Ludovic Rifford
email: rifford@desargues.univ-lyon1.fr.

On the existence of nonsmooth control-Lyapunov functions in the sense of generalized gradients

Keywords: 1991 Mathematics Subject Classification. 93D05, 93D15, 93D20, 93D30, 93D09, 93B05, 34D20, 49J52, 49K24, 70K15 Asymptotic stabilizability, converse Lyapunov theorem, nonsmooth analysis, differential inclusion, Filippov and Krasovskii solutions, feedback. L

Let ẋ = f (x, u) be a general control system; the existence of a smooth control-Lyapunov function does not imply the existence of a continuous stabilizing feedback. However, we show that it allows us to design a stabilizing feedback in the Krasovskii (or Filippov) sense. Moreover, we recall a definition of a control-Lyapunov function in the case of a nonsmooth function; it is based on Clarke's generalized gradient. Finally, with an inedite proof we prove that the existence of this type of control-Lyapunov function is equivalent to the existence of a classical control-Lyapunov function. This property leads to a generalization of a result on the systems with integrator.

Introduction

The object of this paper is to study the control systems of the form ẋ = f (x, u), which admit nice properties of stabilization at the origin. First, using the Kurzweil's result [START_REF] Kurzweil | On the inversion of Lyapunov's second theorem on stability of motion[END_REF], it is straightforward to show that if a control system admits a continuous feedback u : R n → U for which the closed-loop system ẋ = f (x, u(x)) is globally asymptotically stable at the origin, then it possesses a smooth control-Lyapunov function. The converse is false. Many systems which possess a smooth control-Lyapunov function do not admit continuous stabilizing feedbacks. However, as it was shown by Artstein [START_REF] Artstein | Stabilization with relaxed controls[END_REF], such systems admit relaxed continuous feedbacks. If moreover the system is affine in the control, these ones can be taken to be continuous outside the origin. Actually, the relaxed feedbacks which are provided by Artstein are not the only one which allow us to overcome the lack of continuous stabilizing closed-loops. Other kinds of feedbacks coming from notions developped by Filippov [START_REF] Filippov | Differential Equations with Discontinuous Righthand Sides[END_REF] and Krasovskii [START_REF] Krasovskiȋ | Stability of motion. Applications of Lyapunov's second method to differential systems and equations with delay[END_REF] enable us as well to stabilize correctly the systems which admit smooth control-Lyapunov functions. Considering a closed-loop system x → f (x, u(x)) that is not necessarly continuous, they associate with it a differential inclusion which is upper semicontinuous with convex values. Roughly speaking, in the case of Krasovskii, giving a state x 0 it puts together all the possible values of the vector field f (x, u(x)) around x 0 . The corresponding trajectories of this differential inclusion have indeed the same behavior as the perturbed trajectories associated with the discontinuous ordinary differential equation ẋ = f (x, u(x)) (see [START_REF] Hájek | Discontinuous differential equations. I[END_REF][START_REF] Yu | A Lyapunov characterization of robust stabilization[END_REF]). On the other hand, it is not necessary to work with smooth functions. Using Clarke's generalized gradients permits us to extend the classical definition of control-Lyapunov function: we do not require them to be smooth anymore, but only locally Lipschitz. We next refer to them as control-Lyapunov functions in the sense of generalized gradients. In this way, we can give a sense to some locally Lipschitz functions which arises naturally as potential control-Lyapunov functions. We prove in this paper that the existence of a control-Lyapunov function in the sense of generalized gradients is both equivalent to existence of a smooth control-Lyapunov function and to existence of a stabilizing feedback of Krasovskii or Filippov type. Even if the first equivalence is new, we stress that the novelty of this paper is not so much in the results but in the proofs. In particular, we precise that the notion of locally Lipschitz control-Lyapunov function ever appeared in [START_REF] Freeman | Robust Nonlinear Control Design. State-Space and Lyapunov Techniques[END_REF].

In addition, we present a precise example of stabilization problem where the use of a nonsmooth control-Lyapunov function comes out naturally. We prove that if a control system possesses a locally Lipschitz stabilizing feedback, then the corresponding control system with integrator admits a stabilizing feedback which is continuous on the state space. This question concerning the integrator problem has been studied by many authors [START_REF] Byrnes | New results and examples in nonlinear feedback stabilization[END_REF][START_REF] Coron | On the stabilization of some nonlinear control systems: results, tools, and applications[END_REF][START_REF] Coron | Some open problems in control theory[END_REF][START_REF] Coron | A relation between continuous time-varying and discontinuous feedback stabilization[END_REF][START_REF] Sontag | Stability and stabilization: discontinuities and the effect of disturbances[END_REF][START_REF] Tsinias | A Lyapunov description of stability in control systems[END_REF] in the smooth case (i.e. when the dynamics is smooth); however let us notice that we can recognize the nonsmooth cases in the works of Freeman, Kokotovic [START_REF] Freeman | Robust Nonlinear Control Design. State-Space and Lyapunov Techniques[END_REF] and Praly, Teel [START_REF] Teel | A smooth Lyapunov function from a class-KL estimate involving two positive semidefinite functions[END_REF].

Our paper is organised as follows: in Section 2 we describe our results. In section 3 we apply our main theorem to the integrator problem. In section 4, we give the proofs of Theorems 1 and 2 and we conclude with some comments.

Throughout this paper, •, • denotes the inner product in R n , • the euclidean norm, B the open unit ball, and B its closure. We will also denote the distance between x and the set S by d S (x).

IGD, , ,

Definitions and statements of the results

In this paper we are concerned with control systems of the form

ẋ(t) = f (x(t), u(t)) (2.1)
where the state x(t) is in the Euclidean space R n and the control u(t) belongs to the set of control U which is assumed to be a subset of R m . Moreover we will assume that the function f is continuous from R n × U into R n and that it admits an equilibrium point at the origin, i.e. f (0, 0) = 0 (where "0" is some particular control in U ). We recall that the trajectories of (2.1) are absolutely continuous functions x(•) associated with some open loops u : [0, ∞[→ U and solutions of (2.1) almost everywhere.

Our objective is to find a necessary and sufficient condition for the existence of a smooth control-Lyapunov function. Let us give the definition the definition of this concept. Note before that the function

V : R n -→ R ≥0 is positive definite if x = 0 ⇒ V (x) > 0 and V (0) = 0 and that it is proper if lim x →∞ V (x) = ∞.
Definition 2.1. Let V be an application which is assumed to be C 1 from R n into R. It is said to be a control-Lyapunov function (abreviated clf) for the system (2.1) if it is positive definite, proper and if the following condition is satisfied: For any compact set K of R n , there exists a compact subset

U 0 ⊂ U such that ∀x ∈ K \ {0}, min u∈U 0 ∇V (x), f (x, u) < 0. (2.2)
As it has been shown by Kurzweil [START_REF] Kurzweil | On the inversion of Lyapunov's second theorem on stability of motion[END_REF] (and by Massera [START_REF] Massera | Contributions to stability theory[END_REF] in the local case) in the context of classical differential equations, if the system ẋ = g(x) is globally asymptotically stable at the origin then there exists a (classical) Lyapunov function. If the system (2.1) admits a continuous stabilizing feedback, that is, a continuous function u : R n -→ U such that the closed-loop system ẋ = f (x, u(x)) is globally asymptotically stable, then there exists a smooth (C ∞ ) control-Lyapunov function. Unfortunatly, the converse is false; the presence of a smooth control-Lyapunov function does not lead to the existence of continuous stabilizing feedbacks. Let us give the following example (quoted by Sontag in [START_REF] Sontag | Stability and stabilization: discontinuities and the effect of disturbances[END_REF]) to highlight this problem:

ẋ1 = u 2 u 3 ẋ2 = u 1 u 3 ẋ3 = u 1 u 2 , (2.3)
where (u 1 , u 2 , u 3 ) ∈ B is the unit ball of R 3 . The function x → x 2 is clearly a smooth clf for this system and yet it does not admit a continuous stabilizing feedback. As a matter of fact, no point of the form (0, ǫ, ǫ) belongs to the set f (x, u) for x in a neighbourhood of the origin and u in the unit ball B. In other words, the system (2.3) does not satisfy the Brockett's condition which is necessary for the existence of continuous feedback. This example leads us to consider another form of feedback. We will deal with differential inclusions that we proceed to define.

IGD, , ,

Let F : R n → 2 R n be a multivalued map. We shall say that the hypotheses (H) are satisfied if the two following conditions hold: (H1) For any x in R n , F (x) is a nonempty compact convex set. (H2) The multivalued map F is upper semicontinous; that is:

∀x ∈ R n , ∀ǫ > 0, ∃δ > 0 such that x -y < δ ⇒ F (y) ⊂ F (x) + ǫB.
The hypotheses (H) correspond to the usual framework for the study of the differential inclusion ẋ(t) ∈ F (x(t)) a.e.

(2.4) associated with F . For the sake of simplicity, we will sometimes speak about the multivalued map F for talking about its differential inclusion (2.4). We suggest the following references for a detailed study of multivalued maps and differential inclusions [START_REF] Deimling | Multivalued Differential Equations[END_REF], [START_REF] Aubin | Set-valued analysis[END_REF] and [START_REF] Aubin | Differential Inclusions[END_REF].

Knowing this notion we are now able to extend the classical definition of asymptotic stability to the more general case of differential inclusions. Definition 2.2. The differential inclusion (2.4) is Globally Asymptotically Stable (abreviated GAS) provided that no solution exhibits finite time blowup, and provided that the following holds:

(a) Uniform Attraction: For any 0 < r < R, there exists T = T (r, R) > 0 such that for any solution x(•) of (2.4) with x(0) ≤ R, one has x(t) ≤ r ∀t ≥ T.

(b) Uniform Boundedness: There is a continuous nondecreasing function

M :]0, ∞[→]0, ∞[
such that for any solution x(•) of (2.4), one has

x(t) ≤ M ( x(0) ) ∀t ≥ 0. (c) Lyapunov Stability: lim R↓0 M (R) = 0.
Remark 2.3. This definition agrees with the one taken in [START_REF] Clarke | Asymptotic stability and smooth Lyapunov functions[END_REF] and [START_REF] Lin | A smooth converse Lyapunov theorem for robust stability[END_REF]. The concept of differential inclusion will allow us to give a sense to the Cauchy problem ẋ(t) = f (x(t), u(x(t))) when the function x → f (x, u(x)) does not have continuity properties. As a matter of fact, in the general case of non necessarly continuous functions, no theorem insures the existence of solution to the Cauchy problem ẋ(t) = f (x(t), u(x(t))), x(0) = x 0 .

(2.5)

That's why we have to recall an alternative concept of solutions.

Let be given u : R n → U, an application which is not assumed to be continuous. We shall say that the function u is locally bounded if, for any compact set K of R n , there exists IGD, , ,

δ K > 0 such that for any x ∈ K ∀δ ≤ δ K , u(x + δB) is a compact set in R m .
We can associate two multivalued maps with the function u as follows. We set for any x in R n :

K u (x) := co{∩ δ>0 f (x, u(x + δB))}, (2.6)
and We are going to connect the existence of a stabilizing feedback to the one of a control-Lyapunov function. As we said in the introduction, it is not necessary to work with smooth control-Lyapunov functions. Let us introduce some concepts of nonsmooth analysis.

F u (x) := co{ λ(N )=0 δ>0 f (x, u(x + δB \ N ))}, ( 2
Let f : R n -→ R be a locally Lipschitz function, we denote by ∂ P f (x) its proximal subdifferential at x, i.e. the set of vectors ζ ∈ R n such that there exists δ, σ > 0 veryfying

f (y) -f (x) + σ y -x 2 ≥ ζ, y -x ∀y ∈ x + δB.
This object permits us to define the generalized gradient of f at x:

∂f (x) := co{lim ζ k : x k → x, ζ k ∈ ∂ P f (x k )}.
This tool has been introduced by Clarke in 1973. Since this time, a complete calculus has been developed, one that extends all the theorems of the usual smooth calculus. In particular, the generalized gradient of the sum of two functions is included in the sum of the generalized gradients. Remark 2.5. The definition of the generalized gradient given above coincides with the following one which is based on the Rademacher's Theorem:

∂f (x) := co{lim ∇f (x k ) : x k → x, x k ∈ D f },
where D f denotes the set of points where f is differentiable.

We proceed now to adapt the classical defnition of control-Lyapunov function to the case of functions which are only locally Lipschitz.

IGD, , , Definition 2.6. Let V : R n -→ R; it is said to be a control-Lyapunov function in the sense of generalized gradients for the system (2.1) if V is locally Lipschitz, positive definite, proper and if it satisfies the following property: For any compact set K in R n , there exists

U 0 ⊂ U such that ∀x ∈ K \ {0}, ∀ζ ∈ ∂V (x), min u∈U 0 ζ, f (x, u) < 0.
(2.8)

Like the proximal subdifferential in the case of systems which are globally asymptotically controllable (see [START_REF] Clarke | Feedback stabilization and Lyapunov functions[END_REF], [START_REF] Rifford | Existence of Lipschitz and semiconcave control-Lyapunov functions[END_REF]), the generalized gradient constitutes a fantastic tool for the study of systems which possess strong properties of stability. Let us state our main result. Theorem 2.7. The following properties are equivalent:

(i) The system (2.1) admits a control-Lyapunov function in the sense of generalized gradients. (ii) The system (2.1) admits a control-Lyapunov function which is C ∞ on R n . (iii) The system (2.1) admits a stabilizing feedback of Krasovskii type (or of Filippov type). The parts (iii) ⇒ (i) and (iii) ⇒ (ii) are not new. However, we present a proof which differs from the known ones (see Clarke et al. [START_REF] Clarke | Asymptotic stability and smooth Lyapunov functions[END_REF], Lin et al. [START_REF] Lin | A smooth converse Lyapunov theorem for robust stability[END_REF] and more recently Praly and Teel [START_REF] Teel | A smooth Lyapunov function from a class-KL estimate involving two positive semidefinite functions[END_REF]). Instead of constructing a global locally Lipschitz differential inclusion to deduce the existence of a control-Lyapunov function in the sense of generalized gradients, we design locally some control-Lyapunov functions that we glue to obtain a global clf. This approach requires some easy facts of the nonsmooth calculus and makes the proof inedite. Actually this kind of construction could be related to the one used in the proof of the existence of a locally Lipschitz control-Lyapunov function for asymptotically controllable systems, see [START_REF] Rifford | Existence of Lipschitz and semiconcave control-Lyapunov functions[END_REF]. On the other hand, we precise that the proof of (i) ⇒ (ii) is inspired by ideas given in [START_REF] Clarke | Asymptotic stability and smooth Lyapunov functions[END_REF]. However, in our context, we deal with weak control-Lyapunov functions (i.e. we have a minimum over U instead of a maximum in the decrease property (2.8)) that require a more technical proof.

Furthermore, we find the Artstein's result (also given in [START_REF] Sontag | A "universal" construction of Artstein's theorem on nonlinear stabilization[END_REF] and [START_REF] Coron | A relation between continuous time-varying and discontinuous feedback stabilization[END_REF]) in the case of f affine in the control; that is, when the function f is of the form

f (x, u) = f 0 (x) + m i=1 u i f i (x),
where the f i 's are some locally Lipschitz functions from R n into R n . In addition we are able to precise the regularity of the stabilizing feedback given in (iii) under an additional assumption around the origin. Let us state the result. Theorem 2.8. If f is affine in the control and if the property (i) of Theorem 2.7 holds (or equivalently(ii)) then there exists a stabilizing feedback which is continuous on R n \{0}. If moreover for all ǫ > 0, ∃δ > 0 such that if x < δ then the compact set W x of Definition 2.1 can be taken to be included in ǫB, then the stabilizing feedback can be taken to be continuous at the origin.

We present now a nice example where our results apply.

IGD, , ,

Application to the integrator problem

In order to highlight the using of nonsmooth clf, we present a result which concerns the system with integrator associated with the initial system (2.1). This kind of problem has been studied recently in [START_REF] Freeman | Backstepping design with nonsmooth nonlinearities[END_REF][START_REF] Praly | On assigning the derivative of a disturbance attenuation clf[END_REF] where we can recognize the nonsmooth method which will be used. Our proof is inspired by the one given by Byrnes, Isidori [START_REF] Byrnes | New results and examples in nonlinear feedback stabilization[END_REF] and Tsinias [START_REF] Tsinias | Sufficient Lyapunov-like conditions for stabilization[END_REF] in the smooth case; one can also find this proof in the book of Sontag [START_REF] Sontag | Mathematical Control Theory[END_REF]. On the other hand, we stress that Rosier [START_REF] Rosier | Étude de quelques problèmes de stabilisation[END_REF] provides an alternative demonstration of 3.1 in the smooth case; it does not involve control-Lyapunov function. Besides, we suggest more additional readings to the curious reader on other questions related to the integrator problem [START_REF] Coron | On the stabilization of some nonlinear control systems: results, tools, and applications[END_REF][START_REF] Coron | Some open problems in control theory[END_REF]. Theorem 3.1. Let f : R n × U → R n a function which is assumed to be continuous in x and locally Lipschitz in u uniformly in x in a neighbourhood of the origin in R n × R m . If the control system (2.1) possesses a locally Lipschitz stabilizing feedback, then the control system with integrator

ẋ = f (x, u) u = v ∈ R m (3.1)
admits a stabilizing feedback which is continuous on R n .

Proof. Let us denote k the locally Lipschitz feedback which stabilizes our system. By Theorem 2.7, there exists a smooth control-Lyapunov function V on R n verifying in particular:

∀x = 0, ∇V (x), f (x, k(x)) < 0. (3.2) Consider the function W from R n × R m into R ∀(x, z) ∈ R n × R m , W (x, z) = V (x) + 1 2 z -k(x) 2 .
We shall prove that the function W is a control-Lyapunov function in the sense of generalized gradients. Therefore, we will conclude by Theorem 2.8. First of all, it is easy to see that W is locally Lipschitz, positive definite and proper. Let us look at the generalized gradients of W . Take

(x, z) ∈ R n × R m and (ζ 1 , ζ 2 ) ∈ ∂W (x, z) where ζ 1 ∈ R n and ζ 2 ∈ R m . We see immediatly that ζ 2 = z -k(x).
Moreover, by a classical chain rule on the generalized jacobians (see [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF] or [START_REF] Hiriart-Urruty | Les fonctions d'appui de la jacobienne généralisée de Clarke et de son enveloppe plénière[END_REF]), one has

ζ 1 = ∇V (x) + ζ 1 , with ζ 1 ≤ L k z -k(x) , where L k is the Lipschitz constant of k in a neighbourghood of x. We deduce that ∂W (x, z) ⊂ (∇V (x) + L k z -k(x) B, z -k(x)). Consider now (x, z) ∈ R n × R m \ (0, 0) and (ζ 1 , ζ 2 ) ∈ ∂W (x, z). We can write (ζ 1 , ζ 2 ) = (∇V (x) + Ψ, (z -k(x))) with ψ ≤ L k z -k(x) . First case : if z = k(x), then (ζ 1 , ζ 2 ) = (∇V (x), 0) and the decrease condition (2.8) is satisfied by (3.2). Second case : If z -k(x) > 0
We define M a matrix of M n,m (R) as follows:

M := f (x, z) -f (x, k(x))(z -k(x)) t z -k(x) 2 . IGD, , ,
It is straightforward to show that M is bounded by the Lipschitz constant of f in the variable z with x fixed. We set:

v := k(x) -z -M t ∇V (x) + µ where µ := k(x) -z z -k(x) 2 Ψ, f (x, z) .
We can compute by (3.2):

(ζ 1 , ζ 2 ), (f (x, z), v) = -z -k(x) 2 + ∇V (x), f (x, z) -M (z -k(x)) = -z -k(x) 2 + ∇V (x), f (x, k(x)) < 0.
Moreover, if (x, z) is in a small enough neighbourhood V × W of the origin, then by the hypotheses the function f is locally Lipschitz in the variable u uniformly in x; that is there exists a constant

L 0 f such that ∀x ∈ V, ∀(u 1 , u 2 ) ∈ W × W, f (x, u 1 ) -f (x, u 2 ) ≤ L 0 f u 1 -u 2 .
We deduce that the matrix M that we defined above is such that M ≤ L 0 f . If we denote by L 0 k the Lipschitz constant of the function k on W, then we get:

v ≤ z -k(x) + L 0 f ∇V (x) + L 0 k f (x, z)
, where (x, z) belongs to V ×W and v corresponds to v (= v(x, z)) from above. Hence we deduce that the quantity v (= v(x, z)) tends to 0 when the couple (x, z) converge to the origin. This enables us to apply Theorem 2.8 and to obtain the existence of a stabilizing feedback v(•, •) which is continuous on R n .

We will try now to understand why we cannot relax the hypotheses of Theorem 3.1 into continuity of the stabilizing feedback k. First of all, we know a counterexample in this kind of situation; Coron and Rosier provide in [START_REF] Coron | A relation between continuous time-varying and discontinuous feedback stabilization[END_REF] the following control system:

ẋ = f (x, u) = x
V (x, u) = x -u 2 .
Along a trajectory (x(t), u(t)) of the integrator system, it satisfies:

V = x 1 3 -u -2uv ≥ 0 whenever V is nonnegative, v ≤ 1 and |(x, u)| is sufficently small.
But if there would exist a stabilizing feedback continuous at the origin, then it would verify the Lyapunov Stability property (see Definition 2.2). But every point of the set A := {(x, u) : xu 2 ≥ 0} can't reach asymptotically the origin without leaving the part A where V is positive. In other terms, in order to tends to 0 the stabilizing trajectories must move away from the IGD, , , origin. As we are seeing, the integrator system does not possess continuous stabilizing feedbacks because the Lyapunov stability property can't be satisfied. Actually, for (x 0 , u 0 ) close enough to the origin, there does not exist stabilizing trajectories which converge to zero with respect to small controls! The system is not stabilizable at the origin with small controls, how could it be stabilizable by a continuous feedback or even by a continuous "time-varying feedback" (see [START_REF] Coron | A relation between continuous time-varying and discontinuous feedback stabilization[END_REF])?

In a general manner, if we assume that the initial control system (2.1) admits a stabilizing feedback which is only continuous at the origin, then it seems to be difficult to prove that the integrator system (3.1) is controllable at the origin with small controls. As a matter of fact, let be given an initial state (x 0 , u 0 ), if we consider a trajectory x(•) of the GAS system ẋ = f (x, k(x)) on [0, ∞). It converges to the origin and moreover we can compute:

u(t) = dk(x(t)) • f (x(t), k(x(t)).
Thus the control v := u(t) drives the system (3.1) to the origin, but we cannot claim that the controls are sufficiently small around the origin (the term dk(x(t) might be very big).

To conclude, if we omit the assumption of Lipschitz regularity on k, we cannot prove the existence of a stabilizing feedback (for the integrator system) which is continuous at the origin. However, it is in fact possible to get a stabilizing feedback which is continuous (even smooth) outside the origin. In other words, we can produce a control-Lyapunov function which is sufficiently regular (C 1 or locally lipschitz) but which does not verify the small control property (see Artstein [START_REF] Artstein | Stabilization with relaxed controls[END_REF] or the last assumption of Theorem 2.8). Such results have been proven by Tsinias; we refer to [START_REF] Tsinias | A local stabilization theorem for interconnected systems[END_REF] for the case m = 1 and to [START_REF] Tsinias | An extension of Artstein's theorem on stabilization by using ordinary feedback integrators[END_REF] for the multi-input case.

Proofs Theorems 2.7 and 2.8

Let us begin by the most difficult part: (iii) =⇒ (i). In fact we will derive our result from the following theorem. Theorem 4.1. Let F : R n → 2 R n be a multivalued function satisfying the hypotheses (H) and such that the differential inclusion (2.4) is GAS. Then under these assumptions, there exists a function V : R n → R which is locally Lipschitz, positive definite, proper and such that:

∀x = 0, ∀ζ ∈ ∂V (x), ∀v ∈ F (x), ζ, v < 0. (4.1)
Remark 4.2. In this case, the application V is called strong control-Lyapunov function for F . By the same proof as for (i) ⇒ (ii), one can deduce from Theorem 4.1 the existence of a smooth strong control-Lyapunov function F ; we recognize the main result given by Clarke, Ledyaev and Stern [START_REF] Clarke | Asymptotic stability and smooth Lyapunov functions[END_REF].

Let us consider F a multivalued application being GAS; we are going to construct a strong control-Lyapunov function. Let us before change the dynamic to transform it into a bounded dynamic. Lemma 4.3. Let F be a multifunction which satisfies (H), let α : R n → [0, ∞[ be a positive definite continuous function. Then the multifunction IGD, , , defined as follows F α (x) := α(x)F (x) satisfies (H) and is GAS.

We suggest the reading of [START_REF] Clarke | Asymptotic stability and smooth Lyapunov functions[END_REF] for a proof of this result. Up to set F (x) := F α (x) with α(x) := [max{1 + v : v ∈ F (x)}] -1 , we can assume that F is bounded by 1 on R n . Besides, since α does not vanish outside the origin, a Lyapunov function for F is obviously a control-Lyapunov function for F α . The proof of our Theorem requires the following compactness property of solutions of differential inclusions (see e.g. [START_REF] Aubin | Differential Inclusions[END_REF][START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF][START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF][START_REF] Deimling | Multivalued Differential Equations[END_REF]). Lemma 4.4. Let F be a multivalued map satisfying hypotheses (H). Then for any sequence δ k → 0 and sequence x k (•) of absolutely continuous and uniformly bounded functions on [a, b] satisfying

ẋk (t) ∈ co F (x k (t) + δ k B) + δ k B a.e.,
there exists a subsequence x k i (•) converging uniformly to some solution x(•)

of ẋ ∈ F (x) on [a, b].
Give now the proposition which will be fundamental in the construction of the control-Lyapunov function in the sense of generalized gradients. We will invoke it many times with different functions G. with x(0) = x, is locally Lipschitz on R n ; moreover it satisfies the following property L(x n (t))dt converge to V (x), then all stay in the ball (r + bm)B; this means that they are uniformly bounded. Hence we can apply the compactness of trajectories (see [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF]) to deduce that the maximum in the definition of V (x) is attained IGD, , , for some trajectory x(•) of (4.2) with x(0) = x. Besides for any y ∈ rB, by [3, Cor.1,p.121] there exists a trajectory y(•) of (4.2) with y(0) = y which satisfies

∀ζ ∈ ∂V (x), ∀v ∈ G(x), ζ, v ≤ -L(x) + L x,b , (4.3
∀t ∈ [0, b], y(t) -x(t) ≤ e λ G b y -x ,
where λ G is the Lipschitz constant of G on (r + mb)B. If we set λ L the Lipschitz constant of L on the ball (r + mb)B, we obtain

V (x) = b 0 L(x(s))ds ≤ b 0 L(y(s))ds + b 0 λ L y(s) -x(s) ds ≤ V (y) + bλ L e λ G b y -x .
Since we can repeat the same thing by inverting x and y and by varying r, we deduce that the function V is locally Lipschitz on R n . We proceed now to prove inequality (4.3) for x 0 fixed in R n . Let us begin by verifying (4.3) with the elements of the proximal subdifferential. Let ζ ∈ ∂ P V (x 0 ) and v ∈ G(x 0 ). By the famous Michael's Theorem [START_REF] Michael | Continuous selections[END_REF], there exists a continuous map (we can even get it locally Lipschitz, see [START_REF] Aubin | Set-valued analysis[END_REF]) g : R n → R n such that ∀x ∈ R n , g(x) ∈ G(x) and g(x 0 ) = v. Consider x(•) a solution of the classical ordinary differential equation ẋ = g(x(t)) on [0, b] with x(0) = x 0 , it is a trajectory of (4.2). Thus we obtain by definition of V , for any t ∈ [0, b]:

V (x(t)) + t 0 L(x(s))ds - b+t b L(x t (s -t))ds ≤ V (x 0 ).
where xt (•) denotes a trajectory which realizes the maximum in the definition of V (x(t)), that is such that

V (x(t)) = b 0 L(x t (s))ds = b+t t L(x t (s -t))ds.
On the other hand, there exist σ, η > 0 such that

V (y) -V (x 0 ) + σ y -x 0 2 ≥ ζ, y -x 0 ∀y ∈ x 0 + ηB.
Therefore by setting y = x(t), we get for t sufficently small:

ζ, x(t) -x 0 ≤ σ x(t) -x 0 2 + b+t b L(x t (s -t))ds - t 0 L(x(s))ds.
By dividing by t, and passing to the limit we obtain easily

ζ, v ≤ L x,b -L(x 0 ).
This proves our inequality in the case of proximal subgradients. To conclude, we claim that the inequality (4.3) will remain by passing to the limit and then will be satisfied for the elements of the limiting subdifferential ∂ L V (x 0 ). The inequality will also remain by passing to the convex hull. The proposition is proved.

Come back to our proof; set new things from the multivalued function F . Let be given a positive number δ, we shall call F δ the multifunction defined as follows: For any x ∈ R n , ẋ(t) ∈ F δ (x(t)) := co F (x(t) + δB) + δB.

(4.4)

We proceed now to present a lemma which can be found in [START_REF] Clarke | Asymptotic stability and smooth Lyapunov functions[END_REF]; we give its proof for being more self-contained. IGD, , , Lemma 4.7. Let R, T, ǫ be given positive numbers. Then there exists ∆ = ∆(R, T, ǫ) such that the following hold: For any δ ∈ [0, ∆], each solution x(•) of (4.4) with x(0) ≤ R does not blow-up on [0, T ] and satisfies

(a) x(t) ≤ 2M (R) ∀t ∈ [0, T ]. (b) There exists a solution x(•) of (2.4) on [0, T ] such that x(t) -x(t) ≤ ǫ ∀t ∈ [0, T ].
Proof. If there exist sequences δ i ↓ 0 and T i and corresponding solutions x i (•) of (4.4) on [0, T i ] with δ = δ i and x i (0) ≤ R, such that

x i (T i ) = 2M (R), x i (t) < 2M (R) ∀t ∈ [0, T i [. (4.5)
Since the multifunction F is bounded on the ball 2M (R)B, then all the F δ i are bounded too, and even uniformly. Because of fact that x i (T i ) = 2M (R), this implies that the sequence T i does not admit 0 as a cluster point. Hence, we can assume that T i ↑ T ≤ T . Thus, we can apply Lemma 4.4 to deduce the existence of a solution x(•) of (2.4) which is a uniform limit of the x i 's on [0, T ]. Consequently, it satisfies x(T ) = 2M (R). That brings a contradiction to the definition of M (R)! Therefore we have proved the existence of some ∆ > 0 verifying the first part of the lemma. On the other hand, we have shown that any trajectory starting in the ball RB remains in 2M (R)B (whenever t ≤ T ). Then we can always extend it to [0, T ]. Assume now that there exists a sequence δ i ↓ 0 and corresponding trajectories x i (•) of (4.4) on [0, T ] with δ = δ i and x i (0) ≤ R, such that max t∈[0,T ]

x i (t)x(t) ≥ ǫ for any trajectory x(•) of (2.4). The sequence being uniformly bounded (by the first part of the proof), Lemma 4.4 brings a contradiction. Finally, Lemma 4.7 is proved.

We are now able to prove the following. Lemma 4.8. Let 0 < r < R two positive numbers, then there exist ∆ = ∆(r, R) > 0 and T > 0 which satisfy: For any δ ∈ [0, ∆], any x(•) solution of (4.4) with x(0) ≤ R can be extended to [0, ∞) such that

x(t) ≤ 2M (R) ∀t ≥ 0, (4.6)
et x(t) ≤ r ∀t ≥ T . (4.7)
Proof. By Lyapunov stability, we know that M (ǫ) ↓ 0 when ǫ ↓ 0, then there exists 0 < ǫ < r such that 2M (ǫ) < r. By applying Lemma 4.7 with T = T ( ǫ 2 , R), ǫ 2 and R, we get the existence of ∆ such that for any δ ∈ [0, ∆], for any solution x(•) of (4.4) on [0, T ], one has

x(t) ≤ 2M (R) ∀t ∈ [0, T ],
and there exists a solution x(•) of (2.4) such that

x(t) -x(t) ≤ ǫ 2 ∀t ∈ [0, T ].

IGD, , ,

By applying still once Lemma 4.7 with T = T ( ǫ 2 , R) and R = ǫ, we get ∆ such that for any δ ∈ [0, ∆ ], any solution x(•) of (4.4) can be extended to [0, T ] and satisfies

x(t ≤ 2M (ǫ) < r ∀t ∈ [0, T ].
Set ∆ := min{∆, ∆ }. Let us prove that the properties (4.6) and (4.7) are verified for any δ in [0, ∆]. Let us consider such a δ and a solution x(•) of (4.4). It remains in the ball 2M (R) whenever t ≤ T , thus we will be able to extend it to [0, T ]. Moreover, by construction of ∆, there exists a trajectory x(•) of (2.4) such that

x(t) -x(t) ≤ ǫ 2 ∀t ∈ [0, T ].
On the other hand, by definition of T , one has x(T ) ≤ ǫ 2 . Thus x(T ) ≤ ǫ ≤ R. Therefore, we can extend our trajectory on [T, 2T ]. Since x(T ) ≤ ǫ, it remains in 2M (ǫ)B and hence in rB. As before x(2T ) ≤ ǫ. The work done above means that we can repeat the extension of our trajectory on the different intervals [nT, (n + 1)T ] in such a manner that we stay in the ball 2M (R)B for t ≥ 0 and in rB for t ≥ T = T ( ǫ 2 , R). The lemma is proved. We proceed now to construct our control-Lyapunov function in the sense of generalized gradients. For that, we start by setting a sequence of value functions {V p } p≥1 corresponding to the one that appeared in Proposition 4.5. Set for any positive integer p, R p := 2 p . By Lemma 4.8, we can associate with each couple (1, 2M (R p ) + 2) some constants ∆p and Tp such that ∆p+1 ≤ ∆p ≤ ∆(1, R p ) and Tp+1 ≥ Tp ≥ T (1, 2M (R p ) + 2).

Fix now some positive integer p. We define a multifunction F p , and then a new dynamics as follows. Set for any x in R n :

W p x := y : y -x ≤ 1 2 ∆p .
Since the family {W p x } x is a covering of R n , there exists a locally finite open cover {W n

x i } i of R n and {Ψ i } i a subordinated C ∞ partition of unity. We set for any x in R n :

F p (x) := i Ψ i (x)F x i + ∆p 2 B .
By construction, F p satisfies obviously (H1), is locally Lipschitz and its norm is bounded by 1 on R n ; on the other hand we have that for any x in R n :

F (x) ⊂ F p (x) ⊂ coF (x + ∆p B) ⊂ F ∆p (x), (4.8)
and

F p+1 (x) ⊂ F p (x). (4.9)
Let us go back to the construction of our sequence of value functions. We shall apply Proposition 4.5 with each of the dynamics previously defined.

IGD, , ,

We proceed recursively; let us start by defining V 1 . We set for any x in R n :

V 1 (x) := max T1 0 L 1 (x(s))ds : ẋ ∈ F 1 (x(t)) a.e. and x(0) = x ,
where L 1 (y) = d B (y) ∀y ∈ R n . By Proposition 4.5, the value function V 1 is locally Lipschitz on R n and proper (because the dynamics is globally bounded); moreover, it is nonnegative. Besides, if we consider x ∈ R n with x ≤ 2M (R 1 ) + 2, then by Lemma 4.8 for every trajectory x(•) of (4.4) with δ = ∆1 and x(0

) = x, one has x(t) ≤ 1 ∀t ≥ T1 , that is d B (x(t)) = 0.
The property (4.3) allows us to write for any x ∈ R n with x ≤ 2M (R 1 ) + 2:

∀ζ ∈ ∂V 1 (x), ∀v ∈ F (x), ζ, v ≤ -d B (x). (4.10)
Let us now assume that we have defined the functions V 1 , V 2 , • • • , V p , and let us define the function V p+1 . We begin by setting two auxiliary functions; we set for any x ∈ R n :

B p (x) := max{V p (y) : y ≤ x + 1} + 1,
and

L p+1 := max 1 2 , x -2M (R p ) L p + max {0, x -2M (R p )} B p (x).
Define our function V p+1 ; set for any x in R n :

V p+1 (x) := max

∀ζ ∈ ∂V p+1 (x), ∀v ∈ F (x), ζ, v ≤ -L p+1 (x). (4.11)
Furthermore, we have the following lemma. Lemma 4.9.

Let x ∈ R n . (a) If x ≤ R p , then V p+1 (x) ≤ Vp(x) 2 . (b) If x ≥ 2M (R p ) + 2, then V p+1 (x) ≥ V p (x) + 1.

Proof.

(a) Let x ∈ R p B. By Lemma 4.8 (because ∆p+1 ≤ ∆p ≤ ∆(1, R p )) and from (4.8) applied for p + 1, all the trajectories of ẋ ∈ F p+1 (x) starting at x stay in the ball 2M (R p )B. Thus, all along those we have:

L p+1 ≥ L p 2 .
IGD, , ,

In addition to that, by (4.9), these ones are also some trajectories of ẋ ∈ F p (x), and for all t ≥ Tp , they remain in the unit ball, where L p vanishes. Since Tp+1 ≥ Tp , this implies that Tp+1 Tp L p+1 (x(s))ds = 0.

We conclude that

V p+1 (x) ≤ V p (x) 2 . (b) Consider x / ∈ [2M (R p ) + 2]B.
Since the dynamics F p+1 has a norm bounded by 1, all its trajectories x(•) starting at x stay outside the ball [2M (R p ) + 1]B for t ∈ [0, 1]. But from the definitions of L p+1 and B p , we have for any t ∈ [0, 1]:

L p+1 (x(t)) ≥ B p (x(t)) ≥ V p (x) + 1.
Hence we deduce that V p+1 (x)

≥ 1 0 L p+1 (x(s))ds ≥ V p (x) + 1.
Since the construction of our sequence (V p ) p≥1 is achieved, we can define:

∀x ∈ R n , V 0 (x) := max{V p (x) : p ≥ 1}. (4.12)
This new function is well defined. As a matter of fact, the preceding lemma implies that if for some q and s, we have

2M (R q ) + 2 ≤ x ≤ R s , then V 0 (x) = max q≤p≤s V p (x) and if x ≤ 2 = R 1 , then V 0 (x) = V 1 (x).
We conclude that the maximum in the definition of V 0 (x) is attained only for a finite number of V p (x)'s. Moreover, since the functions V p are locally Lipschitz and proper, the function V 0 is itself locally Lipschitz and proper. On the other hand, it is obviously nonnegative and positive outside the closed unit ball B. In addition, by Proposition 2.3.12 of [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF], if ζ belongs to ∂V 0 (x) then it belongs to the convex hull of the ∂V p (x) for which the maximum in (4.12) is attained, that is such that V p (x) = V 0 (x). From these results and by (4.10) and (4.11) we deduce that: We set now a new sequence r p := 1 2 p . As before, we can choose a sequence of couples ( ∆ p , T p ) in such a manner that T p ≥ T (r p , 2).

∀x ∈ R n \ B, ∀ζ ∈ ∂V 0 (x), ∀v ∈ F (x), ζ, v < 0. ( 4
We set as before a sequence F p of dynamics. Then we can set for any p: Finally, we set:

V(x) := V 0 (x) + ∞ p=1 ρ p V p (x),
where ρ p := min{ 1 Mp2 p , 1 Kp2 p }. Still by construction, the function V is positive definite, proper and locally Lipschitz. Besides by a rule on the sum of generalized gradients (see [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF]), we have:

If ζ ∈ ∂V(x) then ζ ∈ ∂V 0 (x) + ∞ p=1 ρ p ∂V p (x).
Hence by linearity of the scalar product, we get by (4.13), (4.14), (4.15) and (4.16): ∀x = 0, ∀ζ ∈ ∂V(x), ∀v ∈ F (x), ζ, v < 0.

Theorem 4.1 is proved.

We proceed now to show now the part (i) =⇒ (ii): We assume that the system (2.1) admits a control-Lyapunov function in the sense of generalized gradients V ; we shall regularize it. Fix x in R n \ {0}. By Definition 2.6, there exists a compact set W x such that

∀y ∈ 2 x B \ {0}, ∀ζ ∈ ∂V (y), min u∈Wx ζ, f (y, u) < 0. ( 4

.17)

We can define the following positive quantity:

ǫ x := -max y∈x+ x 2 B max ζ∈∂V (y) min u∈Wx ζ, f (y, u) > 0. (4.18)
Note that the positivity of the constant ǫ x is a consequence of the upper semicontinuity of the multivalued map y → ∂V (y) and of the compacity of W x . We set also for any y ∈ x + x 2 B: F x (y) := co{f (y, u) : u ∈ W x }.

IGD, , ,

Since F x (y) and ∂V (y) are convex sets, the minimax Theorem (see [START_REF] Aubin | Viability theory[END_REF]) implies by (4.17 Furthermore, since the multivalued map x → ∂V (x) is upper semicontinuous and since F x is continuous, we can find a neighbourhood U x (with 0 / ∈ U x ) of x and a positive real number δ x such that for any y ∈ U x , there exists v ∈ F x (y) satisfying:

∀z ∈ U x + δ x B, ∀ζ ∈ ∂V (z), ζ, v ≤ - ǫ x 2 . ( 4
δ(x) := min x 2 , min δ j : x + x 2 B ∩ U j = ∅ > 0.
The family {x + δ(x) 2 B}

x is an open cover of R n \ {0}; we can extract from it a locally finite open cover {x p + δ(x p ) 2 B} ∞ p=1 corresponding to a C ∞ partition of unity {ψ} ∞ p=1 . For any p, Supp(ψ p ) will denote the support of the function ψ p ; let us recall that it is included in x p + δ(x p ) 2 B. With this stuff, we can set for any p: ǫ p := min

(x p + x p 2 B)∩U i =∅ {ǫ x i } > 0, (4.21)
ǭp := min min

(x p + x p 2 B)∩U i =∅ min x∈U i V (x), ǫ p > 0, (4.22)
q p := max

(x p + x p 2 B)∩U i =∅ max x∈U i max u∈Wx i { ∇ψ i f (x, u) } > 0. (4.23) Let ω : R n → [0, ∞[ be a C ∞ function with support in the closed unit ball B, such that R n ω(x)dx = 1.
For ν > 0 let us define for any x ∈ R n :

V ν (x) := R n V (x + νy)ω(y)dy.
This function is C ∞ and furthermore if S ⊂ R n is compact, then the sequence (V ν ) ν converges uniformly to V when ν tends to 0. Thus for each p, there exists ν p < δ(x p ) 2 such that

|V νp (x) -V (x)| ≤ ǭp 2 p+1 (1 + q p ) ∀x ∈ x p + δ(x p )B. we obtain that the function Ṽ is smooth outside the origin, positive definite, proper and continuous at the origin. Prove that Ṽ satisfies the decrease condition (2.2). From Rademacher's Theorem (see [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF]), since V is locally Lipschitz, it is differentiable almost everywhere in R n , and by Remark 2.5, for every point x where it is differentiable, we have ∇V (x) ⊂ ∂V (x). Fix x ∈ R n \ {0}. By construction of the family {U i } ∞ i=1 , there exists i 0 such that x ∈ U i 0 . So if x ∈ Supp(ψ p ) for some p, then x ∈ x p + δ(x p ) 2 B and then x p + δ(x p ) 2 B ∩ U i 0 = ∅ that implies (since δ(x p ) ≤ x p ):

x p + x p 2 B ∩ U i 0 = ∅.
We deduce by the definition of δ(x p ) that δ(x p ) ≤ δ i 0 and hence that x + δ(x p ) 2 B is included in U i 0 +δ i 0 B. On the other hand, there exists v i 0 ∈ F x i 0 (x) which satisfies (4.20) with x = x i 0 . We obtain by (4.25) and (4.26) that for any p with x ∈ Supp(ψ p ):

∇V νp (x), v i 0 =

≤

-

ǫ x i 0 2 < 0.
Therefore since v i 0 belongs to the convex hull of {f (x, u) : u ∈ W x i 0 }, there exists some f (x, u) which satisfies (2.2).

We have shown that if x ∈ R n \ {0} and if the function V is differentiable at x, then (2.2) is satisfied with a compact set W x i 0 which depends on x but which could be taken to be uniform for all x with x ≤ r . Let us emphasize that since Ṽ is smooth outside the origin and since V is differentiable almost everywhere, our result implies (also by (H1)) that (2.2) is satisfied for any x = 0. However V is not C ∞ at the origin. By Lemma 4. Hence it will satisfy the decrease condition (2.2) and will be a smooth control-Lyapunov function.

It remains to prove (ii) =⇒ (iii). Let us denote by V the function given by the assertion (ii). By definition, for any positive integer i there exists a compact set W i such that ∀y ∈ 2iB \ {0}, min

IGD, , ,

can easily developp all our lemmae in this case. In this way, we obtain some result close to the theorem presented by Praly and Teel in [START_REF] Teel | A smooth Lyapunov function from a class-KL estimate involving two positive semidefinite functions[END_REF].

1 3 1 3

 11 u.(the function f is continuous and Lipschitz in u uniformly in x.) The feedback law u(x) = 2x is a stabilizing feedback and moreover there does not exist other stabilizing feedbacks which are locally Lipschitz at the origin. Let us now looking at the integrator system ẋ = x 1 3u u = v, It does not possess any continuous stabilizing feedback. Coron and Rosier give the following function:

Proposition 4 . 5 .

 45 Let G be a multivalued map from R n into 2 R n satisfying (H1) and which is locally Lipschitz and bounded (by a constant m), let L : R n → R be a locally Lipschitz function and [0, b] be an interval on the real line. Then the value function defined as follows V (x) := max b 0 L(x(t))dt , where the maximum is taken over all trajectories of ẋ(t) ∈ G(x(t)) a.e. (4.2)

4 . 6 .

 46 ) where L x,b denotes the maximum of L(x(b)) over all the trajectories of (4.2) starting at x and defined on [0, b]. Remark Since the multivalued map G is uniformly bounded, the reachable sets of its corresponding differential inclusion at time b are compact and hence the L x,b 's are well-defined. Proof. Let us begin by proving that the value function V is locally Lipschitz on R n ; fix r > 0. Consider x in rB. Since the multifunction G is bounded by m, for any trajectory z(•) of (4.2) with z(0) = z, one has z(t) ≤ z + tm, ∀t ∈ [0, b]. If we consider a sequence of trajectories (x n (•)) n such that the quantities b 0

Tp+1 0 L

 0 p+1 (x(s))ds : ẋ ∈ F p+1 (x(t))a.e. and x(0) = x .Since the functions B p and L p+1 are locally Lipschitz and nonnegative (positive outside B), and since Tp+1 ≥ T (1, 2M (R p+1 ) + 2), Proposition 4.5 implies that V p+1 is locally Lipschitz, proper, nonnegative (positive outside B) and satisfies for any x with x ≤ 2M (R p+1 ) + 2:

. 13 )

 13 On the other hand, it is easy to show that ∀x ∈ B, ∀ζ ∈ ∂V 0 (x), ∀v ∈ F (x), ζ, v ≤ 0.(4.14)

V 0 d

 0 p (x) := max T p Sp (x(s))ds : ẋ ∈ F p (x(t))a.e. and x(0) = x , where S p denotes the ball r p B. Each of these value functions V p is locally Lipschitz and satisfies by the good choice of T p : ∀x ∈ 2B, ∀ζ ∈ ∂V p (x), ∀v ∈ F (x), ζ, v ≤ -d Sp (x). (4.15)

  ) and (4.18) that for any y ∈ x+ x 2 B, there exists v ∈ F x (y) such that ∀ζ ∈ ∂V (y), ζ, v ≤ -ǫ x .(4.[START_REF] Krasovskiȋ | Stability of motion. Applications of Lyapunov's second method to differential systems and equations with delay[END_REF]

(4. 24 ) 2 V

 242 Let us define a functionṼ (x) := p ψ p (x)V νp (x) if x = 0, 0 if x = 0. IGD, , ,It is clearly differentiable on R n \ {0}. Furthermore, since for anyx ∈ R n | Ṽ (x) -V (x)| ≤ p ψ p (x)|V (x) -V νp (x)| ≤ 1 (x).

(4. 25 )

 25 On the other hand, Lebesgue's Theorem (because the function is locally Lipschitz) implies that for anyx ∈ R n \ {0}, ∇V ν (x) = R n ∇V (x + νy)ω(y)dy.(4.26)

  R n ∇V (x + ν p y), v i 0 ω(y)dy ≤ R n -ǫ x i 0 ω(y)dy ≤ -ǫ x i 0 . (4.27)Now, if the function V is differentiable at x, we can compute:IGD, , , ∇ Ṽ (x), v i 0 = ∇V (x), v i 0 + p ψ p (x) ∇V νp (x) -∇V (x), v i 0 + p ∇ψ p (x), v i 0 (V νp (x) -V (x)) = p ψ p (x) ∇V νp (x), v i 0 + p ∇ψ p (x), v i 0 (V νp (x) -V (x)) ≤ p ψ p (x)(-ǫ x i 0 ) + p ∇ψ p (x) v i 0 |V νp (x) -V (x)| ≤ -ǫ x i 0 + p ǭp 2 p+1

  3 in[START_REF] Lin | A smooth converse Lyapunov theorem for robust stability[END_REF], one can find a functionβ : [0, ∞[→ [0, ∞[ which is increasing, positive, C ∞ on ]0, ∞[ and such that the function V (•) = β( Ṽ (•)) is C ∞ on R n .For this new function, we will have∂ V (x) = β ( Ṽ (x))∂ Ṽ (x).

  u∈W i ∇V (y), f (y, u) < 0. (4.28) For each x ∈ R n , we set W x := W [x]+1 (where [x] denotes the integer part of x ). Define for any x ∈ R n : h(x) := inf y∈R n {min u∈Wy ∇V (y), f (y, u) + xy }. (4.29) This new function is 1-Lipschitz and positive definite (by 4.28), and moreover we have by construction: h(x) ≤min u∈Wx ∇V (x), f (x, u) .

  By setting M p the maximum of V p over the ball 2B, we can truncate it as follows:We define for any x ∈ R n ,V p := min{M p , V p (x)}.First we mention that the maximum M p of V p is achieved on the boundary of 2B (this comes from (4.15) and the fact that 0 is a generalized gradient at the local maximum if it is in 2B). Furthermore Let us notice that V p is globally Lipschitz (let us denote its Lipachitz constant by K p ) and bounded by M p . There is in fact a connection between its generalized gradients and those V p : ∀x ∈ B, ∂V p (x) = ∂V p (x) and ∀x / ∈ B, ∂V p (x) ⊂ co{0, ∂V p (x)}.(4.16)

	IGD, , ,

  .20) This construction being done for any x in R n \ {0}, we can extract from the family {U x } x a locally finite open cover {U i } ∞ i=1 of R n \ {0} (where the U i 's are relatively compact open sets whom the closure does not contain the origin) corresponding to families {x i } ∞ i=1 and {δ i } ∞ i=1 . We set for any x in R n \ {0}:

This allows us to define for any x ∈ R n \ {0} the following nonempty set: G(x) := {u ∈ W x : ∇V (x), f (x, u) ≤ -h(x)} .

(4.30)

Consider a selection u(•) of G (extended by 0 at the origin). By the good choice of the compact sets W x , it is locally bounded. Besides, by linearity of the scalar product, we have for any x = 0 and for any v ∈ K u (x),

By Theorem 15.1 of [START_REF] Filippov | Differential Equations with Discontinuous Righthand Sides[END_REF] we conclude that the differential inclusion

is globally asymptotically stable at the origin. Now, if instead of a feedback of Krasovskii type we want to construct a feedback of Filippov type, we have to prove that the multivalued map K which is considered above is measurable. Thus we can extract a measurable selection (see [START_REF] Aubin | Set-valued analysis[END_REF]) which will allow us to set the differential inclusion given by Filippov. Then we conclude with the same proof and hence the proof of Theorem 2.7 is complete. We have still to demonstrate Theorem 2.8.

In the case of an affine system, that is when

we consider the same compact sets W x as above and the same function h given by (4.29). We define for any x ∈ R n \ {0}:

Since the function f is affine in the control, the multivalued map K has nonempty convex compact values. In addition, since f and ∇V are both continuous and since the sets G(x) given by (4.30) are nonempty, we deduce that the multivalued map K is also lower semicontinuous in R n \ {0} (we refer to [START_REF] Deimling | Multivalued Differential Equations[END_REF] and [START_REF] Aubin | Set-valued analysis[END_REF] for the definition of lower semicontinuity). So, we can apply the famous Michael's Theorem (see [START_REF] Michael | Continuous selections[END_REF]) and get the existence of a continuous selection u : R n \ {0} → U . Of course, we set u(0) = 0; by the same proof as for (ii) =⇒ (iii), we get that the dynamical system

is globally asymptotically stable. Now if the last assumption of Theorem 2.8 is satisfied, then the function K can be extended to R n by setting K (0) = {0}. It is with nonempty convex compact values and furthermore it is henceforth lower semicontinuous in R n . We deduce the existence of a continuous selection on R n such that u(0) = 0. Still once we conclude by the proof of (ii) =⇒ (iii) of Theorem 2.7. Theorem 2.8 is proved.

Further comments

In view of the preceding proof, we can easily generalize Theorem 2.7 to the case of stabilization toward a compact set. As a matter of fact, since our proof is mainly based on the compacity of the basin of attraction, we IGD, , ,


Ludovic Rifford
email: rifford@desargues.univ-lyon1.fr.

A MORSE-SARD THEOREM FOR THE DISTANCE FUNCTION ON RIEMANNIAN MANIFOLDS

Keywords: 1991 Mathematics Subject Classification. 53C22, 28A78, 49J52. L. Rifford:

We prove that the set of critical values of the distance function from a submanifold of a complete Riemannian manifold is of Lebesgue measure zero. In this way, we extend a result of Itoh and Tanaka.

Introduction

A well-known and widely used property of critical values of differentiable mappings is given by the Morse-Sard theorem [START_REF] Morse | The behavior of a function on its critical set[END_REF][START_REF] Sard | The measure of the critical values of differentiable maps[END_REF]: if a mapping is C k -smooth with k sufficiently big, then the set of its critical values has the Lebesgue measure zero. In this article, we prove that the Morse-Sard theorem holds when the smooth function is replaced by the distance function from a C ∞ -smooth submanifold in a complete C ∞ -smooth Riemannian manifold of any dimension. Therefore we settle an open question asked by Itoh and Tanaka [START_REF] Itoh | A Sard theorem for the distance function[END_REF] who proved this result in dimension less than 5. Our proof is mainly based on the Yomdin's method which relies on semialgebraic sets, and on a result by Itoh and Tanaka [START_REF] Itoh | The Lipschitz continuity of the distance function to the cut locus[END_REF], which established the Lipschitz continuity of the distance function to the cut locus. Notice that our main theorem can indeed be viewed as a corollary of a more general result about the minimum of smooth functions on compact manifolds. In order to state our results, we present the notion of critical points of the distance function which is not smooth but only Lipschitz continuous.

Let N be a closed C ∞ -smooth submanifold of a complete C ∞ -smooth Riemannian manifold M . The distance function d N (•) from the submanifold N is defined on M by ∀p ∈ M, d N (p) := min{d(p, q) : q ∈ N }.

A unit speed geodesic segment γ : [0, a] → M emanating from N is called a N -segment if t = d N (γ(t)) on [0, a]. A point p ∈ M \ N is called a critical point of the distance function d N if for any unit tangent vector v at p, there exists an N -segment γ : [0, d N (x)] → M through p such that the angle made by v andγ(d N (p)) is not greater than π 2 ; we denote by C(d N ) the set of critical points of the distance function d N . Notice that this definition of critical point is equivalent to say that the point p contains zero in its Clarke generalized gradient ∂d N (p); we will return to this caracterization of critical points at the end of this section. In this article, we prove that the set of all critical values of d N has the Lebesgue measure zero.

Theorem 1. Let N be a closed C ∞ -smooth submanifold of a complete C ∞smooth Riemannian manifold M . Then the set of critical values of the distance function from N is of Lebesgue measure zero.

Remark 1.1. Twenty years ago, Fu proved in [START_REF] Fu | Tubular neighborhoods in Euclidean spaces[END_REF] that the set of all critical values of the distance function from a compact subset in R 3 is of Lebesgue measure zero. Following Whitney [START_REF] Whitney | A function not constant on a connected set of critical points[END_REF], S. Ferry [START_REF] Ferry | When -boundaries are manifolds[END_REF] gave a counterexample to this result in dimension greater than three.

Actually, we are able to improve Theorem 1 whenever the ambient manifold is the Euclidean space R n . Theorem 2. Let N be a closed submanifold of dimension d and of class C k in R n with k > 2d(n + 1). Then the set of critical values of the distance function from N is of Lebesgue measure zero.

We do not know if the lower bound on k that we give in Theorem 2 is optimal. Note that both theorems above imply that for almost all positive numbers c < sup{d N (x)}, the set of all points whose distances from N are c is a Lipschitz hypersurface of M . Actually, Theorem 1 is a consequence of a more general result that we proceed to explain.

Let N be a compact smooth manifold and let φ : R n × N → R be a smooth function. Define the function f : R n → R by ∀x ∈ R n , f (x) := min q∈N {φ(x, q)}.

Though the function f is not smooth, we are able to define some notion of critical points for f . Since the manifold N is compact, it is straightforward to show that f is indeed Lipschitz continuous. Thus by Rademacher Theorem (see [START_REF] Rademacher | Uber partielle und totale differenzierbarkeit von funktionen mehrerer variabeln und über die transformation der doppelintegrale[END_REF]), it is differentiable almost everywhere in R n . Denote by D f the set of points where f is differentiable and define the generalized gradient of f at x ∈ R n as follows:

∀x ∈ R n , ∂f (x) = co lim i→∞ ∇f (x i ) : x i -→ i→∞ x, x i ∈ D f ,
where co{A} denotes the convex hull of A ⊂ R n . This tool was introduced by Clarke [START_REF] Clarke | Generalized gradients and applications[END_REF] in 1973. Since that time, a complete calculus has been developed, one that extends all the theorems of the usual smooth calculus. A point x ∈ R n will be called a critical point of f if 0 ∈ ∂f (x), the critical values of f are the images of its critical points, and we denote by C(f ) the set of critical points of the function f . Since the function f is defined as the minimum of smooth function, we can compute its generalized gradients (we refer the reader to [3, Exercise 9.13 p. 99] for such a result):

∀x ∈ R n , ∂f (x) = co{∇ x φ(x, q) : q ∈ argminf (x)}, (1)
where ∇ x φ(x, q) denotes the derivative of φ in the first variable and where argminf (x) := {q ∈ N : f (x) = φ(x, q)}. In the context of distance functions, as we will see in the proof of Theorem 1, this formula will imply that both definitions of critical points coincide. In consequence, Theorem 1 will be a corollary of the following. As we said, our proofs are strongly based on the Yomdin's proof of Sard theorem; they demonstrate the power of the Yomdin's method. Our paper is organized as follows: in Section 2 we develop preliminaries on semialgebraic sets, and we give the proofs of our results in Section 3.

Throughout this paper, R denotes the set of real numbers, • the Euclidean norm of R n , B n the open ball {x ∈ R n : x < 1}, B n the closure of B and B n (x, r) = x + rB n (resp. B n (x, r) = x + rB n ) the ball (resp. the closed ball) centered at x with radius r. Finally if A is a subset of R n then coA denotes its convex hull.

Preliminaries on semialgebraic sets

2.1. Definitions and properties. We refer the reader to [START_REF] Coste | Ensembles semi-algébriques[END_REF][START_REF] Lojasiewicz | Ensembles semi-analytiques[END_REF] for surveys on semialgebraic sets and for the proofs of the propositions below. Moreover we refer the interested reader to the excellent book [START_REF] Bochnak | Real algebraic geometry, volume 36 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3)[END_REF] for a very detailed study of semialgebraic sets. A set A ⊂ R n is called semialgebraic if it can be obtained by a finite number of the union and the intersection operations from the sets of the form {f = 0}, {g > 0}, where f, g are polynomials on R n . For given such a representation of A we call its set-theoretic formula together with the dimension n of the ambient space and with the degrees of polynomials in it, the diagram of this representation of A. Hence any constant which depends only on the diagram of A can indeed be replaced by another constant which depends only on the number and the degrees of polynomials defining A. We give below three classical propositions on semialgebraic sets; we usually say that these results are consequences of the Tarski-Seidenberg principle.

Proposition 2.1. The number of connected connected components of a semialgebraic set is bounded by a constant which depends only on the number and the degrees of polynomials which define that set.

Proposition 2.2. Let A ⊂ R n be a semialgebraic set. Then the sets Ā, ∂A, each connected component of A, π(A), where π denotes the projection of R n on R k ⊂ R n , are semialgebraic, and the diagram of each of these sets depends only on the diagram of A.

Proposition 2.3. Let A ⊂ R n be a semialgebraic set. If P 1 , • • • , P q are q polynomials on R n of degree less or equal than d, then the set

{(P 1 (a), • • • , P q (a)) : a ∈ A}
is a semialgebraic set in R q ; its diagram depends only on the diagram of A, d and q.

Our results will be strongly based on the following theorem; we refer the reader to Yomdin's papers [START_REF] Yomdin | The geometry of critical and near-critical values of differentiable mappings[END_REF] and [START_REF] Yomdin | Metric properties of semialgebraic sets and mappings and their applications in smooth analysis[END_REF] for the proof of it.

Theorem 4. For any semialgebraic set A ⊂ R n inside the ball of a given radius r, there is a constant K depending only on the number and degrees of polynomials defining A, such that any x, y belonging to the same connected component of A, can be joined by a connected semialgebraic curve s of length not greater than Kr.

The semialgebraic case

: quasi critical values. Let S be a positive integer. Let A 1 , • • • , A S be S compact semialgebraic sets in R m and let P 1 (•, •), • • • , P S (•, •) be S polynomials on R n × R m . We define f P : R n → R by ∀x ∈ R n , f P (x) := min {P s (x, a) : a ∈ A s , s ∈ {1, • • • , S}} . Since the sets A 1 , • • • , A S are
compact, the function f P is locally Lipschitz on R n . Let δ, µ be two positive constants, let us define the notion of (δ, µ)critical points of f P . For every x ∈ R n , we denote by argmin

δ (x) the set of couples (a, s) in R m × R such that s ∈ {1, • • • , S}, a ∈ A s , and P s (x, a) ≤ f P (x) + δ.
We will say that x ∈ R n is a (δ, µ)-critical point of f P if the following property is satisfied:

µB ∩ co ∇ x P s (x, a) : (a, s) ∈ argmin δ (x) = ∅.
We denote by C δ,µ (f P ) the set of (δ, µ)-critical points of the function f P and we prove the following result.

Theorem 5. Let A 1 , • • • , A S be S compact semialgebraic sets in R m and P 1 (•, •), • • • , P S (•, •) be S polynomials in R n × R m . Denote by K P > 0 some uniform Lipschitz constant of the polynomials P 1 , • • • P S on the set B n × (A 1 ∪ • • • ∪ A S )
and denote by M P > 0 some upper bound of the D2

x P 1 , • • • , D 2 x P S on the same set 1 . Then for any constants δ, µ, r such that 0 < δ, µ ≤ r ≤ 1, the set

f P (C δ,µ (f P ) ∩ B n r ) is included in the union of N intervals of length Krµ + 2M P Krδ + K P 2 M P √ δ,
where the constants N and K depend only on the diagrams of the semialgebraic sets A 1 , • • • , A S , on the integer S and of the degrees of the polynomials

P 1 (•, •), • • • , P S (•, •).
Proof.

Lemma 1. The set C δ,µ (f P ) is semialgebraic; its diagram depends only on the diagrams of A 1 , • • • , A S , on S and on the degrees of polynomials

P 1 (•, •), • • • , P S (•, •).
Proof. By the Carathéodory theorem (cf. Theorem 17.1 in [START_REF] Rockafellar | Convex analysis[END_REF]), the set C δ,µ (f P ) can be described as the set of points x ∈ R n which satisfy the two following conditions:

(i) there exist s 1 , • • • , s n+1 in {1, • • • , S}, there exist a 1 , • • • , a n+1 in R m such that, ∀i ∈ {1, • • • , n + 1}, a i ∈ A s i ,
and ∀s ∈ {1, • • • , S}, ∀a ∈ A s , P s i (x, a i ) ≤ P s (x, a) + δ;

(ii) there exists also (λ

1 , • • • , λ n+1 ) in [0, 1] with n+1 i=1 λ i = 1 such that n+1 i=1 λ i ∇ x P s i (x, a i ) 2 ≤ µ 2
Denote by E the set of tuples (x,

s 1 , • • • , s n+1 , a 1 , • • • , a n+1 , λ 1 , • • • , λ n+1 ) in R n × {1 • • • , S} n+1 × (R m ) n+1 × [0, 1] n+1 such that ∀i ∈ {1, • • • , n + 1}, a i ∈ A s i .
Since the sets A 1 , • • • , A S are semialgebraic, the set E can be represented as a finite union (over all possible (n+1)-tuples (s

1 , • • • , s n+1 )) of semialgebraic sets (of the form R n × {s 1 }, • • • {s n+1 } × A s 1 × • • • × A sn+1 × [0, 1] n+1
), so it is a semialgebraic set. Moreover its diagram depends only on the diagrams of the

A 1 , • • • , A S . Let F be the set of tuples (x, s 1 , • • • , s n+1 , a 1 , • • • , a n+1 , λ 1 , • • • , λ n+1 , s, a) in R n × {1 • • • , S} n+1 × (R m ) n+1 × [0, 1] n+1 × {1, • • • , S} × R m such that ∀i ∈ {1, • • • , n + 1}, a i ∈ A s i , a ∈ A s ,
R n × {1 • • • , S} n+1 × (R m ) n+1 × [0, 1] n+1 × {1, • • • , S} × R m into R n × {1 • • • , S} n+1 × (R m ) n+1 × [0, 1] n+1
which suppress the two last "coordinates" of the tuple, that is, which maps the tuple (x,

s 1 , • • • , s n+1 , a 1 , • • • , a n+1 , λ 1 , • • • , λ n+1 , s, a) to the tuple (x, s 1 , • • • , s n+1 , a 1 , • • • , a n+1 , λ 1 , • • • , λ n+1 ). It is straightforward to show that the set E \ π 1 (F ) corresponds to the set of tuples (x, s 1 , • • • , s n+1 , a 1 , • • • , a n+1 , λ 1 , • • • , λ n+1 ) in R n × {1 • • • , S} n+1 × (R m ) n+1 × [0, 1] n+1 which satisfy ∀i ∈ {1, • • • , n + 1}, a i ∈ A s i , (2)
and ∀i ∈ {1, • • • , n + 1}, ∀s ∈ {1, • • • , S}, ∀a ∈ A s , P s i (x, a i ) ≤ P s (x, a) + δ,
which can also be written as,

∀i ∈ {1, • • • , n + 1}, P s i (x, a i ) ≤ f P (x) + δ. (3)
Therefore, if we denote by π 2 the projection

π 2 : R n × {1 • • • , S} n+1 × (R m ) n+1 × [0, 1] n+1 -→ R n (x, s 1 , • • • , s n+1 , a 1 , • • • , a n+1 , λ 1 , • • • , λ n+1 ) -→ x
then (by ( 2), ( 3)) the set C δ,µ (f P ) corresponds to π 2 (E ∩ E ) where the set E denotes the set of tuples (x,

s 1 , • • • , s n+1 , a 1 , • • • , a n+1 , λ 1 , • • • , λ n+1 ) such that n+1 i=1 λ i = 1,
and

n+1 i=1 λ i ∇ x P s i (x, a i ) 2 ≤ µ 2 .
Since the projection of a semialgebraic set is semialgebraic by Proposition 2.2, and since all inequalities that appear in (i)-(ii) are polynomial in the coordinates of their variables, we deduce that C δ,µ (f P ) is semialgebraic and that its diagram depends only upon that of the sets A 1 , • • • , A S , on the integer S and of the degrees of the polynomials

P 1 (•, •), • • • , P S (•, •).
Returning to the proof of Theorem 5. Denote by D i each connected component of C δ,µ (f P ) ∩ B n r and let E i := f P (D i ). Let x, y be two points belonging to the same connected component of D i . By Theorem 4 there exist some constant K > 0 independent on i and a curve S (parametrized by arc-length) of length l ≤ Kr, joining x and y in D i . Set

ρ := µ + √ 2M P δ. Let 0 ≤ t i < t i+1 ≤ l such that t i+1 -t i = ρ-µ
M P ; we are going to prove that

f P (x(t i+1 )) -f P (x(t i )) ≤ ρ(t i+1 -t i ). (4)
We argue by contradiction; so let us assume that

f P (x(t i+1 )) -f P (x(t i )) > ρ(t i+1 -t i ).
Since x(t i ) ∈ C δ,µ (f P ), there exists (ā, s) ∈ argmin δ (x(t i )) such that

x(t i+1 )x(t i ), ∇ x P s(x(t i ), ā) ≤ µ x(t i+1 )x(t i ) .

By Taylor formula at second order, we deduce

P s(x(t i+1 ), ā) ≤ P s(x(t i ), ā) + ∇ x P s(x(t i ), ā), x(t i+1 ) -x(t i ) • • • + M P 2 x(t i+1 ) -x(t i ) 2 ,
which, since P s(x(t i ), ā) ≤ f P (x(t i )) + δ and f P (x(t i+1 )) ≤ P s(x(t i+1 ), ā) implies,

f P (x(t i+1 )) ≤ f P (x(t i )) + δ + µ x(t i+1 ) -x(t i ) + M P 2 x(t i+1 ) -x(t i ) 2 .
Remark that since the curve S is parametrized by arc-length, we have x(t i+1 )x(t i ) ≤ (t i+1t i ). Hence we deduce

f P (x(t i+1 )) -f P (x(t i )) ≤ δ + µ(t i+1 -t i ) + M P 2 (t i+1 -t i ) 2 .
Since we are assuming f P (x(t i+1 ))

-f P (x(t i )) > ρ(t i+1 -t i ), this implies (t i+1 -t i ) ρ -µ - M P 2 (t i+1 -t i ) < δ. (5)
By definition of ρ and t i+1 -t i , the left-hand side of ( 5) equals δ hence we get a contradiction. We conclude that for every interval I = [t i , t i+1 ] ⊂ [0, l] of length ρ-µ M P , we have f P (x(t i+1 ))-f P (x(t i )) ≤ ρ(t i+1 -t i ). In particular, this implies that on each subinterval of [0, l] of the form I = [0, t] with t = k ρ-µ M P and where k is a positive integer, we have

f P (x(t)) -f P (x(0)) ≤ ρt ≤ ρl ≤ ρKr.
The interval [0, l] on which the curve S is defined, can be partitionned into an interval of length of the form k ρ-µ M P and another subinterval of length less than ρ-µ M P . Since the polynomial P is Lipschitz with constant

K P on B n × (A 1 ∪ • • • ∪ A S )
, the function f P has the same Lipschitz constant on B n , hence we deduce that

f P (y) -f P (x) ≤ Krρ + K P ρ -µ M P = Krµ + 2M P Krδ + K P 2 M P √ δ.
Since x and y are arbitrary points in D i and since the number of connected components of C δ,µ (f P ) ∩ B n r depends only on the diagrams of A and P (•, •), we conclude.

Proof of our results

3.1. Proof of Theorem 3. Fix some compact cube T in U and let us prove that meas(f (C(f ) ∩ T )) = 0. Without loss of generality, we can assume that the n-cube T has sides of length 1. By Whitney embedding's theorem (see [START_REF] Sternberg | Lectures on differential geometry[END_REF]Theorem 4.4 p. 63], without loss of generality, we can assume that the manifold N is a compact submanifold of R m of dimension d and of class C k . Therefore the function φ can be smoothly extended to some neighbourhood V of N in R m in such a way that the function

φ : U × V -→ R (x, q) -→ φ(x, q)
is a C k -smooth function. Let V ⊂ V be a compact neighbourhood of N ; for the rest of the proof we denote by K φ the Lipschitz constant of the function φ on T × V and we denote by R 2 φ (resp. R k φ ) the maximum of the norms d(∇ x φ) (resp. d k φ(x, q) ) for x ∈ T and q ∈ V . Furthermore, there exists a covering of N by a finite family of open sets (O i ) i∈I in V such that for each i ∈ I, O i is included in some cube of side 1 and such that there exists a diffeomorphism Ψ i :

C m → O i of class C k which satisfies Ψ i (C m ∩ (R d × {0 m-d })) = O i ∩ N,
where C m is some cube of dimension m in R m with sides of length less or equal than one (in fact for sake of simplicity, we can assume that C m is the open unit cube of R m ). From now we denote by K Ψ the maximum of Lipschitz constants of the mappings Ψ i (i ∈ I) and by R k Ψ the maximum of the d k Ψ i (i ∈ I) over C m (without loss of generality, we can assume that the Ψ i 's are defined on some uniform compact set containing the cube C m ). Let l be a positive integer, set r := 1 l . The cube

C d := C m ∩ (R d × {0 m-d }) can be divided into l d cubes D j (j = 1, • • • , l d ) of side r.
For each i ∈ I and each j, let P i,j be the (k -1)-Taylor polynomial 2 of Ψ i at the center of the cube D j . We have that for any x ∈ D j ,

Ψ i (x) -P i,j (x) ≤ R k Ψ r k . (6)
By Proposition 2.3, for each i, j, the set P i,j (D j ) is semialgebraic in R m . Moreover by definition of K Ψ , there exists c i,j ∈ P i,j (D j ) such that

P i,j (D j ) ⊂ B n (c i,j , K Ψ r) . (7)
For sake of simplicity, let us set for each couple (i, j), E i,j := P i,j (D j ). The cube T in U can be divided in smaller cubes as well. Thus there exists l n cubes T t (t = 1, • • • , l n ) of side r which cover T ; denote by x t the center of each cube T t . For each (i, j, t), let P t i,j (x, q) be the (k -1)-th Taylor polynomial of φ(•, •) at the point (x t , c i,j ). By [START_REF] Itoh | The Lipschitz continuity of the distance function to the cut locus[END_REF] and by the definition of R k φ , we have for every x ∈ T t and for every q ∈ E i,j , |φ(x, q) -

P t i,j (x, q)| ≤ R k φ K k Ψ r k (8)
∇ x φ(x, q) -∇ x P t i,j (x, q) ≤ R k φ K k-1 Ψ r k-1 (9)
d(∇ x φ)(x, q) -d(∇ x P t i,j )(x, q) ≤ R k φ K k-2 Ψ r k-2 . (10)
In this way, we produced one family of polynomials

{P t i,j } (with i ∈ I, j = 1, • • • , l d and t = 1, • • • , l n ) of degree k -1 and one family {E i,j } (with i ∈ I and j = 1, • • • , l d )
of semialgebraic sets which are all defined by polynomials of degree k -1. Let us call a configuration, one (2n + 3)tuple of the form (i

1 , i 2 , • • • , i n+1 , j 1 , j 2 , • • • , j n+1 , t) where t ∈ {1, • • • , l n },
where each i s (s = 1, • • • , n + 1) belongs to I, and where each j s (s = 1, • • • , n + 1) belongs to the set {1, • • • , l d }. We denote by S a configuration which corresponds to some (2n + 3)-tuple of the form above. Notice that there are exactly |I| n+1 l d(n+1) l n such (2n+3)-tuples. We claim the following. Lemma 2. There exist two positive constants M 1 , M 2 such that if the point x belongs to C(f ) ∩ T , then there exists some configuration S such that x is a (M 1 r k , M 2 r k-1 )-critical point of the mapping

f S : T t -→ R x -→ min{P t is,js (x, q s ) : q s ∈ E is,js , s ∈ {1, • • • , n + 1}}. Moreover we have, |f (x) -f S (x)| ≤ M 1 r k . ( 11
)
Proof. First, since x is in T , then there exists t ∈ {1, • • • , l n } such that x ∈ T t . Moreover, since it is a critical value of f , by (1) there exist q1 ,

• • • , qn+1 in the manifold N and λ 1 , • • • , λ n+1 ∈ [0, 1] n+1 with n+1 s=1 λ s = 1 such that n+1 s=1 λ s ∇ x φ(x, qs ) = 0, (12)
with f (x) = φ(x, qs ), ∀s = 1,

• • • , n + 1. ( 13
)
2

The function Ψi is defined from C m ⊂ R m into R m , hence its Taylor polynomial denotes the m-tuple with coordinates each Taylor polynomials of the coordinates of Ψi.

Thus each qs belongs to some set O i with i ∈ I. This means that there exist n + 1 couples (i 1 , j 1 ), • • • , (i n+1 , j n+1 ) and n + 1 points x s in D js (s = 1, • • • , n + 1) such that for every s = 1,

• • • , n + 1, Ψ is (x s ) = qs .
Set for each s = 1, • • • , n + 1, let q s := P is,js (x s ). Each q s belongs to E is,js and moreover by [START_REF] Fu | Tubular neighborhoods in Euclidean spaces[END_REF],

∀s = 1, • • • , n + 1, qs -q s ≤ R k Ψ r k . (14)
By ( 9)-( 10) and ( 12) we obtain,

n+1 s=1 λ s ∇ x P t is,js (x, q s ) ≤ n+1 s=1 λ s ∇ x P t is,js (x, qs ) + n+1 s=1 λ s (∇ x P t is,js (x, q s ) -∇ x P t is,js (x, qs )) ≤ R k φ K k-1 Ψ r k-1 + (R 2 φ + R k φ K k-2 Ψ r k-2 )R k Ψ r k . (15)
Furthermore, by ( 8), ( 13) and ( 14), we have

f S (x) -f (x) = f S (x) -φ(x, q1 ) ≤ f S (x) -φ(x, q 1 ) + K φ R k Ψ r k ≤ P t i 1 ,j 1 (x, q 1 ) -φ(x, q 1 ) + K φ R k Ψ r k ≤ R k φ K k Ψ r k + K φ R k Ψ r k . ( 16
)
On the other hand, if q = P i,j (p) belongs to E i,j then since Ψ i (p) ∈ O i ∩ N we get by [START_REF] Itoh | A Sard theorem for the distance function[END_REF],

|P t i,j (x, q) -φ(x, Ψ i (p))| ≤ |P t i,j (x, q) -φ(x, q)| + K φ R k Ψ r k ≤ R k φ K k Ψ r k + K φ R k Ψ r k . (17)
Hence we deduce that for some q = Pī , j (p) ∈ Eī , j such that f S (x) = P t ī, j (x, q), we have

f (x) -f S (x) ≤ f (x) -P t ī, j (x, q) ≤ φ(x, Ψī(p)) -P t ī, j (x, q) ≤ R k φ K k Ψ r k + K φ R k Ψ r k . (18)
Consequently, by ( 13), ( 17) and [START_REF] Yomdin | The geometry of critical and near-critical values of differentiable mappings[END_REF], we obtain that for every s = 1, • • • , n+ 1,

P t is,js (x, qs ) ≤ [P t is,js (x, qs ) -φ(x, qs )] + φ(x, qs ) ≤ R k φ K k Ψ r k + K φ R k Ψ r k + f (x) ≤ f S (x) + 2(R k φ K k Ψ + K φ R k Ψ )r k . (19)
Set

M 1 := 2(R k φ K k Ψ + K φ R k Ψ ),
and 15) and ( 19), we conclude that x is a (M 1 r k , M 2 r k-1 )-critical point of f S . Moreover by ( 16) and ( 18), we conclude that

M 2 := R k φ K k-1 Ψ + (R 2 φ + R φ k K k-2 Ψ r k-2 )R k Ψ r. By (
|f (x) -f S (x)| ≤ M 1 r k .
The proof of the lemma is complete.

Returning to the proof of Theorem 3. Since the semialgebraic sets E i,j are all defined by a fix number of polynomials of degree k -1, since the polynomials P t i,j have degree k -1 and since their Lipschitz constants and their C 2 norms are uniform on T × V by ( 8)-( 10) (let us call them K P and M P ); by Theorem 5, we deduce that there exists two positive constants N and K such that for each configuration S,

f (C(f S ) ∩ T t ) is included in N intervals of length Krµ + 2M P Krδ + K P 2 M P √ δ,
where δ := M 1 r k and µ := M 2 r k-1 . By [START_REF] Morse | The behavior of a function on its critical set[END_REF] we conclude that for r small enough, we have

meas(f (C(f ) ∩ T )) ≤ |I| n+1 l d(n+1) l n 2K P 2M 1 M P 1 √ l k ≤ |I| n+1 2K P 2M 1 M P l n+d(n+1)-k 2 .
As l tends to infinity, since k > 2n + 2(n + 1)d , we deduce that meas(f (C(f ) ∩ T )) = 0, which conclude the proof of Theorem 3. Since N is a C ∞ -smooth submanifold of M , there exists ǫ > 0 small enough such that the set

N := {x ∈ M : d N (x) = ǫ} is a C ∞ -smooth hypersurface of M , and such that the distance function d N (•) is of class C ∞ on the set {x ∈ M : d N (x) ∈ (0, 2ǫ)}. For such an ǫ > 0, if γ : [0, a] → M is a N -segment joining N to p = γ(a)
with a > ǫ, then the unit speed geodesic segment γ : [ǫ, a] → M is a N -segment joining N to p = γ(a). In particular, this means that the distance functions d N (•) and d N have the same critical points and that

∀p ∈ {x ∈ M : d N (x) > ǫ}, d N (x) = d N (x) + ǫ. (20)
By construction, N is a hypersurface with one normal vector at q ∈ N given by ∇d N (q) of norm 1. Hence, by completeness of M , for every q ∈ N there exists a unique unit speed geodesic γ q (•) : [0, ∞) → M starting at q conclude that both definitions of critical points of d F (d-α/2) coincide. Hence we conclude.

3.3. Proof of Theorem 2. In the Euclidean case, by [START_REF] Bochnak | Real algebraic geometry, volume 36 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3)[END_REF] we have, ∂d N (x) = co xq xq : q ∈ N .

Hence we deduce that x / ∈ N is a critical point of d N (•) if and only if it is a critical point of d N (•) 2 , that is if 0 ∈ co{xq : q ∈ N }.

We conclude that x -→ min{ xq s 2 : q s ∈ E is,js , s ∈ {1, • • • , n + 1}}.

Moreover we have,

|d N (x) 2 -d 2 S (x)| ≤ M 1 r k . ( 22
)
Proof. In this case the function φ(•, •) is already a polynomial, hence we do not have to approximate it by its Taylor polynomial. The calculations are easier than in Lemma 2 above. The proof is let to the reader.

Returning to the proof of Theorem 2. By the same arguments as in the proof of Lemma 3, we deduce that there exists two constants N and K such that for r small enough,

meas(d 2 N (C(d 2 N )) ∩ T ) ≤ |I| n+1 l d(n+1) 2K P 2M 1 M P 1 √ l k ≤ |I| n+1 2K P 2M 1 M P l d(n+1)-k 2 .
As l tends to infinity, since k > 2(n + 1)d , we deduce that measf (C(f ) ∩ T ) = 0, which conclude the proof of Theorem 2.

Theorem 3 .

 3 Let U be an open subsets of R n and let N be a compact manifold of class C k and of dimension d. Let φ : U × N → R be a smooth function of class C k . Then the function f : U → R defined by f (x) := min q∈N {φ(x, q)}, satisfies the following: If k > 2n + 2d(n + 1), then f (C(f )) has Lebesgue measure zero.

  and ∃i ∈ {1, • • • , n + 1} s.t. P s i (x, a i ) > P s (x, a) + δ. By the same argument as above, since P 1 (•, •), • • • , P S (•, •) are all polynomials, the set F is semialgebraic and its diagram depends only on data of the problem. Denote by π 1 the projection from

3. 2 .

 2 Proof of Theorem 1. For each critical point p of the distance function d N (•) we will show that there exists an open neighbourhood U of p such that the Lebesgue measure of d N (U ∩C(d N )) is zero. Then C(d N ) is covered by a union of countably many subsets U i (i ∈ N) such that the Lebesgue measure of each subset d N (U i ∩ C(d N )) is zero. Thus the claim of Theorem 1 is clear. Fix p ∈ C(d N ).

Lemma 3 .

 3 d N (C(d N )) = { √ ρ : ρ ∈ d 2 N C(d2N )}. Since the function ρ → √ ρ is locally Lipschitz on (0, ∞), it suffices to show that the set of all critical values of d N (•) 2 has Lebesgue measure zero. As in the proof of Lemma 3, if we consider N as a submanifold of R m (by Whitney embedding's theorem), we can approximate N by a finite union of some algebraic sets E i,j with i ∈ I and j ∈ {1, • • • , l d }. So if we call configuration, some (2n + 2)-tuple (i 1, • • • , i n+1 , j 1 , • • • , j n+1 ) with each i s (s = 1, • • • , n + 1)belonging to I, and with each j s (s = 1,• • • , n + 1) belonging to the set {1, • • • , n+1}, we enumerate |I| n+1 l d(n+1) such configurations.We have the following lemma. There exists two positive constants M 1 , M 2 such that if x belongs to C(d N (•) 2 ) ∩ T , then there exists some configuration S such that x is a (M 1 r k , M 2 r k-1 )-critical point of the mappingd 2 S : T -→ R

Here if P (•, •) is a polynomial on R n × R m , then D

x P denotes the second order derivative of the polynomial P in the first variable.

with γq (0) = ∇d N (q). The distance from N to the cut point on γ q is given by ρ(q) := sup{t : γ q|[0,t] is a N -segment}. The function ρ : N → (0, ∞] is called the distance function from N to the cut locus. In [START_REF] Itoh | The Lipschitz continuity of the distance function to the cut locus[END_REF], Itoh and Tanaka proved the following (we refer the reader to [START_REF] Itoh | The Lipschitz continuity of the distance function to the cut locus[END_REF] for the definition of the distance function to the cut locus ρ(•) whenever N is an arbitrary submanifold of M ).

The set Q(p) is clearly compact and nonempty. Hence by Theorem 6, there exist some neighbourhood

On the other hand, there exists a uniform normal neighbourhood W of p and α > 0 such that W is contained in a geodesic ball of radius α around each of its points (see for instance [9, Lemma 5.12 p. 78] for such a result).

In particular, this means that the function

For each element q of Q(p), ρ(q) = d. Moreover by construction of N for each element q of Q(p), the unique unit speed geodesic starting at q and orthogonal to N is minimizing on [0, d]. Hence there exists some neighbourhood V ⊂ V of Q(p) such that V ∩ N is a compact C ∞smooth hypersurface with boundary, such that the closure of the (dα/2)front F (dα/2) of N ∩ V (that is, the closure of the set of γ q (dα/2) for any q ∈ V ) is included in W, and such that

Consequently, by (21), the front F (dα/2) is a compact C ∞ -smooth hypersurface with boundary of M , which is included in W. Moreover there exists some neighbourhood U of p which does not intersect F (dα/2) such that for every p ∈ U ,

We deduce that d N (•) and d F (d-α/2) (•) have locally the same critical points and that d