F R É D É R I C Bernardin

Michelle Schatzman

Claude-Henri Lamarque

Second-order multivalued stochastic differential equations on Riemannian manifolds

Keywords: stochastic differential equations, manifolds, friction, differential inclusions, maximal monotone operators, rigid body dynamics

The existence and uniqueness of solutions to multivalued stochastic differential equations of the second order on Riemannian manifolds are proved. The class of problem is motivated by rigid body and multibody dynamics with friction and an application to the spherical pendulum with friction is presented.

Introduction

In the theory of rigid bodies mechanics, a system is described by the configuration manifold M , which is generally the set of isometries of the three-dimensional space; the movement of the body is then a curve on this manifold. The kinetic energy defines a Riemannian metric and the equation of motion is a differential equation on a Riemannian manifold. If a stochastic term is involved in the forces acting on the system, stochastic differential equations must be used and we refer to [START_REF] Ikeda | Stochastic differential equations and diffusion processes[END_REF], [START_REF] Norris | A complete differential formalism for stochastic calculus in manifolds[END_REF] and [START_REF] Emery | Stochastic calculus in manifolds[END_REF] for information. We are interested in multivalued maximal monotone operators, since our physical model includes friction. The friction force f is of physical origin. Mathematically, it belongs to the cotangent space at x to M , since it is a generalized force. Moreover, it must belong to a closed convex subset C(x) of the cotangent space T *

x M , and it is related to the velocity through the principle of maximal dissipation, stated for instance by [START_REF] Ballard | Formulation and well-posedness of the dynamics of rigid bodies with unilateral or frictional constraints[END_REF] under the form ∀g ∈ C(x),f, ẋ xg, ẋ x , where •, • x denotes the scalar product between T *

x M and the tangent space at x, T x M . It is equivalent to writing f ∈ -∂ψ C(x) ( ẋ), where ψ C(x) is a convex homogeneous function of degree 1 which will be defined later, and ∂ denotes the subdifferential. It is well known that the subdifferential of such a convex function is a multivalued maximal monotone operator. This the reason why we are interested in second-order multivalued stochastic differential equations (MSDEs) on Riemannian manifolds. We prove existence and uniqueness under the classical assumptions: Lipschitz continuity and linear growth of the drift and diffusion coefficients (see Mao (1997, p. 51) for classical stochastic differential equations).

Let us observe that it is not easy to give a meaning to a first-order multivalued differential equation on a Riemannian manifold; indeed, assume that in a local chart the equation would be of the type dx dt + A(t, x) f (t).

We have to understand dx/dt as an element of the tangent space at x(t), T x(t) M and, therefore, f (t) must belong to the same space and A(t, x) must be a subset of this space. If we would like to state a monotonicity assumption on A(t, •), it is not difficult to transport the elements of A(t, x 1 ) by parallel transport to the tangent space at x 2 ; however, monotonicity is a concept which requires duality between vector spaces. The difference x 1 -x 2 cannot be assigned an intrinsic meaning and, therefore, the main principle of differential geometry, that is, chart independence, cannot be upheld. This objection fails if, instead of working on a manifold M , we work on its tangent space T M , which is the space of classical mechanics, our present setting.

In a local chart, the simplest equation of mechanics, that is, the equation of geodesics, contains a quadratic term of geometric origin. In order to apply the results of [START_REF] Cépa | Equations différentielles stochastiques multivoques[END_REF], we have to get rid of this quadratic term, and we can do that by a parallel transport of the equation back to the initial position. Unfortunately, we have to pay for this transformation, since now the coefficients of the new equation depend on the history up to time t of the position, i.e. the restriction x /[0,t] of x to [0, t]. The initial formulation is Markovian, the transformation by parallel transport does not have to be Markovian, the construction in the tangent space at the initial point is not Markovian, but the reciprocal transformation which enables us to return to the initial formulation takes care of all these difficulties.

The paper is organized as follows. In § 2, we give definitions and assumptions and define the problem. Working in a local chart is equivalent to working in an open subset of R d ; in § 3 we extend the coefficients of the equation to all of R d and we will denote these coefficients b, ν, σ and Ã. Unfortunately, they are only locally Lipschitz continuous with respect to the velocity v. We define new coefficients which have better properties; typically, σ n (t, x, v) is equal to σ(t, x, v) if v n, and is Lipschitz continuous if v n. There may exist a geometrical proof showing that b is Lipschitz continuous, under appropriate assumptions on the curvature tensor but this proof will not be presented in this article. In § 4, we apply the results of [START_REF] Cépa | Equations différentielles stochastiques multivoques[END_REF] to prove the existence and uniqueness of a solution of our equation with coefficients b n , σ n and ν n . In § 5, we prove convergence for n tending to infinity by stopping-time techniques. In § 6, we patch together the local solutions. Finally, an illustrative example is given in § 7: the spherical pendulum.

Throughout this article, the Einstein summation convention is used, the time derivative of a map f is denoted by ḟ and the value of f at the point t is denoted indifferently by f (t) or f t .

Definitions

We consider a smooth Riemannian manifold M of dimension We suppose the following.

Assumption 2.1. There exist r > 0, r ∈ ]0, r[ and a finite or countable number of points z k in M , such that M can be covered by balls B M (z k , r ) of radius r about z k and each ball of radius r about z k is totally included in one chart of M . Moreover, this covering can be taken to be locally finite.

In particular, this assumption is verified if M is a compact manifold or if its curvature tensor is bounded. A curve on M is a map ρ of class C 1 from [0, T ] to M (T > 0) and we denote by τ (ρ / [a,b] ) the parallel transport along ρ / [a,b] from T ρ(a) M to T ρ(b) M . We recall that τ (ρ / [a,b] ) is an isometry between T ρ(a) M and T ρ(b) M (see [START_REF] Abraham | Foundations of mechanics[END_REF] for a proof) and is independent of the parametrization of ρ. We need also the important particular case of geodesics: assuming that the geodesic from

x 1 ∈ M to x 2 ∈ M is unique τ (x 1 , x 2 ) is the parallel transport from T x 1 M to T x 2 M . We denote by π : T M → M the projection map of the tangent bundle T M of M .
In order to describe the class of maximal monotone operators used here, we observe first that, for each x ∈ M , a maximal monotone graph relative to the tangent scalar product (•, •) x in the fibre T x M can be defined in a straightforward fashion: such a graph is a subset A x of (T x M ) 2 such that for all pairs (v 1 , w 1 ) and

(v 2 , w 2 ) in A x , (v 1 -v 2 , w 1 -w 2 ) x 0 (monotonicity)
and A x is maximal for inclusion among all monotone graphs. In consequence and, without being too formal, there is a bundle of maximal monotone graphs on the manifold M : if T 2 M denotes the bundle whose generic fibre is T 2

x M = (T x M ) 2 , the fibre of the bundle of maximal monotone graphs is simply the set of maximal monotone graphs in T 2

x M . In order to avoid repetitions we will even assume that this A x also depends on the time t ∈ [0, T ] so that we consider for each t ∈ [0, T ] and x ∈ M a maximal monotone graph A (t, x) in T 2

x M . Assumption 2.5 describes the regularity of A (t, x) with respect to t and x. For all t, x and v ∈ T x M , we denote by A (t, x, v) the image by A (t, x) of v, which is a closed convex set. We suppose that the domain of A (t, x) is all T x M for all 0 t T and x ∈ M , i.e. for all t in [0, T ] and all x in M and all v in T x M , A (t, x, v) is not empty. Given w ∈ T x M , we denote by A (t, x, v, w) the projection of w onto A (t, x, v) relative to the metric of T x M .

We denote R d by E. If F and G are two finite-dimensional vector spaces, we denote by L(F, G) the set of linear maps from F into G. For t ∈ [0, T ] and x ∈ M , let the drift ν (t, x) be a map from T x M into T x M and let the diffusion matrix σ (t, x) be a differentiable map from

T x M into L(E, T x M ). For all v ∈ T x M , let us define R(t, x, v) = σ (t, x, v) 2 L(E,T x M ) .
Here σ (t, x, v) and ν (t, x, v) are, respectively, the image of v by σ (t, x) and by ν (t, x). Our assumption on the drift and the diffusion are tailored for a Stratonovich stochastic differential system; for this reason the square of the diffusion matrix appears in assumptions 2.2 and 2.3. We denote by a continuous map from [0; T ] to R + such that lim t→0 (t) = 0. We suppose the following assumptions.

Assumption 2.2. For all bounded open subset U ⊂ M , there exists C 0 (U ) such that for all close enough x and y in U , for all v in T x M and u in T y M , and for all s and t in [0, T ],

∂R ∂v (t, x, v) - ∂R ∂v (s, y, u) C 0 (U )( (|t -s|) + d M (x, y) + v -τ (y, x)u x ).
Assumption 2.3. There exists a constant C 1 such that for all 0 t T and for all (x, v) ∈ T M ,

ν (t, x, v) x + σ (t, x, v) L(E,T x M ) + ∂R ∂v (t, x, v) C 1 (1 + d M (x, x 0 ) + v x ).
Here x 0 is fixed in M .

Assumption 2.4. For all bounded open subset U ⊂ M , there exists C 2 (U ) such that for all close enough x and y in U , for all v in T x M and u in T y M , and for all s and t in [0, T ],

ν (t, x, v) -τ (y, x)ν (s, y, u) x + σ (t, x, v) -τ (y, x) • σ (s, y, u) L(E,T x M ) C 2 (U )( (|t -s|) + d M (x, y) + v -τ (y, x)u x ).
Assumption 2.5. For all bounded open subset U ⊂ M , there exists C 3 (U ) such that for all close enough x and y in U , for all v and w 1 in T x M and u and w 2 in T y M , and for all s and t in [0, T ],

A (t, x, v, w 1 ) -τ (y, x)A (s, y, u, w 2 ) x C 3 (U )( (|t -s|) + d M (x, y) + v -τ (y, x)u x + w 1 -τ (y, x)w 2 x ).
Remark 2.6. Assumption 2.5 has an immediate consequence: A (t, x, v) is locally bounded with respect to v, uniformly with respect to t ∈ [0, T ] and to x in compact subsets of M .

We consider E = (Ω, F, (F t ) t 0 , P) a filtered probability space which verifies the usual conditions (see [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF] for a definition). Let (B t ) t 0 be the standard R d -Brownian motion on E . Let δ be the symbol for the Stratonovich differentiation. Let T 0 be a (F t )-stopping time such that T 0 T almost surely. Let η be a random variable F T 0 -measurable which takes its values in T M , whose secondorder moment, written in a local chart of M , is finite. [START_REF] Emery | Stochastic calculus in manifolds[END_REF] for details), we denote by D ρ(t) v t the Stratonovich covariant differential of v along ρ at time t (see [START_REF] Norris | A complete differential formalism for stochastic calculus in manifolds[END_REF] for a definition). We consider also the MSDE on the manifold M :

If (v t ) t 0 is a T M -valued continuous semi-martingale over the curve ρ : [0, T ] → M (a stochastic process X on a manifold Q is a continuous semi-martingale on Q if f • X is a real continuous semi-martingale for every C ∞ map f : Q → R: see
δx t = v t dt, D ẋ (t) v t + A (t, x t , v t ) dt ν (t, x t , v t ) dt + σ (t, x t , v t )δB t , T 0 t < T, (x (T 0 ), v (T 0 )) = (π(η ), η ) a.s. ⎫ ⎪ ⎬ ⎪ ⎭ (2.1)
The first equation of (2.1) shows that x t is almost surely of class C 1 on [T 0 , T [. For t ∈ [T 0 , T ], we apply parallel transport along the trajectories' x solution of (2.1), and [START_REF] Norris | A complete differential formalism for stochastic calculus in manifolds[END_REF] has proved that the covariant differential is mapped to Stratonovich differential yielding then the following system:

τ (x /[T 0 ,t] ) -1 δx t = (τ (x /[T 0 ,t] )) -1 v t dt, δ((τ (x /[T 0 ,t] )) -1 v t ) + (τ (x /[T 0 ,t] )) -1 A (t, x t , v t ) dt (τ (x /[T 0 ,t] )) -1 ν (t, x t , v t ) dt{(τ (x /[T 0 ,t] )) -1 • σ (t, x t , v t )}δB t , (x (T 0 ), v (T 0 )) = (π(η ), η ) a.s. ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (2.2)
Let V (respectively, V ) a bounded open subset of E (respectively, of M ) such that ϕ : V → V define a local chart of M and put ψ = ϕ -1 . Let x 0 be a point of V and we identify T x 0 M and E = R d . Without loss of generality, we suppose that ϕ(x 0 ) = 0 and T x 0 ϕ = I E , where T y ϕ denotes the tangent mapping to ϕ at the point y ∈ M and I E denotes the identity map of E. Let W ⊂ V be an open set of M such that each pair of points of W can be linked by a unique geodesic of M . We define W = ϕ(W ). We will define now all the transported objects; for all the primed notation, the corresponding non-primed notation denote a transported object. For 0 s t T we denote by

C 1 W ([s, t], V ) the set of the maps ξ : [s, t] → V of class C 1 such that ξ(0) ∈ W . For ξ an element of C 1 W ([s, t], V ), ψ(ξ) is a curve on V , and for v ∈ V , we define b(s, t, ξ, v) = T ψ(ξ t ) ϕτ (ψ(ξ))τ (x 0 , ψ(ξ)(s))v.
(2.3)

We define

A(s, t, ξ, v) = τ (ψ(ξ)(s), x 0 )(τ (ψ(ξ))) -1 A (t, ψ(ξ t ), τ(ψ(ξ))τ (x 0 , ψ(ξ)(s))v), (2.4) ν(s, t, ξ, v) = τ (ψ(ξ)(s), x 0 )(τ (ψ(ξ))) -1 ν (t, ψ(ξ t ), τ(ψ(ξ))τ (x 0 , ψ(ξ)(s))v), (2.5)
and for all z ∈ E, σ (s, t, ξ, v)

z = τ (ψ(ξ)(s), x 0 )(τ (ψ(ξ))) -1 [σ (t, ψ(ξ t ), τ(ψ(ξ))τ (x 0 , ψ(ξ)(s))v)]z.
(2.6)

Remark 2.7. It is convenient to express the transport in A, ν and σ only through b. Let us explain how to do this on ν: if S = T ψ(ξ(t)) ϕ, we introduce SS -1 at two places in the definition of ν (s, t, ξ, v)

: ν(s, t, ξ, v) = τ (ψ(ξ)(s), x 0 )(τ (ψ(ξ))) -1 S -1 Sν (t, ψ(ξ t ), S -1 Sτ (ψ(ξ))τ (x 0 , ψ(ξ)(s))v). (2.7) Since Sτ (ψ(ξ))τ (x 0 , ψ(ξ)(s)) = b(s, t, ξ), (2.7) is equivalent to ν(s, t, ξ, v) = (b(s, t, ξ)) -1 Sν (t, ψ(ξ t ), S -1 b(s, t, ξ, v)).
(2.8) By a legitimate abuse of notation corresponding to the choice of a local chart, we may drop ψ(ξ t ) and S = T ψ(ξ(t)) ϕ in (2.8), yielding thus b(s, t, ξ, v)).

ν(s, t, ξ, v) = b(s, t, ξ) -1 ν (t, ξ t ,
(2.9)

There are analogous formulae for σ and A: b(s, t, ξ, v)).

σ(s, t, ξ, v) = b(s, t, ξ) -1 • σ (t, ξ t ,
(2.10) and b(s, t, ξ, v)).

A(s, t, ξ, v) = b(s, t, ξ) -1 A (t, ξ t ,
(2.11)

We suppose in a first time that π(η) belongs to W almost surely. By defining for all T 0 t T ,

x(t) = ϕ(x t ), v(t) = τ (x (T 0 ), x 0 )(τ (x /[T 0 ,t] )) -1 v t , η = τ (x (T 0 ), x 0 )η, we have dx t = b(T 0 , t, x/[T 0 ,t] , vt ) dt, δv t + A(T 0 , t, x/[T 0 ,t] , vt ) dt ν(T 0 , t, x/[T 0 ,t] , vt ) dt + σ(T 0 , t, x/[T 0 ,t] , vt )δB t , T 0 t, (x(T 0 ), v(T 0 )) = (π(η), η) a.s. ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ (2.12) Finally, we define (η 1 , η 2 ) = (π(η), η) and x, v : [0, T -T 0 ] → E by x(t) = x(t + T 0 ) and v(t) = v(t + T 0 )
and then, by denoting b(0, t, ξ, v) by b (t, ξ, v), ν(0, t, ξ, v) by ν(t, ξ, v), σ(0, t, ξ, v) by σ(t, ξ, v) and A(0, t, ξ, v) by A(t, ξ, v), we seek the restriction to [0, T -T 0 [ of the solution to the following system:

dx t = b(t, x /[0,t] , v t ) dt, δv t + A(t, x /[0,t] , v t ) dt ν(t, x /[0,t] , v t ) dt + σ(t, x /[0,t] , v t )δW t , 0 t T, (x(0), v(0)) = (η 1 , η 2 ) a.s., ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ (2.13)
where (W t ) t 0 is defined by

W t = B t+T 0 -B T 0 . For all t 0, let F t = F T 0 +t . It is well known that (W t ) t 0 is a standard Brownian motion on the filtered probability space E = (Ω, F, (F t ) t 0 , P). We say that (x t , v t ) 0 t T is a solution to (2.13) if (x t , v t ) 0 t T is F t -adapted, dx t = b(t, x /[0,t] , v t ) dt
almost surely and if there exists a stochastic process F t -adapted (K t ) 0 t T taking its values in E with bounded variation on [0, T ] such that K 0 = 0 almost surely and

v t = v 0 + t 0 ν(s, x /[0,s] , v s ) ds + t 0 σ(s, x /[0,s] , v s )δW s -K t and dK t ∈ A(t, x /[0,t] , v t ) dt.
The last inclusion has the following sense: the measure [START_REF] Cépa | Equations différentielles stochastiques multivoques[END_REF] for precisions).

(v t -α t , dK t -β t dt) x 0 is almost surely non-negative on [0, T ] for all choices of continuous maps α, β : [0, T ] → E such that β t ∈ A(t, x /[0,t] , α t ) (see
Remark 2.8. If σ (t, x, v) does not depend on v, it is possible to consider the equation (2.1) in the Itô sense-the formulation is intrinsic-and the assumption 2.2 and the linear growth condition 2.3 for R (t, x, v) are then not necessary.

In the following we will denote by (•, •) the scalar product (•, •) x 0 .

Extension of the problem to E

We are slightly abusive when we say that we extend b, A, ν and σ since we have first to restrict the values of x to a set U x 0 , with U a subset of V . More precisely, given a smooth cut-off function χ such that

χ(y) = 1 if y ∈ U, 0 if y / ∈ V, (3.1)
the function s → χ(x s )x s = xs coincides with x s as long as x s does not leave U .

If we use xs instead of x s in the definition of b, A, ν and σ, we will prove that the corresponding b, ν and σ are not Lipschitz continuous with respect to x or v. Therefore, we will need an extra step, where we project v onto the ball of radius n about 0. Let us go into the technicalities.

The functions b, ν, σ and A have been respectively defined in (2.3), (2.5), (2.6) and (2.4), which depend on the local chart ϕ; therefore all the upcoming constants will also depend on ϕ.

Let U be a fixed bounded neighbourhood of x 0 ∈ V and assume

W ⊂ U ⊂ U ⊂ V . For B an open subset of E containing x 0 , Δ(B) is the set of pairs (t, x) with t ∈ [0, T ] and x a function of class C 1 from [0, t] to B, such that x(0) ∈ W : Δ(B) = {(t, x), t ∈ [0, T ], x ∈ C 1 W ([0, t], B)}. We denote by G = (g ij ) 1 i,j d the metric tensor whose inverse is denoted by G -1 = (g ij ) 1 i,j d
and the functions g ij and g ij are smooth on V . Let (x 1 , . . . , x d ) be the local coordinates of the chart ϕ and e 0 = (∂/∂x 1 , . . . , ∂/∂x d ) be a basis of E. For (t, x) ∈ Δ(V ) and 1 i d, b(t, x, e 0,i ) = e j i (t)e 0,j , where (see [START_REF] Abraham | Foundations of mechanics[END_REF])

e j i (t) = e j i,0 - t 0 Γ j kl (x s )e k i (s) ẋl s ds.
The functions

Γ i lk (x) = Γ i kl (x) = 1 2 g ih (x) ∂g hk ∂x l (x) + ∂g hl ∂x k (x) - ∂g kl ∂x h (x)
are the Christoffel symbols of (M, G). If we denote by

I E the identity matrix of E, we can write b(t, x) T G(x t )b(t, x) = I E (see Abraham & Marsden 1985) and then tr{b(t, x)b(t, x) T } = tr{G -1 (x t )}, which shows the boundedness of (t, x) → b(t, x)
, which can also be seen on the definition of b in (2.3). Let χ be as in (3.1). Throughout this paper we denote by • ∞,t the uniform norm on the space

C 0 ([0, t], E) of continuous maps from [0, t] to E, t > 0. Lemma 3.1. Define ζ on C 0 ([0, t], E) by ζ(u) = χ(u)u. Then ζ is Lipschitz continuous from C 0 ([0, t], E) to itself. Proof . Clear. Let (t, x) ∈ Δ(V ); we first define t U
x , which is the first exit time out of U :

t U x = inf{s t, x(s) / ∈ U } ∧ T.
We now let x = χ(x)x and x coincides with x on the interval [0, t U x ]. Moreover, the following estimates are straightforward:

x

∞,t C ϕ , ẋ ∞,t C ϕ ẋ ∞,t , (3.2) and, if (t, y) ∈ Δ(V ), x -ỹ ∞,t + ẋ -ẏ ∞,t C ϕ ( ẋ ∞,t + ẏ ∞,t )( x -y ∞,t + ẋ -ẏ ∞,t ), (3.3)
with C ϕ depending only on U , V and ϕ. The verification of these estimates is straightforward and left to the reader.

Definition 3.2. We let, for all (t, x)

∈ D(E), b(t, x) = b(t, x), (3.4) ν(t, x) = ν(t, x), (3.5) σ(t, x) = σ(t, x), (3.6) Ã(t, x) = A(t, x).
(3.7)

Lemma 3.3. The maps b, ν and σ grow at most linearly, that is, for all

((t, x), v) ∈ Δ(E) × E, ν(t, x, v) + σ(t, x, v) + b(t, x, v) C ϕ ( x ∞,t + ẋ ∞,t + v ).
Proof . We know that a constant C ϕ exists so that for all ((t,

x), v) ∈ Δ(E) × E, b(t, x, v) C ϕ v and ( b(t, x)) -1 v C ϕ v . Since τ (ψ(x))τ (x 0 , ψ(x)(0)
) is an isometry between E and T ψ(x)(t) M and using assumption 2.3, the result holds.

The functions b, ν and σ are not Lipschitz continuous with respect to x and v. We replace them by Lipschitz continuous approximations as follows. For n ∈ N * , let U n = B(0, n), V n = B(0, n + 1) and χ n be the map associated to these bounded open sets. All constants depending only on n are denoted by C n and by C ϕn if they depend on ϕ and n. For (t, x) ∈ Δ(E), let us define Using lemma 3.1, for all (t, y) ∈ Δ(E), we have

x n (s) = x(0) + s 0 χ n ( ẋ(u)) ẋ(u) du,
x n -y n ∞,t + ẋn -ẏn ∞,t C ϕn ( x(0) -y(0) + ẋ -ẏ ∞,t ). (3.9) If t n x = inf{s t, ẋ(s) n} ∧ T , x n = x on [0, t n x ]. Definition 3.4. We let for all (t, x, v) ∈ Δ(E) × E, b n (t, x, v) = b(t, x n , proj B(0,n) v),
(3.10)

ν n (t, x, v) = ν(t, x n , proj B(0,n) v), (3.11) σ n (t, x, v) = σ(t, x n , proj B(0,n) v), (3.12) A n (t, x, v) = Ã(t, x n , v). (3.13)
We give now several results on the above introduced functions.

Lemma 3.5. The functions b n and b verify

∀(t, x) ∈ Δ(E), ∀v ∈ B(0, n), ∀s t n x , b n (s, x /[0,s] , v) = b(s, x /[0,s] , v). Proof . Indeed by definition of b n , b n (s, x /[0,s] , v) = b(s, x n/[0,s] , proj B(0,n) v) = b(s, x /[0,s] , v), since s t n x = inf{u t, ẋ(u) n}.
Remark 3.6. With the help of remark 2.7 and the definitions of ν n , σ n and A n , the same equality occurs for ν n , σ n and A n .

Lemma 3.7. Operators A(t, x) and A n (t, x) are multivalued, maximal and monotone on E for all (t, x) ∈ Δ(E).

Proof . This is a direct consequence of the fact that the transport operator is an isometry.

Remark 3.8. The functions b n , ν n and σ n grow at most linearly, thanks to lemma 3.3. Lemma 3.9. The maps b n (t, ., .), (b(t, • n ) -1 , ν n (t, ., .) and σ n (t, ., .) are Lipschitz continuous functions: for all

((t, x), (t, y), v, u) ∈ (Δ(E)) 2 × E 2 , b n (t, x, v) -b n (t, y, u) + (b(t, x n )) -1 v -(b(t, y n )) -1 u + ν n (t, x, v) -ν n (t, y, u) + σ n (t, x, v) -σ n (t, y, u) C ϕn ( x -y ∞,t + ẋ -ẏ ∞,t + v -u ). (3.14)
Proof . If we write b(t, x) = E(t) and b(t, y) = F (t), by definition of b we have Ė(t) = L(x, t)E(t) and Ḟ (t) = L(ỹ, t)F (t), where L(ρ, t) maps linearly the set of invertible square matrices of dimension d × d into itself and is defined by

(L(ρ, s)A) j i = -Γ j kl (ρ(s))A k i ρl
s , s t, for all ρ in Δ(V ) and matrix A. We begin by writing

|[L(x, s)E(s) -L(ỹ, s)E(s)] j i | = |Γ j kl (ỹ(s))E k i (s) ẏl s -Γ j kl (x(s))E k i (s) ẋl s | |(Γ j kl (ỹ(s)) -Γ j kl (x(s)))E k i (s) ẏs | + Γ j kl (x(s))E k i (s)( ẏs -ẋs ) C ϕ ( ẏ ∞,t . x -ỹ ∞,t + ẏ -ẋ ∞,t ). ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (3.15)
We comment here on the last line of (3.15): it is the term ẏ ∞,t xỹ ∞,t which prevents b from being Lipschitz continuous with respect to x, since ẏ ∞,t does not have to be bounded. There may exist a geometrical proof showing that b is Lipschitz continuous, under appropriate assumptions on the curvature tensor. Let us define

E n (t) = b(t, x n ) and F n (t) = b(t, y n ). Since E n (t) = I E - t 0 L( x n , s)E n (s) ds and F n (t) = I E - t 0 L( y n , s)F n (s) ds,
we have

E n (s) -F n (s) s 0 L( y n , u)F n (u) -L( y n , u)E n (u) du + s 0 (L( y n , u) -L( x n , u))E n (u) du,
and then sup

0 s t E n (s) -F n (s) sup 0 s t L( y n , s) t 0 sup 0 u s E n (u) -F n (u) ds + C ϕ ( ˙ y n ∞,t x n -y n ∞,t + ˙ y n -˙ x n ∞,t ).
Using (3.2), (3.8) and the continuity of the functions (Γ j kl ) k,l,j , we have sup

0 s t L( y n , s) C ϕn ˙ y n ∞,t C ϕn (n + 1).
Using the Gronwall's lemma and inequalities (3.2), (3.3), (3.8), (3.9), we have

E n (t) -F n (t) C ϕn ( x -y ∞,t + ẏ -ẋ ∞,t ).
This last inequality and the definition of b n lead to:

b n (t, x, v) -b n (t, y, u) C ϕn ( x -y ∞,t + ẋ -ẏ ∞,t + v -u ). Since b(t, x) T G(x t )b(t, x) = I E and x → G(x) is of class C 1 , the last inequality is valid for (b(t, x n )) -1 and (b(t, y n )) -1 instead of b n (t, x, .
) and b n (t, y, .). Using expressions defining ν n and σ n in terms of b n , b -1 n , ν and σ , the assumptions 2.4, 2.3 and inequality b n (t, x, v) C ϕn we obtain easily the Lipschitz continuity of ν n and σ n . Lemma 3.10. For all (t, x) ∈ Δ(E), 0 s t and u ∈ E, the following estimates hold:

b n (s, x /[0,s] , u) -b n (t, x, u) C ϕn |t -s|, b(s, x /[0,s] ) -1 u -b(t, x) -1 u C ϕn |t -s|, ν n (s, x /[0,s] , u) -ν n (t, x, u) C ϕn (|t -s|), σ n (s, x /[0,s] , u) -σ n (t, x, u) C ϕn (|t -s|). ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ (3.16)
Proof . With the notation and intermediary results of lemma 3.9, we can write

b n (s, x /[0,s] , u) -b n (t, x, u) = (b(s, x n/[0,s] ) -b(t, x n )).proj B(0,n) u n b(s, x n/[0,s] ) -b(t, x n ) = n E n (s) -E n (t) C ϕn |t -s| ˙ x n ∞,t C ϕn |t -s|.
Using this last inequality and the assumption 2.4 we obtain the result.

Lemma 3.11. Let (T, x) ∈ Δ(E). Then there exists λ 0 > 0 such that α λ : u →

(I + λA n (u, x /[0,u] , .)) -1 (z) is continuous on [0, T ] for any 0 < λ λ 0 , z ∈ E. Proof . For (t, s) ∈ [0, T ] 2 , we can write z -α λ t λ ∈ A n (t, x /[0,t] , α λ t ) and z -α λ s λ ∈ A n (s, x /[0,s] , α λ s ).
Using the monotonicity, we have then

α λ t -α λ s 2 = λ α λ t -α λ s , α λ t -z λ + z -α λ s λ = 2λ α λ t - α λ t + α λ s 2 , α λ t -z λ + β 1 -λ(α λ t -α λ s , β 1 -β 2 ) + 2λ α λ s + α λ t 2 -α λ s , β 2 - z -α λ s λ λ(α λ t -α λ s , β 2 -β 1 ) 1 2 λ α λ t -α λ s 2 + 1 2 λ β 1 -β 2 2 , with β 1 = proj A n (t,x /[0,t] ,(α λ s +α λ t )/2) 0 and β 2 = proj A n (s,x /[0,s] ,(α λ s +α λ t )
/2) 0. By assumption 2.5, we have

α λ t -α λ s 2 λC ϕn ( 2 (|t -s|) + |α λ t -α λ s | 2 + x t -x s 2 )
and it is possible to choose λ 0 (ϕ, n) > 0 such that for all λ λ 0 , α λ is continuous on [0, T ].

Remark 3.12. We note that λ 0 depends only on n and ϕ.

Lemma 3.13. Let r and r be continuous functions from [0, t] into [0, T ], where t verifies 0 t T . If we set t r = sup 0 s t r(s) and t r = sup 0 s t r (s), we consider (t r , x) and (t r , x ) elements of Δ(E). Let us assume the existence of continuous functions v, v , K and K from [0, t] into E such that (i) K and K are functions with bounded variations on [0, t];

(ii) dK s ∈ A n (r(s), x /[0,r(s)] , v(s)) ds and dK s ∈ A n (r (s), x /[0,r (s)] , v (s)) ds, that is, the measures (v s -α s , dK s -β s ds) and (v s -α s , dK s -β s ds)
are non-negative on [0, t] for any choice of continuous functions α, α , β, β :

[0, t] → E verifying ∀s ∈ [0, t], (α s , β s ) ∈ Gr(A n (r(s), x /[0,r(s)] )) and (α s , β s ) ∈ Gr(A n (r (s), x /[0,r (s)] )),
where Gr(B) denotes the graph of the multivalued operator B defined on the whole space E, i.e. the set {(x, y) ∈ E 2 , y ∈ B(x)}. Then there exists a map C ϕn : [0, t] → R + such that the measure

(v s -v s , dK s -dK s ) + C ϕn (s)( x -x 2 ∞,r(s)∧r (s) + v -v 2 ∞,s + x(r(s)) -x (r (s)) 2 + 2 (|r(s) -r (s)|)) ds (3.17) is non-negative on [0, t].
Remark 3.14. In the case where A n is independent of t and x, the first term in (3.17) is already non-negative; the extra terms account for the dependency on t and x.

Proof . For p ∈ N * and s ∈ [0, t], let us define

α p (s) = (I + A n (r(s), x /[0,r(s)] )/p) -1 ((v(s) + v (s))/2), β p (s) = p((v(s) + v (s))/2 -α p (s)), α p (s) = (I + A n (r (s), x /[0,r (s)] )/p) -1 ((v(s) + v (s))/2),
and

β p (s) = p((v(s) + v (s))/2 -α p (s)). If c : [0, t] → E is continuous, the function γ p : s → (I + 1/pA n (r(s), x /[0,r(s)] )) -1 (c(s))
is continuous for p p 0 , where p 0 depends on n and ϕ. This results from lemma 3.11. We can conclude that α p , α p , β p and β p are continuous on [0, t].

By hypothesis (ii) we can write (v(s)α p (s), dK(s)β p (s) ds) 0 and (v (s)α p (s), dK (s)β p (s) ds) 0.

By summation of these last inequalities, we have

(v(s) -α p (s), dK(s)) + (v (s) -α p (s), dK (s)) -(v(s) -α p (s), β p (s) ds) -(v (s) -α p (s), β p (s) ds) 0. (3.18)
By results of [START_REF] Brézis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF],

lim p→+∞ α p (s) = lim p→+∞ α p (s) = 1 2 (v(s) + v (s)), lim p→+∞ β p (s) = A 0 n (r(s), x /[0,r(s)] , 1 2 (v(s) + v (s)))
and

lim p→+∞ β p (s) = A 0 n (r (s), x /[0,r (s)] , 1 2 (v(s) + v (s))),
where

A 0 n (r(s), x /[0,r(s)] ) (respectively, A 0 n (r (s), x /[0,r (s)]
)) denotes the principal section of A n (r(s), x /[0,r(s)] ) (respectively, of A n (r (t), x /[0,r (t)] )). Inequality (3.18) implies then

(v(s) -v (s), dK (s) -dK(s)) (v(s) -v (s), A 0 n (r(s), x /[0,r(s)] , 1 2 (v(s) + v (s))) -A 0 n (r (s), x /[0,r (s)] , 1 2 (v(s) + v (s)))) 1 2 v(s) -v (s) 2 + 1 2 A 0 n (r(s), x /[0,r(s)] , 1 2 (v(s) + v (s))) -A 0 n (r (s), x /[0,r (s)] , 1 2 (v(s) + v (s)))
2 . This last inequality and assumption 2.5 yields the result.

Existence of solutions in E

In order to obtain the existence of a solution we will use the results of [START_REF] Cépa | Problème de Skorohod multivoque[END_REF], which ensure the existence of solutions to the MSDE of the following type:

dX t + B(X t ) dt α(X t ) dt + β(X t ) dW t ,
where B is a maximal monotone operator on E and α and β verify the classical Lipschitz and linear growth conditions. If the probability space is (Ω, G, G 0 t T , Q) and the initial condition is G 0 -measurable, then the solution is G t -adapted and belongs to L 2 (Ω, C 0 ([0, T ], E)).

Remark 4.1. It is crucial for the following to remark that the results of Cépa (1998) can be extended to random operator B and random maps α and β for which we have the following properties. (iv) For all maps Y, Z :

Ω → E such that Z ∈ B(Y ) almost surely, if Y is G 0 - measurable, then Z is G 0 -measurable.
We give here a lemma which will be useful.

Lemma 4.2. For all c ∈ E, t ∈ [0, T ] and continuous map v : [0, t] → E there exists a unique

C 1 map x : [0, t] → E such that ẋ(s) = b(s, x /[0,s] , v(s)), x(0) = c. Proof . The space Z = C 1 ([0, t], E) is a Banach space when equipped with the norm f = f + df/dt . For f ∈ Z, let L(f ) ∈ Z be defined by L(f )(s) = c + s 0 b(u, f /[0,u] , v(u)) du.
Using lemma 3.9, L m is strictly contracting for large enough m. We infer the existence of the solution on [0, t] by application of the Banach fixed point theorem on Z.

The construction of a solution is performed in several steps. We replace first the v equation by an equation which contains only the initial data, and we can integrate it with the help of Cépa's results. We then integrate the x equation according to lemma 4.2. This gives an approximate solution on an interval [0, t 1 ]. Assuming this interval to be small, we restart the process: we integrate an approximate v equation on [t 1 , t 2 ] by using only the x data from the previous interval, which is possible thanks to Cépa's results. The process is recursive and yields an approximate solution on [0, T ], the approximation parameter being the size of the subintervals. We take this size to be T /2 p , and we compare the approximate solutions for the integers p and q, and we show by stochastic techniques that the expectation of appropriate norms of the difference tends to 0. Theorem 4.3. For all n 1, there exists a solution to the following system

δx n t = b n (t, x n /[0,t] , v n t ) dt, δv n t + A n (t, x n /[0,t] , v n t ) dt ν n (t, x n /[0,t] , v n t ) dt + σ(t, x n /[0,t] , v n t )δW t , 0 t T, (x(0), v(0)) = (η 1 , η 2 ). ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ (4.1)
Proof . To lighten the notations, we denote b n , ν n , σ n and A n by b, ν, σ and A.

Step 1 (Construction of the approximate solution). We denote by s i (t, ξ, v) the square of the norm of the ith row (considered as an E-valued vector) of σ(t, ξ, v) for all (t, ξ) ∈ Δ(E) and v ∈ E. We define then

α(t, ξ, v) = ν(t, ξ, v) + ∂ ∂v {(s 1 (t, ξ, v), . . . , s d (t, ξ, v)) T }. (4.2)
We consider now, for p ∈ N * , the uniform partition 0 < t p 1 < • • • < t p 2 p = T of [0, T ] with step size T /2 p and we construct two processes x p and v p as follows. To simplify the notation we denote t p i by t i . Results of Cépa mentioned in the beginning of the section, remarks 4.1 and 2.6, (3.14), (3.16), assumptions 2.2 and 2.3 give a solution v p of the Itô MSDE on [0, t 1 ]:

dv p t + A(0, η 1 , v p t ) dt α(0, η 1 , v p t ) dt + σ(0, η 1 , v p t ) dW t , v p (0) = η 2 . (4.3)
By definition of α and using the relation between the Itô integral and the Stratonovich integral, equation (4.3) is equivalent to the following Stratonovich MSDE

δv p t + A(0, η 1 , v p t ) dt ν(0, η 1 , v p t ) dt + σ(0, η 1 , v p t )δW t , (4.4) v p (0) = η 2 .
Using lemma 4.2, let x p be the solution on [0,

t 1 ] to ẋp (s) = b(s, x p /[0,s] , v p (s)), x p (0) = η 1 . Suppose (x p , v p ) defined on [0, t 1 ] ∪ • • • ∪ [t i-1 , t i ] = [0, t i ] and consider the solution (v p t ) t i t t i+1 on [t i , t i+1 ] of the MSDE δv p t + A(t i , x p /[0,t i ] , v p t ) dt ν(t i , x p /[0,t i ] , v p t ) dt + σ(t i , x p /[0,t i ] , v p t )δW t , v p (t + i ) = v p (t - i ), and x p the solution on [t i , t i+1 ] to ẋp (s) = b(s, x p /[0,s] , v p s ), x p (t + i ) = x p (t - i ). (4.5) Since (v p t ) 0 t T belongs to L 2 (Ω, C 0 ([0, T ], E)) and b(s, f ) C ϕ for all (s, f ) ∈ Δ(E), (x p t ) 0 t T belongs to L 2 (Ω, C 1 ([0, T ], E)).
Step 2 (the approximate solutions form a Cauchy sequence). We will now show that (

x p t , v p t ) 0 t T is a Cauchy sequence of L 2 (Ω, C 0 ([0, T ], E) × C 1 ([0, T ], E)). Let 1
p q be integers and denote by 0

= t 0 < t 1 < • • • < t 2 q-p < • • • < t 2 q-p+1 < • • • < t 2 q = T the partition of [0, T ] associated to q. We note that 0 < t 2 q-p < t 2.2 q-p < • • • < t k.2 q-p < • • • < t 2 q = T
is the partition associated with p. Let s ∈ [0, T ] and put r 1 (s) = t s2 q and r 2 (s) = t s2 p . We then have

v q s -v p s = s 0 (ν(r 1 (u), x q /[0,r 1 (u)] , v q u ) -ν(r 2 (u), x p /[0,r 2 (u)] , v p u )) du + s 0 (σ(r 1 (u), x q /[0,r 1 (u)] , v q u ) -σ(r 2 (u), x p /[0,r 2 (u)] , v p u ))δW u + K p s -K q s with dK q u ∈ A(r 1 (u), x q /[0,r 1 (u)]
, v q u ) du, 0 u s, and dK p u ∈ A(r 2 (u), x p /[0,r 2 (u)] , v p u ) du, 0 u s. Using Itô's formula, we can write

v q s -v p s 2 = 2 s 0 (v q u -v p u , δv q u -δv p u ) = 2 s 0 (α(r 1 (u), x q /[0,r 1 (u)] , v q u ) -α(r 2 (u), x p /[0,r 2 (u)] , v p u ), v q u -v p u ) du + 2 s 0 σ(r 1 (u), x q /[0,r 1 (u)] , v q u ) -σ(r 2 (u), x p /[0,r 2 (u)] , v p u ) 2 du + 2 s 0 (σ(r 1 (u), x q /[0,r 1 (u)] , v q u ) -σ(r 2 (u), x p /[0,r 2 (u)] , v p u ), v q u -v p u ) dW u + 2 s 0 (v q u -v p u , dK p u -dK q u ),
where the norm of matrices is defined by B 2 = tr{B T B}. Using lemmas 3.10, 3.9, the assumptions 2.2 and 2.3, Cauchy-Schwarz's and Burkholder-Gundy-Davis's inequalities and denoting r(p, q) = sup 0 s T |r 1 (s)-r 2 (s)|, which tends to zero when p and q tend to infinity, we have

E sup 0 s t v q s -v p s 2 C ϕn 2 (r(p, q)) + C ϕn t 0 (E x p -x q 2 ∞,s + E ẋp -ẋq 2 ∞,s + E v p -v q 2 ∞,s ) ds + C ϕn E t 0 [ 2 (r(p, q))+ x p -x q 2 ∞,s + ẋp -ẋq 2 ∞,s + v p -v q 2 ∞,s ] v q s -v p s 2 ds + 2E sup 0 s t s2 q i=1 t i t i-1 (v q u -v p u , dK p u -dK q u ) + t t s2 q (v q u -v p u , dK p u -dK q u ).
Using the inequalities

√ a + b √ a + √ b, ab (1/δ)a 2 + δb 2 (δ > 0)
and lemma 3.13 for the intervals [t i-1 , t i ] (on each of them we put r = r 1 and r = r 2 ), we have

E sup 0 s t v q s -v p s 2 C ϕn 2 (r(p, q)) + C ϕn δE sup 0 s t v q s -v p s 2 + C ϕn δ t 0 (E x p -x q 2 ∞,s + E ẋp -ẋq 2 ∞,s + E v p -v q 2 ∞,s ) ds + C ϕn E sup 0 s t s2 q i=1 t i t i-1 ( 2 (r(p, q)) + x p -x q 2 ∞,u + ẋp -ẋq 2 ∞,u + v p -v q 2 ∞,u ) du + t t s2 q ( 2 (r(p, q)) + x p -x q 2 ∞,u + ẋp -ẋq 2 ∞,u + v p -v q 2 ∞,u ) du .
Choosing δ such that C ϕn δ = 1/2, we can write

E sup 0 s t v q s -v p s 2 C ϕn 2 (r(p, q)) + C ϕn t 0 (E x p -x q 2 ∞,s + E ẋp -ẋq 2 ∞,s + E v p -v q 2 ∞,s ) ds. (4.6)
Moreover, by lemmas 3.10 and 3.9, we have

E sup 0 s t ẋq s -ẋp s 2 C ϕn 2 (r(p, q)) + C ϕn t 0 (E x p -x q 2 ∞,s + E ẋp -ẋq 2 ∞,s + E v p -v q 2 ∞,s ) ds (4.7)
and

E sup 0 s t x q s -x p s 2 C ϕn 2 (r(p, q)) + C ϕn t 0 (E x p -x q 2 ∞,s + E ẋp -ẋq 2 ∞,s + E v p -v q 2 ∞,s ) ds. (4.8)
The inequalities (4.6)-( 4.8) imply that

E( x q -x p 2 + ẋq -ẋp 2 ∞,t + v q -v p 2 ) C ϕn 2 (r(p, q)) + C ϕn t 0 E( x p -x q 2 ∞,s + ẋp -ẋq 2 ∞,s + v p -v q 2 ∞,s ) ds,
and the application of Gronwall's lemma yields

E( x q -x p 2 + ẋq -ẋp 2 ∞,t + v q -v p 2 ) C ϕn 2 (r(p, q))
and the sequence (

x p , v p ) is a Cauchy sequence in L 2 (Ω, C 1 ([0, T ], E)×C 0 ([0, T ], E))
and there exists then a stochastic process (x, v) which is the limit of (x p , v p ).

To complete the proof of existence we must prove that (x, v) verifies the MSDE. For this it remains to show that (K p ) p 1 converges almost surely towards a process (K t ) 0 t T with bounded variation on [0, T ] and verifying dK

t ∈ A(t, x /[0,t] , v t ) dt. Since (x p , v p ) converges in L 2 (Ω, C 1 ([0, T ], E) × C 0 ([0, T ], E))
, there exists a subsequence, which is still denoted by p, such that (x p , v p ) tends almost surely to (

x, v) in C 1 ([0, T ], E) × C 0 ([0, T ], E). It follows that K p tends almost surely to a process K in C 0 ([0, T ], E).
Step 3 (uniform estimation of the total variation of K on [0, T ]). For each p ∈ N * , we consider the above partition 0

< t p 1 < • • • < t p 2 p = T of [0,
T ] and we denote by r p (t) the t p i such that t p i t < t p i+1 . Lemma 4.4. There exists a random variable almost surely non-negative which estimates from above for all p the total variation

|K p | T of K p on [0, T ].
Proof . Let γ be a positive real and

Λ(γ, V ) = {(t, y, v) ∈ [0, T ] × T M, y ∈ V , v y γ}. By remark 2.6, μ = sup (t,y,v)∈Λ(γ,V ) A (t, y, v) y (4.9)
is finite. We apply proposition 4.4 of Cépa (1998) (with a = 0 and γ) to the process (v p , K p ) on each [t p i , t p i+1 ] to obtain for all 0 s t T

γ|K p | t s l i=k γ|K p | t i+1 t i + γ|K p | t k s + γ|K p | t t l l i=k t i+1 t i (v p (u), dK p (u)) + t l t (v p (u), dK p (u)) + t k s (v p (u), dK p (u)) + l i=k μ t i+1 t i v p (u) du + γμ(t i+1 -t i ) + μ t k s v p (u) du + γμ(t k -s) + μ t t l v p (u) du + γμ(t -t l ) = t s (v p (u), dK p (u)) + μ t s v p (u) du + γμ(t -s), (4.10)
where t k and t l verify t k-1 < s t k and t l t < t l+1 . Otherwise, with w p = v p +K p , we have

v p (t) 2 = w p (t) 2 -2 t 0 (v p (u), dK p (u)) + 2 t 0 (w p (u) -w p (t), dK p (u))
and

-v p (s) 2 = -w p (s) 2 + 2 s 0 (v p (u), dK p (u)) -2 s 0 (w p (u) -w p (s), dK p (u)).
By summation of the two last equalities and using the inequality (4.10), we can write

2[γ -sup{ w p (u) -w p (v) , |u -v| |t -s|}]|K p | t s C ϕn (ω),
since v p , K p and w p are almost surely uniformly bounded with p on [0, T ]. Recalling that (w p ) p 1 converges uniformly towards a continuous function w we can conclude by Ascoli's theorem that there exists η ϕn (ω) > 0 such that sup{

w p (u)-w p (v) , |u- v| η ϕn } γ/2 and then |K p | t s C ϕn (ω)/γ. If 0 = s 0 < s 1 < • • • < s N = T is a partition of [0, T ] such that s i+1 -s i η ϕn , then |K p | T N -1 i=0 |K p | s i+1 s i N ϕn C ϕn (ω) γ .
This last result implies that K is almost surely with bounded variation on [0, T ] and for all sequences of processes ϕ p ∈ C([0, T ], E) converging almost surely in C([0, T ], E ) to a process ϕ, then (see [START_REF] Saisho | Stochastic differential equations for multi-dimensional domain with reflecting boundary[END_REF] for more details):

lim p→+∞ t s (ϕ p (u), dK p (u)) = t s (ϕ(u), dK(u)), 0 s t T.
(4.11)

Step 4 (end of the proof ). Let α, β : [0, T ] → E be two continuous functions such that β t belongs to A(t, x /[0,t] , α t ). Let us set β p (t) = proj K p (t) β t with K p (t) = A(r p (t), x p /[0,r p (t)] , α t ). By assumption 2.5,

β p (t) -β(t) C ϕn (ω)( x -x p ∞,r p (t) + x(t) -x p (r p (t)) + (|t -r p (t)|))
and then (β p ) p converges almost surely uniformly to β. Moreover, β p is almost surely continuous on each [t p i , t p i+1 [ and then

t p i+1 t p i (v p (u) -α(u), dK p (u) -β p (u) du) 0.
We deduce from this last inequality, (4.11) and the dominated Lebesgue theorem that

t s (v(u) -α(u), dK(u) -β(u) du) 0, 0 s t T, which is the definition of dK(t) ∈ A(t, x /[0,t] , v t ) dt.

Existence and uniqueness in the local chart (a) Existence

We return to the notation b n , ν n , σ n and A n , and n will tend to infinity. Let τ n be the stopping time defined by τ n (ω) = inf{t T, ẋn t (ω) n or v n t (ω) n}. Clearly, (τ n ) n 1 is a non-decreasing sequence of stopping times and (x n t , v n t ) = (x m t , v m t ) almost surely on [0, τ n ] for all n m. We note then τ = lim n→+∞ τ n and we put, for 0

t < τ(ω), x(t, ω) = x n (t, ω) and v(t, ω) = v n (t, ω),
where n is such that t τ n (ω). Therefore, by theorem 4.3 and lemma 3.5, (x, v) verifies:

δx t = b(t, x /[0,t] , v t ) dt δv t + Ã(t, x /[0,t] , v t ) dt ν(t, x /[0,t] , v t ) dt + σ(t, x /[0,t] , v t )δW t , 0 t < τ, (x(0), v(0)) = (η 1 , η 2 ). ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ (5.1) Lemma 5.1. Almost surely, τ = T .
Proof . It suffices to prove for all t T , lim n→+∞

P{τ n t} = 0.

Let t < T , n 1. Since for all 0

s t ∧ τ n , ẋs = b(s, x /[0,s] , v s ), ẋs C ϕ v s , (5.2)
and therefore, since t < T ,

x s η 1 + C ϕ v ∞,s . (5.3)
We write

v s∧τ n 2 = η 2 2 + 2 s∧τ n 0 (v u , δv u ) = η 2 2 + 2 s∧τ n 0 (v u , α(u, x /[0,u] , v u )) du + 2 s∧τ n 0 tr{σ(u, x /[0,u] , v u ) T σ(u, x /[0,u] , v u )} du -2 s∧τ n 0 (v u , dK u ) + 2 s∧τ n 0 (v u , σ(u, x /[0,u] , v u )) dW u .
Let 0 u v T . With the same notation of lemma 4.4 and by (4.10),

γ|K p | v u v u (v p (s), dK p (s)) + μ v u v p (s) ds + γμ(v -u),
and then, when p tends to infinity,

- v u (v(s), dK(s)) μ v u v(s) ds + γμ(v -u).
(5.4) Lemma 3.3 gives the growth properties of b, ν and σ and then, by assumption 2.3, the growth property of α. Using this lemma, (5.4) with u = 0 and v = s ∧ τ n , the Cauchy-Schwarz inequality and writing μ v(s) (μ 2 + v(s) 2 )/2, we have

v s∧τ n 2 η 2 2 + C ϕ + C ϕ s∧τ n 0 v u 2 du + C ϕ s∧τ n 0 (1 + x ∞,u + ẋ ∞,u + v ∞,u ) 2 du + 2 s∧τ n 0 (v u , σ(u, x /[0,u] , v u )) dW u .
Taking the suprema and the expectation in the last inequality, using the assumptions 2.3 and 2.4, inequality ab ((1/ε)a 2 + εb 2 )/2, ε > 0, the Burkhölder-Gundy-Davis inequalities, (5.2) and (5.3)

E sup 0 s t v s∧τ n 2 C ϕ + C ϕ t 0 E sup 0 u s v u∧τ n 2 ds + C ϕ E 1/2 t 0 v s 2 (1 + v 2 ∞,s ) ds
and then

E sup 0 s t v s∧τ n 2 C ϕ + C ϕ t 0 E sup 0 u s v u∧τ n 2 ds + 1 2 E sup 0 s t v s∧τ n 2 ,
and the Gronwall lemma yields

E v t∧τ n 2 C ϕ and by (5.2), E ẋt∧τ n 2 C ϕ . Now, E( v t∧τ n 2 + ẋt∧τ n 2 ) = Ω ( v t∧τ n 2 + ẋt∧τ n 2 ) dP {τ n t} ( v t∧τ n 2 + ẋt∧τ n 2 ) dP n 2 P(τ n t)
and then

P(τ n t) C ϕ n 2 ,
which completes the proof.

Theorem 5.2. There exists a solution to the system

⎧ ⎪ ⎨ ⎪ ⎩ δx t = b(t, x /[0,t] , v t ) dt, δv t + Ã(t, x /[0,t] , v t ) dt ν(t, x /[0,t] , v t ) dt + σ(t, x /[0,t] , v t )δW t , 0 t < T, (x(0), v(0)) = (η 1 , η 2 ).
Proof . Since τ = T , the construction of (x, v) from (x n , v n ) and theorem 4.3 give the result.

By definition of b, Ã, ν and σ, if we put

τ U = inf{t < T, x(t) / ∈ U } ∧ T , (x t , v t ) 0 t<τ U verifies ⎧ ⎪ ⎨ ⎪ ⎩ δx t = b(t, x /[0,t] , v t ) dt, δv t + A(t, x /[0,t] , v t ) dt ν(t, x /[0,t] , v t ) dt + σ(t, x /[0,t] , v t )δW t , 0 t < τ U , (x(0), v(0)) = (η 1 , η 2 ).
Remark 5.3. We can remark that the process (x t , v t ) 0 t<τ U is F t -adapted.

(b) Uniqueness

We have the following proposition.

Proposition 5.4. Let (x, v) and (x , v ) be solutions to (4.1), then x = x almost surely and v = v almost surely.

Proof . Indeed, by Itô's formula,

v t -v t 2 = 2 t 0 (v s -v s , δv s -δv s ) = 2 t 0 (ν n (s, x /[0,s] , v s ) -ν n (s, x /[0,s] , v s ), v s -v s ) ds +2 t 0 σ n (s, x /[0,s] , v s ) -σ n (s, x /[0,s] , v s ) 2 ds +2 t 0 (σ n (s, x /[0,s] , v s ) -σ n (s, x /[0,s] , v s ), v s -v s )δW s + 2 t 0 (v s -v s , dK s -dK s ).
(5.5)

Using lemma 3.9, the inequality E x tx t 2

C ϕn E v tv t 2 and lemma 3.13, (5.5) becomes

E sup 0 s t v s -v s 2 C ϕn (t) t 0 E sup 0 u s v u -v u 2 ds,
and the Gronwall lemma yields the result.

(c) Independence on the chart

Let V 1 and V 2 be two open subsets of M such that V = V 1 ∩ V 2 = ∅
and let x 0 ∈ V . We denote by E the tangent space of M at the point x 0 . Let W be an open subset of M such that x 0 ∈ W ⊂ W ⊂ V and each pair of points of W can be linked by a unique geodesic of M . Let

ϕ 1 : V 1 ⊂ M → V 1 ⊂ E, ψ 1 = ϕ -1 1 , ϕ 2 : V 2 ⊂ M → V 2 ⊂ E, ψ 2 = ϕ -1
2 two local charts of M such that ϕ 1 (x 0 ) = ϕ 2 (x 0 ) = 0 and T x 0 ϕ 1 = T x 0 ϕ 2 = I E . We denote by b i , ν i , σ i and A i , 1 i 2 the maps associated to ϕ i . For all open subsets U of M verifying W ⊂ U ⊂ U ⊂ V and for all second-order random variables η on T M such that π(η) ∈ W , we put U = ϕ(U ) and denote by (x i , v i ), (1 i 2), the solution to

dx i t = b i (t, x i /[0,t] , v i t ) dt, δv i t + A i (t, x i /[0,t] , v i t ) dt ν i (t, x i /[0,t] , v i t ) dt + σ i (t, x i /[0,t] , v i t )δW t , (x i (0), v i (0)) = (ϕ i (η), τ (π(η)x 0 )η), ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ (5.6) until the exit time τ i U of U . Let τ U = τ 1 U ∧ τ 2 U .
To prove the independence on the chart, it suffices to show that x 2 t = ϕ 2 (ψ 1 (x 1 t )) and v 2 t = v 1 t for all t < τ U . To do this, by uniqueness of (5.6), we will prove that (ϕ 2 (ψ 1 (x 1 t )), v 1 t ) is a solution of (5.6) with i = 2. For t < τ U , let us define y t = ϕ 2 (ψ 1 (x 1 t )) and write ẏ

(t) = T ψ 1 (x 1 t ) ϕ 2 (T x 1 t ψ 1 ẋ1 t ) = (T ψ 2 (x 2 )(t) ϕ 2 τ (ψ 2 (x 2 /[0,t] ))τ (x 0 , π(η)))(τ (π(η), x 0 )(τ (ψ 1 (x 1 /[0,t] ))) -1 T x 1 t ψ 1 ẋ1 t ) = b 2 (t, x 2 /[0,t] )(b 1 (t, x 1 /[0,t] )) -1 ẋ1 t = b 2 (t, x 2 /[0,t] , v 1 t ).
To complete the proof, it remains to show that ν 2 (t, y /[0,t] ) = ν 1 (t, x 1 /[0,t] ) and similar equalities for σ 2 and A 2 . By definition of ν 2 , ν 1 , x 1 and x 2 , we can write, for 0 t < τ U and v ∈ E:

ν 2 (t, y /[0,t] , v) = τ (π(η), x 0 )(τ (ψ 2 (y /[0,t] ))) -1 ν (t, ψ 2 (y)(t), τ(ψ 2 (y)τ (x 0 ,π(η))v) = τ (π(η), x 0 )(τ (ψ 1 (x 1 /[0,t] ))) -1 ν (t, ψ 1 (x 1 )(t), τ(ψ 1 (x 1 /[0,t] )τ (x 0 ,π(η))v) = ν 1 (t, x 1 /[0,t] , v
) and the calculations are similar for σ 2 and A 2 .

In § 3, we have shown the existence and uniqueness in a local chart, and in this section we have proved that two local solutions starting with the same local initial data obtained thanks to several local charts define the same solution on the manifold.

Global existence on the manifold

We have the following result. Proposition 6.1. Let T be a strictly positive number, and let A , σ and ν satisfy assumptions 2.2-2.5. Let E = (Ω, F , (F t ) t 0 , P) be a filtered probability space verifying the usual conditions. Let B t be the standard R d Brownian motion on E , and let T 0 be an F t -stopping time with values in [0, T ] almost surely. Let η be an F T 0 -measurable random variable which takes its values in T M and whose second-order moment is finite. There exists a unique F t -adapted solution (x t , v t ) to the following system:

δx t = v t dt, D ẋ t v t + A (t, x t , v t ) dt ν (t, x t , v t ) dt + σ (t, x t , v t )δB t , T 0 t T, (x (T 0 ), v (T 0 )) = (π(η ), η ), a.s. ⎫ ⎪ ⎬ ⎪ ⎭ (6.1)
Proof . We keep the notation used until now. Under the assumption that π(η ) takes its values almost surely in W , we have obtained the existence of a solution (x t , v t ) T 0 t<τ U of (2.1) for all open subsets U of M containing the closed set W and which can be described by a unique chart of M . Moreover, (x S , v S ) is F S -measurable for all F t -stopping time S such that T 0 S T almost surely. We now suppose that η can take its values in all T M .

A consequence of assumption 2.1 is that M can be covered by an, at most, countable number of open sets W i such that (i) for each pair of points of W i there is a unique geodesic between them;

(ii) π -1 W i is trivializable; (iii) W i is included in one of the balls B M (z k , r ). Let W i now be a measurable partition of M such that W i is included in W i . We define Ω i = (πη ) -1 W i
and η i a random variable on W i which coincides with η on Ω i . Let (x i , v i ) be the solution to (2.1) with initial condition η i at time T 0 . Let K(i) be the set of indices k

P N P P T F s O F + R F r Figure 1
. The weight is P , its normal and tangential components are, respectively, P N and P T , F r is the friction force, F s is the stochastic force and F + R is the sum of the reaction of the constraints and the tension of the stem.

such that W i is included in B M (z k , r ). There is for each k ∈ K(i) an exit time σ k , which is the largest time for which x i/[T 0 ,σ k [ belongs to B M (z k , r), and we let σi = max{σ k , k ∈ K(i)}.

We define now a F t -stopping time

τ (T 0 , η ) = i 1 Ω i σi ; then (x , v ) is obtained by patching together the (x i , v i ): (x , v ) = i 1 Ω i (x i , v i ),
which is well defined for all stopping times S ∈ [T 0 , τ(T 0 , η )[. In particular (x S , v S ) is F S -measurable. We are going to estimate τ (T 0 , η ) from below: on {τ (T 0 , η ) < T }, x (τ (T 0 , η )) is almost surely at least at a distance rr from x (T 0 ). By the triangle inequality,

τ (T 0 ,η ) T 0 v (s) x (s) ds r -r . (6.2)
Denoting by τ 1 = τ (T 0 , η ), we obtained thus a solution (x t , v t ) T 0 t<τ 1 of (2.1) with initial condition η ∈ T M at time T 0 . We can apply the previous process with η = v (τ 1 ) and T 0 = τ 1 and define then a F t -stopping time τ 2 and so on. We construct thus a non-decreasing sequence (τ n ) n of stopping time, which tends to a stopping time τ . By similar calculations to those presented in lemma 5.

Application to the pendulum

We consider a spherical pendulum (see figure 1) consisting of a material point with unit mass attached to a rigid stem of length 1 and of negligible mass, which is connected at its other end to a fixed point O by a connection authorizing only angular displacements. In our units, the gravity is 1. In addition to its weight, the pendulum is submitted, at its mobile end, to a dry friction of Coulomb type (with coefficient μ) and a random force acting in the tangent plane of the sphere. We will now define precisely our model of stochastic and dry friction forces. The initial conditions are (x 0 , v 0 ) ∈ T S 2 . In the present case, the Riemannian metric is the metric induced by the Euclidean metric of R 3 and, therefore, we will abuse notation and denote the scalar product in the tangent space at x to S 2 without an index x, the same convention holding for the norm of tangent vectors.

Let H 1 and H 2 be two sections of T S 2 which satisfy the following conditions. Observe that (7.2) implies the continuity of H 1 and H 2 ; therefore, according to the theorem of the hairy sphere, each of the fields H 1 and H 2 has to vanish at some point of S 2 .

Let (W 1 , W 2 ) be a two-dimensional Brownian motion; we formally write the stochastic part of the exterior forces as F s (t, x) dt = dW 1 (t)H 1 (x) + dW 2 (t)H 2 (x).

Thanks to assumptions (7.1) and (7.2), the linear mapping

σ (x)(v 1 , v 2 ) = v 1 H 1 (x) + v 2 H 2 (x)
satisfies assumptions 2.3 and 2.4 of § 2.

If t → x(t) is a motion on the sphere S 2 , we differentiate twice the relation x(t) 2 = 1 and we find x • ẍ = -| ẋ| 2 , which means that the normal component of the acceleration is a N (x, v) =v 2 x.

The normal component of the weight at x is P N (x) = p n (x)x, with p n (x) equal to x 3 . Therefore, the sum of the reaction of the support and of the tension of the stem is R(x, v) + F (x, v) = x(x 3 + v 2 ).

The friction force is then given by the multivalued relation

F r (x, v) ∈ ⎧ ⎪ ⎨ ⎪ ⎩ {0} if p n (x) -v 2 , -μ{(p n (x) + v 2 )v/ v } if p n (x) > -v 2 and v = 0, μ{u ∈ T x S 2 , u
p n (x), } if p n (x) > 0 and v = 0.

(7.3)

Let us check now that -F r is a monotone operator in the fibre, i.e. for all w ∈ -F r (x, v) and w ∈ -F r (x, v ), the following inequality holds:

(ww , vv ) 0.

(7.4)

If v and v are both in case 1 or 3 of (7.3), it is plain that (7.4) holds. If w is in case 2 or 3, and w is in case 1, the conclusion is still clear; therefore, we just have to check the inequality when v and v are in case 2, or when v is in case 2 and v is in case 3. If neither v nor v vanish and p n (x) >min( v 2 , v 2 ), then

(w -w , v -v ) = p n (x) v v - v v , v -v + ( v v -v v , v -v ).
An elementary calculation shows that the above expression is non-negative. If v is in the second case and v in the third, we have to study the expression

(p n (x) + v 2 ) v v -u, v ,
where u p n (x). It is once again straightforward to verify that this expression is non-negative.

If we put ν (x, v) = P T (x) = -x + x 3 x, which is the tangential component of the weight, the equation of the dynamic applied to the pendulum takes the form: dx t = v t dt, D ẋ(t) v t + A (x t , v t ) dt ν (x t , v t ) dt + σ (x t ) dW t , 0 t < T, (x(0), v(0)) = (x 0 , v 0 ).

⎫ ⎪ ⎬

⎪ ⎭ (7.5) Remark 7.2. Observe that we could have considered more general forces than the weight, which can be easily included in this theory.

Thanks to remark 2.8 and, since assumptions 2.3-2.5 are clearly verified, we then have the existence and uniqueness of solutions to equation (7.5).

Conclusion

In this work a new result on stochastic differential equations is given: the existence and uniqueness of solutions to MSDEs on Riemannian manifolds are obtained. This work generalizes the results of [START_REF] Cépa | Problème de Skorohod multivoque[END_REF] devoted to the Euclidean case. To simplify the expression of Stratonovich's covariant differential in the equation relative to velocities, that is, to eliminate the quadratic terms, which offers the possibility to approximate equations by classical (with Lipschitz continuous assumptions) MSDE on Euclidean space, we used the parallel transport along the solution curve, but we had to pay by losing Lipschitz continuity of b. Then we had to approximate b by Lipschitz continuous maps b n and this allowed us to construct the expected solution. From the application viewpoint the result obtained in this article finds its interest in the modelling of mechanical systems submitted to stochastic forcing and friction; the example of the stochastic spherical pendulum with friction fits into this framework.

To approximate the solution it is possible to adapt a numerical scheme developed for classical stochastic differential equations with convex constraints in [START_REF] Pettersson | Projection scheme for stochastic differential equations with convex constraints[END_REF] and for general MSDE in [START_REF] Bernardin | Multivaled stochastic differential equations: convergence of a numerical scheme[END_REF]. Numerical simulation of this class of models is the topic of future work.

  d and we denote by d M the Riemannian metric on M and by B M (x, ε) (respectively BM (x, ε)) the open ball {y ∈ M, d M (x, y) < ε} (respectively the closed ball {y ∈ M, d M (x, y) ε}).

  (i) There exists a constant C > 0 such that for all x ∈ E and y ∈ E,α(x) + β(x) C(1 + x ) a.s., α(x)α(y) + β(x)β(y) C xy a.s.(ii) There exists a constant R > 0 and M > 0 such that ∀x ∈ E, x R =⇒ (∀y ∈ B(x), y M ) a.s. (iii) The maps α and β are, respectively, G 0 ⊗ B(E), B(E)-measurable and G 0 ⊗ B(E), B(E d )-measurable, where B(O) denotes the Borel σ-algebra of the topological set O.

  1, we can show that, for t < T and for all n, Moreover, by (6.2) applied on each [τ i , τ i+1 [, we have for all stopping time S ∈ [T 0 , T [,

		E sup 0 s t	v s∧τ n	2 x (s)	C.
	S
	C E	v (s) x (s) ds	n(r -r ) dP,
	T 0				{τ n S}
	and then
		lim n→+∞	P({τ n S}) = 0,
	which proves that τ = T .

Assumption 7.1.

∀x ∈ S 2 , H 1 (x) x + H 2 (x) x C, (7.1) ∀x, y ∈ S 2

in the same open hemisphere

H 1 (x)τ (y, x)H 1 (y) x + H 2 (x)τ (y, x)H 2 (y) x Cd S 2 (x, y). (7.2)

Proc. R. Soc. Lond. A (2004)

The authors express their thanks to Professor J. Picard and Professor A. Lachal for their suggestions and encouragement.


Frédéric Mazenc
email: frederic.mazenc@l2s.centralesupelec.fr

Michael Malisoff
email: malisoff@lsu.edu.

New Prediction Approach for Stabilizing Time-Varying Systems under Time-Varying Input Delay

Keywords: Delay, robustness, stability, time-varying Mazenc

We provide a new sequential predictors approach for the exponential stabilization of linear time-varying systems with pointwise time-varying input delays. Our method circumvents the problem of constructing and estimating distributed terms in the stabilizing control laws, and allows arbitrarily large input delay bounds. We illustrate our method using a pendulum dynamics.

I. INTRODUCTION

This work continues our search (which we began in [START_REF] Mazenc | Stabilization of nonlinear time-varying systems through a new prediction based approach[END_REF]) for predictive methods for time-varying systems that can be applied without computing Lie derivatives and without computing distributed terms, and which compensate for arbitrarily long input delays. Our work is motivated by the ubiquity of input delays across engineering, and the challenges that can occur when building delay tolerant controls, if one applies traditional methods that have distributed terms. See, e.g., [START_REF] Bekiaris-Liberis | Nonlinear Control Under Nonconstant Delays[END_REF] and [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF], and [START_REF] Kamalapurkar | Time-varying input and state delay compensation for uncertain nonlinear systems[END_REF], [START_REF] Merad | Isometric torque control for neuromuscular electrical stimulation with time-varying input delay[END_REF], [START_REF] Sharma | Predictor-based compensation for electromechanical delay during neuromuscular electrical stimulation[END_REF] for electromechanical input delays in muscle response in neuromuscular electrical stimulation (or NMES). For constant coefficient linear systems, it often suffices to use linear matrix inequalities (or LMIs) to build delay tolerant controls, but many important linear systems are time-varying. For instance, when tracking reference trajectories under input delays and linearizing around the reference trajectories, one obtains time-varying linear systems, even if the original system is time invariant.

Traditional input delay compensation methods can roughly be grouped into three categories of approaches. One approach is to solve the control problem with the input delay set to 0, and to then look for upper bounds on the input delay that the resulting closed loop system can tolerate without sacrificing the desired stability properties. Two advantages of this socalled emulation approach are that (a) it makes it possible to use relatively simple controls for undelayed systems (such as Lie derivative feedback design methods and other approaches from [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF]) and (b) the strict Lyapunov functions one often obtains from solving the feedback design problem for the corresponding undelayed system can often be transformed into Lyapunov-Krasovskii functionals, which can in turn be used to compute bounds on the input delays that the system can tolerate. See [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF] for ways to transform strict Lyapunov functions for undelayed systems into Lyapunov-Krasovskii functionals for the corresponding input delayed systems.

Another approach is the reduction model method, where the control is expressed implicitly as a solution of an integral equation, which can lead to challenging problems of numerically computing control values, especially when the system has uncertainty or the delay is time-varying. Prediction is another useful method, where the state values in the controls are replaced by numerical predictions of the state values. Standard prediction or reduction model methods can compensate for arbitrarily long input delays, but the distributed terms in their controls may make them harder to implement than emulation; see [START_REF] Artstein | Linear systems with delayed controls: a reduction[END_REF], [START_REF] Bekiaris-Liberis | Compensation of time-varying input and state delays for nonlinear systems[END_REF], [START_REF] Mazenc | Lyapunov-Krasovskii functionals and application to input delay compensation for linear time-invariant systems[END_REF], and [START_REF] Witrant | Remote stabilization via communication networks with a distributed control law[END_REF].

This note provides a new sequential predictors approach for the exponential stabilization of time-varying systems with time-varying pointwise input delays, and so builds on recent significant works such as [START_REF] Najafi | Closed-loop control of dead time systems via sequential subpredictors[END_REF] (which used LMI methods for time invariant linear systems to build sequential predictors) and [START_REF] Léchappé | Predictive Control and Estimation of Uncertain Systems with an Input Delay[END_REF] (which extended [START_REF] Najafi | Closed-loop control of dead time systems via sequential subpredictors[END_REF] by studying constant coefficient linear systems with time-varying delays, which is a smaller class of systems than the time-varying systems we consider here). It also extends our work [START_REF] Mazenc | Stabilization of nonlinear time-varying systems through a new prediction based approach[END_REF], which was confined to constant delays. Since we do not use distributed terms or Lie derivatives in our control, our work is very different from the classical reduction model or prediction approaches that have been used by M. Krstic and others (as in [START_REF] Bresch-Pietri | Robust compensation of a chattering time-varying input delay[END_REF] and [START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF]).

We use several dynamic extensions, with each one having the same dimension as the original system. Since we do not use distributed terms, our work differs from [START_REF] Ahmed-Ali | Global exponential sampled-data observers for nonlinear systems with delayed measurements[END_REF] and other works that use several dynamic extensions and distributed terms. Our result gives closed form expressions for the control and explicit lower bounds on the number of required dynamic extensions in terms of the time-varying delays and their derivatives. Our work is mainly a theoretical and methodological development. However, we illustrate our work using a pendulum dynamics that was also studied in [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF], where we used a predictive approach that led to distributed controls, but [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF] did not cover time-varying delays.

We use standard notation and definitions. Throughout the sequel, the dimensions are arbitrary, unless otherwise noted. For simplicity, we omit arguments of functions when they are clear from the context, and we always assume that the initial times t 0 for our solutions of our time-varying systems are t 0 = 0, but we can write analogs of our results for general choices of t 0 ≥ 0. We use | • | to denote the usual Euclidean norm and the induced matrix norm, and |φ | ∞ (resp., |φ | I ) denotes the essential supremum (resp., supremum over any interval I ) for bounded measurable functions φ .

II. MAIN RESULT

We study systems of the form

ẋ(t) = A(t)x(t) + B(t)u(t -h(t)), (1)
where the state x and the control u are valued in R n and R , respectively, and h is a time-varying delay. Throughout this section, we make two assumptions:

Assumption 1: The function h : R → [0, ∞) is C 1 and bounded from above by a constant c h > 0. Also, its derivative ḣ is bounded from below, and ḣ is bounded from above by a constant l h ∈ (0, 1), and ḣ has a global Lipschitz constant n h > 0.

Assumption 2: The functions A and B are bounded and continuous, and there is a known bounded continuous function

K : [0, ∞) → R ×n such that ẋ(t) = [A(t) + B(t)K(t)]x(t) (2)
is exponentially stable to 0. We also assume that the initial functions are constant on [-c h , 0]. In terms of the known delay h(t) and an integer m > 1 that we specify later, we then introduce the functions

Ω i (t) = t -i m h(t) and θ i (t) = Ω -1 m-i+1 (Ω m-i (t)) (3)
for all i ∈ {0,...,m}, and we inductively define the functions 4) Such functions exist by Assumption 1, which implies that the Ω i 's are strictly increasing and that their ranges are all of R. The required inverses in (3) can be computed in practice using standard programs. In terms of the positive constants

R 1 (t) = θ1 (t) and R i (t) = θi (t)R i-1 (θ i (t)) for i > 1. (
u c = n h c h (1-l h ) 2 + l h 1-l h (5) and b 1 = 1 + 1 + u c m m |A| ∞ 1 + u c m m |A| ∞ , b 2 = 1 + 1 + u c m m |A| ∞ 2 , and b 3 = b 2 1 + u c m , (6)
and with I n ∈ R n×n denoting the identity matrix, we then prove:

Theorem 1: If Assumptions 1-2 hold and m ∈ N satisfies max 2, 4 b 1 √ 2 + b 3 c h 1-l h < m, (7)
then we can find positive constants µ 1 and µ 2 such that for all solutions of (1) in closed loop with the control

u(t) = K(t + h(t))z m (t), (8)
where z m is the last n components of the system

ż1 (t) = R 1 (t)A(θ 1 (t))z 1 (t) + R 1 (t)B(θ 1 (t))u(Ω m-1 (t)) + L 1 (t)[z 1 (θ -1 1 (t)) -x(t)] żi (t) = R i (t)A(G i (t))z i (t) + R i (t)B(G i (t))u(Ω m-i (t)) + L i (t)[z i (θ -1 i (t)) -z i-1 (t)], i ∈ {2,. .., m} (9)
with the choices

L i (t) = -I n -R i (t)A(G i (t)) and G i = Ω -1 m • Ω m-i , (10) we have |(x(t), E (t))| ≤ µ 1 |(x, E )| [-c h ,0] e -µ 2 t ( 11
)
for all t ≥ 0, where the vector E is defined by

E (t) = (z 1 (t)- x(θ 1 (t)), z 2 (t) -z 1 (θ 2 (t)),. .., z m (t) -z m-1 (θ m (t))).
Remark 1: The estimate [START_REF] Mazenc | Stabilization of nonlinear time-varying systems through a new prediction based approach[END_REF] means that the combined (x, E ) dynamics are uniformly globally exponentially stable to 0. Three key features of the preceding construction are (a) our allowing arbitrarily large delay bounds c h > 0, (b) the fact that our control [START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF] has no distributed terms and no Lie derivatives, and (c) the fact that only the bottom n components of the dynamic extension ( 9) are needed to compute the control values [START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF].

III. SKETCH OF PROOF OF THEOREM 1

We use this lemma, which we prove in the appendix:

Lemma 1: The θ i 's in (3) and the constant u c in (5) satisfy | θi (t) -1| ≤ u c m and |θ -1 i (t) -t| ≤ c h 1-l h 1 m (12)
for all t ≥ 0 and i ∈ {1, 2,. .., m}.

By Lemma 1 and our definition of the R i 's in (4), we get

|R i (t)| ≤ 1 + u c m m ( 13
)
for all i, where throughout the proof, all inequalities and inequalities are for all t ≥ 0, unless otherwise indicated. In terms of the states z i of (9), we define ξ i (t) = z i-1 (θ i (t)) for i ≥ 1, where z 0 = x. The rest of the proof has four parts. Part 1: Error Dynamics. We derive a useful formula for the dynamics of the error E = (E 1 ,. .., E m ), where

E i (t) = z i (t) -ξ i (t) (14)
for all i ∈ {1,...,m}, so

E i (t) = z i (t) -z i-1 (θ i (t)
) for all i ∈ {1,. .., m}. Our formulas (3) for the Ω i 's and θ i 's give ξ1 [START_REF] Mazenc | Lyapunov-Krasovskii functionals and application to input delay compensation for linear time-invariant systems[END_REF] and our choice of the z 1 dynamics in [START_REF] Léchappé | Predictive Control and Estimation of Uncertain Systems with an Input Delay[END_REF] give

(t) = R 1 (t)A(θ 1 (t))ξ 1 (t) +R 1 (t)B(θ 1 (t))u (Ω m-1 (t)) . Hence,
Ė1 (t) = R 1 (t)A(θ 1 (t))E 1 (t) + L 1 (t)E 1 (θ -1 1 (t)). (15
) For i > 1, we can use [START_REF] Malisoff | Robustness of adaptive control under time delays for three-dimensional curve tracking[END_REF]

to get G i-1 • θ i = G i , so (4) gives ξi (t) = θi (t) {R i-1 (θ i (t)) [A(G i (t))z i-1 (θ i (t)) + B(G i (t))u(Ω m-i+1 (θ i (t)))] +L i-1 (θ i (t))[z i-1 (θ -1 i-1 (θ i (t))) -z i-2 (θ i (t))]} = R i (t)B(G i (t))u(Ω m-i+1 (θ i (t))) + θi (t)L i-1 (θ i (t))[z i-1 (θ -1 i-1 (θ i (t))) -z i-2 (θ i (t))] + R i (t)A(G i (t))ξ i (t). (16) Since Ω m-i+1 (θ i (t)) = Ω m-i (t), we deduce that for i ≥ 2, Ėi (t) = L i (t)[z i (θ -1 i (t)) -z i-1 (t)] -θi (t)L i-1 (θ i (t))[z i-1 (θ -1 i-1 (θ i (t))) -z i-2 (θ i (t))] + R i (t)A(G i (t))E i (t) = R i (t)A(G i (t))E i (t) + L i (t)E i (θ -1 i (t)) -θi (t)L i-1 (θ i (t))E i-1 (θ -1 i-1 (θ i (t))) . (17)
To summarize, we have the following error dynamics:

         Ė1 (t) = R 1 (t)A(G 1 (t))E 1 (t) + L 1 (t)E 1 (θ -1 1 (t)) Ėi (t) = R i (t)A(G i (t))E i (t) + L i (t)E i (θ -1 i (t)) -θi (t)L i-1 (θ i (t))E i-1 (θ -1 i-1 (θ i (t))) for i = 2,. .., m (18)
Part 2: Fundamental System. Motivated by the triangular structure of [START_REF] Sharma | Predictor-based compensation for electromechanical delay during neuromuscular electrical stimulation[END_REF], we next study stability properties of the fundamental system ṡi

(t) = R i (t)A(G i (t))s i (t) + L i (t)s i (θ -1 i (t)) = -s i (t) + [I n + R i (t)A(G i (t))] s i (t) -s i (θ -1 i (t)) (19)
for a fixed i. Observe that θ -1 i (t) ≤ t for all t ≥ 0, by our choices (3) of the Ω i 's. Since θ -1 i (t) ≥ 0 for all t ≥ c h , it follows that

ṡi (t) = -s i (t) -L i (t) t θ -1 i (t) R i (l)A(G i (l))s i (l)dl + L i (t) t θ -1 i (t) (I n + R i (l)A(G i (l)))s i (θ -1 i (l))dl (20)
for all t ≥ c h . We will use the function

Q(s i ) = |s i | 2 /2.
By our choices (6) of the constants b 1 and b 2 , our first bound in [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF] from Lemma 1, and ( 13), it follows that the time derivative of Q along all trajectories of (20) satisfies

Q(t) ≤ -|s i (t)| 2 + b 1 |s i (t)| t t- c h 1-l h 1 m |s i (l)|dl + b 2 |s i (t)| t t- c h 1-l h 1 m |s i (θ -1 i (l))|dl ≤ -|s i (t)| 2 + b 1 |s i (t)| t t- c h 1-l h 1 m |s i (l)|dl + b 2 |s i (t)| θ -1 i (t) θ -1 i t- c h 1-l h 1 m |s i (r)|| θi (r)|dr. (21)
Here and in what follows, all inequalities and equalities are for all t ≥ max{c h , 2c h /(m(1l h ))}, unless otherwise indicated. Hence, since θ -1 i (t) ≤ t for all t ≥ 0, [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF] gives

Q(t) ≤ -|s i (t)| 2 + b 1 |s i (t)| t t- c h 1-l h 1 m |s i (l)|dl + b 3 |s i (t)| t t- 2c h 1-l h 1 m |s i (r)|dr. (22)
Since [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF] gives

2c h m(1-l h ) < 1 b 1 / √ 2+b 3 , (23)
we can use Young's inequality twice (i.e., s i (t)r ≤ 1 4 |s i (t)| 2 + r 2 for suitable r ≥ 0) and then Jensen's inequality (applied to the convex integrand function M (s) = |s| 2 ) to get

Q(t) ≤ -1 2 |s i (t)| 2 + (b 1 / √ 2) 2 b 1 / √ 2+b 3 t t- c h 1-l h 1 m |s i (l)| 2 dl + b 2 3 b 1 / √ 2+b 3 t t- 2c h 1-l h 1 m |s i (r)| 2 dr ≤ -1 2 |s i (t)| 2 + b 1 √ 2 + b 3 t t- 2c h 1-l h 1 m |s i (r)| 2 dr, (24)
where the first application of Young's and Jensen's inequalities that we used to obtain the first inequality in (24) was

b 1 |s i (t)| t t- c h 1-l h 1 m |s i (l)|dl ≤ 1 4 |s i (t)| 2 + b 1 t t- c h 1-l h 1 m |s i (l)|dl 2 ≤ 1 4 |s i (t)| 2 + b 2 1 c h 1-l h 1 m t t- c h 1-l h 1 m |s i (l)| 2 dl ≤ 1 4 |s i (t)| 2 + (b 1 / √ 2) 2 b 1 / √ 2+b 3 t t- c h 1-l h 1 m |s i (l)| 2 dl, (25)
and the second one was analogous, and then we used the fact that (r 2 + s 2 )/(r + s) ≤ r + s for all r ≥ 0 and s > 0, with the choices r = b 1 / √ 2 and s = b 3 .

Since our lower bound on m in ( 7) also allows us to find a constant λ > 1 that is close enough to 1 so that

4 b 1 √ 2 + b 3 λ c h 1-l h < m, (26)
it follows from (24) that we can also find a constant c 0 > 0 such that the time derivative of

Q (s i,t ) = Q(s i (t)) + λ b 1 √ 2 + b 3 t t- 2c h 1-l h 1 m t w |s i (r)| 2 drdw
along all solutions of the fundamental system [START_REF] Witrant | Remote stabilization via communication networks with a distributed control law[END_REF] satisfies

Q ≤ -1 2 |s i (t)| 2 + b 1 √ 2 + b 3 2c h 1-l h λ m |s i (t)| 2 +(1 -λ ) b 1 √ 2 + b 3 t t- 2c h m(1-l h ) |s i (r)| 2 dr ≤ -c 0 Q (s i,t ), (27)
where the second inequality in (27) used the bound

t t- 2c h 1-l h 1 m t w |s i (r)| 2 drdw ≤ 2c h (1-l h )m t t- 2c h 1-l h 1 m |s i (r)| 2 dr
and s i,t () = s i (t + ) for all ∈ [-2c h /((1l h )m), 0]. This concludes our analysis of the fundamental system [START_REF] Witrant | Remote stabilization via communication networks with a distributed control law[END_REF].

Part 3: Exponential Stability of the Error Dynamics (18) This part entails finding positive constants ω i > 0 and η i > 0 such that along all solutions of the dynamics for

E (t) = E 1 (θ -1 1 • θ m (t)), E 2 (θ -1 2 • θ m (t)),. .., E m (t) , the function Q m (E t ) = ω 1 Q (E 1,t ) +.. .+ ω m-1 Q (E m-1,t ) + ω m Q (E m,t ) satisfies Q m (E t ) ≤ -η 1 Q m (E t ) . ( 28
)
This can be done by an inductive argument, using our lower and upper bounds on ḣ from Assumption 1. For instance, we can use the triangle inequality to prove that along all solutions for the (E 1 (t), E 2 (t)) dynamics, we have

d dt Q (E 2,t ) ≤ -θ c 0 2 Q (E 2,t ) + 2 θ 2 θ c 0 | θ2 | 2 ∞ |L 1 | 2 ∞ Q (E 1,t ) d dt Q (E 1,t ) ≤ -θ c 0 Q (E 1,t ), (29)
where the positive constants θ and θ are chosen such that θ ≤ (θ -1 i • θ m ) (t) ≤ θ for all t ≥ 0 and all i ∈ {1, 2,. .., m}. Hence, the function

Q 2 (E 1,t , E 2,t ) = Q (E 2,t ) + 1 + 2 θ | θ2 | ∞ |L 1 | ∞ θ c 0 2 Q (E 1,t ) satisfies Q 2 ≤ -θ c 0 2 Q E 1,t + Q E 2,t , ( 30
)
and so is a Lyapunov-Krasovskii functional for the (E 1 (t), E 2 (t)) dynamics, and this process can be continued. Part 4: Exponential Stability of (1) in Closed Loop with [START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF]. By our error variable formulas ( 14) and E i (t) = z i (t)z i-1 (θ i (t)) for all i ∈ {1,. .., m}, we have

z m (t) = ξ m (t) + E m (t) = z m-1 (θ m (t)) + E m (t) = z m-2 (θ m-1 (θ m (t))) + E m-1 (θ m (t)) + E m (t), (31)
and therefore we can argue inductively to get

z m (t) = x(θ 1 (...θ m-1 (θ m (t))...)) + E 1 (θ 2 (...θ m (t)...)) + ... + E m (t). ( 32
)
Then our formulas for the θ i 's give z m (t) = x(Ω -1 m (t)) + E 1 (θ 2 (...θ m (t)...)) + ... + E m (t). (33) Hence, our control (8) can be written as

u(t -h(t)) = K(t)[x(Ω -1 m (t -h(t))) + E 1 (θ 2 (...θ m (t -h(t)...))) + ... + E m (t -h(t))]. (34)
Since Ω m (t) = th(t), the closed loop system (1) is then

ẋ(t) = (A(t) + B(t)K(t))x(t) +{B(t)[K(t)E 1 (θ 2 (...θ m (t -h(t)...))) +... + K(t)E m (t -h(t))]}. (35)
Also, Assumption 2 provides a C 1 bounded function P such that the Lyapunov function V (t, x) = x P(t)x has a quadratic lower bound in |x| and satisfies V ≤ -|x(t)| 2 along all solutions of (2) (using [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF]Theorem 4.14]). Then the triangle inequality gives

V ≤ -1 2 |x(t)| 2 + 2|P| 2 ∞ |δ | 2 ∞ ≤ -1 2|P| ∞ V (t, x(t)) + 2|P| 2 ∞ |δ | 2 ∞ ≤ -1 2|P| ∞ V (t, x(t)) + 2m (|P| ∞ |B| ∞ |K| ∞ ) 2 × |E 1 (θ 2 (...θ m (t -h(t)...)))| 2 + .. . + |E m (t -h(t))| 2 (36)
along all solutions of (35), where δ is the quantity in curly braces in (35). One then can show that the function

Q + E t = V (t, x (t)
)

+ 1 + 2m M η 1 (|P| ∞ |B| ∞ |K| ∞ ) 2 m ∑ i=1 2 ω i Q m E M i (t) (37)
admits a constant d 0 > 0 such that the Lyapunov-Krasovskii

decay condition Q + (E t ) ≤ -d 0 Q + (E t ) (38)
holds along all solutions of the dynamics for

E (t) = x(t), E M 1 (t) ,. .., E M m (t) , (39)
where

M i (t) = θ -1 m • θ i • θ i+1 • .. . • θ m (t -h(t)) = Ω 1 • Ω -1 m-i+1 •Ω m (t) for i = 1, 2,. .., m-1 and M m (t) = t -h(t) = Ω m (t)
, by choosing positive constants M and M such that M ≤ Ṁi (t) ≤ M for all i ∈ {1, 2,. .., m} and t ≥ 0. The final uniform global exponential stability estimate for the (x, E ) dynamics follows from the quadratic structure of V and Q, the invertibility of the functions

H i = θ -1 i • θ m • M i
for all i, the existence of positive constants h a and h b such that H -1 i (t) ≥ h a th b for all i and t ≥ 0 (which follows from our bounds on ḣ from Assumption 1 and the Mean Value Theorem), and the subadditivity of the square root.

Remark 2: Since Ω m (t) = t -h(t), our control is u(t -h(t)) = K(t)x(t)+ K(t)[E 1 (θ 2 (...θ m (t h(t)...))) + ... + E m (t -h(t))], (40)
which may at first seem to contradict our assumption that the current value of the state x(t) is not available to use in the control. However, our formulas (3) for the θ i 's imply that

θ 2 • .. . • θ m (t -h(t)) = θ -1 1 (t) for all t ≥ 0. This gives E 1 (θ 2 (...θ m (t -h(t)...))) = z 1 (θ -1 1 (t)) -ξ 1 (θ -1 1 (t)) = z 1 (θ -1 1 (t)) -x(t), (41)
so two terms in (40) containing x(t) cancel. Hence, our control does not require current values x(t).

IV. APPLICATION TO PENDULUM

Consider the model ṙ1 (t) = r 2 (t)

ṙ2 (t) = -g l sin(r 1 (t)) + 1 ml 2 v(t -h(t)) (42)
of the simple pendulum with a time-varying delay h(t), where g = 9.8 m/s is the gravity constant, l is the pendulum length in meters, m is the pendulum mass, and v is the input. As in [START_REF] Mazenc | Stabilization of nonlinear time-varying systems through a new prediction based approach[END_REF] and [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF], our control objective is to track a given C 1 reference trajectory (r 1,s (t), r 2,s (t)) such that ṙ1,s (t) = r 2,s (t). Using the error variables ri = r ir i,s (t) for i = 1, 2 and the change of feedback

u(t -h(t)) = 1 ml 2 v(t -h(t)) -ṙ2,s (t) -g l sin(r 1,s (t)), (43)
we obtain the tracking system

     ṙ1 (t) = r2 (t) ṙ2 (t) = g l [sin(r 1,s (t)) -sin(r 1 (t) + r 1,s (t))] + u(t -h(t)). (44)
The work [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF] showed that when r 1,s (t) = ωt and ω > 0 is a large enough constant and h = 1, the linearization ẋ1 (t) = x 2 (t)

ẋ2 (t) = -g l cos(ωt)x 1 (t) + u(t -h) (45)
of ( 44) at 0 has the globally exponentially stabilizing distributed control u defined by u(t) = -0.6x 1 (t) -0.4x 2 (t) -t t-1 0.6(tr -1) + 0.4 u(r)dr.

(46)

Our work [START_REF] Mazenc | Stabilization of nonlinear time-varying systems through a new prediction based approach[END_REF] designed a globally asymptotically stabilizing sequential predictor control for the original nonlinear system (44) for cases where h is constant. However, it is not obvious how to extend the analysis from [START_REF] Mazenc | Stabilization of nonlinear time-varying systems through a new prediction based approach[END_REF] to time-varying delays. Therefore, we apply Theorem 1 to its linearization (45) using the time-varying delay h(t) = 1 + α sin(t), where α ∈ (0, 1) is a known constant, but analogous reasoning applies for any delay h(t) satisfying Assumption 1 above. To this end, first note that for any choice of the constant ω > 0, Assumption 2 is satisfied using

A(t) =   0 1 -g l cos(ωt) 0   , B =   0 1   , and
K(t) = g l cos(ωt) -1 -1 (47)
and then |A| ∞ = max{g/l, 1}. Therefore, our requirement (7) on m ≥ 2 from Theorem 1 holds if

m 4 -1+α 1-α max{g/l,1} √ 2 1+max g l , 1 1+ u c m m 1+ u c m m + 1 + max g l , 1 1 + u c m m 2 1 + u c m > 0. (48)
Condition (48) can easily be checked for suitable choices of α, and l. For instance, if we pick α = 1/7, any ω > 0, any l > g, and the choice m = 47, then we get the value 0.376057 for the left side of (48), so Theorem 1 applies with m = 47.

V. CONCLUSIONS AND FUTURE WORK

We designed uniformly globally exponentially stabilizing sequential predictor controllers for a large class of timevarying linear systems under general conditions on the timevarying input delays. By allowing arbitrarily large bounds on the input delays, we overcame a possible challenge in applying emulation methods to systems that can have long input delays. We provided a formula for finding the required numbers of sequential predictors that are needed to realize our stability objectives.

We hope to extend our work to systems with uncertainties, and to adaptive systems, where dynamic extensions identify unknown parameters. We also hope to cover nonlinear systems with time-varying delays, by building sequential predictors for the linear parts, and proving that the controls ensure local stabilization of the original nonlinear systems. Finally, we plan to merge our methods with the work [START_REF] Malisoff | Robustness of adaptive control under time delays for three-dimensional curve tracking[END_REF], by proving robust forward invariance under state constraints and time-varying delays.

APPENDIX: PROOF OF LEMMA 1

Fix any i ∈ {1, 2,. .., m} and t ≥ 0. Our formula θ i (t) = Ω -1 m-i+1 (Ω m-i (t)) from (3) gives

θ i (t) = t + Ω -1 m-i+1 Ω m-i+1 (t) + 1 m h(t)
-Ω -1 m-i+1 (Ω m-i+1 (t)) .

(A.1)

Hence, the Mean Value Theorem provides a w ∈ R (depending on t and i in general) such that θ i (t) = t + which gives the second inequality in [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF], by the invertibility of θ i .

To check the first inequality in [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF], first note that θi (t) =

1-m-i m ḣ(t) 1-m-i+1 m ḣ(Ω -1 m-i+1 (Ω m-i (t))) = 1-m-i+1 m ḣ(t) 1-m-i+1 m ḣ(Ω -1 m-i+1 (Ω m-i (t))) + 1 m ḣ(t) 1-m-i+1 m ḣ(Ω -1 m-i+1 (Ω m-i (t)))
.

(A.4)

Since Ω m-i (t) = Ω m-i+1 (t) + 1 m h(t), the function The lemma now follows from our formula (5) for u c .

G i (t) = Ω -1 m-i+1 Ω m-i+1 (t) + 1 m h(t) (A.5) is such that θi (t) = 1-m-i+1 m ḣ(t) 1-m-i+1 m ḣ(G i (t))

( 1 -

 1 (t) in the first numerator in (A.6) as ḣ(t) = ( ḣ(t) -ḣ(G i (t)) + ḣ(G i (t)).It follows that| θi (t) -1| ≤ m-i+1 m n h |Gi(t)-Ω -1 m-i+1 (Ω m-i+1 (t))| 1-constantn h for ḣ. Hence, our choice of G i and the Mean Value Theorem applied to Ω -1 m-i+1 gives | θi (t) -1| ≤ m-i+1 m n h |Ωm-i+1(t)+ 1 m h(t)-Ω m-i+1 (t)|

Supported by NSF-ECCS Grant 1408295, and by the Roy Paul Daniels Professorship #3 in the LSU College of Science.


Stellarium software and the occultation of Aldebaran observed by Copernicus Amelia Carolina Sparavigna

Politecnico di Torino Here we propose an example of the use of Stellarium software for simulating a phenomenon of 1497, the lunar occultation of Aldebaran observed by Nicolaus Copernicus. We will also use CalSKY for the phase of the moon.

Stellarium is a free open source planetarium which possesses a remarkable visual rendering of the sky and of the horizon. It is easy to use, in particular in choosing time and location of observations. We used it in some previous papers to simulate the sky seen by ancient stargazers [START_REF] Sparavigna | The Stellarium Planetarium for the Simulation of the Astronomical Landscapes of Ancient Stargazers[END_REF][START_REF] Sparavigna | A Possible Role of Alpha Crucis in the Astronomical Landscape of Silbury Hill[END_REF][START_REF] Sparavigna | Maria Reiche's Line to Archaeoastronomy[END_REF][START_REF] Sparavigna | The Nazca Frigatebird and Fomalhaut[END_REF]: simulations were quite useful in understanding the links of the ancient anthropized environment with the night sky and stars. The examples proposed in [START_REF] Sparavigna | The Stellarium Planetarium for the Simulation of the Astronomical Landscapes of Ancient Stargazers[END_REF][START_REF] Sparavigna | A Possible Role of Alpha Crucis in the Astronomical Landscape of Silbury Hill[END_REF][START_REF] Sparavigna | Maria Reiche's Line to Archaeoastronomy[END_REF][START_REF] Sparavigna | The Nazca Frigatebird and Fomalhaut[END_REF] are concerning prehistoric times. However, there are many phenomena that we can find in historical records. Here we propose one of them, simulated by means of Stellarium: the lunar occultation of Aldebaran observed by Copernicus and mentioned in his book De Revolutionibus Orbium Coelestium. It is an example which uses another remarkable feature of the software, related to the calendar. In [START_REF] Team | Stellarium and Sky Visualization[END_REF], we read the following sentences about Stellarium: "Go to the date October 4, 1582 using the 'Date/time window'. Fast forward the time using the bottom menu and keep your eye on the date you see. Did you find anything interesting? Try entering any date between October 5th and October 14th of 1582. Stellarium doesn't allow you to enter these dates, because they don't exist!! This is because of the calendar reforms by Pope Gregory in 1582". The Pope introduced the Gregorian calendar as a refinement to the Julian calendar that was in use at that time. Moreover, it was necessary to have an agreement between the calendar and the dates of equinoxes; to have this agreement it was decided to remove ten days from the calendar, so that October 5th became October 15th 1582. Therefore, if we are using Stellarium and we have a phenomenon recorded with a Julian date, we simply need to put this date in the software to see the sky of that day. Let us consider then the observation made by Copernicus of the lunar occultation of Aldebaran. It was on March 9, 1497 (Julian calendar). For the simulation we need also to know the location. Nicolaus Copernicus was born in Thorn, Poland in 1473. He studied mathematics and astronomy at the University of Krakow. Thanks to his uncle's influence, Copernicus became a canon of the Church. In this manner, using the income gained from his position, he had the possibility to travel in Italy for further studies. So he spent some years in Bologna, Padua, and Ferrara. While he was in Bologna, he lived at home of his professor, the astronomer Domenico Maria Novara [START_REF] Gariboldi | Copernico[END_REF]. It was in Bologna that Copernicus made his first astronomical observations, that he made with the naked eye. In fact, Copernicus died in 1543, more than fifty years before Galileo used, for the first time, a telescope to observe the sky (Galileo discovered the satellites of Jupiter in 1610). However, it is possible that Copernicus used a sighting tube. This instrument was a tube without lenses that an observer could use to look through. The sighting tube was effective in blocking out the light around and allowing the observer to better concentrate on a single object. From his observations and studies, Copernicus concluded that the Earth rotates daily about its axis and that every planet, including Earth, are revolving around the Sun. In the De Revolutionibus Orbium Coelestium, Book IV, Chapter XXVII (we use the edition of 1617, Nicolai Copernici Astronomia instaurata, libris sex comprehensa, qui de Reuolutionibus orbium caelestium inscribitur, excudebat Wilhemus Iansonius, Google e-book), Copernicus discusses the problem of the parallax of the moon. He reports the observations he made in Bologna, with his professor Novara of the abovementioned occultation on 9 March 1497 (he also observed a conjunction of Saturn and the moon on 4 March 1500 [START_REF]Dipartimento di Astronomia, Museo della Specola[END_REF]). Let us read the words written by Copernicus on the occultation in the Chapter entitled "Confirmatio eorum, quae circa Lunae parallaxes sunt exposita". He tells: "Quod igitur parallaxes Lunae sic expositae conformes sint apparentijs, pluribus alijs experimentis possumus affirmare, quale est hoc quod habuimus Bononiae septimo Idus Martij post occasum Solis, anno Christi 1497. Consideravimus enim, quod Luna occultatura stellam fulgentem Hyadum, quam Palilicium vocant Romani, quo expectato, vidimus stellam applicatam parti corporis Lunaris tenebrosi; jamque delitescentem inter cornua Lunae in horae quintae noctis, propinquiorem vero Austrino cornu per trientem, quasi latitudinis sive diametri Lunae." Copernicus is using the Julian Calendar and tells that the observation was made in the evening, seven days before the Ides of March. For the star Aldebaran, he is using the Latin name Palicium. He continues to discuss the phenomenon, telling also that "manifestum erat, quod centrum Lunae secundum visum praecedebat stellam dimidia diametri, & idcirco locus ejus visus in longitudine partium 2 scrup. 36. In latitudine part. 5 scrup. 2 fere. Fuerunt igitur à principio annorum Christi anni AEgyptij. 1497 dies 76, horae 23 Bononiae, …". Here we can also appreciate the difference between the Julian Calendar and the Gregorian Calendar we use today. Copernicus explains that the observation was on the evening of 76th day of year 1497. Today, March 9 is the 68th day of year 2017. Let us use Stellarium to simulate. The result is given in the following Figure 1. The occultation was due West. In the evening, Aldebaran was approaching the moon, as shown by the snapshots. In Stellarium, about 22:20, we have visible only the moon. Then, also the hour is in agreement with the words written by Copernicus, who told that the occultation was at the 23th hour of the day.

Copernicus is also describing the lunar phase. For what concerns the moon then, let us use another software. It is CalSKY, available at www.calsky.com/cs.cgi (we used it in [START_REF] Sparavigna | Astronomy and Tidal Analysis Applied to the Study of Julius Caesar's Commentarii[END_REF]). We obtain that Saturday 4 March 1497, it was New Moon. Saturday 11 March 1497. Moon: First Quarter. This is the northernmost first quarter moon of the year. Former more northern first quarter moon was at 18.3.1491. Next more northern first quarter moon is at 1.3.1498. Saturday 18 March 1497. Moon: Full Moon. Saturday 25 March 1497. Moon: Last Quarter. This is the southernmost last quarter moon of the year. Former more southern last quarter moon was at 3.3.1491. Next more southern last quarter moon is at 14.3.1498. In fact, Copernicus tells that he and Novara saw "stellam applicatam parti corporis Lunaris tenebrosi". The star was approaching the dark part of the moon (they were observing due West, the illuminated part of the moon was towards the sun), closer to the lower horn, "propinquiorem vero Austrino cornu". Besides the lunar occultation of Aldebaran, described in De Revolutionibus, we can also use Stellarium to see the conjunction of Saturn and the moon on 4 March 1500 (Figure 2). This phenomenon is not described in the abovementioned book [START_REF] Swerdlow | Mathematical Astronomy in Copernicus' De Revolutionibus: In Two Parts[END_REF]. Using CalSKY, we have that Sunday 8 March 1500 was the first quarter moon and therefore Copernicus was observing the sky in the same condition of the lunar occultation of Aldebaran.

The two examples shown in the Figures 1 and2 demonstrate the use of Stellarium for the simulation of astronomical events of which we have location and date, of Julian or Gregorian calendar (in the case the moon is involved, the software CalSKY is useful too). In fact, the simulation can help us in understanding the text written by Copernicus, such as in all the other cases of which we have a detailed description in historical records.

Figure 1 :

 1 Figure 1: Observation due West (O, Ovest), from Bologna, March 9, 1497. Using Stellarium, we can see Aldebaran approaching the moon, as shown by the two snapshots.

Figure 2 :

 2 Figure 2: Observation due West (O, Ovest), from Bologna, March 4, 1500. Using Stellarium, we can see the motion of Saturn with respect to the moon. Here in the image, the two bodies are almost coincident.