D Chadrin

F Degrutere

G Veidig

J Seyssaud
email: jeremy.seyssaud@cea.fr

N Mahjoubi

P Devaux

Investigation Robot on Cables -15250

The Decontamination and Decommissioning (D&D) Operations take place in hostile environments to the human intervention. In this context, robotic systems are a solution to achieve these operations while minimizing human intervention. Among these operations, the monitoring of lengthwise plant has to be performed in places where the radioactive contaminations make hostile environments. These monitoring operations need to measure several physics parameters in several locations in a precise and safe manner. These repetitive tasks can be made by robots that are simple, small sized and easily adaptable. In this scope, a robotic system on cables has been designed to control and monitor the inside of lengthwise nuclear facilities. In lengthwise installations, the using of cables allows to increase the covered zone of the robots which is an essential quality in such places. Unlike cable-driven parallel robots used in workspaces of 2D or 3D controlled by redundant tensioning motors, the current robot moves in a 1D workspace. Thus a tractors cable controlled by one tensioning motor is used to drive the robot on carrier cables. In this paper, we will describe the first results of this work by bringing out the innovation of our system. Then, we will discuss the perspective to upgrade the actual system whose applications could be perfectly imagined outside the nuclear D&D industry.

INTRODUCTION

The French Nuclear Safety Authority fixes a typical radiological upper limit for radiation workers exposure to 20 mSv per year [START_REF] De | ASN report on the state of nuclear safety and radiation protection in France in 2013[END_REF]. When this limit is reached, a work should stop any activities inside of a nuclear installation. This limitation implies a need of important manpower and associated risks in order to deal with decommissioning dockyards and subsequently, an increase of the related costs. Considering this regulatory limit and the high dose rates in some nuclear installation, the use of robotic systems to assist or replace human operations seems to be a solution to conduct these operations in a safe manner [START_REF] Bogue | Robots in the nuclear industry: a review of technologies and applications[END_REF]. Among these operations in particular, conducting monitoring and control campaigns inside of lengthwise installations [START_REF] Agency | Maintenance, surveillance and inservice inspection in nuclear power plants -Safety Guide[END_REF]; thus, independent of the radiation protection constraint, justifying the use of robots, these occasional or periodic campaigns need a large amount of measurements. Further, in view of the large size of the facilities, robots will allow us to reduce the cost of these monitoring campaigns, increase the accuracy of the measurements and ensure the robustness and repeatability of such operations. The most mobile robots typically use wheeled, tracked or legged to move on the ground. One of the drawbacks of this kind of mobile robots is their difficulty to access the entire site due to range limitations. Indeed, most the decommissioning sites are large in terms of size, most monitoring operations can encounter a difficulty, or to be caught on obstacles. Furthermore, the power supplies are also responsible of the range limitations of this kind of robots. Indeed, if the robots is tethered (where a power cord is trailed behind the robot), the cord has a limited length and may snag on obstacles. Likewise, if the robots carry a power source (such as a battery) with them, it is often heavy and has to be recharged regularly. With the aim of minimizing such drawbacks and increasing the efficiency of the robots operating in large facilities, we have to increase its range of mobility [START_REF] Bruzzone | Review article: locomotion systems for ground mobile robots in unstructured environments[END_REF]. Thus, one solution is to use cables deployed in height inside the facility to monitor. The use of cables allows increasing the ability of the robots to operate in large areas while avoiding the interference with the other activities of the facility. The actuator of the robot is a motor placed outside the facility. This type of robot is linked to motor by traction cables to ensure its movement on carrier cables. In this paper, we will present an illustration of these developments, highlighting the innovative aspects of this solution. Then, we will focus on discussing developments of this solution to meet future requirements inspections.

APPLICATION TO A TELEVISUAL INSPECTION SYSTEM

The original need behind these developments is the inspection of the concrete structure of a nuclear installation [START_REF] Masson | Marcoule UP1 reprocessing plant first steps towards decommissioning[END_REF]. These inspections focus on the upper internal parts located at a height of 5 meters. On Figure 1, we can see a representation of this kind of installation on which we can note the lengthwise aspect of the facility. The chosen solution is a robot on cable that can browse the site on its greater length (between 40 and 65 meters). The periodical inspection must evaluate visually the concrete structure and detect the eventual surface defects. The monitoring of the detected failures will allow following their growth. A robot using cables to move inside this kind of facility has two advantages. First of all, this technology allows us to inspect the entire installation in expedited manner despite its large size. Secondly, it avoids the interference with obstacle elements arranged on the ground or with others mobile ground robots used for maintenance tasks. Our system of inspection is formed by two batches of equipment. The first one is installed permanently on the civil engineering structures prior to any inspection. It is used to introduce the inspection robot into the installation and to define its path inside it. The second batch is formed of equipment introduced temporarily inside the installation to monitor. Thus we will present our robot system by distinguishing the equipment according their properties permanent or temporary.

Permanent Equipment

The permanent equipment articles are formed by a pair of intervention boxes (Figure 2) located on either side of the installation and of cables. The intervention boxes contain the mechanical devices that are used to strain the cables and govern the movement of the inspection robot. Given the expected nuclear application, the design of the intervention boxes must perform a containment function when the robot is manipulated for the introduction and extraction of the installation. The implementation of these intervention boxes requires performing coring within the walls of the facility that are then blocked by biological protections sized to allow the subsequent intervention of operators nearby in a safe manner. The other permanently installed equipments are the cables used to support and guide the inspection robot when an inspection is performed. These textile cables have a low linear density (4 to 5 times lower than the steel) and a high tensile strength modulus. These mechanical properties allow straining the cables to maintain a sufficient rigidity to reduce the deflection under the weight of the inspection robot. The structure of these cables and its selected materials reflect its high quality in terms of good tensile strength (about 39.2 kilonewtons similar to a steel cable of equivalent diameter1 ). Furthermore, this property ensures an optimal operation of the inspection robot by securing its path inside the facility avoiding any contact with the structure. That is why the impact of ionizing radiation on the characteristic of tensile strength has been looked in detail particular by performing various radiation tests. In operation, these cables are tensioned to a tension of 2200 dekanewtons in a radiating environment. The results of these radiation tests are reported in In this scope, a set of cables have been irradiated with a cumulative dose equivalent to an exposition of 1, 5 and 15 years with respectively 1270, 6350 and 19050 Gy. Tensile testings have been performed on this set of cables until the break of specimens to evaluate the tensile strength at failure. The type of cables used is the one that the tensile strength after irradiation is consistent with that of non-irradiated cable (in accordance with the manufacturer's data).

In addition, the structure of theses textile cables ensures both safety factors relating to the intervention in the nuclear industry (electrical and fire risk). Indeed textiles are insulators which forewarned of the risk of spread lightning. A special attention has been put to this aspect due to the fact that the intervention boxes are situated outside the installation. The flame propagation is also eliminated due to the particular structure of the cables that avoid the fire risk.

Temporary Equipment

In addition to these two devices installed prior to any investigation (intervention boxes and cables), this inspection system is composed of three devices used temporarily. It is an inspection robot, a storage box associated and a supervisory staff to communicate with the inspection robot. The role of the inspection robot is to carry the set of sensors in the area to inspect. It is placed on the supporting cables previously stretched in the installation. The so-called "tractors" and "carriers" cables are those mentioned above.

The design of the truck of the inspection robot is performed in a way that combines both stiffness and light weight of the structure. This structure houses all the relevant sensors to the mission without that its own weight be a large percentage of the global weight supported by the cables. The truck is fixed on the cable by means of a fly-off to prevent the fall of the robot in case of failure of one of the two "carries" cables system. Figure 3 shows a configuration of the robot equipped with a digital camera. This inspection robot must be inserted and removed safely from the nuclear facility where the investigation should be made. For this purpose, a storage case with a metal frame and Plexiglas walls has been developed (Figure 4). Its design allows preserving the containment facility during docking or leaving the investigation robot ensuring tightness. It is equipped with round glove that allow maintenance on the robot and control the level of contamination of its structure.

Figure 4 -Storage case

Once the robot introduced in the facility, communication and data transfer between the robot and the checkpoint control robot is done via radio link. For this purpose, a radio antenna is positioned at the intervention box when the robot is introduced and removed during its extraction. However, in case of failure of the communication system, it is always possible to control the robot via the wired connection to bring it back to the intervention box.

On this basis, the perspectives for the actual system will be focused on the integration of additional equipment to meet the needs of specific radiological mapping.

FUTURE WORK

Following this first work that led to integrate the operational constraints, a study is underway to improve this inspection system on cables by the adding of additional sensors. This approach is conducted in the interest of standardization of this system especially by having a capacity of adaptability to the specific cases encountered and with a lower cost. However the notion of standardization is complex, given the diversity of cases that need to be addressed in terms of installation configuration and measures to be performed. Thus this standardization effort will be treated at the appropriate level of system decomposition. In addition to this standardization effort, the modularity is an important property that the inspection system must provide. The modularity is the ability to integrate and use technology independently. It should be translated both at the hardware and software level. At the hardware level, the future developments have to be conducted in the objective to reach the optimal in terms of weight and size because a compact size and a low weight allow it to be deployed quickly, and is easy to clean up and store.

The sizing of the equipment should be examined according to two aspects. The first aspect concerns the choice of equipment that we need to integrate for an inspection mission. Indeed, this choice will dictate the size and weight of the robot fixed to the carrier cables. The second aspect concerns the need to adapt the components of the robot to a nuclear environment. As often stated, the weakest parts of robots system are the electronic components and particularly the semiconductors that should be investigated more precisely. The reliability under gamma ray radiation is apprehended first by using the manufacturer's data concerning the behavior of components under radiation when those are available or by performing our own gamma ray radiations. Then the required of a thickness lead plate to protect the electronic components from gamma rays is discussed. The hardening of the structure must be undertaken by taking into account the implication on the mobility and the maintainability of the robot.

In the initial project, the following sensors have been integrated for inspection tasks in a waste storage facility:

• A digital camera with a laser pointer. A series of photo exposed parts of the structure is used to assess the surface of the concrete. This allows to visualize cracks of 0.3 millimeters of length and depth • An accelerometer. This sensor is used to find the angles of pitching and rolling of the inspection robot. It can also measure the acceleration of the robot which allows calculating its position during an inspection. • Thermo-hygrometer • Radiometer More and more, the decommissioning scenarios are evaluated using virtual reality modeling. This approach allows taking into account the specific constraints of an operation without expose workers. Theses simulations can include the data obtained from a gamma camera (location of a source, activities). The radiological characterization of large facilities is an example of use of inspection robots on cables in order to optimize the operations that should take place into it. The radiological characterization of large facilities is an example of use of inspection robots on cables in order to optimize the operations that will for diversifying the environment of application by developing its modularity and for allowing the integration of rich features embedded in the system.

Figure 1 -

 1 Figure 1 -Representation of a storage site for radioactive waste

Figure 2 -

 2 Figure 2 -Intervention boxes

Figure 3 -

 3 Figure 3 -Inspection robot placed in the "carriers" cables

Table 1 .

 1

	Equivalent time exposure	Cumulative dose	Tensile strength
	(in year)	(in Gy)	(in daN)
	1	1270	4000.4 ± 2%
	5	6350	4166 ± 3%
	15	19050	4284 ± 5%

Table 1 -Results of tensile strength on irradiated cables

 1

The cable has a 7.7 mm diameter.

take place inside. The integration of a gamma camera must be performed taking into account particularly the principle of minimizing the mass of the inspection robot [START_REF] Gal | CARTOGAM: a portable gamma camera[END_REF][START_REF] Gmar | GAMPIX: A new generation of gamma camera[END_REF].

Figure 5 -Source location by a gamma camera

Nowadays, 3D scanner technologies of large facility are available. These technology can be used to create CAD models of the inside installation. The collect data from a 3D scanner can then be used in a 3D virtual reality showroom facility. We find here the concept of virtual simulation of the dismantling scenarios. Furthermore, when we add to this virtual model, dataset as dose rates and source locations obtained by a gamma camera, we obtain a better knowledge of the initial state of a nuclear facility. These elements are important in the objectives to reduce cost and guarantee the schedules of D&D operations. A large amount of 3D scanner devices are available on the market with specific characteristics. And the integration of such technology has to done with the objectives to always minimize the size and weight of the inspection robot.

The use of robot systems implies the need to monitor its internal components. For instance, the temperature of the motors must to be known during an operation to avoid failure of the motor. In the case of our inspection robot as noted previously, the motor used to move the robot is outside the facility to inspect. However as noted previously, the behaviors of the electronic components have to be particularly monitored. This is why we schedule to integrate a dosimeter to prevent any failure of the electronic and thus anticipate its maintenance.

CONCLUSIONS

In this paper, the development of an inspection robot on cables suited for monitoring tasks in nuclear facilities is presented. The use of a robot on cables has various advantages: capacity to monitor lengthwise facilities while avoiding the possible constraints on the ground and with a secure path whose positions are perfectly controlled. This system ensures robustness and repeatability of measurements in terms of location while achieving a high level of safety and security. If the use presented in this paper is for an application in a hostile environment due to the radiation and contamination, we can perfectly imagine use this system in a less hostile environment. Indeed this system is suited for monitoring tasks in lengthwise facilities or with hard access and in which measurements are difficult to realize. The presented illustration provides a first level of standardization in severe environments: cables, transfer carriage and containment equipment. Today, the goal is to improve the current system for example in term of range or


Vincent Andrieu
email: vincent.andrieu@gmail.com

Bayu Jayawardhana
email: bayujw@ieee.org

Laurent Praly
email: laurent.praly@mines-paristech.fr

† B Jayawardhana
email: b.jayawardhana@rug.nl

‡ L Praly

Transverse exponential stability and applications

We investigate how the following properties are related to each other: i) -A manifold is "transversally" exponentially stable; ii) -The "transverse" linearization along any solution in the manifold is exponentially stable; iii) -There exists a field of positive definite quadratic forms whose restrictions to the directions transversal to the manifold are decreasing along the flow. We illustrate their relevance with the study of exponential incremental stability. Finally, we apply these results to two control design problems, nonlinear observer design and synchronization. In particular, we provide necessary and sufficient conditions for the design of nonlinear observer and of nonlinear synchronizer with exponential convergence property.

Introduction

The property of attractiveness of a (non-trivial) invariant manifold is often sought in many control design problems. In the classical internal model based output regulation [START_REF] Isidori | Output regulation of nonlinear systems[END_REF], it is known that the closed-loop system must have an attractive invariant manifold, on which, the tracking error is equal to zero. In the Immersion & Invariance [START_REF] Astolfi | Invariance: A new tool for stabilization and adaptive control of nonlinear systems[END_REF], in the sliding-mode control approaches, or observer designs [START_REF] Andrieu | On the existence of Kazantzis-Kravaris / Luenberger Observers[END_REF], obtaining an attractive manifold is an integral part of the design procedure. Many multi-agent system problems, such as, formation control, consensus and synchronization problems, are also closely related to the analysis and design of an attractive invariant manifold, see, for example, [START_REF] De Persis | On the internal model principle in the coordination of nonlinear systems[END_REF][START_REF] Scardovi | Synchronization in networks of identical linear systems[END_REF][START_REF] Wieland | An internal model principle is necessary and sufficient for linear output synchronization[END_REF].

The study of stability and/or attractiveness of invariant manifolds and more generally of sets has a long history. See for instance [35, §16] and the references therein. In this paper, we focus on the exponential convergence property by studying the system linearized transversally to the manifold. We show that this attractiveness property is equivalent to the existence of positive definite quadratic forms which are decreasing along the flow of the transversally linear system. For constant quadratic forms and when the system has some specific structure, the latter becomes the Demidovich criterion which is a sufficient, but not yet necessary, condition for convergent systems [START_REF] Pavlov | Convergent dynamics, a tribute to Boris Pavlovich Demidovich[END_REF][START_REF] Pavlov | Uniform Output Regulation of Nonlinear Systems: A convergent Dynamics Approach[END_REF][START_REF] Rüffer | Convergent systems vs. incremental stability[END_REF]. On the other hand, if we consider the standard output regulation theory as pursued in [START_REF] Isidori | Output regulation of nonlinear systems[END_REF], the attractiveness of the invariant manifold is established using the center manifold theorem which corresponds to the stability property of the linearized system at an equilibrium point. Due to the lack of characterization of an attractive invariant manifold, most of the literature on constructive design for nonlinear output regulator is based on various different sufficient conditions that can be very conservative. In these regards, our main results can potentially provide a new framework for control designs aiming at making an invariant manifold attractive.

The paper is divided into two parts. In Subsection 2.1, we study a dynamical system that admits a transverse exponentially stable invariant manifold. In particular, we establish equivalent relations between:

(i) the transverse exponential stability of an invariant manifold;

(ii) the exponential stability of the transverse linearized system; (iii) the existence of field of positive definite quadratic forms the restrictions to the transverse direction to the manifold of which are decreasing along the flow. We illustrate these results by considering a particular case of exponential incremental stable systems in Subsection 2.2. Here, incremental stability refers to the property where the distance between any two trajectories converges to zero (see, for example, [START_REF] Angeli | A Lyapunov approach to incremental stability properties[END_REF][START_REF] Fromion | Connecting nonlinear incremental Lyapunov stability with the linearizations Lyapunov stability[END_REF][START_REF] Angeli | Further results on incremental input-to-state stability[END_REF]). For such systems, the property (i) ⇔ (iii) is used to prove that the exponential incremental stability property is equivalent to the existence of a Riemannian distance which is contracted by the flow.

In the second part of the paper, we apply the equivalence results to two different control problems: nonlinear observer design and synchronization of nonlinear multi-agent systems. In both problems, a necessary condition is obtained. Based on this necessary condition, we propose a novel design for an observer, in Subsection 3.1, and for a synchronizer, in Subsection 3.2.

In Subsection 3.1, we reinterpret the three properties (i), (ii) and (iii) in the context of observer design. This allows us to revisit some of the results obtained recently in [START_REF] Sanfelice | Convergence of nonlinear observers on R n with a Riemannian metric (part i). Automatic Control[END_REF] and [START_REF] Andrieu | Observability necessary conditions for the existence of observers[END_REF] and, more importantly, to show that the sufficient condition given in [START_REF] Sanfelice | Convergence of nonlinear observers on R n with a Riemannian metric (part i). Automatic Control[END_REF] is actually also a necessary condition to design an exponential (local) full-order observer.

Finally, in Subsection 3.2, we solve a nonlinear synchronization problem. In particular, we give some necessary and sufficient conditions to achieve (local) exponential synchronization of nonlinear multi-agent systems involving more than two agents. This result generalizes our preliminary work in [START_REF] Andrieu | On the transverse exponential stability and its use in incremental stability, observer and synchronization[END_REF]. Moreover, under an extra assumption, we show how to obtain a global synchronization for the two agents case.

It is worth noting that our main results are applicable to other control problems beyond the two control problems mentioned before. In a recent paper by Wang, Ortega & Su [START_REF] Wang | On Parameter Convergence of Nonlinearly Parameterized Adaptive Systems: Analysis via Contraction and First Lyapunovs Methods[END_REF], our results have been applied to solve an adaptive control problem via the Immersion & Invariance principle.

Main result 2.1 Transversally exponentially stable manifold

Throughout this section, we consider a system in the form ė = F (e, x) , ẋ = G(e, x)

where e is in R ne , x is in R nx and the functions F : R ne × R nx → R ne and G : R ne × R nx → R nx are C 2 . We denote by (E(e 0 , x 0 , t), X(e 0 , x 0 , t)) the (unique) solution which goes through (e 0 , x 0 ) in R ne × R nx at time t = 0. We assume it is defined for all positive times, i.e. the system is forward complete.

The system (1) above can be used, for example, to study the behavior of two distinct solutions X(x 1 , t) and X(x 2 , t) of the system defined on R n by ẋ = f (x)

(2)

Indeed, we obtain an (e, x)-system of the type (1) with

F (e, x) = f (x + e) -f (x) , G(e, x) = f (x) . (3)
This is the context of incremental stability that we will use throughout this section to illustrate our main results.

In the following, to simplify our notations, we denote by B e (a) the open ball of radius a centered at the origin in R ne .

We study the links between the following three notions. TULES-NL (Transversal uniform local exponential stability)

The system (1) is forward complete and there exist strictly positive real numbers r, k and λ such that we have, for all (e 0 , x

0 , t) in B e (r) × R nx × R ≥0 , |E(e 0 , x 0 , t)| ≤ k|e 0 | exp(-λt) . (4)
UES-TL (Uniform exponential stability for the transversally linear system)

The system

˙ x = G( x) := G(0, x) (5)
is forward complete and there exist strictly positive real numbers k and λ such that any solution ( E( e 0 , x 0 , t), X( x 0 , t)) of the transversally linear system

˙ e = ∂F ∂e (0, x) e , ˙ x = G( x) (6)
satisfies, for all ( e 0 , x

0 , t) in R ne × R nx × R ≥0 , | E( e 0 , x, t)| ≤ k exp(-λt)| e 0 | . (7)
ULMTE (Uniform Lyapunov matrix transversal equation) For all positive definite matrix Q, there exists a continuous function P : R nx → R ne×ne and strictly positive real numbers p and p such that P has a derivative d G P along G in the following sense

d G P ( x) := lim h→0 P ( X( x, h)) -P ( x) h (8)
and we have, for all x in R nx ,

d G P ( x) + P ( x) ∂F ∂e (0, x) + ∂F ∂e (0, x) P ( x) ≤ -Q (9) p I ≤ P ( x) ≤ p I . (10)
In other words, the system (1) is said to be TULES-NL if the manifold E := {(e, x) : e = 0} is exponentially stable for the system (1), locally in e and uniformly in x; and it is said to be UES-TL if the manifold Ẽ := {( x, e) : e = 0} of the linearized system transversal to E in ( 6) is exponentially stable uniformly in x.

Concerning the ULMTE property, condition ( 9) is related to the notion of horizontal contraction introduced in [11, Section VII]). However a key difference is that we do not require the monotonicity condition [START_REF] Demidovich | Dissipativity of a system of nonlinear differential equations in the large[END_REF] to hold in the whole manifold R ne × R nx but only along the invariant submanifold E. In this case the corresponding horizontal Finsler-Lyapunov function V : (R nx × R ne ) × (R nx × R ne ) that we get takes the form V ((x, e), (δ x , δ e )) = δ e P (x)δ e .

In the case where the manifold is reduced to a single point, i.e. when the system (1) is simply ė = F (e) with an equilibrium point at the origin (i.e. F (0) = 0) then

• the TULES-NL property can be understood as the local exponential stability of the origin;

• the UES-TL notion translates to the exponential stability of the linear system ˙ e = ∂F ∂e (0) e; and

• the ULMTE concept is about the existence of a positive definite matrix P solution to the Lyapunov equation P ∂F ∂e (0)+ ∂F ∂e (0) P = -Q where Q is an arbitrary positive definite matrix. In this particular case it is well known that these three properties are equivalent.

For the example of incremental stability, as mentioned before, the three properties of TULES-NL, UES-TL and ULMTE can be understood globally as follows : 2) is globally exponentially incrementally stable. Namely there exist two strictly positive real numbers k and λ such that for all

P1 (TULES-NL) System (
(x 1 , x 2 ) in R n × R n we have, for all t in R ≥0 , |X(x 1 , t) -X(x 2 , t)| ≤ k|x 1 -x 2 | exp(-λt). ( 11
)

P2 (UES-TL)

The manifold E = {(x, e), e = 0} is globally exponentially stable for the system

ė = ∂f ∂x (x)e , ẋ = f (x) (12)
Namely there exist two strictly positive real numbers k e and λ e such that for all (e, x) in R n ×R n , the corresponding solution of [START_REF] Fromion | Connecting nonlinear incremental Lyapunov stability with the linearizations Lyapunov stability[END_REF] satisfies

|E(e, x, t)| ≤ k e |e| exp(-λ e t) , ∀t ∈ R ≥0 .
P3 (ULMTE) There exists a positive definite matrix Q in R n×n , a C 2 function P : R n → R n×n and strictly positive real numbers p and p such that P has a derivative d f P along f in the sense of [START_REF] Pavlov | Convergent dynamics, a tribute to Boris Pavlovich Demidovich[END_REF] ,and satisfies ( 21) and [START_REF] Lewis | Differential equations referred to a variable metric[END_REF].

In this context it is known that P3 ⇒ P1. Actually asymptotic incremental stability for which Property P1 is a particular case is known to be equivalent to the existence of an appropriate Lyapunov function. This has been established in [START_REF] Yoshizawa | Extreme stability and almost periodic solutions of functional-differential equations[END_REF][START_REF] Teel | A smooth Lyapunov function from a class-${\ mathcal {KL}}$ estimate involving two positive semidefinite functions[END_REF][START_REF] Angeli | A Lyapunov approach to incremental stability properties[END_REF] or [START_REF] Rüffer | Convergent systems vs. incremental stability[END_REF] for instance. In our context, this Lyapunov function is given as a Riemannian distance. We shall show below that, as for the case of an equilibrium point, we have also P1 ⇒ P2 ⇒ P3, (see Proposition 4), namely incremental exponential stability implies the existence of a Riemannian distance for which the flow is contracting.

In studying the equivalence relation between TULES-NL,UES-TL and ULMTE, we are not interested in the possibility of a solution near the invariant manifold to inherit some properties of solutions in this manifold, such as, the asymptotic phase, the shadowing property, the reduction principle, etc., nor in the existence of some special coordinates allowing us to exhibit some invariant splitting in the dynamics (exponential dichotomy). This is the reason that, besides forward completeness, we assume nothing for the in-manifold dynamics given by :

ẋ = G(x) = G(0, x) .
So, for not misleading our reader, we prefer to use the word "transversal" instead of "normal" as seen for instance in the various definitions of normally hyperbolic submanifolds given in [14, §1].

In order to simplify the exposition of our results and to concentrate our attention on the main ideas, we assume everything is global and/or uniform, including restrictive bounds. Most of this can be relaxed with working on open or compact sets, but then with restricting the results to time intervals where a solution remains in such a particular set.

2.1.1

TULES-NL "⇒" UES-TL

In the spirit of Lyapunov first method, we have the following result. Proposition 1. If Property TULES-NL holds and there exist positive real numbers ρ, µ and c such that, for all

x in R nx , ∂F ∂e (0, x) ≤ µ , ∂G ∂x (0, x) ≤ ρ (13)
and, for all (e, x) in B e (kr) × R nx ,

∂ 2 F ∂e∂e (e, x) ≤ c , ∂ 2 F ∂x∂e (e, x) ≤ c , ∂G ∂e (e, x) ≤ c , (14)
then Property UES-TL holds.

The proof of this proposition is given in Appendix .1. Roughly speaking, it is based on the comparison between a given e-component of a solution E( e 0 , x 0 , t) of ( 6) with pieces of e-component of solutions E( e i , x i , t-t i ) of solutions of ( 1) where e i , x i are sequences of points defined on E( e 0 , x 0 , t). Thanks to the bounds ( 13) and ( 14), it is possible to show that E and E remain sufficiently closed so that E inherit the convergence property of the solution E. As a consequence, in the particular case in which F does not depend on x, the two functions E and E do not depend on x either and the bounds on the derivatives of the G function are useless.

UES-TL "⇒" ULMTE

Analogous to the property of existence of a solution to the Lyapunov matrix equation, we have the following proposition on the link between UES-TL and ULMTE notions. Proposition 2. If Property UES-TL holds and there exists a positive real number µ such that

∂F ∂e (0, x) ≤ µ ∀x ∈ R nx , (15)
then Property ULMTE holds.

The proof of this proposition is given in Appendix .2. The idea is to show that, for every symmetric positive definite matrix Q, the function P : R nx → R ne×ne given by

P ( x) = lim T →+∞ T 0 ∂ E ∂ e (0, x, s) Q ∂ E ∂ e (0, x, s)ds (16)
is well defined, continuous and satisfies all the requirements of the property ULMTE. The assumption ( 15) is used to show that P satisfies the left inequality in [START_REF] Filippov | Differential Equations with Discontinuous Right Hand Sides[END_REF]. Nevertheless this inequality holds without (15) provided the function s → ∂ E ∂ e (0, x, s) does not go too fast to zero.

∂ 2 F ∂e∂e (e, x) ≤ c , ∂ 2 F ∂x∂e (e, x) ≤ c , ∂G ∂e (e, x) ≤ c , (17)
then Property TULES-NL holds.

The proof of this proposition can be found in Appendix .3. This is a direct consequence of the use of V (e, x) = e P (x)e as a Lyapunov function. The bounds [START_REF] Jouffroy | Some ancestors of contraction analysis[END_REF] and [START_REF] Lewis | Metric properties of differential equations[END_REF] are used to show that, with equation ( 9), the time derivative of this Lyapunov function is negative in a (uniform) tubular neighborhood of the manifold {(e, x), e = 0}.

Revisiting the exponential incremental stable systems

Incremental stability of an autonomous system (2) is the property that a distance between any two solutions of (2) converges asymptotically to zero. The characterization of it has been studied thoroughly, for example, in [START_REF] Angeli | A Lyapunov approach to incremental stability properties[END_REF][START_REF] Fromion | Connecting nonlinear incremental Lyapunov stability with the linearizations Lyapunov stability[END_REF][START_REF] Angeli | Further results on incremental input-to-state stability[END_REF]. In [START_REF] Angeli | A Lyapunov approach to incremental stability properties[END_REF][START_REF] Angeli | Further results on incremental input-to-state stability[END_REF], a Lyapunov characterization of incremental stability (δ-GAS for autonomous systems and δ-ISS for non-autonomous ones) is given based on the Euclidean distance between two states that evolve in an identical system. A variant of this notion is that of convergent systems discussed in [START_REF] Pavlov | Convergent dynamics, a tribute to Boris Pavlovich Demidovich[END_REF][START_REF] Rüffer | Convergent systems vs. incremental stability[END_REF]. All these studies are based on the notion of contracting flows which has been widely studied in the literature and for a long time, see, for example, [START_REF] Lewis | Metric properties of differential equations[END_REF][START_REF] Lewis | Differential equations referred to a variable metric[END_REF][START_REF] Hartman | Ordinary differential equations[END_REF][START_REF] Demidovich | Dissipativity of a system of nonlinear differential equations in the large[END_REF][START_REF] Németh | Geometric aspects of Minty-Browder monotonicity[END_REF][START_REF] Lohmiller | On contraction analysis for non-linear systems[END_REF][START_REF] Forni | A differential Lyapunov framework for contraction analysis[END_REF]. These flows generate trajectories between which an appropriately defined distance is monotonically decreasing with increasing time. See [START_REF] Jouffroy | Some ancestors of contraction analysis[END_REF] for a historical discussion on the contraction analysis and [START_REF] Sontag | Contractive systems with inputs[END_REF] for a partial survey.

The big issue in this view points is to find the appropriate distance which may be a difficult task. The results in Section 2 may help in this regard with providing an explicit construction of a Riemannian distance.

Precisely, let P be a C 2 function defined on R n the values of which are symmetric matrices satisfying p I ≤ P (x)

≤ p I ∀x ∈ R n (19)
The length of any piece-wise C 1 path γ : [s 1 , s 2 ] → R n between two arbitrary points x 1 = γ(s 1 ) and x 2 = γ(s 2 ) in R n is defined as :

L(γ) s2 s1 = s2 s1 dγ ds (σ) P (γ(σ)) dγ ds (σ) dσ (20)
By minimizing along all such path we get the distance d(x 1 , x 2 ). Then, thanks to the well established relation between (geodesically) monotone vector field (semigroup generator) (operator) and contracting (non-expansive) flow (semi-group) (see [START_REF] Lewis | Metric properties of differential equations[END_REF][START_REF] Hartman | Ordinary differential equations[END_REF][START_REF] Brezis | Opérateur maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF][START_REF] Isac | Scalar and asymptotic scalar derivatives: theory and applications[END_REF] and many others), we know that this distance between any two solutions of (2) is exponentially decreasing to 0 as time goes on forward if we have

d f P (x) + P (x) ∂f ∂x (x) + ∂f ∂x (x) P (x) ≤ -Q ∀x ∈ R n , ( 21
)
where Q is a positive definite symmetric matrix and

d f P (x) = lim h→0 P (X(x, h)) -P (x) h . (22)
For a proof, see for example [START_REF] Lewis | Metric properties of differential equations[END_REF]Theorem 1] or [15, Theorems 5.7 and 5.33] or [24, Lemma 3.3] (replacing f (x) by x + hf (x)).

In this context, using the main results of our previous section, we can show that, if we have exponential incremental stability, then there exists a function P meeting the above requirements. Specifically, we have the following proposition.

Proposition 4 (Incremental stability). Assume the system (2) is forward complete with a function f which is C 3 with bounded first, second and third derivatives. Let X(x, t) denotes its solutions. Then we have P1 ⇒ P2 ⇒ P3 (and therefore P1 ⇔ P2 ⇔ P3).

In other words, exponential incremental stability property is equivalent to the existence of a Riemannian distance which is contracted by the flow and can be used as a δ-GAS Lyapunov function. Note also that, despite the fact that the main results in Section 2 are local, when we restrict ourselves to the incremental stability problem, we can obtain a global result.

Proof : P1 ⇒ P2 ⇒ P3: Consider the system (3) and let n x = n e = n. The boundedness of the first derivative of f implies the forward completeness of the corresponding systems (1) and ( 5). Moreover the inequalities ( 13), ( 14) and ( 15) with r = +∞ follow from the assumption of boundedness of the derivatives of f .

As a consequence P1⇒ P2 follows from Proposition 1 and P2⇒ P3 from Proposition 2. Note however that it remains to show that P defined in ( 16) is C 2 . This is obtained employing the boundedness of the first, second and third derivatives of f . Indeed, note that we have for all (t, x) ∂ E ∂e (0, x, t) = ∂X ∂x (x, t). So to show that P is C 1 it suffices to show that the mapping t → ∂ 2 X ∂xi∂x (x, t) goes exponentially to zero as time goes to infinity. Note that this is indeed the case since given a vector v in R n and i in {1, . . . , n} the mapping

ν(t) = ∂ 2 X ∂xi∂x (x, t)v is solution to ν = ∂f ∂x (X(x, t))ν+ n j=1 ∂ 2 f ∂x j ∂x (X(x, t)) ∂X j ∂x i (x, t) ∂X ∂x (x, t)v
Hence, from ( 21), [START_REF] Lewis | Differential equations referred to a variable metric[END_REF] and the fact that f has bounded second derivatives, it yields the existence of a positive real number c such that

˙ ν P (X(x, t))ν ≤ -ν Qν + c|ν| ∂X ∂x (x, t) 2 .
Since t → ∂X ∂x (x, t) exponentially goes to zero as time goes to infinity, it implies that ν exponentially goes to zero. Hence, P is C 1 . Employing the bound on the third derivative and following the same route, it follows that P is C 2 . 2

Applications

In this section, we apply Propositions 1, 2 and 3 in two different contexts: full order observer and synchronization.

3.1 Nonlinear observer design

Consider a system ẋ = f (x) , y = h(x) . (23)
with state x in R n and output y in R p augmented with a state observer of the particular form

ẋ = f (x) + K(y, x) (24)
with state x in R n and where

K(h(x), x) = 0 ∀x . ( 25
)
Assuming the functions f , h and K are C 2 , we are interested in having the manifold {(x, x) : x = x} exponentially stable for the overall system

ẋ = f (x) , ẋ = f (x) + K(y, x) . ( 26
)
When specified to this context, the properties TULES-NL, UES-TL and ULMTE are

Exponentially convergent observer (TULES-NL): The system ( 26) is forward complete and there exist strictly positive real numbers r, k and λ such that we have, for all (x, x, t) in

R n × R nx × R ≥0 satisfying |x 0 -x0 | ≤ r, we have |X(x 0 , t) -X(x 0 , x0 , t)| ≤ k|x 0 -x0 | exp(-λt) . (27)
UES-TL FOR OBSERVER The system ẋ = f (x) is forward complete and there exist strictly positive real numbers k and λ such that any solution ( E( e 0 , x 0 , t), X(x 0 , t)) of the transversally linear system

˙ e = ∂f ∂x (x) + ∂K ∂y (h(x), x) ∂h ∂x (x) e , ẋ = f (x) (28)
satisfies, for all ( e 0 , x

0 , t) in R n × R n × R ≥0 , | E( e 0 , x 0 , t)| ≤ k exp(-λt)| e 0 | . (29)
ULMTE FOR OBSERVER : For all positive definite matrix Q, there exists a continuous function P : R n → R n×n and strictly positive real numbers p and p such that we have, for all x in R n , p I ≤ P (x) ≤ p I ,

d f P (x) + 2Sym P (x) ∂f ∂x (x) + ∂K ∂y (h(x), x) ∂h ∂x (x) (30)
≤ -Q .

where Sym(A) = A + A .

Propositions 1, 2 and 3 give conditions under which these properties are equivalent. But these properties assume the data of the correction term K. Hence, by rewriting UES-TEL and TULES-NL in a way in which the design parameter K disappears, these propositions give necessary conditions for the existence of an exponentially convergent observer.

Property UES-TL involves the existence of an observer with correction term depending on x for the time-varying linear system resulting from the linearization along a solution to the system [START_REF] Pavlov | Uniform Output Regulation of Nonlinear Systems: A convergent Dynamics Approach[END_REF]

, i.e. ˙ e = ∂f ∂x (x) e , ỹ = ∂h ∂x (x) e (31)
seeing ỹ as output. As a consequence of Proposition 1, a necessary condition for Property UES-TL to hold and further, when some derivatives are bounded, for the existence of an exponentially convergent observer is that the system (23) be infinitesimally detectable in the following sense

Infinitesimal detectability We say that the system ( 23) is infinitesimally detectable if every so- A similar necessary condition has been established in [START_REF] Andrieu | Observability necessary conditions for the existence of observers[END_REF] for a larger class of observers but under an extra assumption (the existence of a locally quadratic Lyapunov function).

lution of ẋ = f (x) , ˙ e = ∂f ∂x (
Example 1: Consider the planar system

ẋ1 = x 3 2 , ẋ2 = -x 1 , y = x 1 . (32)
We wish to know whether or not it is possible to design an exponentially convergent observer for this nonlinear oscillator in the form of ( 24). The linearized system is

˙ e 1 = 3x 2 2 e 2 , ˙ e 2 = -e 1 , ỹ = e 1 ( 33
)
This system is not detectable when the solution, along which we linearize, is the origin which is an equilibrium of (32). Consequently the system (32) is not infinitesimally detectable on R 2 and so there is no exponentially convergent observer on R 2 . Fortunately the subset {x ∈ R 2 :

x 2 1 2 + x 4 2
4 ≥ }, with > 0, is invariant and ( 32) is infinitesimally detectable in it.

To design a correction term K for an exponentially convergent observer, we use the property that

L(y, x) = ∂K ∂y (y, x)
should be an observer gain for the linear system [START_REF] Wang | On Parameter Convergence of Nonlinearly Parameterized Adaptive Systems: Analysis via Contraction and First Lyapunovs Methods[END_REF]. So we start our design by selecting L. We pick

L(y, x) = -3x 2 2 -3x 2 2 + 1 This gives (see (28)) A(x) = ∂f ∂x (x) + ∂K ∂y (h(x), x) ∂h ∂x (x) = 3x 2 -1 1 0 -1
The transition matrix generated by A(X(x, t)) when X(x, t) is a solution of (32) is

Φ(t, 0) = exp (-I(t)) 1 I(t) 0 1
where

I(t) = 3 t 0 X 2 (x, s) 2 ds .
Since X 2 (x, t) is periodic, (29) holds when the initial condition x is in the compact invariant subset

C = {x ∈ R 2 : 1 ε ≥ x 2 1 2 + x 4 2 4 ≥ } . ( 34
)
Then, according to Propositions 2 and 3, and in view of [START_REF] Rüffer | Convergent systems vs. incremental stability[END_REF], we obtain an exponentially convergent observer by choosing K as

K(y, x) = y x1 L(s, x)ds = -3x 2 2 -3x 2 2 + 1 (y -x 1 ) .
Similarly, Property ULMTE involves the existence of P and K such that (30) holds. By restricting this inequality on quadratic forms to vectors which are in the kernel of ∂h ∂x , we obtain as a consequence of Propositions 1 and 2 that a necessary condition for Property ULMTE to hold and further, when some derivatives are bounded, for the existence of an exponentially convergent observer is that the system (23) be R-detectable (R for Riemann) in the following sense.

R-Detectability

We say that the system ( 23) is R-detectable if there exist a continuous function P : R n → R n×n and positive real numbers 0 < p ≤ p and 0 < q such that P has a derivative d f P along f in the sense of ( 22) and we have

p I ≤ P (x) ≤ p I ∀x ∈ R n (35)
and

v d f P ( x)v + 2v P ( x) ∂f ∂x (x)v ≤ -q v P (x)v (36)
holds for all (x, v) in R n × R n satisfying ∂h ∂x (x)v = 0. A similar necessary condition has been established in [START_REF] Sanfelice | Convergence of nonlinear observers on R n with a Riemannian metric (part i). Automatic Control[END_REF], where only asymptotic and not exponential convergence is assumed. In that case, the condition allows p and q to be zero.

Further it is established in [START_REF] Sanfelice | Convergence of nonlinear observers on R n with a Riemannian metric[END_REF] that when the R-detectability holds then

K(y, x) = k P (x) -1 ∂h ∂x (x) T (y -h(x))
gives, for k large enough, a (locally) exponentially convergent observer.

Example 1 continued: For the system (32), the necessary R-Detectability condition is the existence of P = P 11 P 12 P 12 P 22 satisfying in particular [START_REF] Yoshizawa | Extreme stability and almost periodic solutions of functional-differential equations[END_REF] which is (see [START_REF] Wieland | An internal model principle is necessary and sufficient for linear output synchronization[END_REF])

∂P 22 ∂x 1 (x) x 3 2 - ∂P 22 ∂x 2 (x) x 1 + 6P 12 (x) x 2 2 < -q P 22 (x) ∀x ∈ C (37)
We view this as a condition on P 22 only since whatever P 12 is, we can always pick P 11 to satisfy [START_REF] Yoshizawa | Stability theory by Liapunov's second method[END_REF]. Note also that we can take care of any term with x 2 2 in factor by selecting P 12 appropriately. With this, it can be shown that it is sufficient to pick P 22 in the form

r(x) 2 ≤ P 22 (x) = r(x) + x 1 x 2 r(x) + x 2 2 ( ≤ 3r(x)) ,
where the presence of r defined below is justified by homogeneity considerations

( 4ε ≤) r(x) = 2x 2 1 + x 4 2 ≤ 4 ε .
This motivates us to design

P 12 (x) = - 5 24
x 2 2 r(x) - 1 3 r(x)
In this case, the left hand side in the inequality (37) is

- x 2 1 √ r - 1 4 x 4 2 r(x) + 2x 1 x 2 -2 r(x)x 2 2 ≤ - 1 4 r(x) r(x) ≤ - r(x) 12 P 22 (x) ≤ - √ ε 6 P 22 (x) .
Finally, by choosing

P 11 (x) = 2 + P 12 (x) 2 P 22 (x) -r(x) 4 ,
it can be shown that we obtain

ε I ≤ min 1, r(x) 4 I ≤ P (x) ≤ 3 max{1, r(x)} I ≤ 12 ε .
Hence [START_REF] Yoshizawa | Stability theory by Liapunov's second method[END_REF] holds on C. From this, the correction term

K(y, x) = kP 22 (x) P 11 P 22 (x) -P 12 (x) 2 P 22 (x) -P 12 (x) (y -x 1 )
gives a (locally) convergent observer on C.

Exponential synchronization

Finally, we revisit the synchronization problem as another class of control problems that can be dealt with the results in Section II. We consider here the synchronization of m ≥ 2 identical systems given by ẇi = f

(w i ) + g(w i )u i , i = 1, . . . , m , (38)
In this setting, all systems have the same drift vector field f and the same control vector field g : R n → R n×p , but not the same controls in R p . The state of the whole system is denoted w = (w 1 , . . . w m ) in R mn . We define also the diagonal subset of R mn

D = {(w 1 , . . . , w m ) ∈ R mn , w 1 = w 2 • • • = w m }
Given w in R mn , we denote the Euclidean distance to the set D as |w| D . The synchronization problem that we consider in this section is as follows.

Definition 1. The control laws u i = φ i (w), w = (w 1 , . . . w m ), i = 1 . . . , m solve the local uniform exponential synchronization problem for (38) if the following holds:

1. φ is invariant by permutation of agents. More precisely, given a permutation π : {1, . . . m} → {1, . . . m} φ πi (w 1 , . . . , w m ) = φ i (w π1 , . . . , w πm )

2. φ is zero on D:

φ(w) = 0 ∀w ∈ D , (39)
3. and the set D is uniformly exponentially stable for the closed-loop system, i.e., there exist positive real numbers r w , k and λ > 0 such that, for all w in R mn satisfying |w| D < r w ,

|W (w, t)| D ≤ k exp(-λt) |w| D , (40)
holds for all t in the domain of existence of the solutions W (w, t) going through w at t = 0.

When r w = ∞, it is called the global uniform exponential synchronization problem.

In this context, we assume that every agent shares an information (which will be designed later) to all other agents (in which case, it forms a complete graph) and it has local access to its state variables.

It is possible to rewrite the property of having the manifold D exponentially stable as property TULES-NL. As it has been done in the observer design context, employing Propositions 1 and 2 and by rewriting properties UES-TL and ULMTE it is possible to give equivalent characterization of the synchronization property. By rewriting these conditions in a way in which the control law disappears, these properties give necessary conditions to achieve exponential synchronization.

Proposition 5 (Necessary condition). Consider the systems in (38) and assume the existence of control laws u i = φ i (w), i = 1, . . . m that solve the uniform exponential synchronization of (38). Assume moreover that g is bounded and f , g and the φ i 's have bounded first and second derivatives. Then the following two properties hold. Q1: The origin of the transversally linear system

˙ e = ∂f ∂x ( x) e + g( x)u , ˙ x = f ( x) , (41)
is stabilizable by a (linear in e) state feedback.

Q2: For every positive definite matrix Q, there exist a continuous function P : R n → R n×n and positive real numbers p and p such that inequalities [START_REF] Lewis | Differential equations referred to a variable metric[END_REF] are satisfied, P has a derivative d f P along f in the sense of [START_REF] Pavlov | Convergent dynamics, a tribute to Boris Pavlovich Demidovich[END_REF], and

d f v P (x)v + 2v P (x) ∂f ∂x (x)v ≤ -v Qv (42)
holds for all (v, x) in R 2n satisfying v P (x)g(x) = 0.

Proof : First of all note that the vector fields having bounded first derivatives, it implies that the system is complete. Consider (i, j, k, l), 4 integers in {1, . . . , m} and consider a permutation π : {1, . . . , m} → {1, . . . , m} such that π i = k and π j = . Note that k = if and only if i = j. Note that the invariance by permutation implies

φ k (w) = φ i (w π1 , . . . , w πm ) .
Hence, it follows that ∂φ k ∂w (w) = ∂φ i ∂w j (w π1 , . . . , w πm ) and if we consider w in D, this implies

∂φ k ∂w (w) = ∂φ i ∂w j (w) , i = j , k = ∂φ j ∂w j (w) = ∂φ i ∂w i (w)
By denoting e = (e 2 , . . . e m ) with e i = w i -w 1 , i = 2, . . . m and x = w 1 , we obtain an (e, x)-system of the type (1) with

F (e, x) = (F i (e, x)) i=2,...m (43) F i (e, x) = f (x + e i ) -f (x) +g(x + e i ) φi (e, x) -g(x) φ1 (e, x) , (44)
G(e, x) = f (x) + g(x) φ1 (e, x) , (45)
where we have used the notation φi (e, x) = φ i (x, x + e 2 , . . . , x + e n )

Note that we have

|e| 2 ≤ (m -1)|w| 2 D , (46)
and

|w| 2 D ≤ |e| 2 + (m -1) m i=1 w 1 -w i m 2 ≤ 1 + m -1 m 2 |e| 2 (47)
Hence, (40) implies for all (e, x)

with |e| ≤ mrw √ m 2 +m-1 |E(e, x, t)| ≤ (m -1) 1 + m -1 m 2 k exp(-λt) |e|,
It follows from the assumptions of the proposition that Property TULES-NL is satisfied with r = mrw √ m 2 +m-1 and that inequalities ( 13) and ( 14) hold. We conclude with Proposition 1 that Property UES-TL is satisfied also. So, in particular, there exist positive real numbers k and λ such that any e i component of ( E( e 0 , x 0 , t), X( x 0 , t)) solution of (6) satisfies, for all ( e 0 , x 0 , t) in R mn × R mn × R ≥0 ,

| E i ( e 0 , x, t)| ≤ k exp(-λt)| e 0 | . (48)
On another hand, with (39), we obtain :

∂F i ∂e i (0, x) = ∂f ∂x ( x) + g( x) ∂ φi ∂e i (0, x) - ∂ φ1 ∂e i (0, x) . (49)
And, when j = i, it yields

∂F i ∂e j (0, x) = g( x) ∂ φi ∂e j (0, x) - ∂ φ1 ∂e j (0, x) = 0 . (50)
Consequently, any solution of the system

˙ e i = ∂f ∂x ( x) + g( x) ∂ φi ∂e i (0, x) - ∂ φ1 ∂e i (0, x) e i ,
and ˙ x = f ( x) can be expressed as an e i component of ( E( e 0 , x 0 , t), X( x 0 , t)) solution of ( 6) Since these solutions satisfy (48), Property Q1 does hold.

Finally we consider the system with state (e i , x) in

R 2n ėi = Fi (e i , x) , ẋ = G(e i , x) = f (x) (51)
with Fi (e i , x) = F i ((0, e i , 0), x). The previous property and Proposition 2 imply that Property ULMTE is satisfied for system (51). So in particular we have a function P satisfying the properties in Q2 and such that we have, for all

(v, x) in R n × R n , v d f P ( x)v +2v P ( x) ∂f ∂x ( x) + g( x) ∂ φi ∂e i (0, x) - ∂ φ1 ∂e i (0, x) v ≤ -v Qv which implies (42) when v P (x)g(x) = 0. 2
Example 2: As an illustrative example consider the case in which the system is given by by m agents w i in R 2 with individual dynamics

ẇi1 = w i2 + 2 sin(w i2 ) , ẇi2 = a + u i , (52)
where a is a real number. Because of a singularity when 1 + 2 cos(w i2 ) = 0, this system is not feedback linearizable per se. Hence the design of a synchronizing controller may be involved. In order to check if local synchronization in the sense of Definition 1 is possible, the necessary conditions of Proposition 5 may be tested. The transversally linear system is

˙ e = 0 1 + 2 cos(x 0 + at) 0 0 e + 0 1 u . ( 53
)
When a = 0 and x 0 = 2π 3 , this system is not stabilizable by any feedback law. Hence in this case, with Proposition 5, there is no exponentially synchronizing control law in the sense of Definition 1 satisfying (39) in particular.

Similar to the analysis of incremental stability in the previous section and observer design in [START_REF] Sanfelice | Convergence of nonlinear observers on R n with a Riemannian metric (part i). Automatic Control[END_REF] , by using a function P satisfying the property Q2 in Proposition 5, we can obtain sufficient conditions for the solvability of uniform exponential synchronization of (38).

We do this under an extra assumption which is that, up to a scaling factor, the control vector field g is a gradient field with P as Riemannian metric.

Proposition 6 (Local sufficient condition). Assume f has bounded first and second derivatives, and g is bounded and has bounded first and second derivatives. Moreover, assume that 1. there exist a C 2 function U : R n → R and a bounded C 2 function α : R n → R p which has bounded first and second derivative such that

∂U ∂x (x) = P (x)g(x)α(x) ; ( 54
)
holds for all x in R n ; and 2. there exist a positive definite matrix Q, a C 2 function P : R n → R n×n with bounded derivative, and positive real numbers p, p and ρ > 0 such that ( 19) is satisfied and

v d f P (x)v + 2v P (x) ∂f ∂x (x)v -ρ ∂U ∂x (x)v 2 ≤ -v Qv , (55) holds for all (x, v) in R n × R n .
Then there exist a real number such that with the control laws u i = φ i (w) given by

φ i (w) = α(w i )   m j=1 U (w j ) m -U (w i )   ( 56
)
and ≥ and if the closed loop system is complete then the local uniform exponential synchronization of (38) is solved.

Note that, for its implementation, the control law (56) requires that each agent i communicates U (w i ) to all the other agents. Proof : First of all, note that the control law φ i is invariant by permutation due to its structure. Let e = (e 2 , . . . , e m ) with e i = x 1 -x i and x = x 1 . We obtain an (e, x)-system of the type (1) with F and G as defined in (43-44-45) with φ as control input. For this system, we will show that property ULMTE is satisfied. Consider the function P m : R n → R (m-1)n×(m-1)n defined as a block diagonal matrix composed of (m -1) matrices P . i.e. P m (x) = Diag(P (x), . . . , P (x)). Note that with property (49) and (50), it yields that ∂F ∂e (0, x) is also (m -1) block diagonal. Hence, we have

d G P m ( x) + P m ( x) ∂F ∂e (0, x) + ∂F ∂e (0, x) P m ( x) = Diag{R( x)), . . . , R( x))} where R( x) = d f P ( x)v + P ( x) ∂f ∂x ( x) -g( x)α( x) ∂U ∂x ( x) + ∂f ∂x ( x) -g( x)α( x) ∂U ∂x ( x) P ( x) .
With (55), this gives

v R( x)v ≤ -v Qv + (k -2 ) ∂U ∂x ( x)v 2 .
for all ( x, v) in R n × R n . Hence, picking > k 2 , inequality (9) holds. To apply proposition 3, it remains to show that inequalities (15), ( 17) and ( 18) are satisfied. Note that employing the bounds on the functions P , f , g, α and there derivatives, it is possible to get a positive real number c (depending on ) such that for all i in 2, . . . , m and all (e, x)

∂F i ∂e i (e, x) ≤ c m j=1 U (x + e i ) m -U (x + e i ) + c , ∂F i ∂e i ∂x (e, x) ≤ c m j=1 U (x + e i ) m -U (x + e i ) + c , ∂F i ∂e i ∂e j (e, x) ≤ c m j=1 U (x + e i ) m -U (x + e i ) + c , m j=1 U (x + e i ) m -U (x + e i ) ≤ c|e| , ∀(e, x)
So we fix η positive and pick c = c2 η + c. The above shows that inequalities ( 17) and ( 18) are satisfied. With Proposition 3, we conclude that Property TULES-NL holds. Hence e = 0 is (locally) exponentially stable manifold. With inequalities ( 46) and (47) this implies that inequality (40) holds. 2

In this result it is important to remark that there is no guarantee that the control law given here ensures completeness of the solution. Note however, that on the manifold |w| D = 0, the trajectories satisfy ẋ = f (x) which is a complete system Example 2 (continued): We come back to the example (52) in the case where a = 1. We note that the linear system (53) is stabilizable by a feedback in the form u = -(1 + 2(cos(x 0 + t)) 2 3 e Indeed, the solution of (53) with the previous feedback satisfies

˙ e = (1 + cos(x 0 + t)) 0 1 -2 -3 e .
Hence its solution are E(e, x, t) = ψ(x, t) e, where ψ is the generator of this time varying linear system given as

ψ(x, t) = exp (t + 2 sin(x + t) 0 1 -2 -3 = ϕ(x, t) -2 -1 + 2ϕ(x, t) -1 + ϕ(x, t) 2(1 -ϕ(x, t)) 2 -ϕ(x, t)
with ϕ(x, t) = e t+2 sin(t+x) . Consequently, we get that e goes exponentially to zero. Hence Property Q 1 is satisfied. We can then introduce the matrix P solution to Q2 and given in [START_REF] Isidori | Output regulation of nonlinear systems[END_REF] as

P (x) = +∞ 0 ψ(x, s) ψ(x, s)ds . ( 57
)
This matrix is positive definite and satisfies property Q2. So we may want to use it for designing an exponentially synchronizing control law. With decomposing the 2 × 2 matrix P as P (w i ) = P 11 (w i2 ) P 12 (w i2 ) P 12 (w i2 ) P 22 (w i2 ) ,

we obtain P (w)g(w) = P 12 (w i2 ) P 22 (w i2 ) . Note that it can be shown (numerically) that

P 12 (w i2 ) = ∞ 0 4 ϕ(w i2 , t) 2 - 9 ϕ(w i2 , t) 3 + 5 ϕ(w i2 , t) 4 dt > 0 .
It follows the that function α(w) = 1 P12(wi2) is well defined and setting

U (w i ) = w i1 + wi2 0 P 22 (s) P 12 (s) ds
property (54) is satisfied. Hence, for this example, picking a sufficiently large real number, the control law (56) ensures local exponential synchronization of m agents. We have checked this via simulation for the case m = 5, = 3. The time evolution of the solution with w i (0), i = 1, . . . , 5 chosen randomly according to a uniform distribution on [0, 10] is shown in Figure 1a for w i1 and 1b for w i2 . As in the context of the observer design given in [START_REF] Sanfelice | Convergence of nonlinear observers on R n with a Riemannian metric (part i). Automatic Control[END_REF], a global result can be obtained by imposing a further constraint on P . Specifically, the notion we need to introduce is the following. Definition 2 (Totally Geodesically Set). Given a C 2 function P defined on R n the values of which are symmetric positive definite matrices, a C 1 function ϕ : R n → R + and a real number φ, the (level) set S = {x ∈ R n , ϕ(x) = φ} is said to be totally geodesic with respect to P if, for any

(x, v) in S × R n such that ∂ϕ ∂x (x)v = 0, v P (x)v =
(x, v) in R n ×R n such that ∂U ∂x (x) v = 0 1 2 v d f P (x)v + v P (x) ∂f ∂x (x)v ≤ -λ v P (x)v , (59)
3. For all Ū in R, the set S = {x ∈ R n , U (x) = Ū } is totally geodesic with respect to P . Then there exists a function : R 2n → R + , invariant by permutation such that, with the controls given by

φ i (w) = (w)α(w i ) (U (w j ) -U (w i )) ,
with (i, j) ∈ {(1, 2), (2, 1)} the following holds and for all w in R 2n ,

|W (w, t)| D ≤ k|w| D exp - λ 2 t , ( 60
)
where t is any positive real number in the time domain of definition of the closed loop solution.

The proof of this result is given in Appendix .4. It borrows some ideas of [START_REF] Sanfelice | Convergence of nonlinear observers on R n with a Riemannian metric (part i). Automatic Control[END_REF]. However, different from [START_REF] Sanfelice | Convergence of nonlinear observers on R n with a Riemannian metric (part i). Automatic Control[END_REF], we have here a global convergence result. This follows from the fact that in the high gain parameter , the norm of the full state space can be used (and not only the norm of the estimate as in the observer case ).

Note that nothing is said about the domain of existence of the solution.

Conclusion

We have studied the relationship between the exponential stability of an invariant manifold and the existence of a Riemannian metric for which the flow is "transversally" contracting. It was shown that the following properties are equivalent 1. A manifold is "transversally" exponentially stable;

2. The "transverse" linearization along any solution in the manifold is exponentially stable;

3. There exists a field of positive definite quadratic forms whose restrictions to the transverse direction to the manifold are decreasing along the flow.

As an illustrative example for these equivalence results, we have revisited the property of exponential incremental stability where we can obtain a global result. The characterization of transverse exponential stability has allowed us to investigate a necessary condition for two different control problems of nonlinear observer design and of synchronization of nonlinear multi-agent systems which leads to a novel constructive design for each problem. Recent result by others has also shown the applicability of our results beyond these two control problems. Although the main results hold for local uniform transverse exponential stability, we show that global results can also be obtained in some particular cases. The extension of all the results to the global case is currently under study.

Acknowledgement

The authors would like to thank the anonymous reviewers for their helpful comments.

.

Proof of Proposition 1

Proof : Let us start with some estimations. Let z = e -e. Along solutions of ( 1) and ( 6), we have

ż = F (e, x) - ∂F ∂e (0, x) e = ∂F ∂e (0, x)z + ∆(x, e, x) with the notation ∆(e, x, x) = F (e, x) - ∂F ∂e (0, x)e ,
= [F (e, x) -F (e, x)]

+ F (e, x) -F (0, x) - ∂F ∂e (0, x)e .
Note that the manifold E := {(e, x) : e = 0} being invariant, it yields

F (0, x) = 0 ∀x . (61)
With Hadamard's lemma, (61) and ( 14), we obtain the existence of positive real numbers c 1 and c 2 such that, for all (e, x, x) in B e (kr

) × R nx × R nx , |∆(e, x, x)| ≤ c 1 |e| 2 + c 2 |e||x -x| .
This, with [START_REF] Isac | Scalar and asymptotic scalar derivatives: theory and applications[END_REF], gives, for all (e, e, x, x) in B e (kr

) × R ne × R nx × R nx , 1 ˙ |z| ≤ µ|z| + c 1 |e| 2 + c 2 |e||x -x| . ( 62
)
1 Here the notation ˙ |z| is abusive. The function x → |x| is not C 1 but only Lipschitz. Nevertheless given a vector field f an upper right Dini Lie derivative, i.e. lim sup h→0 +

|x+hf (x)|-|x| h

does exist and, by the triangle inequality, we have

-|f (x)| ≤ lim sup h→0 + |x + hf (x)| -|x| h ≤ |f (x)| .
So here and in the following ˙ |x| denotes this upper right Dini Lie derivative.

Similarly (1), ( 6) and [START_REF] Hirsch | Invariant manifolds[END_REF] give, for all (e, x, x) in B e (kr

) × R nx × R nx , ˙ |x -x| ≤ |G(e, x) -G(0, x)| + |G(0, x) -G(0, x)| , ≤ c|e| + ρ|x -x| . (63)
Now let r be a positive real number smaller than r and S be a positive real number both to be made precise later on. Let e 0 in B e ( r) and x 0 in R nx be arbitrary and let ( E( e 0 , x 0 , t), X( x 0 , t)) be the corresponding solution of [START_REF] Astolfi | Invariance: A new tool for stabilization and adaptive control of nonlinear systems[END_REF]. Because of the completeness assumption on (1), the linearity of ( 6) and the fact that (0, X( x 0 , t)) is solution of both ( 1) and ( 6), ( E, X) is defined on [0, +∞). We denote :

e i = E( e 0 , x 0 , iS) , x i = X( x 0 , iS) ∀i ∈ N 2
and consider the corresponding solutions (E( e i , x i , s), X( e i , x i , s)) of (1). By assumption, they are defined on [0, +∞) and, because of (4), if e i is in B e (r), then E( e i , x i , s) is in B e (kr) for all positive times s, making possible the use of inequalities ( 62) and (63). Finally, for each integer i, we define the following time functions on [0, S]

Z i (s) = |E( e i , x i , s) -E( e 0 , x 0 , s + iS)| , W i (s) = |X( e i , x i , s) -X( x 0 , s + iS)| . Note that we have Z i (0) = W i (0) = 0.
From the inequalities (63), and (7), we get, for each integer i such that e i is in B e (r), and for all s in [0, S],

W i (s) ≤ c s 0 exp(ρ(s -σ))|E( e i , x i , s)|dσ , ≤ c s 0 exp(ρ(s -σ))k exp(-λσ)dσ| e i | , ≤ ck exp(-λs) exp((ρ + λ)s) -1 ρ + λ | e i | .
Similarly, using (62) and Grönwall inequality we get

Z i (s) ≤ c s 0 exp(µ(s -σ))|E( e i , x i , σ)| (|E( e i , x i , σ)|+ W i (σ)) dσ ≤ γ(s) | e i | 2 ∀s ∈ [0, S] ,
where we have used the notation,

γ(s) = c s 0 exp(µ(s -σ))k exp(-2λσ)× × k + ck exp(-λσ) exp((ρ + λ)σ) -1 ρ + λ dσ .
With all this, we have obtained that, if we have e j in B e (r) for all j in {0, . . . , i}, then we have also,

2 N denotes the set of integers.

20

for all s in [0, S] and all j in {0, . . . , i},

| E( e 0 , x 0 , s + jS)| = | E( e j , x j , s)| , ≤ |E( e j , x j , s)| + |Z j (s)| , ≤ k exp(-λs) + γ(s)| e j | | e j |
Now, given a real number ε in (0, 1), we select S and r to satisfy :

k exp(-λS) ≤ min{k, 1 -ε} 2 , r ≤ min r, min{k, 1 -ε} 2 sup s∈[0,S] γ(s) .
Then, for all e j smaller in norm than r, we have

| E( e 0 , x 0 , s + jS)| ≤ (1 -) | e j |
So, since e 0 is in B e ( r), it follows by induction that we have :

| e i | = | E( e 0 , x 0 , iS)| ≤ (1 -ε) i r ≤ r ∀i ∈ N .
Since, with [START_REF] Isac | Scalar and asymptotic scalar derivatives: theory and applications[END_REF], we have also ˙ | e| ≤ µ| e|, we have established, for all s in [0, S] and all i in N,

| E( e 0 , x 0 , s + iS)| ≤ exp(µs)(1 -) i | e 0 |
and therefore, for all ( e 0 , x

0 , t) in B e (a) × R nx × R ≥0 , | E( e 0 , x 0 , t)| ≤ exp(µS)(1 -) t-S S | e 0 | .
By rearranging this inequality and taking advantage of the homogeneity of the system [START_REF] Astolfi | Invariance: A new tool for stabilization and adaptive control of nonlinear systems[END_REF] in the e component, we have obtained [START_REF] Brezis | Opérateur maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF] with k = exp(µS) 1-and λ = -ln(1-) S .

2

.

Proof of Proposition 2

Proof : Let ( E( e 0 , x 0 , t), X( x 0 , t)) be the solution of (6) passing through an arbitrary pair ( e 0 , x 0 ) in R ne × R nx . By assumption, it is defined on [0, +∞). For any v in R ne , we have

∂ ∂t ∂ E ∂ e (0, x 0 , t)v = ∂F ∂e (0, X( x 0 , t)) ∂ E ∂ e (0, x 0 , t)v .
Uniqueness of solutions then implies, for all ( e 0 , x 0 , t) in R ne ×R nx ×R ≥0 , E( e 0 , x 0 , t) = ∂ E ∂ e (0, x 0 , t) e 0 and our assumption [START_REF] Brezis | Opérateur maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF] gives, for all ( e 0 , x 0 , t) in

R ne × R nx × R ≥0 , ∂ E ∂ e
(0, x 0 , t) e 0 ≤ k| e 0 | exp(-λt) and therefore

∂ E ∂ e (0, x 0 , t) ≤ k exp(-λt) ∀( x 0 , t) ∈ R nx × R ≥0 .
This allows us to claim that, for every symmetric positive definite matrix Q, the function P : R nx → R ne×ne given by ( 16) is well defined, continuous and satisfies

λ max {P ( x)} ≤ k 2 2 λ λ max {Q} = p ∀ x ∈ R nx .
Moreover we have

∂ ∂t   v ∂ E ∂ e (0, x 0 , t) -1   = -v ∂ E ∂ e (0, x 0 , t) -1 ∂F ∂e (0, X( x 0 , t)) ,
With [START_REF] Isac | Scalar and asymptotic scalar derivatives: theory and applications[END_REF], this yields

v ∂ E ∂ e (0, x 0 , t) -1
≤ exp(µt) |v| and implies

[v v] 2 ≤ v ∂ E ∂ e (0, x 0 , t) -1 2 ∂ E ∂ e (0, x 0 , t)v 2 ≤ 1 λ min {Q} v ∂ E ∂ e (0, x 0 , t) -1 2 × × v ∂ E ∂ e (0, x 0 , t) Q ∂ E ∂ e (0, x 0 , t)v ≤ |v| 2 exp(2µt) λ min {Q} v ∂ E ∂ e (0, x 0 , t) Q ∂ E ∂ e (0, x 0 , t)v This gives p = 1 2µ λ min {Q} ≤ λ min {P ( x)} ∀ x ∈ R nx .
Finally, to get (9), let us exploit the semi group property of the solutions. We have for all ( e, x) in R nx × R ne and all (t, r) in R 2

≥0

E( E( e, x, t), X( x, t), r) = E( e, x, t + r) .

Differentiating with respect to e the previous equality yields Consequently, this yields,

∂ E ∂ e ( E(
P ( X( x, h)) = lim T →+∞ T 0 ∂ E ∂e (0, X( x, h), s) Q ∂ E ∂ e (0, X( x, h), s)ds = lim T →+∞ ∂ E ∂ e (0, X( x, h), -h) × × T 0 ∂ E ∂ e (0, x, s + h) Q ∂ E ∂ e (0, x, s + h)ds × × ∂ E ∂ e (0, X( x, h), -h)
But we have :

lim h→0 ∂ E ∂ e (0, X( x, h), -h) -I h = - ∂F ∂e (0, x) ,
+ T 0 ∂ E ∂ e (0, x, s) Q ∂ ∂s ∂ E ∂ e (0, x, s) ds = ∂ E ∂ e (0, x, T ) Q ∂ E ∂ e (0, x, T ) -Q .
Since lim T and lim h commute because of the exponential convergence to 0 of ∂ E ∂ e (0, x, s), we conclude that the derivative (8) does exist and satisfies [START_REF] Demidovich | Dissipativity of a system of nonlinear differential equations in the large[END_REF]. On the other hand, using Hadamard's Lemma and ( 18), we get :

F (e, x) - ∂F ∂e (0, x)e ≤ c|e| 2 , |G(e, x) -G(0, x)| ≤ c|e| ∀(e, x) ∈ B e (η) × R nx .
These inequalities together with [START_REF] Filippov | Differential Equations with Discontinuous Right Hand Sides[END_REF] and [START_REF] Jouffroy | Some ancestors of contraction analysis[END_REF] imply, for all (e, x) in

B e (η) × R nx , ˙ V (e, x) ≤ - λ min {Q} p -2c(1 + c) p p |e| V (e, x) .
It shows immediately that (4) holds with r, k and λ satisfying :

r < p p min η, λ min {Q} 2pc(1 + c) , k = p p , λ = λ min {Q} 2p -rc(1 + c) p p .

.4 Proof of Proposition 7

The result holds when w is in D or when U is constant (since (59) holds for all v). So, in view of [START_REF] Sanfelice | Convergence of nonlinear observers on R n with a Riemannian metric (part i). Automatic Control[END_REF]Proposition A.2.1], we can assume without loss of generality that ∂U ∂w has nowhere a zero norm and, in the following, we restrict our attention to R 2n \D. In R 2n \D the dynamics of w is

ẇi = f (w i ) + (w)g(w i )α(w i ) 2 j=1 [U (w j ) -U (w i )]
With the C 2 matrix function P we define the Riemannian length of a piece-wise C 1 path γ : [s 1 , s 2 ] → R n , between w 1 = γ(s 1 ) and w 2 = γ(s 2 ) as in [START_REF] Lohmiller | On contraction analysis for non-linear systems[END_REF] and the corresponding distance d(w 1 , w 2 ) by minimizing along all such path Because of [START_REF] Lewis | Differential equations referred to a variable metric[END_REF] and the fact that P is C3 , Hopf-Rinow Theorem implies the metric space we obtain this way is complete, and, given any w 1 in R n and w 2 in R n , there exists a C 3 normalized 3 minimal geodesic γ * , solution of (58), such that

w 1 = γ * (s 1 ) , w 2 = γ * (s 2 ) , d(w 1 , w 2 ) = L(γ * ) s2 s1 = s 2 -s 1 . (64)
Following [START_REF] Sanfelice | Convergence of nonlinear observers on R n with a Riemannian metric (part i). Automatic Control[END_REF] = A(w) + C(w)

where

A(w) = (w) [U (w 1 ) -U (w 2 )] dγ * ds (s 2 ) P (w 2 )g(w 2 )α(w 2 ) -(w) [U (w 2 ) -U (w 1 )]
dγ * ds (s 1 ) P (w 1 )g(w 1 )α(w 1 )

C(w) = dγ * ds (s 2 ) P (w 2 )f (w 2 ) - dγ * ds (s 1 ) P (w 1 )f (w 1 )
But, with (54), we obtain :

A(w) = -(w) [U (w 2 ) -U (w 1 )] × × ∂U ∂x (w 2 ) dγ * ds (s 2 ) + ∂U ∂x (w 1 ) dγ * ds (s 1 )
Also, with the Euler-Lagrange form of the geodesics equation (58), we get :

C(w) = s2 s1 d ds dγ * ds (s) P (γ * (s)) f (γ * (s)) + dγ * ds (s) P (γ * (s)) d ds f (γ * (s)) ds , = s2 s1 1 2 ∂ ∂x dγ * ds (s) P (x) dγ * ds (s) x=γ * (s) f (γ * (s)) + dγ * ds (s) P (γ * (s)) ∂f ∂x (γ * (s)) dγ * ds (s) ds .
Here the integrand is nothing but the left hand side of (59). With a compactness argument 5 we can show that condition (59) in Proposition 7 is equivalent to the existence of a smooth function ν : R n → R + such that, for all (x, v),

1 2 v d f P (x)v + v P (x) ∂f ∂x (x)v ≤ -λ v P (x)v + ν(x) ∂U ∂x (x)v 2 .
Hence, the geodesic being normalized, we have :

C(w) + λ s2 s1 dγ * ds (s) P (γ * (s) dγ * ds (s)ds = C(w) + λ d(w 1 , w 2 ) ≤ B(w) ,
with the notation :

B(w) = s2 s1 ν(γ * (s)) ∂U ∂x (γ * (s)) dγ * ds (s) 2 ds .
From A and B we define two C 2 functions a and b by dividing by d(w 1 , w 2 ) = s 2 -s 1 . Namely, we define :

a γ * (w, r) = U (γ * (r + s 1 )) -U (w 1 ) r × (66)
× ∂U ∂x (γ * (r + s 1 )) dγ * ds (γ * (r + s 1 )) + ∂U ∂x (w 1 ) dγ * ds (s 1 ) b γ * (w, r) = 1 r r+s1 s1 ν(γ * (s)) ∂U ∂x (γ * (s)) dγ * ds (s) 2 ds . (67)
They are defined on R 2n \D× ]0, d(w 1 , w 2 )] and depend a priori on the particular minimizing geodesic γ * we consider. We extend by continuity (in r ) their definition to R 2n \D × [0, d(w 1 , w 2 )] by letting

a γ * (w, 0) = 2 ∂U ∂x (γ * (s 1 )) dγ * ds (s 1 ) 2 , b γ * (w, 0) = ν(γ * (s 1 )) ∂U ∂x (γ * (s 1 )) dγ * ds (s 1 ) 2 .
In this way, for any pair (w 1 , w 2 ) in R 2n \D and any minimizing geodesic γ * between w 1 and w 2 , -the function r → (a γ * (w, r), b γ * (w, r)) is defined and C 1 6 on [0, d(w 1 , w 2 )], 5 The following two properties are equivalent a) v f (x)v < 0 for all v with |v| = 1 and all x satisfying g(x)v = 0 b) there exists ν such that v f (x)v -ν(x)|g(x)v| 2 ≤ 0 for all v with |v| = 1 and all x. Proof b) ⇒ a) is trivial. For the converse, let C be an arbitrary compact set, if b) does not hold for some η C and all x in C, there exist x i and v i with

|v i | = 1 satisfying v i f (x i )v i ≥ i|g(x i )v i | 2 .
With compactness this implies the existence of xω and vω with |vω| = 1 satisfying g(xω)vω = 0 and v ω f (xω)vω ≥ 0. This contradicts a). 6 This comes from this general result. Let f be a C 2 function defined on a neighborhood of 0 in R, where it is 0.

The function ϕ defined as ϕ

(r) = f (r) r if r = 0 and ϕ(0) = f (0) is C 1 .
Indeed it is clearly C 2 everywhere except may be at 0. Its first derivative is ϕ

(r) = f (r)-rf (0) r 2
. It is also continuous -we have :

1 d(w 1 , w 2 ) d dt L(Γ(s, t)) s2 s1 t=0 + λ ≤ b γ * (w, d(w 1 , w 2 )) - (
a γ * (w, d(w 1 , w 2 )) = U (γ * (s 2 )) -U (w 1 ) d(w 1 , w 2 ) × × ∂U ∂x (γ * (s 2 )) dγ * ds (γ * (s 2 )) + ∂U ∂x (w 1 ) dγ * ds (s 1 )
is non positive, then there exists r 0 in in ]0, d(w 1 , w 2 )] such that either U (γ * (r 0 + s 1 )) = U (w 1 ). But the level sets of U being totally geodesic and γ * being a minimizing geodesic between w 1 and w 2 and therefore between w 1 and γ * (r 0 + s 1 ), it follows from at 0 since lim r→0 ϕ(r) = f (0) = ϕ(0). Its first derivative at 0 exists if lim r→0 ϕ(r)-ϕ(0) r

= lim r→0 f (r)-rf (0) r 2
exists. But this is the case, since f being C 2 , we have This implies that ∂U ∂x (γ * (s 1 )) dγ * ds (s 1 ) and ∂U ∂x (γ * (r 0 +s 1 )) dγ * ds (r 0 +s 1 ) have opposite signs and so the function r → ∂U ∂x (γ * (r + s 1 )) dγ * ds (r + s 1 ) must vanish on ]0, r 0 [. Again this implies U (w 2 ) = U (w 1 ) and and a γ * (w, d(w 1 , w 2 )) and b γ * (w, d(w 1 , w 2 )) are zero.

f (r) -rf (0) r 2 = 1 r 2 r 0 [f (s) -f (0)]ds = 1 r 2 r 0 s 0 f (t)dtds = 1 r 2 r 0 f (t)[r -t]dt which leads to ϕ (0) = 1 2 f (0). We have also f (r) -rf (r) r 2 = - 1 r 2 r 0 sf (
2

Now, to each pair of integers (i 1 , i 2 ), we associate the compact set

C i1i2 = (w 1 , w 2 ) ∈ R 2n : i 1 ≤ d(w 1 , 0) ≤ i 1 + 1 , i 2 ≤ d(w 2 , 0) ≤ i 2 + 1 .
Lemma 2. For any pair (i 1 , i 2 ), there exists a real number i1i2 such that, for all (w 1 , w 2 ) in C i1i2 \ D and all larger or equal to i1i2 , we have : The sequence (w 1k , w 2k ) k∈N has a cluster point (w 1ω , w 2ω ). To keep the notations simple, we still denote by k the index of the subsequence for which we have convergence to this point. By compactness, C 1 property and boundedness, there exists a real number M and an integer k * such that, for all k larger than k * , we have With Lemma 1, we get : b γ * ω (w 1ω , w 2ω , d(w 1ω , w 2ω )) = 0 So, as in the previous case, we get a contradiction of (70).

a
To complete the proof of Proposition 7, it is sufficient to observe that for any pair (w 1 , w 2 ) in R 2n , we can find (i 1 , i 2 ) such that C i1i2 contains it. It is then sufficient to pick (w 1 , w 2 ) larger or equal to i1i2 to obtain, with (68),

D + d(

  x) e , ∂h ∂x (x) e = 0 defined on [0, +∞) satisfies lim t→+∞ | E(e, x, t)| = 0.

10 5 Figure 1 :

 51 Figure 1: Numerical simulation results of the interconnected five agents as considered in Example 1. The left-figure shows the trajectories of the first state variable of each agent, w i1 , i = 1 . . . 5 while the right-figure shows those of the second state variable of each agent w i2 , i = 1 . . . 5. The simulation results show that the proposed contraction-based distributed control law is able to synchronize the states w i , i = 1 . . . 5.

2

 2

. 3 Proof of Proposition 3 Proof:

 33 Consider the function V (e, x) = e P (x)e. Using (9), the time derivative of V along the solutions of the system (1) is given, for all (e, x), by ˙ V (e, x) = 2e P (x)F (e, x) + ∂e P (•)e ∂x (x)G(e, x) ≤ -e Qe + 2e P (x) F (e, x) -∂F ∂e (0, x)e + ∂e P (•)e ∂x (x) [G(e, x) -G(0, x)] .

b 2 . 2 +

 22 (w, d(w 1 , w 2 )) -a(w, d(w 1 , w 2 )) ≤ λ For the sake of getting a contradiction, assume there exist a pair (i 1 , i 2 ) and a sequence (w 1k , w 2k , γ * k ) k∈N of points and minimizing geodesic satisfyingd(w 1k , w 2k ) = 0 w 1k = γ * k (0) w 2k = γ * k (d(w 1k , w 2k )) = w 1k + d(w 1k ,w 2k ) b γ * k (w 1k , w 2k , d(w 1k , w 2k )) (70) ≥ k a γ * k (w 1k , w 2k , d(w 1k , w 2k )) ≥ 0 . Because thesequence (b γ * k (w 1k , w 2k , d(w 1k , w 2k ))) k∈N is bounded, we have lim k→∞ a γ * k (w 1k , w 2k , d(w 1k , w 2k )) = 0 . (71)

Case 1 :

 1 w 1ω = w 2ω = w ω . Assume we have lim k→∞ d(w 1k , w 2k ) = lim k→∞ d(w 1k , w ω ) = lim k→∞ d(w 2k , w ω ) = 0 .

2 ≥ -M 1 k

 21 + 1 d(w 1k , w 2k ) compactness w 2k -w 1k d(w 1k ,w 2k ) has a cluster point we denote v ω . With again k as index for the subsequence (of the subsequence!), we havev ω = lim k→∞ w 2k -w 1k d(w 1k , w 2k )But with (69), this gives alsov ω = lim k→∞ k (w 1k , w 2k , d(w 1k , w 2k )) = 0 Since a γ * k (w 1k , w 2k , d(w 1k , w2k)) is non negative, this contradicts (70). Case 2 : w 1ω = w 2ω . Assume now we have lim k→∞ d(w 1k , w 1ω ) = lim k→∞ d(w 2k , w 2ω ) = 0 , d(w 1ω , w 2ω ) = 0 .(w 1k , w 2k ) is in the compact set C i1i2 and γ * k is a minimal geodesic at least on [0, d(w 1p , w 2p )]. So, from :p |w 1 -w 2 | ≤ d(w 1 , w 2 ) ≤ p |w 1 -w 2 | ∀(w 1 , w 2 ) ∈ C × C ,we get, for all s in [0, d(w 1k , w 2k )],p |γ * k (s) -w 1k | ≤ d(γ * k (s), w 1k ) ≤ d(w 1k , w 2k ) ≤ D i1i2whereD i1i2 = sup (w1,w2)∈Ci 1 i 2 d(w 1 , w 2 )We have also|w 1k | ≤ |w 2k -w 1k | + |w 2k | ≤ d(w 2k , w 1k ) + d(w 2k , 0) √ p , ≤ D i1i2 + (i 2 + 1) √ p .With the completeness of the metric, this implies that γ * k : [0, D i1i2 ] → R n takes its values in a compact set independent of the index k and is a solution of the geodesic equation. It follows, from instance from [10, Theorem 5, §1], that there exist a subsequence (of the subsequence !) again with k as index and a solution γ * ω of the geodesic equation satisfying,γ * ω (0) = w 1ω , γ * ω d(w 1ω , w 2ω )) = w 2ω .and limk→∞ γ * k (s) = γ * ω (s) uniformly in s ∈ [0, D i1i2 ] .Also each γ * k being minimizing between w 1k and w 2k , γ * ω is minimizing between w 1ω and w 2ω (see[26, Lemma II-.4.2]). With the definitions (66) and (67) of a γ * (w 1 , w 2 , d(w 1 , w 2 )) and b γ * (w 1 , w 2 , d(w 1 , w 2 )) and with (71) we obtain a γ * ω (w 1ω , w 2ω , d(w 1ω , w 2ω )) = lim k→∞ a γ * k (w 1k , w 2k , d(w 1k , w 2k )) , = 0 , b γ * ω (w 1ω , w 2ω , d(w 1ω , w 2ω )) = lim k→∞ b γ * k (w 1k , w 2k , d(w 1k , w 2k )) .

  If Property ULMTE holds and there exist positive real numbers η and c such that, for all (e, x) in B e (η) × R nx ,

	2.1.3 ULMTE "⇒" TULES-NL
	Proposition 3. ∂P ∂x	(x) ≤ c ,

  For the case of two agents only, we have the following. Proposition 7 (Global sufficient condition for m = 2). Assume 1. there exist a C 3 function U : R n → R which has bounded first and second derivatives, and a C 1 function α : R n → R p such that, for all x in R n , (54) is satisfied; 2. there exist a positive real number λ, a C 3 function P : R n → R n×n and positive real numbers p and p, such that inequalities (19) hold and we have, for all

	any geodesic γ, i.e. a solution of
	d ds	dγ * ds	(s) P (γ * (s)) l							(58)
						=	1 2	∂ ∂x	dγ * ds	(s) P (x)	dγ * ds	(s)	x=γ * (s)	,
	with γ(0) = x and dγ ds (0) = v satisfies
			∂ϕ ∂x	(γ(s))	dγ ds	(s) = 0 , ∀s .
							1 ,

  , for each s in [s 1 , s 2 ] consider the C 1 function t → Γ(s, t) solution of , t) = W 1 (w, t) , Γ(s 2 , t) = W 2 (w, t)and so, for each t, s ∈ [s 1 , s 2 ] → Γ(s, t) is a C 2 path between W 1 (w, t) and W 2 (w, t). From the first variation formula (see[START_REF] Spivak | A Comprehensive Introduction to Differential Geometry[END_REF] Theorem 6.14] for instance 4 ), we have

	∂Γ ∂t	(s, t) = f (Γ(s, t))
	+ (W (w, t)) g(Γ(s, t))α(Γ(s, t))×
					2
					×	[U (W j (w, t)) -U (Γ(s, t))]
					j=1
	with initial condition
			Γ(s, 0) = γ * (s) .	(65)
	With (39), we have
		Γ(s 1 d dt	L(Γ(s, t))	s2 s1	t=0

  w) a γ * (w, d(w 1 , w 2 )) . For any pair (w 1 , w 2 ) in R 2n \D and any minimizing geodesic γ * between w 1 and w 2 , a γ * (w, d(w 1 , w 2 )) is non negative, and if it is zero, the same holds for b γ * (w, d(w 1 , w 2 )). For any pair w in R 2n \ D, the function r → a γ * (w, r) is defined and continuous on [0, d(w 1 , w 2 )].

	Also
	Lemma 1. Proof : If the real number	a γ * (w, 0) = 2	∂U ∂x	(γ * (s 1 ))	ds dγ *	(s 1 )	2
	is zero, the level sets of U being totally geodesic, we have
	∂U ∂x	(γ * (r + s 1 ))	dγ * ds	(r + s 1 ) = 0 ,
		U (γ

* (r + s 1 )) = U (w 1 ) , ∀r ∈ [0, d(w 1 , w 2 )] .

and so U (w 2 ) = U (w 1 ) and a γ * (w, d(w 1 , w 2 )) and b γ * (w, d(w 1 , w 2 )) are zero.

If instead that real number is positive and

  the proof of)[START_REF] Sanfelice | Convergence of nonlinear observers on R n with a Riemannian metric (part i). Automatic Control[END_REF] Proposition A.3.2] that γ * takes its values in the level set {x : U (x) = U (w 1 )} at least on [s 1 , r 0 + s 1 ]. This implies , r 0 +s 1 ] and consequently in [s 1 , s 2 ]. This yields U (w 2 ) = U (w 1 ) and a γ * (w, d(w 1 , w 2 )) and b γ * (w, d(w 1 , w 2 )) are zero;

					∂U ∂x	(γ * (s))	dγ * ds	(s) = 0
	for all s in [s 1 or we have
	∂U ∂x	(γ * (r 0 + s 1 ))	dγ * ds	(γ * (r 0 + s 1 )) +	∂U ∂x	(γ * (s 1 ))	dγ * ds	(s 1 )
									= 0 .
									s)ds
	This implies
							lim r→0	ϕ (r) = ϕ (0)
	and therefore ϕ is continuous.

(

  γ * k (w 1k , w 2k , d(w 1k , w 2k )) ≥ a γ * k (w 1k , w 2k , 0) -M d(w 1k , w 2k ) M d(w 1k , w 2k ) b γ * k (w 1k , w 2k , dw 1k , w 2k )) ≤ b γ * k (w 1k , w 2k , 0) + M d(w 1k , w 2k ) ≤ ν(w 1k )

				≥ 2	∂U ∂x	(w 1k )	dγ * k ds	(0)
						∂U ∂x	(w 1k )	dγ * k ds	(0)	2	+ M d(w 1k , w 2k )
	This implies
	-	λ 2k	+	ν(w 1k ) k	-2	∂U ∂x	(w 1k )	dγ * k ds	(0)

2

-

  w 1 , w 2 ) ≤ |W 1 (w 1 , w 2 , t) -W 2 (w 1 , w 2 , t)| ≤ d (W 1 (w 1 , w 2 , t) -W 2 (w 1 , w 2 , t)) ,

	d dt		L(Γ(s, t))	s2 s1	t=0	≤ -	λ 2	d(w 1 , w 2 )
	This gives :
	p ≤ exp -	λ 2	t d(w 1 , w 2 ) ,
	≤ exp -	λ 2	t	p |w 1 -w 2 | .		2

This means that γ * satisfies dγ * ds (s) P (γ * (s)) dγ * ds (s) = 1 .

In[START_REF] Spivak | A Comprehensive Introduction to Differential Geometry[END_REF] Theorem 6.14], the result is stated with γ * C ∞ note however that C 2 is enough.