Samir Adly

Daniel Goeleven

A stability theory for second-order nonsmooth dynamical systems with application to friction problems

Keywords: 47J20, 49J40, 34A60, 34G25, 35K85, 74H99 Variational inequalities, Unilateral dynamic systems, Nonsmooth Mechanics, Lyapunov stability, LaSalle invariance principle, Convex analysis, Friction problems in Mechanics

A LaSalle's Invariance Theory for a class of first-order evolution variational inequalities is developed. Using this approach, stability and asymptotic properties of important classes of secondorder dynamic systems are studied. The theoretical results of the paper are supported by examples in nonsmooth Mechanics and some numerical simulations.

Résumé

Dans cet article, la théorie d'invariance de LaSalle est généralisée pour une classe d'inéquations variationnelles d'évolution du premier ordre. Des résultats de stabilité (au sens de Lyapunov) et d'attractivité sont ensuite obtenus pour des systèmes dynamiques du second ordre non réguliers. Une extension du théorème de Lagrange aux systèmes conservatifs non réguliers est également proposée. Enfin, quelques exemples et simulations numériques illustrent les principaux résultats théoriques.

Introduction

The stability of stationary solutions of unilateral dynamic systems constitutes a very important topic in Mathematics and Engineering which has recently attracted important research interest (see, e.g., [START_REF] Alvarez | An invariance principle for discontinuous dynamic systems with applications to a Coulomb friction oscillator[END_REF][START_REF] Brogliato | Absolute stability and the Lagrange-Dirichlet theorem with monotone multivalued mappings[END_REF][START_REF] Brogliato | On the control of nonsmooth complementarity dynamical systems[END_REF][START_REF] Brogliato | Some perspectives on the analysis and control of complementarity systems[END_REF][START_REF] Brogliato | Nonsmooth Mechanics[END_REF][START_REF] Goeleven | On the stability of stationary solutions of first-order evolution variational inequalities[END_REF][START_REF] Goeleven | Stability and Unstability Matrices for Linear Evolution Variational Inequalities[END_REF][START_REF] Martins | Dynamic stability of finite linearly elastic systems with unilateral contact and Coulomb friction[END_REF][START_REF] Quittner | On the principle of linearized stability for variational inequalities[END_REF][START_REF] Quittner | An instability criterion for variational inequalities[END_REF]).

The aim of this paper is to provide a mathematical theory applicable to the study of dynamic systems of the form

M
where t 0 ∈ R is fixed, Φ : R l → R ∪ {+∞} is a proper convex and lower semicontinuous function and M, C, K ∈ R m×m , H 1 ∈ R m×l , H 2 ∈ R l×m are given matrices. Here ∂Φ denotes the convex subdifferential of Φ.

The model in [START_REF] Alvarez | An invariance principle for discontinuous dynamic systems with applications to a Coulomb friction oscillator[END_REF] plays an important role in Unilateral Mechanics. Indeed, the motion of various systems (with m degrees of freedom) having frictional contact can be written in the compact form [START_REF] Alvarez | An invariance principle for discontinuous dynamic systems with applications to a Coulomb friction oscillator[END_REF]. Then M is the mass matrix of the system, C is the viscous damping matrix and K is the stiffness matrix. The vector q is a vector in which the generalized coordinates are listed. Generally the matrices M, C and K are symmetric and positive semidefinite matrices. In many cases M and K can be positive definite and H 2 = H T 1 . The term H 1 ∂Φ(H 2 .) has been introduced in order to model the unilaterality of the contact induced by friction forces. Indeed, it is now well known that contact with friction can be described by a relation of the form:

f ∈ -H 1 ∂Φ H 2 dq dt , ( 2
)
where f denotes the vector of friction forces. Indeed, friction force which opposes motion, is a complicated combination of all the force components that are distributed along the mechanical links like flat surfaces, bearings, etc. Friction characteristics can also be influenced by lubrication, temperature, a possible gear mechanical system, etc. It has been observed that experimental friction characteristics versus velocities approximated by making use of spline polynomial functions may include vertical segments. If a "graph" ( dq i dt , -f i ) is monotone then it can usually be recovered by a subdifferential relation of the form f i ∈ -∂ϕ i ( dq i dt ) where ϕ i is a convex function. This is, for example, the case of the famous Coulomb model (see, e.g., [START_REF] Brogliato | Nonsmooth Mechanics[END_REF][START_REF] Goeleven | Variational and Hemivariational Inequalities[END_REF][START_REF] Goeleven | Variational and Hemivariational Inequalities[END_REF][START_REF] Martins | Dynamic stability of finite linearly elastic systems with unilateral contact and Coulomb friction[END_REF][START_REF] Panagiotopoulos | Inequality Problems in Mechanics and Applications[END_REF]). Most discrete systems are made of point masses connected to each others and a whole formulation of the friction dynamic leads usually to a mathematical model like the one given in [START_REF] Brogliato | Absolute stability and the Lagrange-Dirichlet theorem with monotone multivalued mappings[END_REF].

In this paper, we give also some results applicable to the model:

M d 2 q dt 2 (t) + Π q(t) ∈ -H 1 ∂Φ H T 1 dq dt (t) , a.e. t t 0 , (3)
where Π ∈ C 1 (R m ; R) and M ∈ R m×m is assumed symmetric and positive definite.

The model in [START_REF] Brogliato | On the control of nonsmooth complementarity dynamical systems[END_REF] concerns mechanical systems involving conservative forces Q = -Π (q) where Π is the potential energy of the system.

It seems that the literature does not yet propose a general mathematical approach to study the stability of problems [START_REF] Alvarez | An invariance principle for discontinuous dynamic systems with applications to a Coulomb friction oscillator[END_REF] and [START_REF] Brogliato | On the control of nonsmooth complementarity dynamical systems[END_REF]. This is the aim of this work.

In this paper, we give conditions on the data M, K, C, H 1 , H 2 and Φ so as to ensure the existence and uniqueness of a solution q(•; t 0 , q 0 , q0 ) of (1) satisfying given initial conditions q(t 0 ) = q 0 and dq dt (t 0 ) = q0 . Then we give conditions ensuring that any stationary solution of (1) is stable (in the sense of Lyapunov). Finally, we discuss some asymptotic properties of the model. More precisely, we give conditions ensuring that lim τ →+∞ d q(τ ; t 0 , q 0 , q0 ), W = 0 and lim τ →+∞ dq dt (τ ; t 0 , q 0 , q0 ) = 0, where W := { q ∈ R m : K q ∈ -H 1 ∂Φ(0)} denotes the set of stationary solutions of [START_REF] Alvarez | An invariance principle for discontinuous dynamic systems with applications to a Coulomb friction oscillator[END_REF].

To prove such results, we first give conditions ensuring that problem [START_REF] Alvarez | An invariance principle for discontinuous dynamic systems with applications to a Coulomb friction oscillator[END_REF] can be reduced to a first-order evolution variational inequality. Next we develop a theory extending LaSalle invariance principle (see, e.g., [START_REF] Rouche | Equations Différentielles Ordinaires[END_REF]) to first-order evolution variational inequalities.

Sections 2 and 3 concern this class of first-order dynamic systems. In Section 2, we recall a stability theorem which has been recently proved in [START_REF] Goeleven | On the stability of stationary solutions of first-order evolution variational inequalities[END_REF]. In Section 3, we prove some general invariance theorem applicable to a large class of first-order evolution variational inequalities. In Section 4, we use the results of Sections 2 and 3 so as to discuss the stability of the system in [START_REF] Alvarez | An invariance principle for discontinuous dynamic systems with applications to a Coulomb friction oscillator[END_REF].

The results of Sections 2 and 3 are also used in Section 5 so as to prove a theorem extending the famous Lagrange's theorem (see, e.g., [START_REF] Merkin | Introduction to the Theory of Stability[END_REF]) to the model in [START_REF] Brogliato | On the control of nonsmooth complementarity dynamical systems[END_REF].

Finally, some illustrative small-sized examples in Mechanics are presented in Section 6.

First-order dynamic systems

In this section, we deal with the following general class of first-order dynamic systems. Let ϕ : R n → R ∪ {+∞} be a proper convex and lower semicontinuous function. The notations D(ϕ) and D(∂ϕ) stand for the domain of ϕ and the domain of the subdifferential ∂ϕ of ϕ, respectively, i.e.,

D(ϕ)

:= x ∈ R n : ϕ(x) < +∞ and D(∂ϕ) := x ∈ R n : ∂ϕ(x) = ∅ . Recall that D(∂ϕ) ⊂ D(ϕ), D(∂ϕ) = D(ϕ).
Let F : R n → R n be a continuous operator such that for some ω 0, F + ωI is monotone.

Here I denotes the identity mapping on R n . For (t 0 , x 0 ) ∈ R × D(∂ϕ), we consider the problem P (t 0 , x 0 ):

Find a function t → x(t) (t t 0 ) with x ∈ C 0 ([t 0 , +∞); R n ), dx dt ∈ L ∞ loc (t 0 , +∞; R n ) and such that ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ x(t) ∈ D(∂ϕ), t t 0 , dx dt (t) + F x(t) , v -x(t) + ϕ(v) -ϕ x(t) 0, ∀v ∈ R n , a.e. t t 0 , x(t 0 ) = x 0 . ( 4
)
Here •, • denotes the Euclidean scalar product in R n . The corresponding norm is denoted by • . It follows from standard convex analysis that (4) can be rewritten equivalently as the differential inclusion:

dx dt (t) + F x(t) ∈ -∂ϕ x(t) . ( 5
) Remark 1. Note that if F : R n → R n is Lipschitz continuous with Lipschitz constant k > 0 then F is continuous and F + kI is monotone.
Let us first specify some conditions ensuring the existence and uniqueness of the initial value problem P (t 0 , x 0 ). The following existence and uniqueness result is essentially a consequence of Kato's theorem [START_REF] Kato | Accretive Operators and Nonlinear Evolutions Equations in Banach Spaces, Part 1, Nonlinear Functional Analysis[END_REF]. We refer the reader to [6, Corollary 2.2] for the details.

Theorem 1. Let ϕ : R n → R ∪ {+∞} be a proper convex and lower semicontinuous function and let F : R n → R n be a continuous operator such that for some ω 0, F + ωI is monotone. Let t 0 ∈ R and x 0 ∈ D(∂ϕ) be given. Then there exists a unique

x ∈ C 0 ([t 0 , +∞); R n ) such that dx dt ∈ L ∞ loc t 0 , +∞; R n , (6)
x is right-differentiable on [t 0 , +∞),

x(t 0 ) = x 0 , (8)

x(t) ∈ D(∂ϕ), t t 0 , ( 9
) dx dt (t) + F x(t) , v -x(t) + ϕ(v) -ϕ x(t) 0, ∀v ∈ R n , a.e. t t 0 . ( 10
)
Suppose that the assumptions of Theorem 1 are satisfied and denote by x(•; t 0 , x 0 ) the unique solution of problem P (t 0 , x 0 ). We prove below that for t t 0 fixed, the application x(t; t 0 , •) is uniformly continuous on D(∂ϕ). This property will be used later in Section 3. Let us first recall some Gronwall inequality that is used in our next result (see, e.g., Lemma 4.1 in [START_REF] Showalter | Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations[END_REF]).

Lemma 1. Let T > 0 be given and let a, b ∈ L 1 (t 0 , t 0 + T ; R) with b(t) 0 a.e. t ∈ [t 0 , t 0 + T ]. Let the absolutely continuous function w : [t 0 , t 0 + T ] → R + satisfy: (1 -α)w (t) a(t)w(t) + b(t)w α (t), a.e. t ∈ [t 0 , t 0 + T ],
where 0 α < 1. Then

w 1-α (t) w 1-α (t 0 )e t t 0 a(τ ) dτ + t t 0 e t s a(τ ) dτ b(s) ds, ∀t ∈ [t 0 , t 0 + T ].
Theorem 2. Let the assumptions of Theorem 1. Let τ t 0 be fixed. The application

x(τ ; t 0 , •) : x 0 → x(τ ; t 0 , x 0 )
is uniformly continuous on D(∂ϕ).

Proof. Let τ t 0 be fixed. Let ε > 0 be given and set,

δ := ε √ e 2ω(τ -t 0 )
. We claim that if x 0 , x 0 ∈ D(∂ϕ), x 0x 0 δ then x(τ ; t 0 , x 0 )x(τ ; t 0 , x 0 ) ε. Indeed, let us set x(t) := x(t; t 0 , x 0 ) and x (t) := x(t; t 0 , x 0 ). We know that dx dt

(t) + F x(t) , v -x(t) + ϕ(v) -ϕ x(t) 0, ∀v ∈ R n ,
) 15
We may apply Lemma 1 with 2 to get:

T > τ -t 0 , α = 0, b(•) = 0, a(•) = 2ω and w(•) = x (•) -x(•)
x (t) -x(t) 2 x 0 -x 0 2 e 2ω(t-t 0 ) , ∀t ∈ [t 0 , t 0 + T ]. (16)
It follows that

x (τ ) -x(τ ) δ e 2ω(τ -t 0 ) = ε. 2
Suppose now in addition to the assumptions of Theorem 1 that

0 ∈ D(∂ϕ), F (0) ∈ -∂ϕ(0). (17)

Then

x(t; t 0 , 0) = 0, ∀t t 0 , i.e., the trivial stationary solution 0 is the unique solution of problem P (t 0 , 0). We may now define as in [START_REF] Goeleven | On the stability of stationary solutions of first-order evolution variational inequalities[END_REF] the stability of the trivial solution. The stationary solution 0 is called stable if small perturbations of the initial condition x(t 0 ) = 0 lead to solutions which remain in the neighborhood of 0 for all t t 0 , precisely: Definition 1. The equilibrium point x = 0 is said to be stable in the sense of Lyapunov if, for every ε > 0, there exists η = η(ε) > 0 such that for any x 0 ∈ D(∂ϕ) with x 0 η the solution x(•; t 0 , x 0 ) of problem P (t 0 , x 0 ) satisfies x(t; t 0 , x 0 ) ε, ∀t t 0 .

If in addition the trajectories of the perturbed solutions are attracted by 0 then we say that the stationary solution is asymptotically stable, precisely: Definition 2. The equilibrium point x = 0 is asymptotically stable if (1) it is stable and (2) there exists δ > 0 such that for any x 0 ∈ D(∂ϕ) with x 0 δ the solution x(•; t 0 , x 0 ) of problem P (t 0 , x 0 ) fulfills lim t →+∞ x(t; t 0 , x 0 ) = 0.

Note that the equilibrium point x = 0 is said attractive (respectively globally attractive) as soon as part (2) of Definition 2 holds (respectively for any x 0 ∈ D(∂ϕ)).

Let us now recall a general abstract theorem of stability in terms of generalized Lyapunov functions V ∈ C 1 (R n ; R). The following result is a particular case of the one proved in [START_REF] Goeleven | On the stability of stationary solutions of first-order evolution variational inequalities[END_REF]. Here, for r > 0, we denote by B r the closed ball of radius r, i.e., B r := {x ∈ R n : x r}.

Theorem 3. Let the assumptions of Theorem 1 together with condition [START_REF] Showalter | Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations[END_REF]. Suppose that there exists σ > 0 and

V ∈ C 1 (R n ; R) such that (1) V (x) a( x ), x ∈ D(∂ϕ) ∩ B σ , with a : [0, σ ] → R satisfying a(t) > 0, ∀t ∈ (0, σ ); (2) V (0) = 0; (3) F (x), V (x) + ϕ(x) -ϕ(x -V (x)) 0, x ∈ D(∂ϕ) ∩ B σ .
Then the trivial solution of (9)-( 10) is stable.

Various applications of Theorem 3 can be found in [START_REF] Goeleven | On the stability of stationary solutions of first-order evolution variational inequalities[END_REF][START_REF] Goeleven | Stability and Unstability Matrices for Linear Evolution Variational Inequalities[END_REF]. For example, engineering systems described by a SPR transfer function and a feedback branch containing a sector static nonlinearity are discussed in [START_REF] Goeleven | Stability and Unstability Matrices for Linear Evolution Variational Inequalities[END_REF].

We end this section by remarking that some of the hypothesis stated in Theorem 3 can also be used to obtain some additional information on the set of stationary solutions of ( 9)- [START_REF] Kato | Accretive Operators and Nonlinear Evolutions Equations in Banach Spaces, Part 1, Nonlinear Functional Analysis[END_REF].

Let us here denote by S(F, ϕ) the set of stationary solutions of ( 9)- [START_REF] Kato | Accretive Operators and Nonlinear Evolutions Equations in Banach Spaces, Part 1, Nonlinear Functional Analysis[END_REF], that is:

S(F, ϕ) := z ∈ D(∂ϕ): F (z), v -z + ϕ(v) -ϕ(z) 0, ∀v ∈ R n .
Condition [START_REF] Showalter | Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations[END_REF] ensures that 0 ∈ S(F, ϕ). Let V ∈ C 1 (R n ; R) be given. We set:

E(F, ϕ, V ) := x ∈ D(∂ϕ): F (x), V (x) + ϕ(x) -ϕ x -V (x) = 0 . ( 18
)
Let us end this section by showing that condition (3) in Theorem 3 has some consequences on the qualitative properties of the stationary solutions of ( 9)- [START_REF] Kato | Accretive Operators and Nonlinear Evolutions Equations in Banach Spaces, Part 1, Nonlinear Functional Analysis[END_REF].

Proposition 1. Let the assumptions of Theorem 1 together with condition [START_REF] Showalter | Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations[END_REF]. Let Ψ be a subset of R n . Suppose that there exists

V ∈ C 1 (R n ; R) such that (1) F (x), V (x) + ϕ(x) -ϕ(x -V (x)) 0, x ∈ D(∂ϕ) ∩ Ψ . Then S(F, ϕ) ∩ Ψ ⊂ E(F, ϕ, V ).
Proof. Let z ∈ Ψ ∩ S(F, ϕ) be given. We have z ∈ D(∂ϕ) ∩ Ψ and

F (z), v -z + ϕ(v) -ϕ(z) 0, ∀v ∈ R n . ( 19
) Setting v = z -V (z) in (19), we get F (z), V (z) + ϕ(z) -ϕ z -V (z) 0.
Then using assumption (1), we obtain:

F (z), V (z) + ϕ(z) -ϕ z -V (z) = 0. 2 Proposition 2.
Let the assumptions of Theorem 1 together with condition [START_REF] Showalter | Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations[END_REF]. Suppose that there exists σ > 0 and

V ∈ C 1 (R n ; R) such that (1) F (x), V (x) + ϕ(x) -ϕ(x -V (x)) 0, x ∈ D(∂ϕ) ∩ B σ ; (2) E(F, ϕ, V ) ∩ B σ = {0}.
Then the trivial stationary solution of ( 9)-( 10) is isolated in S(F, ϕ).

Proof. We claim that B σ ∩ S(F, ϕ) = {0}. Indeed, setting Ψ := B σ and using assumption (2) together with Proposition 1, we obtain:

B σ ∩ S(F, ϕ) = Ψ ∩ S(F, ϕ) ⊂ Ψ ∩ E(F, ϕ, V ) = {0}. 2
The following results can be proved by the same arguments as the ones used in the proof of Propositions 1 and 2.

Proposition 3. Let the assumptions of Theorem 1 together with condition (17) hold. Suppose that there exists

V ∈ C 1 (R n ; R) such that (1) F (x), V (x) + ϕ(x) -ϕ(x -V (x)) 0, x ∈ D(∂ϕ).

Then S(F, ϕ) ⊂ E(F, ϕ, V ).

Proposition 4. Let the assumptions of Theorem 1 together with condition [START_REF] Showalter | Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations[END_REF]. Suppose that there exists

V ∈ C 1 (R n ; R) such that (1) F (x), V (x) + ϕ(x) -ϕ(x -V (x)) 0, x ∈ D(∂ϕ); (2) E(F, ϕ, V ) = {0}.
Then S(F, ϕ) = {0}, i.e., the trivial stationary solution of (9)-( 10) is the unique stationary solution of (9)- [START_REF] Kato | Accretive Operators and Nonlinear Evolutions Equations in Banach Spaces, Part 1, Nonlinear Functional Analysis[END_REF].

The invariance theorem

Let the assumptions of Theorem 1 hold. For x 0 ∈ D(∂ϕ), we denote by γ (x 0 ) the orbit γ (x 0 ) := x(τ ; t 0 , x 0 ): τ t 0 and by Λ(x 0 ) the limit set,

Λ(x 0 ) := z ∈ R n : ∃{τ i } ⊂ [t 0 , +∞); τ i → +∞ and x(τ i ; t 0 , x 0 ) → z . We say that a set D ⊂ D(∂ϕ) is invariant provided that x 0 ∈ D ⇒ γ (x 0 ) ⊂ D.
Here we denote by

d(s, M) the distance from a point s ∈ R n to a set M ⊂ R n , that is d(s, M) := inf m∈M s -m . Remark 2. Let x 0 ∈ D(∂ϕ) be given. (i) It is clear that γ (x 0 ) ⊂ D(∂ϕ), Λ(x 0 ) ⊂ D(∂ϕ).
(ii) It is easy to check that

Λ(x 0 ) ⊂ γ (x 0 ). (iii) If γ (x 0 ) is bounded, then Λ(x 0 ) = ∅.
Indeed, if γ (x 0 ) is bounded, then we may find a sequence x(τ i ;

t 0 , x 0 )(τ i t 0 ) such that x(τ i ; t 0 , x 0 ) → z ∈ R n . It results that z ∈ Λ(x 0 ). (iv) If γ (x 0 ) is bounded, then lim τ →+∞ d x(τ ; t 0 , x 0 ), Λ(x 0 ) = 0.
Indeed, if we suppose the contrary then we can find ε > 0 and {τ i } ⊂ [t 0 , +∞) such that τ i → +∞ and d(x(τ i ; t 0 , x 0 ), Λ(x 0 )) ε. The sequence x(τ i ; t 0 , x 0 ) is bounded and along a subsequence, we may suppose that x(τ i ; t 0 , x 0 ) → x * . Thus x * ∈ Λ(x 0 ). On the other hand, we get the contradiction d(x * , Λ(x 0 )) ε.

(v) The set of stationary solutions S(F, ϕ) is invariant. Indeed, if x 0 ∈ S(F, ϕ) then x(τ ; t 0 , x 0 ) = x 0 , ∀t t 0 , and thus γ (x 0 ) = {x 0 } ⊂ S(F, ϕ).
Thanks to Theorem 2, we can prove that the set Λ(x 0 ) is invariant by using standard topological arguments (see, e.g., [START_REF] Rouche | Equations Différentielles Ordinaires[END_REF]). Theorem 4. Let the assumptions of Theorem 1. Let x 0 ∈ D(∂ϕ) be given. The set Λ(x 0 ) is invariant.

Proof. Let z ∈ Λ(x 0 ) be given. There exists {τ i } ⊂ [t 0 , +∞) such that τ i → +∞ and x(τ i ; t 0 , x 0 ) → z. Let τ t 0 be given. Using Theorem 2, we obtain x(τ ; t 0 , z) = lim i→∞ x τ ; t 0 , x(τ i ; t 0 , x 0 ) .

Then remarking that x(τ ; t 0 , x(τ i ; t 0 , x 0 )) = x(τt 0 + τ i ; t 0 , x 0 ), we get x(τ ; t 0 , z) = lim i→∞ x(τ -t 0 +τ i ; t 0 , x 0 ). Thus setting w i := τ -t 0 +τ i , we see that w i t 0 , w i → +∞ and x(w i ; t 0 , x 0 ) → x(τ ; t 0 , z). It results that x(τ ; t 0 , z) ∈ Λ(x 0 ).

Our goal is now to prove an extension of the LaSalle Invariance Principle applicable to the first-order evolution variational inequality given in (4). Lemma 2. Let the assumptions of Theorem 1. Let Ψ be a compact subset of R n . We assume that there exists

V ∈ C 1 (R n ; R) such that (1) F (x), V (x) + ϕ(x) -ϕ(x -V (x)) 0, x ∈ D(∂ϕ) ∩ Ψ . Let x 0 ∈ D(∂ϕ) be given. If γ (x 0 ) ⊂ Ψ then there exists a constant k ∈ R such that V (x) = k, ∀x ∈ Λ(x 0 ).
Proof. Let T > 0 be given. We define the mapping V * : [t 0 ; +∞) → R by the formula

V * (t) := V x(t; t 0 , x 0 ) .
The function x(•) ≡ x(•; t 0 , x 0 ) is absolutely continuous on [t 0 , t 0 + T ] and thus V * is a.e. strongly differentiable on [t 0 , t 0 + T ]. We have:

dV * dt (t) = V x(t) , dx dt (t) , a.e. t ∈ [t 0 , t 0 + T ].
We know (by hypothesis) that We know that

x(t) ∈ D(∂ϕ) ∩ Ψ, ∀t t 0 , and dx dt (t) + F x(t) , v -x(t) + ϕ(v) -ϕ x(t) 0, ∀v ∈ R n , a.e. t t 0 . (20) Setting v = x(t) -V (x(t)) in (20), we obtain: dx dt (t), V x(t) -ϕ x(t) + ϕ x(t) -V x(t) -F x(t) , V x(
x ∈ C 0 ([t 0 , t 0 + T ]; R n ), dx dt ∈ L ∞ (t 0 , t 0 + T ; R n ) and V ∈ C 1 (R n ; R). It follows that V * ∈ W 1,1 (t 0 , t 0 + T ; R n )
and applying, e.g., Lemma 3.1 in [START_REF] Goeleven | On the stability of stationary solutions of first-order evolution variational inequalities[END_REF], we obtain that V * is decreasing on [t 0 , t 0 + T ]. The real T has been chosen arbitrary and thus

V * is decreasing on [t 0 , +∞). Moreover V * is bounded from below on [t 0 , +∞) since γ (x 0 ) ⊂ Ψ and V is continuous on the compact set Ψ . It results that lim τ →+∞ V x(τ ; t 0 , x 0 ) = k, for some k ∈ R.
Let y ∈ Λ(x 0 ) be given. There exists

{τ i } ⊂ [t 0 , +∞) such that τ i → +∞ and x(τ i ; t 0 , x 0 ) → y. By continuity lim i→+∞ V x(τ i ; t 0 , x 0 ) = V (y).
Therefore V (y) = k. Here y has been chosen arbitrary in Λ(x 0 ) and thus

V (y) = k, ∀y ∈ Λ(x 0 ). 2 Lemma 3.
Let the assumptions of Theorem 1. We assume that there exists

V ∈ C 1 (R n ; R) such that (1) F (x), V (x) + ϕ(x) -ϕ(x -V (x)) 0, x ∈ D(∂ϕ).
Let a ∈ R be given and set:

Ψ := x ∈ R n : V (x) a .
The set D(∂ϕ) ∩ Ψ is invariant.

Proof. Let x 0 ∈ D(∂ϕ) ∩ Ψ be given. Then x 0 ∈ D(∂ϕ) and V (x 0 ) a. If τ t 0 then x(τ ; t 0 , x 0 ) ∈ D(∂ϕ) and as in the proof of Lemma 2, we check that

V (x(•; t 0 , x 0 )) is decreasing on [t 0 , +∞). Thus V x(τ ; t 0 , x 0 ) V x(t 0 ; t 0 , x 0 ) = V (x 0 ) a.
It results that

γ (x 0 ) ⊂ D(∂ϕ) ∩ Ψ . 2
Theorem 5 (Invariance theorem). Suppose that the assumptions of Theorem 1 hold. Let Ψ ⊂ R n be a compact set and

V ∈ C 1 (R n ; R) a function such that (1) ϕ(•) -ϕ(• -V (•)) is lower semicontinuous on D(∂ϕ) ∩ Ψ ; (2) F (x), V (x) + ϕ(x) -ϕ(x -V (x)) 0, x ∈ D(∂ϕ) ∩ Ψ ; (3) D(∂ϕ) is closed.
We set:

E Ψ (F, ϕ, V ) := E(F, ϕ, V ) ∩ Ψ
and we denote by M the largest invariant subset of E Ψ (F, ϕ, V ). Then, for each

x 0 ∈ D(∂ϕ) such that γ (x 0 ) ⊂ Ψ , we have lim τ →+∞ d x(τ ; t 0 , x 0 ), M = 0.
Proof. Here γ (x 0 ) is bounded and thus (see Remark 2, (iii) and (iv)) Λ(x 0 ) is nonempty and

lim τ →+∞ d x(τ ; t 0 , x 0 ), Λ(x 0 ) = 0.
Let us now check that Λ(x 0 ) ⊂ E Ψ (F, ϕ, V ). We first note that

Λ(x 0 ) ⊂ γ (x 0 ) ⊂ D(∂ϕ) ∩ Ψ = D(∂ϕ) ∩ Ψ.
From Lemma 2, there exists k ∈ R such that V (x) = k, ∀x ∈ Λ(x 0 ). Let z ∈ Λ(x 0 ) be given. Using Theorem 4, we see that x(t; t 0 , z) ∈ Λ(x 0 ), ∀t t 0 and thus

V x(t; t 0 , z) = k, ∀t t 0 .
It results that

d dt V x(t; t 0 , z) = 0, a.e. t t 0 . ( 22
)
Setting x(•) ≡ x(•; t 0 , z), we check as in the proof of Lemma 2 that

V x(t) , dx dt (t) -F x(t) , V x(t) (23) -ϕ x(t) + ϕ x(t) -V x(t) , a.e. t t 0 . ( 24
)
From ( 22) and ( 24) we deduce that

F x(t) , V x(t) + ϕ x(t) -ϕ x(t) -V x(t) 0, a.e. t t 0 .
Using assumption (1), we see that the mapping, t → F x(t; t 0 , z) , V x(t; t 0 , z) + ϕ x(t; t 0 , z)ϕ x(t; t 0 , z) -V x(t; t 0 , z) is lower semicontinuous on [t 0 , +∞) and thus taking the lim inf as t → t 0 , we obtain:

F (z), V (z) + ϕ(z) -ϕ z -V (z) 0.
This together with assumption (2) ensure that z ∈ E Ψ (F, ϕ, V ).

Finally Λ(x 0 ) ⊂ M since Λ(x 0 ) ⊂ E Ψ (F, ϕ, V ) and Λ(x 0 ) is invariant (see Theorem 4
). The conclusion follows. 2

Remark 3. Note that the conditions of Theorem 5 ensure that

S(F, ϕ) ∩ Ψ ⊂ M. Indeed, Proposition 1 yields S(F, ϕ) ∩ Ψ ⊂ E Ψ (F, ϕ, V ) and S(F, ϕ) ∩ Ψ is invariant.
Corollary 1. Suppose that the assumptions of Theorem

1 hold. Let V ∈ C 1 (R n ; R) be a function such that (1) ϕ(•) -ϕ(• -V (•)) is lower semicontinuous on D(∂ϕ); (2) F (x), V (x) + ϕ(x) -ϕ(x -V (x)) 0, x ∈ D(∂ϕ); (3) V (x) → +∞ as x → +∞, x ∈ D(∂ϕ); (4) D(∂ϕ) is closed.
Let M be the largest invariant subset of E(F, ϕ, V ). Then, for each x 0 ∈ D(∂ϕ), the orbit γ (x 0 ) is bounded and

lim τ →+∞ d x(τ ; t 0 , x 0 ), M = 0.
Proof. Let x 0 ∈ D(∂ϕ) be given. We set 3) and ( 4) ensure that D(∂ϕ) ∩ Ψ is bounded and closed. Thus Ψ is compact. Lemma 3 ensures that Ψ is invariant. Here x 0 ∈ Ψ and thus γ (x 0 ) ⊂ Ψ . It results that γ (x 0 ) is bounded. Moreover, from Theorem 5, we obtain:

Ψ := {x ∈ R n : V (x) V (x 0 )} and Ψ = Ψ ∩ D(∂ϕ). The set Ψ is closed. Assumptions (
lim τ →+∞ d x(τ ; t 0 , x 0 ), M * = 0,
where M * is the largest invariant subset of E Ψ (F, ϕ, V ). It is clear that M * ⊂ M and the conclusion follows. 2 Corollary 2. Let the assumptions of Theorem 1 together with condition [START_REF] Showalter | Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations[END_REF]. Suppose that there exists V ∈ C 1 (R n ; R) such that (1) V (x) a( x ), x ∈ D(∂ϕ), with a : R + → R satisfying a(0) = 0, a strictly increasing on

R + ; (2) V (0) = 0; (3) ϕ(•) -ϕ(• -V (•)) is lower semicontinuous on D(∂ϕ); (4) F (x), V (x) + ϕ(x) -ϕ(x -V (x)) 0, x ∈ D(∂ϕ); (5) D(∂ϕ) is closed; (6) E(F, ϕ, V ) = {0}.
Then the trivial solution of (9)-( 10) is (a) the unique stationary solution of (9)-( 10), (b) asymptotically stable, (c) globally attractive, i.e., for each x 0 ∈ D(∂ϕ), lim t →+∞ x(t; t 0 , x 0 ) = 0.

Proof. Assertion (a) is a consequence of Proposition 4. The stability is a direct consequence of Theorem 3. Moreover, we may apply Corollary 1 with M = {0} (since E(F, ϕ, V ) = {0}) to obtain that for any x 0 ∈ D(∂ϕ) the limit [START_REF] Showalter | Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations[END_REF]. Suppose that there exists σ > 0 and

V ∈ C 1 (R n ; R) such that (1) V (x) a( x ), x ∈ D(∂ϕ) ∩ B σ , with a : [0, σ ] → R satisfying a(t) > 0, ∀t ∈ (0, σ ); (2) V (0) = 0; (3) ϕ(•) -ϕ(• -V (•)) is lower semicontinuous on D(∂ϕ) ∩ B σ ; (4) F (x), V (x) + ϕ(x) -ϕ(x -V (x)) 0, x ∈ D(∂ϕ) ∩ B σ ; (5) D(∂ϕ) is closed; (6) E(F, ϕ, V ) ∩ B σ = {0}.
Then the trivial solution of (9)-( 10) is (a) isolated in S(F, ϕ), (b) asymptotically stable.

Proof. Assertion (a) is a direct consequence of Proposition 2. The stability follows from Theorem 3. The stability ensures the existence of δ > 0 such that if

x 0 ∈ D(∂ϕ) ∩ B δ , then γ (x 0 ) ⊂ B σ .
Applying Theorem 5 with Ψ = B σ , we obtain for

x 0 ∈ D(∂ϕ) ∩ B δ that lim t →+∞ d x(t; t 0 , x 0 ), M = 0,
where M is the largest invariant subset of E Ψ (F, ϕ, V ). It is clear that assumption [START_REF] Goeleven | On the stability of stationary solutions of first-order evolution variational inequalities[END_REF] yields M = {0}. The attractivity and assertion (b) follow. 2 Corollary 4. Let the assumptions of Theorem 1 together with condition [START_REF] Showalter | Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations[END_REF]. Assume that D(∂ϕ) is closed and suppose that there exists σ > 0 such that

F (x), x + ϕ(x) -ϕ(0) > 0, x ∈ D(∂ϕ) ∩ B σ , x = 0.
Then the trivial stationary solution of (9)-( 10) is (a) isolated in S(F, ϕ) and (b) asymptotically stable.

Proof. This follows from Corollary 3 that we may apply with

V ∈ C 1 (R n ; R) defined by V (x) = 1/2 x 2 , x ∈ R n . 2

Second-order dynamic systems

In this section, we deal with the following class of second-order dynamic systems: Let Φ : R l → R ∪ {+∞} be a proper convex and lower semicontinuous function. Let M, C, K ∈ R m×m , H 1 ∈ R m×l and H 2 ∈ R l×m be given matrices. For (t 0 , q 0 , q0 ) ∈ R × R m × R m with H 2 q0 ∈ D(∂Φ), we consider the problem P (t 0 , q 0 , q0 ): Find a function t → q(t) (t t 0 ) with q ∈ C 1 ([t 0 , +∞); R m ), and such that

d 2 q dt 2 ∈ L ∞ loc (t 0 , +∞; R m ), ( 25
) dq dt is right-differentiable on [t 0 , +∞), (26)
q(t 0 ) = q 0 , (27) dq dt (t 0 ) = q0 , ( 28
)
H 2 dq dt (t) ∈ D(∂Φ), t t 0 , ( 29
) M d 2 q dt 2 (t) + C dq dt (t) + Kq(t) ∈ -H 1 ∂Φ H 2 dq dt (t) , a.e. t t 0 . ( 30
)
The model in (30) can be used in Mechanics to describe the motion of various systems having frictional contact. For such problems, m is the number of degrees of freedom of the system, M is the mass matrix of the system, C is the viscous damping matrix of the system and K is the stiffness matrix. The term H 1 ∂Φ(H 2 .) is used to model the unilaterality of the contact induced by friction forces. The Euclidean scalar product in R m is denoted by •, • m and the corresponding norm by • m . The subordinate matrix norm is also denoted by • m . In this section, we also use the notations I m and 0 p×q to denote the m × m identity matrix and the p × q null matrix respectively.

Theorem 6 (Existence and uniqueness). Suppose that the following assumptions are satisfied:

(1) M is nonsingular;

(2) there exists a matrix R ∈ R m×m , symmetric and nonsingular such that:

R -2 H T 2 = M -1 H 1 ;
(3) there exists y 0 = H 2 R -1 x 0 (x 0 ∈ R m ), at which Φ is finite and continuous.

Let t 0 ∈ R, q 0 , q0 ∈ R m with H 2 q0 ∈ D(∂Φ). Then there exists a unique q ∈ C 1 ([t 0 , +∞); R m ) satisfying conditions (25)-(30).

Proof. Let us here for a function f use the notations f = d 2 f dt 2 and ḟ = df dt . We first remark that (30), i.e.,

M q + C q + Kq ∈ -H 1 ∂Φ(H 2 q) is equivalent to q + M -1 C q + M -1 Kq ∈ -M -1 H 1 ∂Φ(H 2 q). Hence, R q + RM -1 CR -1 R q + RM -1 KR -1 Rq ∈ -RM -1 H 1 ∂Φ H 2 R -1 R q . ( 31
)
Setting z = Rq in (31), we get:

z + RM -1 CR -1 ż + RM -1 KR -1 z ∈ -RM -1 H 1 ∂Φ H 2 R -1 ż .
Using now assumption (2), we obtain:

z + RM -1 CR -1 ż + RM -1 KR -1 z ∈ -R -1 H T 2 ∂Φ H 2 R -1 ż . ( 32
)
Let us here define the function χ : R m → R ∪ {+∞} by the formula

χ(w) = Φ • H 2 R -1 (w), ∀w ∈ R m .
It is clear that χ is convex and lower semicontinuous. Moreover, thanks to assumption (3), χ is proper and we have (see, e.g., Proposition 2.4.5 in [START_REF] Panagiotopoulos | Inequality Problems in Mechanics and Applications[END_REF]):

∂χ(w) = R -1 H T 2 ∂Φ H 2 R -1 w , ∀w ∈ R m .
Thus (32) reduces to

z + RM -1 CR -1 ż + RM -1 KR -1 z ∈ -∂χ(ż). (33)
We note also that ( 27), ( 28) and ( 29) can be written here respectively in term of the variable z as z(t 0 ) = Rq 0 , ż(t 0 ) = R q0 and ż(t) ∈ D(∂χ), ∀t t 0 . Moreover, R q0 ∈ D(∂χ) since H 2 q0 ∈ D(∂Φ). Let us now set:

x 1 := z, x 2 := ż, x = x 1 x 2 . ( 34
)
It is clear that (33) is equivalent to the following first-order system:

ẋ1 -x 2 = 0, ẋ2 + RM -1 CR -1 x 2 + RM -1 KR -1 x 1 ∈ -∂χ(x 2 ).
It results that problem P (t 0 , q 0 , q0 ) can be written as follows:

ẋ + Ax ∈ -∂ϕ(x), x(t 0 ) = x 0 ,
where the matrix A ∈ R n×n (n = 2m) is defined by:

A = 0 m×m -I m RM -1 KR -1 RM -1 CR -1 , ( 35
)
the vector x 0 ∈ R n is given by:

x 0 = Rq 0 R q0 , ( 36
)
and the proper, convex and lower semicontinuous function ϕ : R n → R ∪ {+∞} is defined by:

ϕ(x) := χ(x 2 ). ( 37
)
The result is thus a direct consequence of Theorem 1 (with

F (•) = A). Indeed, A is Lipschitz continuous (see Remark 1). 2
Let the assumptions of Theorem 6 and let us now denote by q(•; t 0 , q 0 , q0 ) the unique solution of problem P (t 0 , q 0 , q0 ).

The set W of stationary solutions of ( 29)-( 30) is given by:

W = q ∈ R m : K q ∈ -H 1 ∂Φ(0) .
We suppose that

0 ∈ D(∂Φ). ( 38
) Remark 4. (i) If 0 ∈ ∂Φ(0) then it is clear that 0 ∈ W; (ii) if 0 ∈ D(∂Φ) and K is nonsingular then W = -K -1 H 1 ∂Φ(0); (iii) if ∂Φ(0) = {0} then W = ker K;
(iv) If Φ (0) exists and K is nonsingular then the trivial stationary solution of ( 29)-( 30) is the unique stationary solution of ( 29)-(30). Indeed, here we have

W = {-K -1 H 1 Φ (0)}.
We consider the stability of a stationary solution with respect to the "generalized coordinates" q 1 , . . . , q m and the "generalized velocities" dq 1 dt , . . . , dq m dt . More precisely, we say that a stationary solution q ∈ W is stable provided that for any ε > 0 there exists η(ε) > 0 such that for any q 0 ∈ R m , q0 ∈ R m , H 2 q0 ∈ D(∂Φ) with q 0 -q 2 m + q0 2 m η the solution q(•; t 0 , q 0 , q0 ) of problem P (t 0 , q 0 , q0 ) satisfies q(t; t 0 , q 0 , q0 ) -q

2 m + dq dt (t; t 0 , q 0 , q0 ) 2 m ε, ∀t t 0 .
If there exists a δ > 0 such that for any q 0 ∈ R m , q0 ∈ R m , H 2 q0 ∈ D(∂Φ) with q 0 -q 2 m + q0 2 m δ the solution q(•; t 0 , q 0 , q0 ) of problem P (t 0 , q 0 , q0 ) satisfies the limits: lim t →+∞ q(t; t 0 , q 0 , q0 ) -q m = 0 (39) and lim t →+∞ dq dt (t; t 0 , q 0 , q0

) m = 0, ( 40
)
then we say that the stationary solution q is attractive. If the limits in (39) and (40) hold for any q 0 ∈ R m , q0 ∈ R m , H 2 q0 ∈ D(∂Φ) then we say that the stationary solution q is globally attractive. Finally, a stable and attractive stationary solution is said asymptotically stable.

Theorem 7 (Stability). Let the assumptions of Theorem 6 together with condition (38).

Suppose in addition that

(1) RM -1 CR -1 is positive semidefinite;

(2) RM -1 KR -1 is symmetric and positive definite.

Then W = ∅ and any stationary solution q ∈ W of (29)-( 30) is stable.

Proof. Condition (38) ensures that ∂Φ(0) = ∅ and assumption (2) entails that K is nonsingular.

Thus W = -K -1 H 1 ∂Φ(0) = ∅.
Let q ∈ W be given. Setting Q := q -q, we see that the question of stability of q reduces to the one of the trivial stationary solution of the system:

M Q + C Q + KQ + K q ∈ -H 1 ∂Φ(H 2 Q).
(41)

Setting x 1 := RQ, x 2 := R Q and x := (x 1 x 2 ) T , we check as in the proof of Theorem 6 that the system in (41) can be written as follows:

ẋ + F (x) ∈ -∂ϕ(x)
where

F (x) = Ax + F , A = 0 m×m -I m RM -1 KR -1 RM -1 CR -1 , F = 0 m×1 RM -1 K q , ϕ(x) = χ(x 2 ) := Φ • H 2 R -1 (x 2 ) and ∂ϕ(x) = 0 m×1 R -1 H T 2 ∂Φ(H 2 R -1 x 2 )
.

The mapping F (•) is Lipschitz continuous. Moreover, condition (17) holds since q ∈ W ⇔ K q ∈ -H 1 ∂Φ(0) ⇔ RM -1 K q ∈ -RM -1 H 1 ∂Φ(0) ⇔ RM -1 K q ∈ -R -1 H T 2 ∂Φ(0) ⇔ RM -1 K q ∈ -∂χ(0) ⇔ F ∈ -∂ϕ(0).
Let us now check that all the assumptions of Theorem 3 are satisfied.

Let V ∈ C 1 (R n ; R) (n = 2m) be given by V (x) = 1 2 RM -1 KR -1 x 1 , x 1 m + 1 2 x 2 2 m .
It is clear from hypothesis (2) that assumption (1) of Theorem 3 is satisfied. Assumption (2) of Theorem 3 is also clearly satisfied.

We have:

V (x) = RM -1 KR -1 x 1 x 2 . Thus Ax, V (x) + F , V (x) + ϕ(x) -ϕ x -V (x) = RM -1 CR -1 x 2 , x 2 m + RM -1 K q, x 2 m + χ(x 2 ) -χ(0).
Assumption (1) yields

RM -1 CR -1 x 2 , x 2 m 0. (42)
Moreover, q ∈ W and thus RM -1 K q ∈ -∂χ(0). It results that

RM -1 K q, x 2 + χ(x 2 ) -χ(0) 0. ( 43
)
The inequalities in (42) and ( 43) ensure that hypothesis (3) of Theorem 3 is satisfied. The conclusion is thus a consequence of Theorem 3. 2

It is easy to see from the proof of Theorem 7 that the following variant can also be stated.

Theorem 8. Let the assumptions of Theorem 6 together with condition (38). Let q ∈ W be a stationary solution of (29)-(30). Suppose that

(1) RM -1 CR -1 z + RM -1 K q, z m + Φ(H 2 R -1 z) -Φ(0) 0, z ∈ R m ;
(2) RM -1 KR -1 is symmetric and positive definite.

Then q is stable.

It follows from Remark 4 that an equilibrium point q is in general not isolated in W. The concept of attractivity is for such case not really appropriated. It is then worthwhile to verify if the trajectories of the perturbed solutions are attracted by W.

Theorem 9 (Attractivity of W). Let the assumptions of Theorem 6 together with condition (38). Suppose also that

(1) RM -1 KR -1 is symmetric and positive definite;

(2) RM -1 CR -1 z, z m + Φ(H 2 R -1 z) -Φ(0) > 0, z ∈ R m \{0}; (3) D(∂Φ) is closed.
Then (a) for any q 0 ∈ R m , q0 ∈ R m , H 2 q0 ∈ D(∂Φ), the orbit Ω(q 0 , q0 ) := q(τ ; t 0 , q 0 , q0 ) dq dt (τ ; t 0 , q 0 , q0 ) T : τ t 0 is bounded and (b) the following asymptotic properties hold:

lim τ →+∞ d q(τ ; t 0 , q 0 , q0 ), W = 0 and lim τ →+∞ dq dt (τ ; t 0 , q 0 , q0 ) = 0.
Proof. From the proof of Theorem 6, we know that the study of our problem reduces to the one of the first-order system

ẋ + Ax ∈ -∂ϕ(x),
where A is defined in (35) and ϕ is given by (37).

Let us first check that all assumptions of Corollary 1 are satisfied with V ∈ C 1 (R n ; R) (n = 2m), defined as in the proof of Theorem 7, i.e.,

V (x) = 1 2 RM -1 KR -1 x 1 , x 1 m + 1 2 x 2 2 m .
We have ϕ(x)ϕ(x -V (x)) = χ(x 2 )χ(0) and the application x → ϕ(x)ϕ(x -V (x)) is thus lower semicontinuous. It results that hypothesis (1) of Corollary 1 is satisfied. We have

Ax, V (x) + ϕ(x) -ϕ x -V (x) = RM -1 CR -1 x 2 , x 2 m + χ(x 2 ) -χ(0).
Assumption (2) ensure that hypothesis (2) of Corollary 1 holds. It is clear that hypothesis (3) of Corollary 1 is satisfied. Finally hypothesis (4) of Corollary 1 follows from assumption (3) which ensures that

D(∂ϕ) = R m × D(∂(Φ • H 2 R -1 )) is closed.
Here, we have:

E(A, ϕ, V ) = x ∈ D(∂ϕ): RM -1 CR -1 x 2 , x 2 m + χ(x 2 ) -χ(0) = 0 .
Using assumption (2), we get:

E(A, ϕ, V ) = (x 1 , 0): x 1 ∈ R m . ( 44
)
Corollary 1 ensures that for any

x 0 ∈ D(∂ϕ), the orbit γ (x 0 ) is bounded. If q 0 ∈ R m , q0 ∈ R m , H 2 q0 ∈ D(∂Φ) then R q0 ∈ D(∂χ).
It results that the conclusion of Corollary 1 with x 0 = (Rq 0 R q0 ) T means that the set Ω(q 0 , q0 ) is bounded. This gives part (a) of our result. Corollary 1 ensures also that lim

τ →+∞ d x(τ ; t 0 , x 0 ), M = 0,
where M is the largest invariant subset of E(A, ϕ, V ). We may apply Proposition 3 to see that S(A, ϕ) ⊂ E(A, ϕ, V ). From Remark 2(v), we know also that S(A, ϕ) is invariant. Thus S(A, ϕ) is an invariant subset of E(A, ϕ, V ). We prove now that S(A, ϕ) is the largest invariant subset of E(A, ϕ, V ). Since, S(A, ϕ) ⊂ E(A, ϕ, V ), by (44) we have:

S(A, ϕ) = (x 1 , 0): RM -1 KR -1 x 1 , h m + χ(h) -χ(0) 0, ∀h ∈ R m .
Let us set

N := x 1 ∈ R m : RM -1 KR -1 x 1 ∈ -∂χ(0) .
Then, we may write

S(A, ϕ) = N × {0}.
Let D be any invariant subset of E(A, ϕ, V ) and let z ∈ D be given.

RM -1 KR -1 z 1 , v 2 m + χ(v 2 ) -χ(0) 0, ∀v 2 ∈ R m . Thus z = (z 1 , z 2 ) ∈ N × {0}.
It results that D ⊂ S(A, ϕ) and S(A, ϕ) is well the largest invariant subset of E(A, ϕ, V ). Thus

Recall that in terms of the vector q = R -1 x 1 of "generalized coordinates" and the vector q = R -1 x 2 of "generalized velocities" we have

RM -1 KR -1 x 1 ∈ -∂χ(0) ⇔ Kq ∈ -MR -2 H T 2 ∂Φ(0) = -H 1 ∂Φ(0). Thus the limit in (47) reads lim τ →+∞ d q(τ ; t 0 , q 0 , q0 ), W = 0. ( 49
)
On the other hand, the limit in (48) gives: lim τ →+∞ q(τ ; t 0 , q 0 , q0 ) = 0. (50)

Part (b) of our result is thus proved. 2

Remark 5. (i) Note that if 0 ∈ ∂Φ(0) then assumption (2) in Theorem 9 is satisfied provided that either RM -1 CR -1 is positive definite or RM -1 CR -1 is positive semidefinite and {z ∈ R m :

Φ(H 2 R -1 z) = Φ(0)} = {0}. (ii) If H T 2 =
H 1 and M is symmetric and positive definite then the matrix R = M 1/2 satisfies assumption (2) of Theorem 6. Then conditions (1) and ( 2) in Theorem 7 hold if and only if C is positive semidefinite and K is symmetric and positive definite. Indeed, here RM

-1 CR -1 •, • m = CM -1/2 •, M -1/2 • m and RM -1 KR -1 •, • m = KM -1/2 •, M -1/2 • m .
(iii) The conditions discussed in Remark 5(ii) are usually satisfied as soon as concrete applications in Mechanics are considered.

(iv) Assumption (1) in Theorem 9 implies that K is nonsingular. Hence W = -K -1 H 1 ∂Φ(0).

(v) Suppose that the assumptions of Theorem 9 hold. Suppose in addition that ∂Φ(0) = {0}. Then W = {0} and thus the trivial solution of ( 29)-( 30) is (a) the unique stationary solution of ( 29)-( 30), (b) stable and (c) globally attractive. In particular, the results in (b) and (c) ensure that the trivial solution of (29)-( 30) is asymptotically stable

Nonsmooth conservative systems

In this section, we consider a mechanical system whose state can be described by m generalized independent coordinates q = (q 1 . . . q m ) T . The kinetic energy of the system is:

T = 1 2 M dq dt , dq dt m ,
where M ∈ R m×m is symmetric and positive definite. The generalized forces are denoted by Q. We suppose that

Q = Q 1 + Q 2 ,
where Q 1 are conservative forces, i.e.,

Q 1 = -Π (q)
with Π ∈ C 1 (R m ; R) denoting the potential energy of the system and

Q 2 ∈ -H 1 ∂Φ H T 1 dq dt ,
where Φ : R l → R ∪ {+∞} is a proper convex and lower semicontinuous function and H 1 ∈ R m×l is a given matrix.

Then the trivial stationary solution of (29)-( 30) is (a) isolated in X and (b) asymptotically stable.

Proof. (a) It is clear that assumptions (1), ( 2) and ( 5) imply that X ∩ { q ∈ R m : q m σ } = {0}. (b) We know that the study of our problem reduces to the one of the first-order system in (57). As in Theorem 11, we consider the function V ∈ C 1 (R n ; R) (n = 2m) given by:

V (x) = Π M -1/2 x 1 + 1 2 x 2 2 m .
All the assumptions of Theorem 11 are here satisfied and the stability of the trivial stationary solution of the system ẋ +

F (x) ∈ -∂ϕ(x) is ensured. Let σ := σ / M -1/2 m . From Definition 1, there exists δ > 0 such that if x 0 ∈ D(∂ϕ) ∩ B δ then γ (x 0 ) ⊂ B σ .
Let us first check that all assumptions of the Invariance Theorem 5 are satisfied with the compact set Ψ := B σ . Indeed, the application

x → ϕ(x) -ϕ x -V (x) = Φ H T 1 M -1/2 x 2 -Φ(0)
is lower semicontinuous, hypothesis [START_REF] Goeleven | Variational and Hemivariational Inequalities[END_REF] ensures that D(∂ϕ) is closed and we have seen in the proof of Theorem 11 that

F (•), V (•) + ϕ(•) -ϕ(• -V (•)) 0.
Theorem 5 ensures that for x 0 ∈ D(∂ϕ) ∩ B δ , we have:

lim τ →+∞ d x(τ ; t 0 , x 0 ), M = 0,
where M is the largest invariant subset of E Ψ (F, ϕ, V ).

Using assumption (6), we obtain:

E Ψ (F, ϕ, V ) = x ∈ D(∂ϕ) ∩ Ψ : Φ H T 1 M -1/2 x 2 = Φ(0) = (x 1 , 0): x 1 ∈ R m , x 1 σ .
Let D be any invariant subset of E Ψ (F, ϕ, V ) and let z ∈ D be given.

, z) + M -1/2 Π M -1/2 x 1 (t; t 0 , z) , v 2 -x 2 (t; t 0 , z) m + Φ H T 1 M -1/2 v 2 -Φ H T 1 M -1/2 x 2 (t; t 0 , z) 0, ∀v 2 ∈ R m , a.e. t t 0 .
However, γ (z) ⊂ D ⊂ E Ψ (F, ϕ, V ) and thus x 2 (t; t 0 , z) = 0, ∀t t 0 . We deduce that x 1 (•; t 0 , z) = z 1 , ∀t t 0 and

M -1/2 Π M -1/2 z 1 , v 2 m + Φ H T 1 M -1/2 v 2 -Φ(0) 0, ∀v 2 ∈ R m , a.e. t t 0 . Thus Π (M -1/2 z 1 ) ∈ -H 1 ∂Φ(0). Assumption (1) yields Π (M -1/2 z 1 ) = 0. Recalling that z 1 σ since z ∈ B σ , we obtain M -1/2 z 1 ∈ B σ . Assumption (
= - γ k , + γ k .
This set defines a steady zone due to friction. It is also easy to check that both assumptions of Theorems 7 and 9 are satisfied. It results that each stationary solution q ∈ W is stable. Moreover, lim τ →+∞ d u(τ ; t 0 , q 0 , q0 ), W = 0 and lim τ →+∞ u(τ ; t 0 , q 0 , q0 ) = 0. Some numerical results (m = 1, k = 1, c = 0.2, γ = 1) are given in Figs. 2, 3 and 4 so as to illustrate and support this last theoretical result.

Example 2. We consider the model given in Fig. 5. A mass m > 0 is restrained by a vertical spring with stiffness constant k V > 0 in parallel with a damper with coefficient of viscous damping c V > 0 and some inclined device formed by a spring with stiffness constant k I > 0 in parallel with a nonlinear damper whose characteristic (feedback forcespeed) is described by a monotone set-valued graph ∂Φ as the one depicted in Fig. 6. The angle of inclination is denoted by θ ∈ (0, π/2). The horizontal and vertical displacement of the mass m are respectively denoted by u N and u T .

The model describing the motion of this system is of the form given in (30) with

M = m 0 0 m , K = k I sin 2 θ -k I sin θ cos θ -k I sin θ cos θ k V + k I cos 2 θ , C = 0 0 0 c V , H 1 = sin θ -cos θ , H 2 = H T 1 , q = u T u N
and with Φ : R → R as depicted in Fig. 6. Here D(∂Φ) = R, ∂Φ(0) = {0} and Φ(x) > 0, ∀x = 0. It is clear that all the assumptions of Theorem 6 hold with

R = √ m 0 0 √ m .
The set of stationary solutions reduces here to {0} since K is nonsingular and ∂Φ(0) = {0}. Thus the trivial stationary solution is here the unique stationary solution. We see that RM -1 CR -1 is positive semidefinite and RM -1 KR -1 is symmetric and positive definite. We may apply Theorem 7 and conclude that the trivial stationary solution is stable.

Let us now check that Theorem 9 can also be applied. It remains to verify that assumption (2) in Theorem 9 holds. We have:

RM -1 CR -1 z, z 2 + Φ H 2 R -1 z -Φ(0) = c V m |z 2 | 2 + Φ 1 √ m sin(θ )z 1 - 1 √ m cos(θ )z 2 .
It is thus clear that

RM -1 CR -1 z, z 2 + Φ H 2 R -1 z -Φ(0) 0.
Suppose now that

RM -1 CR -1 z, z 2 + Φ H 2 R -1 z -Φ(0) = 0.
Then |z 2 | 2 = 0 and Φ(m -1/2 sin(θ )z 1m -1/2 cos(θ )z 2 ) = 0. This yields z 2 = 0 and next z 1 = 0. Assumption (2) of Theorem 9 is thus satisfied. Theorem 9 ensures that the trivial stationary solution is globally attractive.

In conclusion, the trivial stationary solution is (a) the unique stationary solution, (b) stable and (c) globally attractive. Properties (b) and (c) entail that the trivial stationary solution is asymptotically stable.

A numerical simulation is given in Fig. 7.

Example 3. Let us consider the system of Fig. 8. Here m > 0 denotes the mass of a mass point, l > 0 is the length of the rod and k > 0 is the stiffness of the spiral spring. The angle θ determines the position of the system. The friction force f at the horizontal cylindrical support is given by the model f ∈ -∂Φ( θ) where Φ : R → R is a convex and lower semicontinuous function. The motion of the system is governed by the model:

ml 2 θ + kθ -mgl sin(θ ) ∈ -∂Φ( θ). (65)
Here q = (θ ), M = (ml 2 ), H 1 = (1) and

Π(θ) = 1 2 kθ 2 -mgl 1 -cos(θ ) .
We see that Π(0) = Π (0) = 0. Moreover if k > mgl, (66) then there exists σ 1 > 0 such that Π(θ) > 0, |θ | σ 1 , θ = 0. It is also clear that there exists σ 2 > 0 such that Π (θ ) = 0, |θ | σ 2 , θ = 0. Suppose that Φ is of the form given in Example 1. Then 0 ∈ ∂Φ(0) and Theorem 11 can be applied to ensure that the trivial stationary solution of (65) is stable. A numerical simulation is given in Fig. 9.

If Φ is of the form depicted in Example 2, then ∂Φ(0) = {0} and Φ(x) > 0, x = 0. It is clear that all the assumptions of Theorem 12 are satisfied. The trivial stationary solution of (65) is thus in this case asymptotically stable.

2 Corollary 3 .

 23 lim τ →+∞ x(τ ; t 0 , x 0 ) = 0 holds. Assertions (b) and (c) follow. Let the assumptions of Theorem 1 together with condition

2

 2 τ ; t 0 , x 0 ), S(A, ϕ) = 0.This implies that lim τ →+∞ d x 1 (τ ; t 0 , x 0 ), (τ ; t 0 , x 0 ) = 0.

2 6. 1 .

 21 Fig. 1. Example 1.

W

Fig. 2 .

 2 Fig. 2. Example 1.

Fig. 3 .

 3 Fig. 3. Example 1.

Fig. 4 .

 4 Fig. 4. Example 1.

Fig. 5 .

 5 Fig. 5. Example 2.

Fig. 6 .

 6 Fig. 6. Multivalued monotone mapping ∂Φ.

Fig. 7 .

 7 Fig. 7. Example 2.

Fig. 8 .

 8 Fig. 8. Example 3.

Fig. 9 .

 9 Fig. 9. Example 3.

  , z) = 0, a.e. t t 0 from which we deduce that x 1 (•; t 0 , z) = z 1 , ∀t t 0 . Then (46) yields

					The function x(•; t 0 , z)
	satisfies:
		dx 1 dt	(t; t 0 , z) -x 2 (t; t 0 , z), v 1 -x 1 (t; t 0 , z)	m	0, ∀v 1 ∈ R m , a.e. t t 0 , (45)
	and
	dx 2 dt	(t; t 0 , z) + RM -1 KR -1 x 1 (t; t 0 , z) + RM -1 CR -1 x 2 (t; t 0 , z), v 2 -x 2 (t; t 0 , z)	m
	+ χ(v 2 ) -χ x 2 (t; t 0 , z)	0, ∀v 2 ∈ R m , a.e. t t 0 .	(46)
	However, γ (z) ⊂ D ⊂ E(A, ϕ, V ) and thus x 2 (t; t 0 , z) = 0, ∀t t 0 . Thus (45) reduces to dx 1 dt (t; t 0

The motion of the system is governed by the second-order Lagrange equations:

More precisely, for (t 0 , q 0 , q0 ) ∈ R × R m × R m with H T 1 q0 ∈ D(∂Φ), we consider the problem L(t 0 , q 0 , q0 ): Find a function t → q(t) (t t 0 ) with q ∈ C 1 ([t 0 , +∞); R m ), and such that

Let us first show the existence and uniqueness of solution to problem (51)-(56).

Theorem 10 (Existence and uniqueness). Let the following assumptions satisfied:

(1) M is symmetric and positive definite;

(2) Π is Lipschitz continuous;

(3) there exists y 0 = H T 1 M -1/2 x 0 (x 0 ∈ R m ), at which Φ is finite and continuous.

Let

we see as in the proof of Theorem 6 that problem L(t 0 , q 0 , q0 ) can be written as follows:

with

where the matrix A ∈ R n×n (n = 2m) is defined by:

the mapping F : R n → R n is defined by:

the vector x 0 ∈ R n is given by:

and the proper, convex and lower semicontinuous function ϕ : R n → R ∪ {+∞} is defined by:

Here F is Lipschitz continuous. The result is thus a direct consequence of Theorem 1. 2

Suppose now that the conditions of Theorem 10 are satisfied and denote by q(•; t 0 , q 0 , q0 ) the unique solution of problem L(t 0 , q 0 , q0 ).

The set X of stationary solutions of (55)-( 56) is given by:

Suppose that

Then 0 ∈ X .

The following result ensures that if at the position of the trivial equilibrium the potential energy has a (strict) local minimum then this trivial equilibrium is stable.

Theorem 11 (Stability). Let the assumptions of Theorem 10 together with condition (62).

Suppose in addition that there exists σ > 0 such that

Then the trivial stationary solution of (29)-( 30) is stable.

Proof. We consider the first-order problem in (57) and we define the function V ∈ C 1 (R n ; R) (n = 2m) by setting:

It is clear from assumption (1) that V (0) = 0. Moreover, setting σ := σ / M -1/2 m , using assumption (2) and recalling that M -1/2 is nonsingular, we see that

Then using a standard result concerning positive definite functions (see, e.g., criterion 3.6 in [START_REF] Rouche | Equations Différentielles Ordinaires[END_REF]), we obtain the existence of a continuous and strictly increasing function a : [0, σ ] → R; t → a(t) such that a(0) = 0 and V (x) a( x ), x ∈ B σ .

We have:

and

All the assumptions of Theorem 3 are satisfied and the conclusion follows. 2

The next result shows that in some particular but important cases we can ensure the asymptotic stability of the trivial stationary solution.

Theorem 12 (Asymptotic stability). Let the assumptions of Theorem 6. Suppose in addition that there exists σ > 0 such that


Didier Eric Benoit

Didier Coquin

Hideyuki Sawada

Distributed data fusion applied to human gesture measurement

I. INTRODUCTION

Gesture measurement is a complex process that needs to be performed with several sensor technologies in order to improve the final result. In many approaches, a data glove is used to give data about the hand gesture but it cannot access to the hand position or the hand orientation. It is then necessary to add another sensors that are able to capture at least the position and the orientation of the hand such as 6Dof (6 degree of freedom) sensors or video cameras. Furthermore, the video camera is able to give some pieces of information about the arm motion. This additional data is useful to recognize a gesture.

Increasing the number of sources can be an improvement only if the measurement results can be combined. In order to simplify the fusion process, all measurements are expressed as fuzzy descriptions or fuzzy distributions of possibility. this common representation allows to combine etherogenous sources as shown in [START_REF] Benoit | [END_REF] and [START_REF] Sawada | [END_REF].

The fusion of the redundant information is performed by a possibilistic aggregation of measurements. This approach allows one to have sources with different confidences. The fusion of complementary information is performed by an aggregation based on fuzzy rules. This general approach based on a fuzzy representation of measurements permits the distribution of the fusion process on a distributed network of sensors.

This paper first presents the definition of the fuzzy description of a measurement and its application to the fuzzy representation of hand and arm gesture. Then the fusion process based on complementary information and redundant information is presented. Finally, an application to the driving of a mobile robot is shown.

II. MEASUREMENT

A. Fuzzy description of measurement

The link between a physical state and its linguistic representation is characterized by a symbolism defined by the triplet <X,L,R>whereX is the set of physical states, L is the lexical set used to represent measurement results and R is a relation on XxL. Two mappings can be extracted from this relation: The description mapping denoted d associates a subset of L to any item of X,andthemeaning mapping denoted m associates a subset of X to any item of L. This two mapping are linked with the following equation.

(1)

The R relation can be a fuzzy relation. Then, the translation of a physical state into its linguistic representation is called a fuzzy linguistic description mapping or simply a fuzzy description mapping. It transforms an object x of the set of physical states X into a fuzzy subset of linguistic terms called the fuzzy description of x. The dual mapping, called the fuzzy meaning mapping, associates a fuzzy subset of X to each term l of the lexical Set L. This fuzzy subset is the fuzzy meaning of l. This two mapping are linked with the following equation:

(2)

B. Hand gestures

A fuzzy description of hand postures add been proposed in [Allevard et al. 2003]. Like in this paper, fingers configurations are given numerically by the Cyberglove©'s bending sensors.

x ∀ X ∈ a ∀ L ∈ x ,, ma () ∈ ad x () ∈ ⇔ x ∀ X ∈ a ∀ L ∈µ ma () x () µ dx () a () = () ,,
Distributed data fusion applied to human gesture measurement.

E. Benoit (1) , D. Coquin (1) ,H.Sawada (2) (1) LISTIC-ESIA, Université de Savoie, B.P. 806, F-74016 Annecy cedex, France Tel: +33 450 096 540, Fax: +33 450 096 559 E-mail: didier.coquin, eric.benoit @univ-savoie.fr

(2) Kagawa University 2217-20 Hayashi-cho, Takamatsu, Kagawa, 761-0396 JAPAN E-mail: sawada@eng.kagawa-u.ac.jp The numerical to lexical conversion is made using a fuzzy partition of the dataglove measurement spaces. The fuzzy glove then provides a fuzzy description of each finger configuration and their relative position. The robustness and efficiency of the recognition system depends on the definition of the partitions. Different methods can be used for its construction (empirically, by interpolation, by clustering methods, etc.). The calibration of the fuzzy glove can be made by modifying this partition.

A preliminary study has shown that each finger can take five different configurations shown in Fig. 2 Depending on the complexity of the sign language to be recognized, either those five linguistic values are used or only the two simple values bent and straight. The relative finger positions can be either separated or together. The recognition of the hand postures is performed by using fuzzy rules which express human knowledge about how a sign is performed. For example, the sign pointing will be described by: index straight and other finger bent; all fingers together. Such a recognition system is readable and understandable. A new gesture can be added to the vocabulary very simply by giving its linguistic description or by asking the system to learn this description from a small set of examples.

The final lexical set can be small. for example we had use:

L={flat, ok, thumbUp, folded, pointing, cut}.

(3)

With this lexical set, a representation of a hand posture can be the lexical fuzzy subset: d(x)={ 0/flat,0 / ok,0 . 9/ thumbUp,0. 1 / folded,0/ pointing, 0/cut}. ( 4)

C. Arm gestures

The recognition of arm gesture (Fig. 4) is performed with a digital video camera. The general process of the method is composed of four main parts: a preprocessing step, a feature extraction step, a training step and a classification step. into its constitutive steps such as the preparation and the stroke [Quek, 1994]. Each gesture will be represented by a particular dynamic signature. The unknown gesture features are be compared with the features stored in the training data. The best match leads to the recognition of the unknown gesture. To compare two dynamic signatures, the Baddeley's distance is computed within the smallest frame containing the skeletons [START_REF] Coquin | [END_REF].

Baddeley distance [Baddeley, 1992] is defined between two binary images A and B having the same support S,by:

(5) with s =( x,y) a pixel, exponent E =2 ,a n dd A (s)i st h e shortest distance between s and the skeleton. Card(S)isthe number of element of support S.

The fuzzy description is obtained by the translation of the distance betweem the unknown gesture features and the features stored in the training data. A membership function is defined for each gesture. The d Min (g) parameter is computed as the max distance between the items of the set of samples signatures of the g gesture. The d Max parameter is defined as the max distance between the items of the set of samples signatures.

The lexical set used to describe arm gesture include both static and dynamic gestures:

L ={ goesBack, goesAhead, goesUp, goesDown, shake-sHand, staysUp}.

(6)

III. FUSION PROCESS

A. Fusion of complementary information

Sources are considered as complementary when the pieces of information concern different physical phenomenons. In this case, measurements from each sources are described separately with a symbolism <X i ,L i ,R i > for each source i.

Each lexical sets is specific to each sensor. Then fuzzy descriptions are fused using a fuzzy aggregation based on a set of fuzzy rules. The final lexical set is defined to represent a full gesture. A set of term used for mobile robot control are chosen.

L={stop, ahead, right, left, behind, unknown}( 7 )

The unknown term is added for all combination of hand posture and arm gesture that can not be used. For example, when the hand is folded and the arm is goesUp,t h e resulting gesture is not use to control the robot. An other approach will be to find a term for all gestures. The term that describe last gesture will be rebelSign. -------------------d

DAB , ()

1 card S ()
A s () d B s () - E sS ∈ ¦ 1 E --- = 0 d Max d Min (g) 0 1 d distance membership degree µ g (d) X i X j physical phenomenon source i perception source j perception R i R j physical phenomenon L i L j L fusion REM2005

B. Fusion of redundant information

Sources are considered as redundant when the pieces of information concern the same physical phenomenons. Each measurement process is based on a symbolism <X i ,L i ,R i >. When sources are partially redundant, it is supposed that the X i sets intersect enough to be described into a unique lexical set L. In this case, the descriptions issued from the different sources are represented by fuzzy subsets of the same lexical set. In our application, the chosen lexical set is the one that can represent the full gesture (see Eq. ( 7) and Fig. 10). As the lexical set is not specific to a source, 2 different physical states can be described by a source with the same term. For example, in the stop the right and the left gesture, the hand is kept flat. Then hand posture is the same and can not be distinguished with the dataglove source. Considering the semantic of the membership degree µ d(x) (a)ofaterma to the fuzzy description d(x), it can be considered as the possibility degree that a represents the state x. The representation processes do not warranty the fuzzy descriptions to be possibility distributions. Generally the fuzzy descriptions are not normalized to 1. A solution is to normalize each description. An other one is to accept non normalized distributions to represent unreliable descriptions.

Considering two symbolisms <X 1 ,L,R 1 >a n d< X 2 ,L,R 2 >, and the associated lexical descriptions d 1 and d 2 of the same physical entity. The aggregation of the fuzzy descriptions computed with these two symbolisms can be performed with a triangular norm i.e. a Tnorm T:

µ d (a)=µ d1 (a)Tµ d2 (a)( 8 )
This simple approach gives good results when both sources are reliable, because in this case both sources are agree to describe a gesture at least one common term. In a most general case the reliability of sources are not equivalent. The possibility theory proposes a solution to take this difference into account as for example in [START_REF] Desachy | Numeric and symbolic data fusion: A soft computing approach to remote sensing images analysis[END_REF]. To know how two sources must be fuzzed, it's only needed to know wich is more reliable.

Consider d 1 as more reliable than d 2 . First a consistency degree between sources is computed:

(9)
then the description of the gesture is defined by: (10)

When both sources are reliable h(d 1 , d 2 )=1andEq.10is equivalent to Eq. 8 with T = min. Depending on the fusion to perform, other fusion processes are proposed in [Dubois and Prade, 1992] and [Oussalah, 2000].

IV. IMPLEMENTATION

This approach is applied to the control of a mobile robot with gestures. Each instrument runs as an UDP server and sends its data with a broadcast. Each frame includes the data and source identification, the indentification of the lexical set, and the data values i.e. a vector of membership degrees. The fusion board catches the frames emitted by the glove and the camera and send the result on the network. The robot catches the frames emitted by the fusion board.

hd 1 d 2 , ( )
sup aL ∈ min µ d 1 a ()µ d 2 a () , () () = µ d a () min µ d 1 a ()max µ d 2 a ()1-hd 1 d 2 , ( )
The fusion board is able to select its mode between the complementary one and the redundant one. If the same lexical set is used by all sources, it switches into the redundant mode. If lexical sets are different and if it includes a set of rules using these lexical sets, it switches into the complementary mode. In the complementary mode, adding a new source need to create a new set rules to take this one into account. In the redundant mode, adding a new source is taken into account by an extension of Eq. 10 to n sources. In this last case the knowledge is distributed over the set of sensors. Indeed, each sensor gives a fuzzy subset that depends on the final application.

V. CONCLUSION

In the domain of gesture recognition with multiple sources of measurements, the nature of sources can be very different and may be difficult to combine. The fuzzy subset theory and the possibility theory give an framework for the aggregation of etherogeneous data. Futermore it allows to take into account the difference of reliability between sources. This paper shows how pieces of information from various sources can be fuzed considering their complementarity or their redundance.

Fig. 1

 1 Fig. 1 Picture of the Cyberglove, name of the sensors.

Fig. 2

 2 Fig. 2 Picture of the five key configurations of a finger.To linguistically the configuration of a finger, two sensors are used: the finger base sensor and finger medial sensor. Two monodimensional partitions have then to be defined on the numerical spaces of those two sensors. The corresponding lexical sets are respectively: L BaseIndex = {Folded, Straight}a n dL MedialIndex ={ Folded, Half, Straight}. The final lexical set used to describe the whole index finger is: L Index ={Folded, Straight, Square, Round, Claw}. The set of rules is given in the table of Fig.3

Fig. 3 Fig. 3

 33 Fig. 3 Set of rules for Index

Fig. 4

 4 Fig. 4 Samples of an arm gesture sequence that means stop. The preprocessing step focuses on the gesture. The preprocessing chain consists in the following operations: binary image computation, binary image enhancement, and hand region extraction.

Fig. 5

 5 Fig. 5 Dynamic signature In order to perform the gesture classification, the gesture alphabet has to be known. A training stage is required. Using a set of samples for each gesture, a training set of data is generated for both the static and the dynamic recognition. Using different representations in space of the same gesture gives invariance to small translations and rotations. the training set consists of m training sequences (dynamic signatures) for each hand gesture within the gesture alphabet. Using n gestures in the gesture alphabet and having m sets of example signatures for each one, there are n.m sample signatures for the training sequence.

Fig. 6

 6 Fig. 6 Fuzzification of the distance between the dynamic signatures of the measured gesture and a training gesture g.

Fig. 7

 7 Fig. 7 Relations between sets with complementary sources.

Fig. 9

 9 Fig. 9 Relations between sets when sources are redundant.

Fig. 10

 10 Fig. 10 Example of fusion with redundant sources

  Fig. 11 Architecture of the application.

  June 30 th -July1 st 2005 FRANCE Fig. 8 Example of fusion with complementary sources

  The sensors are connected to different computing systems and these systems are linked to an ethernet network and communicate with the UDP/IP protocol. The data glove is connected to a Jstik© board that is able to run java J2ME-Cldc programs. The robot is connected to another Jstik© board. The video camera is connected to a computer. A separate board runs the fusion program.


Dr Ana Isabel Perez Jimenez

Dr Lylian Challier

Dr Eric Aït-Yahiatène

Prof Jérôme E Delacotte

Eric Labbé

Olivier Buriez
email: olivier.buriez@ens.fr

Dr O Labbé

Buriez

Selective Electrochemical Bleaching of the Outer Leaflet of Fluorescently Labelled Giant Liposomes

Keywords: Electrochemical bleaching, Fluorescence, Lipid Vesicles, Confocal Microscopy, Diffusion of phospholipids

Electrochemistry and confocal fluorescence microscopy were successfully combined to selectively bleach and monitor the fluorescence of NBD-labelled phospholipids of giant liposomes. Three types of giant unilamellar vesicles have been investigated, the fluorescent phospholipids being localized either mainly on their outer-, inner-, or both inner/outer leaflets. We established that only the fluorescent lipids incorporated in the outer leaflet of the vesicles underwent electrochemical bleaching upon reduction. The relative fluorescence intensity decay was quantified all along the electrochemical extinction through an original Fluorescence Loss In Electrobleaching (FLIE) assay. As expected, the re-organization of the fluorescent phospholipids followed diffusion-driven dynamics. This was also evidenced by comparison with Fluorescence Loss In Photobleaching (FLIP) and the corresponding numerical model. The value of the lateral diffusion coefficient of phospholipids was found to be similar to that obtained by other methods reported in literature. This versatile and selective bleaching procedure appears reliable to explore important biological and pharmacological issues.

Introduction

Combination of electrochemistry with confocal or wide field fluorescence microscopy techniques is of high interest in the fields of analytical chemistry, biophysics and chemical physics. On the one hand, fluorescence microscopy allows imaging and fluorescence quantification. On the other hand, electrochemistry can be used to switch fluorescence on and off. [1] The coupling of fluorescence microscopy with electrochemistry has notably proved to be a powerful approach in view of obtaining both time-and spatially-resolved information at the molecular level. Besides, several works reported in this field focus on the electrochemical establishment of pH or concentration gradients. In the first case, the main strategy relies on the use of pH-sensitive fluorescent probes. This approach was well developed in the 1990s by Engstrom and coworkers to monitor, with a wide field fluorescence microscope, reactions producing or consuming protons at the level of an electrode. [2][3][4] Following the work reported on the use of confocal Raman microscopy to map molecular diffusion at microelectrodes, [5,6] Unwin and colleagues successfully imaged in 2002, for the first time, three-dimensional pH gradients at electrode surfaces thanks to the use of a confocal laser scanning microscope which improves the axial and lateral resolutions. [7] In this case, the local pH changes due to the electrochemical reduction of benzoquinone to hydroquinone were evidenced thanks to the presence of fluorescein which displayed pH-dependent fluorescent signal. Confocal pH measurements in solution were also performed later by Boldt et al. in the presence of a scanning electrochemical microscope. [8] Contrary to pH-gradients, the establishment of concentrationgradients is mainly based on the use of molecules possessing both fluorescent and redox properties. In this case, the measured fluorescence is directly dependent on the potential applied. For instance, the diffusion concentration profile resulting from the oxidation of the fluorescent tris(bipyridine) ruthenium(II) into the non-fluorescent tris(bipyridine)-ruthenium(III) could be mapped by confocal microscopy. [9] Very recently, Doneux and coworkers successfully coupled electrochemistry with confocal microscopy to investigate the electrochemical reactivity of the resazurin-resorufin fluorogenic couple and to reconstruct in 3D the diffusion layer of the fluorescent probe. [10] Though decoupled from the electrochemical reaction of interest, we can also cite the work reported by Zhang and colleagues who developed an imaging tool called "fluorescence-enabled electrochemical microscopy", the principle of which being to associate a redox reaction to a fluorogenic reporter reaction through a closed bipolar electrode. [11][12][13] Combination of fluorescence microscopy with electrochemistry can be also interesting in the presence of adsorbed fluorescent probes. For instance, Bizzotto et al. have used in situ fluorescence to investigate the reductive desorption of self-assembled monolayers made of fluorescently tagged alkylthiols. [14,15] The technique relies on the fact that light emission from a fluorophore may be quenched when placed in close proximity ( 50 nm) to a bulk metallic surface. A fluorescent signal is therefore observed only when the adsorbed molecule becomes desorbed and separated from the electrode surface. More recently, the same group combined electrochemical measurements with in situ fluorescence imaging to monitor the poration / absorption / incorporation process of liposomes (100 nm in diameter), made of DOPC phospholipids, into a fluorescent adsorbed monolayer of octadecanol-covered surface. [16] Fluorescent redox probes can be also trapped in biomolecular architectures such as liposomes which are used as biomembrane and protocell models. [START_REF] New | liposomes: A Practical Approach[END_REF][START_REF] Sackmann | Structure and Dynamics of Membranes: From Cells to Vesicles[END_REF] On the other hand, liposomes with diameters of a few micrometers and above are particularly attractive systems due to their accessibility to optical microscopy and various micromanipulation techniques. [START_REF] Luisi | Giant Vesicles[END_REF][START_REF] Messina | [END_REF] In this context, Bard and Zhan combined scanning electrochemical microscopy and fluorescence to probe the outside and the inside of individual giant liposomes (15-50 µm in diameter) in which the fluorescent tris(bipyridine)ruthenium(II) was encapsulated. [21] However, our literature survey did not highlight any works involving redox fluorescent probes bound to phospholipids diffusing in lipid membranes. Several confocal microscopybased methods have been developed to switch off fluorescence locally in order to obtain physiological and physico-chemical information about cellular dynamics. [START_REF] Pawley | Handbook of Biological Confocal Microscopy[END_REF] Among them, Fluorescence Recovery After Photobleaching (FRAP) and Fluorescence Loss In Photobleaching (FLIP) are the two most popular fluorescence microscopy techniques. [START_REF] Axelrod | [END_REF][24][25][26] Interestingly, these methods are based on a photobleaching step which allows to switch the fluorescence off by high intensity illumination with a focused laser beam. However, considering the energy of the laser beam, care must be taken to avoid disruption of the cellular structure. [27,28] On the other hand, photobleaching-based methods cannot be used to investigate events occurring at the scale of a single lipid bilayer thickness due to the poor laser beam focusing resolution (µm) compared to the lipid bilayer thickness (4 nm). Under these conditions, discrimination between the outer and the inner leaflet of a membrane is therefore not possible. Yet, such discrimination are essential notably for the investigation of mechanisms related to the entrance and exit of active molecules across lipid membranes, a hot topic in the development of innovative pharmaceutical strategies.

In the present contribution, we focused on the development of an original combination between electrochemistry and confocal microscopy to monitor the electrochemical bleaching of fluorescently-labelled phospholipids diffusing in giant unilamellar vesicles membranes (GUVs: 10 -50 µm in diameter) deposited on a transparent conductive surface. For the first time, we show that electrochemistry is a powerful tool to selectively bleach the fluorescence emitted by NBD (7-nitrobenz-2-oxa-1,3-diazole) a well-known redox fluorescent probe especially used to investigate the translocation of cell-penetrating peptides. [29][30][31][32][33] Importantly, compared to photobleaching techniques such as FRAP and FLIP, the electrochemical bleaching process does not affect the fluorescent probes localized "inside" the vesicles thus providing a methodology to discriminate between the outer and the inner membrane leaflets. A comparison of this original Fluorescence Loss In Electrobleaching (FLIE) procedure with FLIP established that the re-organization of the fluorescent phospholipids followed lateral diffusion dynamics, as assessed by the excellent agreement between the experimental fluorescence decay and a model established for FLIP experiments. This procedure therefore appears relevant to explore a variety of biological issues.

Results and Discussion

Development and optimization of a device allowing the preparation and the observation of asymmetrical fluorescent liposomes. The electrochemical reduction of giant unilamellar vesicles (GUVs) essentially tagged on their outer leaflets with NBD (7-nitrobenz-2-oxa-1,3-diazole), used as a model redox fluorescent probe, was first investigated. Accordingly, a first series of GUVs made of 1,2-dioleoyl-snglycero-3-phosphoglycerol (DOPG) phospholipids, a classical anionic phospholipid used to prepare endosome models, and of a small fraction (5 mol. %) of 1,2-dipalmitoyl-sn-glycero-3phosphoethanolamine-N-(7-nitro-2-1,3-benzoxadiazol-4-yl) (NBD-PE) was prepared. The preparation method for such asymmetrical liposomes was inspired from the works reported by Weitz, [START_REF] Pautot | Proc. Natl. Acad. Sci[END_REF] Baigl, [START_REF] Yamada | [END_REF] and Ces [36] who developed protocols to generate giant liposomes from an oil/water interface. Briefly, lipid-coated water/sucrose-in-oil (W/O) droplets (destined for the inner leaflet) were passed through an oil-water/glucose (O/W) column with an interfacial monolayer of lipids to form the vesicle outer-leaflet. The different densities between sucrose and glucose solutions allowed the W/O droplets crossing the O/W phase, forming thus giant vesicles at the bottom surface of the chamber (Figure 1; see the Supporting Information for full experimental details). [34, 36].

Therefore, we developed an original and specific cell made of poly(dimethylsiloxane) (PDMS) allowing not only the in situ preparation and observation of GUVs by fluorescence confocal microscopy, but also the electrochemical reduction of NBD-PE phospholipid probes (see details in S.I.). As shown in Figure 2A, a PDMS well was bonded onto a glass microscope slide previously coated with a thin layer of ITO (10 nm in thickness), a conductive and transparent electrode material commonly used in spectro-electrochemistry experiments. Nevertheless, it was observed that electrochemical properties of ITO were modified after prolonged polarizations (500 seconds) at a cathodic potential of -0.7 V/Ag/AgCl corresponding to the value at which the NBD nitro group is reduced (vide infra). According to recent reported works, [37][38][39] the cathodic polarization of ITO thin films leads to morphological, optical, and conductivity changes of the material due to the reduction of its metallic sites. We overcame this drawback by covering the ITO surface with a thin gold layer (20 nm in thickness). Chronoamperograms obtained in the presence of such ITO/Au surfaces remained much more stable than those obtained in their absence (see S.I.). Cyclic voltammetry of NBD-glycine, used as model, was performed in the same cell to determine the reduction potential value of the fluorescent probe under the same conditions as those used to bleach NBD-tagged giant vesicles. Accordingly, and in agreement with similar nitro derivatives, [40,41] the first reduction wave of NBD was observed at -0.7 V/Ag/AgCl (S.I.). Although light transmission of the NBD probe through the ITO/Au bilayer was only half that transmitted through an unmodified glass microscope cover slide (see S.I.), it was intense enough to visualize NBD-tagged GUVs (Figure 2B). As sketched in Figure 2A, a Pt wire and a Ag/AgCl wire (0.5 mm in diameter each) used as the counter and the reference electrodes, respectively, were inserted into the well to complete the electrochemical device. In Figure 2B is represented a typical confocal microscopy image focused at the equatorial plane of a giant unilamellar vesicle composed of DOPG and of a small fraction (5 mol. %) of NBD-PE fluorescent phospholipids on its outer leaflet.

To improve the stability and integrity of vesicles (some of them were found to be opened on the gold surface i.e. having an around 10 micrometers pore size) the electrode surface was modified with a biotin monolayer (see details in S.I.). [42] This modification affected neither the light transmission nor the electrochemical properties of the gold layer.

Electrochemical bleaching of the outer leaflet of NBDlabelled giant liposomes -An original approach called

Fluorescence Loss In Electrobleaching (FLIE). Our approach was validated by a first series of experiments performed in the presence of outer leaflet NBD-labelled vesicles. Typically, after verifying both the integrity of the vesicle and its contact with the electrode surface (some vesicles did not reach the bottom of the well) a confocal image was taken at the equatorial plane of the GUV. The working electrode was then polarized at a constant cathodic potential value of -0.9 V/Ag/AgCl (i.e. 200 mV more negative than the reduction peak of the NBD nitro group). The potentiometric pulse was interrupted after 30 seconds, and a new confocal microscopy image was taken, still focused at the equatorial plane of the vesicle (i.e., out of the electrochemicallybleached area). This sequence was repeated about 20 times on the same vesicle to follow fluorescence variations at the vesicle equatorial plane as a function of the polarization time. Figure 3 shows a typical series of confocal microscopy images obtained as a function of the reduction time and focused at the equatorial plane of a GUV containing DOPG phospholipids and a small fraction (5 mol %) of fluorescent phospholipids (NBD-PE) on its outer leaflet. Clearly, the fluorescence intensity decreased with the number of potentiometric pulses, or in other words with the progress of the reduction reaction. Importantly, fluorescence extinction was not due to a photo-bleaching process since constant fluorescence intensities were obtained under open circuit conditions (compare A and B in Figure 3). This also indicates that excited NBDfluorophores were not quenched near the thin metal surface (20 nm thickness) as often observed at the surface of thick metallic materials. [14,15,43,44] On the other hand, polarization of the working electrode at -0.9 V also triggers water and oxygen reduction (see the cyclic voltammogram of NBD in the S.I.). Reduction of NBD could possibly be achieved according to a H2 mediated process, and/or OH  that is known to switch off the NBD fluorescence through a pH jump. [45,46] However, the pH increase possibly triggered through the production of hydroxyl anions would be compensated by the generation of protons at the anode through water oxidation since the auxiliary and working electrodes are not separated in the well. Additionally, we showed that bubbling H2 in a NBD-glycine containing solution did not quench the fluorescence. Furthermore, it was impossible to quench a vesicle which was not in contact with the electrode surface, a parameter verified through z stacking in confocal microscopy, demonstrating that no electrogenerated species diffusing in the bulk of the solution was able to switch NBD fluorescence off. This point also rules out the possible indirect reduction of NBD by the superoxide anion (O2  ) or any transient A B Time species that would be produced in case of oxygen reduction. Moreover, at the vesicle/electrode interface, the oxygen concentration is expected to be very small when the GUV is in contact with the electrode surface. Also, it is very unlikely that superoxide anions may easily reach lipid-NBD probes located in a negatively charged (DOPG) environment. Under these conditions, the global fluorescence decrease observed at the GUV equatorial plane can be ascribed to the direct (heterogeneous) electrochemical reduction of the NBD probes in contact with the electrode surface. Note that similar fluorescence extinction decays were obtained at gold bare surfaces indicating that the electron transfer efficiency was not affected by the presence of a biotin monolayer (see S.I.). At this stage it is noteworthy that our approach may be viewed as analogue to the Fluorescence Loss In Photobleaching (FLIP) technique in the sense that the latter consists in switching off the fluorescence by a high intensity illumination, with a focused laser beam, while fluorescence is monitored out of the bleached area. [START_REF] Axelrod | [END_REF][24][25][26] However, the additional interest of an electrochemical switching allows to ascertain that only molecules located in the outer leaflet can be switched (see below). By analogy with the FLIP technique, this quenching method can therefore be assimilated to a bleaching process where the laser beam is replaced by an electrode, thus prompting us to introduce the FLIE acronym (for Fluorescence Loss In Electrobleaching) to qualify this method. The initial electrochemical bleaching only occurs in the region corresponding to the surface of the GUV in contact with the electrode. Yet, the depletion caused by electrolysis generated a concentration gradient triggering a diffusion of fluorescent phospholipids towards the fluorophoredepleted region of the GUV. [47,48] Moreover, when the polarization of the electrode was performed continuously, as in FLIP experiments, the fluorescence intensity decay overlapped the one obtained for successive polarizations (see S.I.), indicating that the fluorescence decay observed at the equatorial plane did not arise from the period of relaxation between two successive electrolyses. Indeed, this time (few seconds) is much shorter than the characteristic time of diffusion-driven relaxation (roughly hundreds of seconds; see below).

Discrimination between the outer and the inner membrane leaflets.

The qualitative electrochemical bleaching of the outer leaflet of NBD-labelled giant liposomes led us then to test our approach in view of a possible discrimination between the outer and the inner membrane leaflets. Accordingly, the fluorescence extinction was then quantitatively investigated as a function of the NBD fluorescent probe localization: (a) either mainly on the outer, (b) mainly on the inner, (c) or on both inner/outer leaflets. Fluorescence variations at the equatorial plane of GUVs were corrected from background signals and photo-bleaching (see details in S.I.) and quantified as a function of the number of potentiostatic polarizations and the total time elapsed for the electrochemical reduction. Results are summarized in Figure 4, where F 0 and F are the corrected fluorescence intensities at time 0 and at time t, respectively. Three different behaviors were obtained depending on the fluorescent probe main localization. First, almost no fluorescence extinction was obtained in the presence of innerleaflet NBD-labelled vesicles demonstrating that electrochemical reduction of fluorescent probes that are inside the vesicle cannot be achieved. As already verified, the slight fluorescence loss (around 10 %) was not due to a photobleaching process. Actually, a small fraction of fluorescent phospholipids may have been incorporated in the outer leaflet of the vesicle during their preparation. [START_REF] Pautot | Proc. Natl. Acad. Sci[END_REF][START_REF] Yamada | [END_REF] Conversely, a significant fluorescence intensity decrease was observed in the case of outer-leaflet NBD-labelled vesicles. Almost 80 % of the fluorescence was indeed quenched after 600 seconds of electrochemical reduction/bleaching. As a symmetrical situation of the preparation of inner-leaflet labelled vesicles, outer-leaflet tagged vesicles finally presented 20% of the fluorescent phospholipids inserted in the inner leaflet, a proportion consistent with reported values in ref. [START_REF] Pautot | Proc. Natl. Acad. Sci[END_REF] . Accordingly, fluorescence extinction of both inner-and outer-leaflet NBDlabelled vesicles led to an average fluorescence extinction of 50 %. Similarly to the two previous series of experiments, some dissymmetry in the NBD-PE repartition between the two leaflets most likely happened, but they compensated. Indeed, the normalized fluorescence intensity at t = 600 s laid in the 0.32-0.69 range with a mean value of 0.50 ± 0.033 (S.E.). Note that one could wonder whether the inter-leaflet heterogeneity in terms of repartition of fluorescent phospholipids NBD-PE could arise from the dynamics of phospholipid exchange between the inner and outer leaflets. Actually, the characteristic time for lipid flip-flop is far too low to be taken into account within the time window of the electrolyses. [49] Importantly, and contrary to the FLIP technique, these results confirmed that electrochemical bleaching only concerned fluorescent phospholipids localized on the outer-leaflet of vesicles allowing thus discrimination between both sides of lipid bilayers.

Fluorescent lipids re-organization upon electrochemical bleaching. The strong analogy between the FLIE assay and the FLIP technique prompted us to carry out a more thorough comparison of both techniques. It was notably important to verify whether the re-organization of the fluorescent phospholipids observed during the FLIE assay followed the expected diffusiondriven dynamics. In this context, FLIP experiments were first performed experimentally on the same GUV population and the resulting fluorescence decay simulated to determine the phospholipid lateral diffusion coefficient. In FLIP experiments presented in the following, photo-bleached zones were localized at the top of vesicles (radius a). The same results were obtained when photobleaching was performed at the bottom of vesicles i.e. where the membrane is in contact with the electrode surface. The region to be bleached was defined (disk with ableach radius) and then confocal microscopy observations were performed at the equatorial plane of GUVs allowing subsequent fluorescence loss monitoring during photo-bleaching. The bleaching consisted in sequences of 20 laser pulses (3 s duration each). The vesicle equatorial plane was imaged between successive sequences of laser pulses. A typical fluorescence decay obtained under these conditions is shown in Figure 5. The fluorescence loss at the equator was then simulated, assuming a lateral diffusion-directed refilling of the bleached region. The model relies on the numerical solution of the diffusion equation solved on the surface of the oblate spheroidal vesicles (see S.I. for details). The diffusion coefficient value (D = 4 µm 2 .s -1 ), obtained through the fitting procedure, was consistent with those reported for the reorganization of similar phospholipids from other methods featuring fluorescence correlation spectroscopy (FCS), [50] single particle tracking, [51] or FRAP [52] . The numerical fit of FLIP data presented in Figure 5, for inner/outer labelled vesicles, was then compared with FLIE data for outer-leaflet labelled vesicles since the latter technique was shown not to bleach the inside of vesicles. To be relevant the comparison of FLIP and FLIE was made with vesicles possessing similar ableach / a (bleached surface radius / vesicule radius) ratios. The value of ableach which corresponds to the membrane surface in contact with the electrode was determined from confocal images. Accordingly, these ratios were equal to 0.66 and 0.71 in FLIP and FLIE techniques, respectively. As shown in Figure 6, a good fit between experimental and simulated data was obtained confirming that the FLIE technique also generates fluorophore gradients which trigger lateral diffusion of the lipids. Interestingly, these results show that FLIE performed on vesicles could be used to estimate diffusion coefficient values. Moreover, since FLIE only affects probes located in the outer leaflet, one can envision to study phospholipid dynamics even in asymmetrical vesicles without fast acquisition requirement as in FCS, FRAP or single particle tracking-based methods.

Conclusions

We have established that electrochemistry is a powerful tool to selectively bleach the fluorescence emitted by redox-active fluorophores localized on the outer leaflet of a lipid bilayer. Compared to photobleaching techniques such as FRAP and FLIP, the electrochemical bleaching process does not affect the fluorescent probes localized inside the vesicles thus providing an exploration tool allowing discrimination between the outer and the inner membrane leaflets. This electrochemical process also discards the use of chemical reducing agents that may undergo slow internalization within vesicles. [53] Furthermore, the FLIE procedure appeared reliable to explore the re-organization dynamics of a single leaflet in real time. The capacity to totally bleach the fluorophores located in the outer leaflet also enables the possibility to monitor their passage between inner and outer leaflets (e.g. flip-flop of lipids) with confocal microscopy. This versatile approach appears very promising in the exploration of biological and pharmacological issues related to the internalization or release of analytes across lipid membranes.

Experimental Section

Lipids. 1,2-dioleoyl-sn-glycerol-3-phospho-(1'-rac-glycerol) sodium salt (DOPG, 10 mg / mL in chloroform) and 1,2-dipalmitoyl-sn-glycero-3phosphoethanolamine-N-(7-nitro-2-1,3-benzoxadiazol-4-yl) ammonium salt (NBD-PE, 1 mg / mL in chloroform, λex=460 nm, λem=535 nm) were purchased from Avanti Polar Lipids. Chemicals. Mineral oil Bio-ultra, D-(+)-Glucose, Sucrose, and Biotin were obtained from Sigma Aldrich. Phosphate buffer saline (PBS) tablets (10 mM phosphate, 150 mM sodium chloride; pH 7.4) were purchased from GIBCO Invitrogen Corporation. All chemicals were used without further modification. Solutions. Water was highly purified (resistivity = 18 M.cm; Milli-Q system; Millipore, Billerica, MA, USA). Glucose and sucrose solutions were prepared by dissolving 6.3 g and 12 g of each compound, respectively, in 50 mL of PBS, to reach a final concentration of 0.7 M and an osmolarity around 990 mOsm. Device fabrication. The experimental device shown in Figure 2 is based on a home-made well inspired by the work reported by Yamada et al. [START_REF] Yamada | [END_REF] Details are given in the S.I. Giant Unilamellar Vesicles (GUVs). The preparation of GUVs was based on the work reported by Yamada et al. who developed a protocol to generate liposomes from an oil/water interface. [START_REF] Yamada | [END_REF] For full experimental details see the S.I. Electrochemical experiments. Just before each experiment, and after the vesicles preparation, the electrochemical device shown in Figure 2 was completed by inserting, through the PDMS well, a Ag/AgCl wire (0.5 mm in diameter) and a platinum wire (0.5 mm in diameter) used as the reference and the counter electrode, respectively. Then, a series of electrochemical reduction were performed by applying a potential value of -0.9 V (µAUTOLAB Type III). A picture of the remaining fluorescence at the equatorial plane of the GUV was taken after every 30 seconds until 300 s then every 60 seconds until 600 seconds. After the end of each experiment, the vesicle integrity was verified. Details are given in the S.I. part. Imaging of GUVs. Giant vesicles were observed by phase-contrast and confocal microscopy (Zeiss LSM 710), using EC Plan-Neofluar 10x (numerical aperture 0.3) and EC Plan-Neofluar 20x (numerical aperture 0.5) objectives. More details are given in the S.I. part.

Image analysis and fluorescence quantification. Images were analyzed with the ImageJ software (http://rsb.info.nih.gov/ij/). The average fluorescence intensity at the equatorial plane of each GUV was measured using the Azimuthal Average plugin. See S.I. for fluorescence quantification details. FLIP experiments. FLIP experiments were performed with the same confocal fluorescence microscope as that used for GUVs imaging (Zeiss LSM 710). Photo-bleaching experiments were carried out with a circular spot using the 458 nm lines from a 40 mW argon laser operating at 100 % power. In FLIP experiments, focus was made on the top of the vesicle (radius a), where it appears as a uniform disk. The region to be bleached was defined (disk with radius ableach), and focus was then made at the equatorial plane of the GUV allowing subsequent fluorescence loss monitoring during bleaching. The bleaching was performed with sequences of 20 laser pulses (3 s total duration). The vesicle equator was imaged between successive sequences of laser pulses. Fluorescence images were acquired at the GUVs equatorial plane as a function of time, at a low laser intensity (0.2 % laser power) under the following conditions: 2 or 3x optical zoom, 512 x 512 pixel resolution, depth 8 bits, nominal speed of 9 (total scan speed <1 s). The pinhole was set at 1 Airy, corresponding to an optical section of 5 μm. Details on data analysis and the corresponding numerical model are given in the S.I.

Figure 1 .

 1 Figure 1. Schematic illustration showing the asymmetric GUV generation via phase transfer. Adapted from refs.[34, 36].

  water/sucrose-in-oil (W/O) droplets (inner leaflet) Asymetric vesicle falling at the bottom surface of the chamber Interfacial monolayer of lipid to form the vesicle outer-leaflet Oil-water/glucose (O/W) interface.

Figure 2 .

 2 Figure 2. A) Scheme of the experimental device allowing preparation as well as visualization of giant vesicles before and during the electrochemical extinction of fluorescence. B) Typical fluorescence (top) and transmission (bottom) microscopy images focused at the equatorial plane of a giant unilamellar vesicle (50 µm in diameter) made of DOPG and of NBD-PE on its outer leaflet (5 mol. %) and settled at the bottom of the PDMS well as sketched in A).

Figure 3 .

 3 Figure 3. Typical series of confocal microscopy images focused at the equatorial plane of a GUV (50 µm in diameter) containing DOPG phospholipids and a small fraction of fluorescent phospholipids (NBD-PE; 5 mol. %) on its outer leaflet obtained in (A) the presence and (B) the absence of cathodic polarization (-0.9 V/Ag/AgCl) of the ITO/Au surface. For simplification, only pictures recorded every minute (until 8 minutes) are shown in (A) and (B).

Figure 4 .

 4 Figure 4. Fluorescence variations as a function of polarization time (E = -0.9 V/Ag/AgCl) obtained at a biotinylated electrode surface for () inner (N = 5), () inner/outer (N = 13), and () outer (N = 5) leaflet NBD-labelled DOPG GUVs. The numbers N between parentheses are those of the GUV analyzed in each case. The error bars represent the standard error (S.E.).

Figure 5 .

 5 Figure 5. Fluorescence intensity decay obtained during a FLIP experiment of outer+inner leaflet NBD-labelled giant vesicle (65 µm in diameterableach / a = 0.66)black points (experimental data) and corresponding modelization (green dashed line) of lateral diffusion driven decay of fluorescence; D = 4 µm 2 /s, µ=0.8, α=0.8, mean deviation 7x10 -4 per point. µ and α are defined in Supporting Information.

Figure 6 .

 6 Figure 6. Comparison between the numerical model of fluorescence intensity decay obtained through the FLIP assay (green dashed line) and the fluorescence decay obtained during FLIE experiments involving outer leaflet NBD-labelled GUVs (red points) also presented in Figure 4.

Acknowledgements

This work was supported by the CNRS (UMR 8640 -PASTEUR), the French Ministry of Research, the Ecole Normale Supérieure, Université P. et M. Curie, and the Agence Nationale de la Recherche (grant number: ANR-12-BS08-0002-01 "ELIPTIC"). The authors thank Conacyt for support of A.I.P.J. and Prof.Dr. Christian Amatore for fruitful discussions and helpful advices.