START_REF] Krishna | Uphill diffusion in multicomponent mixtures[END_REF]).

Some other relations can be obtained from [START_REF] Sasada | Spectral gap for stochastic energy exchange model with non-uniformly positive rate function[END_REF]. By multiplying the first equation by p ss (x) and subtracting the second we have

p ss (x)J p -J e = κ(T ss (x))∂ x T ss (x) = -J Q (x), (32)
while the second derivative of T ss (x) must satisfy

J p ∂ x p ss (x) = κ(T ss (x))∂ xx T ss (x) + κ (T ss (x)) (∂ x T ss (x)) 2 . ( 33
)
Equations ( 31) predict a maximum for the temperature profile T ss (x) inside the interval (0, 1) higher than the boundary temperatures T L , T R . In fact, without losing generality, assume τ R = τ > 0 and τ L = 0, then p ss (x) ≥ 0, J p < 0 and p ss (x) is strictly increasing from p ss (0) = 0 to p ss (1) = τ /γ. Consequently, from (32), we can have only one stationary point for T ss , and by [START_REF] Varadhan | Nonlinear diffusion limit for a system with nearest neighbor interactions II, Asymptotic problems in probability theory: stochastic models and diffusions on fractals[END_REF] it must be a maximum, that we denote by x max , that must satisfy p ss (x max )J p = J e . This implies that J p and J e are of the same sign and, if J e = 0, the strict increase property of p ss implies that x max must be inside the interval (0, 1). Since there are no other stationary points, the maximal temperature T ss (x max ) must be higher than the temperatures at the boundaries.

From [START_REF] Varadhan | Nonlinear diffusion limit for a system with nearest neighbor interactions II, Asymptotic problems in probability theory: stochastic models and diffusions on fractals[END_REF], a flex point x f lex of T ss (x) must satisfy the relation

-D p (T ss (x f lex )) (∂ x p ss (x f lex )) 2 = J p ∂ x p ss (x f lex ) = κ (T ss (x f lex )) (∂ x T ss (x f lex )) 2
It follows that such flex points can exist only around values where κ(T ) is a strictly decreasing function of T .

Another qualitative property of the solution can be seen from the first of [START_REF] Sasada | Spectral gap for stochastic energy exchange model with non-uniformly positive rate function[END_REF]. i.e. we have that ∂ x p ss (x) = -J p D p (Tss(x)) . There is a numerical evidence that D p (T ) is a decreasing function of T , so we have that ∂ x p ss (x) is proportional to T ss (x) α for some α. It follows that

∂ 2 x p ss (x) = J p (D p ) (T ss (x)) D p (T ss (x)) 2 ∂ x T ss (x), (34)
that implies a flex point for p ss (x) at the point of maximum temperature.

The energy current J e can present a negative linear response with respect to T R -T L , due to the decrease of D p (T ) as a function of the temperature. In fact, in the case τ R > 0, τ L = 0 and T R ≥ T L , we have that

∂ x p 2 ss (x) 2
> 0, increasing T R will increase the whole profile T ss (x). This may create a positive increase to

-D p (T ss (x))∂ x p 2 ss (x) 2
larger that the negative increase of J q = -κ(T ss (x))∂ x T ss (x). From the stationary equation we can compute the entropy production of the stationary state, that we can define as

Σ = T -1 R -T -1 L J e -γ -1 T -1 R τ R -T -1 L τ L J p (35)
It turns out that Σ > 0 and that it is equal to

Σ = 1 -1 κ(T ss (x)) T 2 ss (x) (∂ x T ss (x)) 2 + D p (T ss (x)) T ss (x) (∂ x p ss (x)) 2 dx. (36)
Notice that this expression coincide with the time derivative of Clausius entropy S given in [START_REF] Iubini | Coupled transport in rotor models[END_REF]. Dynamical simulations of the rotors chain in the stationary state under an exterior torque τ were first perfomed in [START_REF] Iacobucci | Negative thermal conductivity of chains of rotors with mechanical forcing[END_REF]. The resulting stationary profiles of T ss (•) and p ss (•) are reported in Figures 1 and2 respectively, for different choices of boundary temperatures, and τ L = 0 and τ R = τ .

The profiles of temperatures in Figure 1 present a maximum inside the interval with temperatures much higher than at the boundaries. Two flex points are presents that get closer as the temperature at the border decreases.

In Figure 2 are the corresponding profiles of p ss (x). The maximum of the temperature corresponds to the flex point of p ss , in agreement with (34). In Figure 3 is the energy current J e as function of τ R = τ (with τ L = 0) for different sizes of the system N . In black is the case of same temperature, while in red the curve when T R is rised. Notice in Figure 3 that for τ large enough the curves cross, which implies a negative response to the temperature gradient. This is in agreement with the remark made above, as consequence of the decrease of D p (T ) with T .

The plots in Figures 1, 2 and 3 are obtained by direct dynamical simulations of the NESS. In [START_REF] Iacobucci | Stationary non-equilibrium states in rotors models[END_REF] we attempt to solve numerically equations [START_REF] Sasada | Spectral gap for stochastic energy exchange model with non-uniformly positive rate function[END_REF] after having estimated D p (T ) and κ(T ) with dynamical simulations of the system in equilibrium. Agreement of the corresponding plots will confirm the correctness of the heuristic coming from the linear response theory.

For a general review about up-hill diffusion, see [START_REF] Krishna | Uphill diffusion in multicomponent mixtures[END_REF]. This phenomenon can also appear in models with phase transitions, even if there is only one conserved quantity [START_REF] Colangeli | Microscopic models for uphill diffusion[END_REF].

Stationary states for harmonic chains with random dynamics

Mathematical rigorous results can be obtained for the NESS of the harmonic oscillators dynamics with random exchange of velocities between nearest neighbor particles, such that kinetic energy is conserved. The conserved quantities are the volume and the total energy, and the non-stationary evolution is governed in the diffusive scaling by [START_REF] Komorowski | Macroscopic evolution of mechanical and thermal energy in a harmonic chain with random flip of velocities[END_REF]. In [START_REF] Komorowski | An open microscopic model of heat conduction: evolution and non-equilibrium stationary states[END_REF] we study this dynamics when the system has N harmonic oscillators connecting N + 1 particles, with Langevin heat baths attached at the first left particle and the last right particle, and a constant force τ is attached to the last right particle (see Figure 4.2). The system is driven out of equilibrium by the presence of the external force τ , and the stochastic part of the dynamics has only equilibrium states with 0 average velocities. Thanks to the stochastic dynamics in the bulk, the NESS exists. For τ = 0 this NESS was studied in [START_REF] Bernardin | Transport Fourier's Law for a Microscopic Model of Heat Conduction[END_REF]. In [START_REF] Komorowski | An open microscopic model of heat conduction: evolution and non-equilibrium stationary states[END_REF] we prove the hydrodynamic limit in the NESS, and the stationary profiles of volume stretch r ss (x) and temperature T ss (x) satisfy the equations

∂ 2 x r ss (x) = 0, ∂ 2 x T ss (x) = -2 (∂ x r ss (x)) 2 ,
r ss (0) = 0, T ss (0) = T L , r ss (1) = τ, T ss (1) = T R .

(37) These can be explicitly solved obtaining r ss (x) = τ x and Reprinted from [START_REF] Iacobucci | Negative thermal conductivity of chains of rotors with mechanical forcing[END_REF].

T ss (x) = τ 2 x(1 -x) + (T R -T L )x + T L , x ∈ [0, 1],
q 0 q 1 q n q x-1 q x q x+1 r x T L T R τ Figure 4
. Chain of harmonic oscillators with random flip pf velocities sign, heat bath and tension applied on teh right hand side. Reprinted from [START_REF] Komorowski | An open microscopic model of heat conduction: evolution and non-equilibrium stationary states[END_REF].

that shows again an heating phenomena at the center of the system. The stationary energy current can also be calculated and gives

J e = - 1 4γ (T R -T L ) - τ 2 2γ . (38)
So if uphill diffusion is possible, no negative response to temperature gradient can happen in this system, as these were due to the non-linearities in the temperature of the diffusivities in the rotor chain. Also there are noflex points in these profiles of temperature, as also these were due to the temperature dependence of the thermal conductivity.

5. The two steps approach: weak coupling limits, kinetic limits and hydrodynamic limits

As we mentioned in Section 3, one of the major difficulties in order to obtain the diffusive equations in the hydrodynamic limit is that it involves a simultaneous scaling of space and time. One way to simplify the problem is to introduce a small parameter in the dynamics that makes the interaction weak (weak coupling) or the collisions rarefied (kinetic limit), in order to break the procedure in two steps:

-a weak coupling or kinetic limit where it is obtained an autonomous mesoscopic stochastic dynamics, -a subsequent hydrodynamic limit for these stochastic dynamics.

Weak coupling limit

In the case of dynamics that conserve macroscopically only energy, some progress have been obtained in the past years in the weak coupling limit, even though the full program is not yet complete.

Consider the generic Hamiltonian dynamics qi (t) = p i (t), ṗi (t) = δV (q i+1 (t) -q i (t)) -δV (q i (t) -q i-1 (t)) + U (q i (t)),

where δ > 0 is a small parameter, q i ∈ M where M is some d-dimensional manifold, and U is a potential defined on M . With q i+1 -q i is intended some distance defined on M . When δ = 0, there will be no interaction between the particles and no exchange of energy. When δ > 0 there will be an instantaneous energy current between the particles given by

δj e i,i+1 (t) = -δp i (t) • V (q i+1 (t) -q i (t)). (40)
Notice that in any equilibrium the average of j e is null. In order to see a energy diffusion in the limit as δ → 0, we have to look at a time scale δ -2 t. In fact

e i (δ -2 t) = δ δ -2 t 0 (j e i-1,i (s) -j e i,i+1 (s)) ds (41)
Let us assume that a central limit theorem is valid for the uncoupled dynamics (δ = 0), and somehow stable for small δ > 0. The first step consists in proving that e i (δ -2 t) converges in law to an autonomous stochastic dynamics of energies:

e i (δ -2 t) -→ δ→0 E i (t), (42)
where the E i (t) satisfy the stochastic differential equations

dE i (t) = dJ i-1,i (t) -dJ i,i+1 (t), dJ i,i+1 (t) = α(E i (t), E i+1 (t)) dt + √ 2σ 2 (E i (t), E i+1 (t)) dB i (t), (43)
where B i (t) are independent standard Wiener processes, and σ 2 are the variances of the energy currents j e i,i+1 (t) in the CLT of the uncoupled dynamics:

σ 2 (E 1 , E 2 ) = ∞ 0 j e 1,2 (t)j e 1,2 (0) E1,E2 dt. (44)
Here • E1,E2 denotes the expectation with respect to the uncoupled dynamics of two particles in the microcanonical equilibrium at fixed energies E 1 and E 2 . The functions α(E 1 , E 2 ) are antisymmetric and are defined by

α(E 1 , E 2 ) = (∂ E1 -∂ E2 ) σ 2 (E 1 , E 2 ) + σ 2 (E 1 , E 2 ) Z (E 1 ) Z(E 1 ) - Z (E 2 ) Z(E 2 ) , (45)
where Z(E) is the volume on the microcanonical manifold of energy E of the single uncoupled particle. The equations (44) define a stochastic dynamics reversible with respect to the stationary measures:

dν β = i Z(E i )e -βEi Zβ dE i (46)
A proof of this first step, i.e. the limit (42), would require that the uncoupled dynamics is chaotic enough such that a CLT theorem is valid and is stable for small perturbations.

In [START_REF] Dolgopyat | Energy transfer in a fast-slow Hamiltonian system[END_REF] this is proven for particles moving (deterministically) as geodesic flow in a manifold M with strictly negative curvature, with dynamical system techniques. For anharmonic oscillators with a stochastic noise acting on the velocities and conserving energy, the first step has been proven in [START_REF] Liverani | Toward the Fourier law for a weakly interacting anharmonic crystal[END_REF], using hypocoercive estimates. One particular case of [START_REF] Liverani | Toward the Fourier law for a weakly interacting anharmonic crystal[END_REF] is the harmonic case (V and U quadratic), where σ

2 (E 1 , E 2 ) = γ -2 E 1 E 2 , and α(E 1 , E 2 ) = γ -2 (E 1 -E 2 )
, where γ is the intensity of the noise in the dynamics.

The second step consists in obtaining the hydrodynamic limit for the energy evolution of the stochastic dynamics given by (43), i.e. that we have the convergence of the empirical distributions

i δ i (dx)E i ( -2 t) -→ →0 E(t, x)dx, (47)
where

∂ t E = ∂ x D(E)∂ x E , (48)
with the energy diffusivity of the stochastic dynamics given by D = C-1 v κ, where Cv is the variance of the energies E i under dν β and the conductivity κ is given by the corresponding Green-Kubo formula

κ(β -1 ) = β 2 i ∞ 0 α(E i (t), E i+1 (t)), α(E 0 (0), E 1 (0)) β dt. (49)
The relation between κ(T ) and the thermal conductivity of the original dynamics with δ > 0, i.e. κ δ (T ) defined as in [START_REF] Iacobucci | Negative thermal conductivity of chains of rotors with mechanical forcing[END_REF], is studied in [START_REF] Bernardin | Green-Kubo Formula for Weakly Coupled Systems with Noise[END_REF], and it turns out that

κ δ (T ) = κ(T )δ + o(δ). (50)
The reversible stochastic dynamics ( 43) is a version of the conservative Ginzburg-Landau dynamics considered by Varadhan in [START_REF] Varadhan | Nonlinear diffusion limit for a system with nearest neighbor interactions II, Asymptotic problems in probability theory: stochastic models and diffusions on fractals[END_REF]. With respect to [START_REF] Varadhan | Nonlinear diffusion limit for a system with nearest neighbor interactions II, Asymptotic problems in probability theory: stochastic models and diffusions on fractals[END_REF], this dynamics is restricted on R Z + , and we call the energy Ginzburg-Landau dynamics. In order to apply the method of [START_REF] Varadhan | Nonlinear diffusion limit for a system with nearest neighbor interactions II, Asymptotic problems in probability theory: stochastic models and diffusions on fractals[END_REF] we need a lower bound on the spectral gap for the generator on the corresponding finite dimensional dynamics, i.e. it should be bounded below by CN -2 , where N is the dimension of the system, for some constant C than may depend on the energy but not on the dimension. When this spectral gap bound can be proven, the second step can be perfomed [START_REF] Liverani | Diffusive scaling in energy Ginzburg-Landau dynamics[END_REF]. This is the case for the (43) arising from the weak coupling limit in the anharmonic chain with noise (cf. [START_REF] Liverani | Toward the Fourier law for a weakly interacting anharmonic crystal[END_REF]) with some conditions on the pinning potential U . In fact if Hess U (0) > 0, it follows that σ 2 (E 1 , E 2 ) = E 1 E 2 G(E 1 , E 2 ), with G ≥ c > 0. This implies that the generator of (43) has a spectral gap bound (this can be proven following the argument in [START_REF] Olla | Macroscopic energy diffusion for a chain of anharmonic oscillators[END_REF]). For the energy Ginzburg Landau dynamics emerging from the deterministic dynamics of the geodesic flows [START_REF] Dolgopyat | Energy transfer in a fast-slow Hamiltonian system[END_REF], we have that

σ 2 (E 1 , E 2 ) ∼ E 1 E 2 E 3/2 1 + E 3/2 2 . ( 51
)
Unfortunately under a behaviour like (51) it is not clear that a spectral gap bound will hold, it certainly depends on the energy. At this time the step 2 remains an open problem when starting from a purely Hamiltonian deterministic dynamics.

Kinetic limit

A different two step approach consists in studying models where energy is exchanged between particles through collisions that are rarefied because of constraints in the geometry of the system. Time is scaled in such way that on a unit time there is, in average, a finite number of collisions per unit time. In this sense the first step is similar to the Boltzmann-Grad limit.

A typical model considered in this approach (cf. [6] [14]) is given by a chain of Sinai's billiards, where each particle is confined in one billiard, but it can collide with a particle in a neighbour billiard through a small window. The size δ of this window is the small parameter of the limit. The energy is the only conserved quantity, and it is exchanged through these collision. Because of the chaoticity of the billiard, after time rescaling as δ → 0, the difference in the collision times, conditioned on the energy of each billiard, became independent. Consequently it is expected that the energies per particle e i (δ -1 t) converge to a Markov jump process E i (t) whose generator is given by

LF ({E j }) = i 1 0 dαΛ(E i , E i+1 , α) [F (T i,i+1,α {E j }) -F ({E j })] (T i,i+1,α {E j }) k = α(E i + E i+1 )δ (k=i) + (1 -α)(E i + E i+1 )δ (k=i+1) + E k δ (k =i,i+1) . (52)
This means that at random times, exponentially distributed with intensity Λ(E i , E i+1 , α), the total energy of the site i and i + 1 is redistributed in the two sites with proportion α and 1 -α. The precise form of the rate function Λ can be found in [14] and [START_REF] Sasada | Spectral gap for stochastic energy exchange model with non-uniformly positive rate function[END_REF], but what matters is that Λ(E i , E i+1 , α) ∼ (E i + E i+1 ) 1/2 .

Unlike the weak coupling limit, no rigorous results about this first step exist at the moment starting from the deterministic dynamics. Some partial attemps and a detailed discussion of the problem can be found in [START_REF] Bálint | On the limiting Markov process of energy exchanges in a rarely interacting ball-piston gas[END_REF].

The second step will be the hydrodynamic limit on the stochastic dynamics generated by (52) in order to obtain the diffusive equation for the energy. This is a non-gradient dynamics where in principle could be applied Varadhan's approach [START_REF] Varadhan | Nonlinear diffusion limit for a system with nearest neighbor interactions II, Asymptotic problems in probability theory: stochastic models and diffusions on fractals[END_REF]. With respect to the stochastic dynamics emerging in the weak coupling case from a purely mechanical model, there exists a proof of the spectral gap bound (cf. [START_REF] Sasada | Spectral gap for stochastic energy exchange model with non-uniformly positive rate function[END_REF]), necessary in order to apply Varadhan's method. Still there are other intrinsic difficulties that did not allow yet to prove the hydrodynamic limit.

Figure 1 .Figure 2 .

 12 Figure 1. Stationary temperature profiles in rotors dynamics, with τ R = τ, τ L = 0 and different values of temperatures for the Langevin heat bath. Reprinted from [15].

Figure 3 .

 3 Figure 3. Energy current J e as function of τ R = τ , for τ L = 0. Reprinted from [15].

Acknowledgements

This article contains ideas and results developped in collaboration with Cedric Bernardin, François Huveneers, Alessandra Iacobucci, Tomasz Komorowski, Joel Lebowitz, Carlangelo Liverani, Makiko Sasada, Marielle Simon, Gabriel Stoltz.

This work is supported by the grant ANR-15-CE40-0020-01 LSD of the French National Research Agency.