F H Clarke

Yu S Ledyaev

L Rifford

R J Stern

Feedback Stabilization and Lyapunov Functions

Keywords: Asymptotic stabilizability, discontinuous feedback law, system sampling, locally Lipschitz Lyapunov function, nonsmooth analysis, robustness 0 Mathematical Subject Classification: 93D05, 93D20, 34D20, 26B05

Given a locally defined, nondifferentiable but Lipschitz Lyapunov function, we construct a (discontinuous) feedback law which stabilizes the underlying system to any given tolerance. A further result shows that suitable Lyapunov functions of this type exist under mild assumptions. We also establish a robustness property of the feedback relative to measurement error commensurate with the sampling rate of the control implementation scheme.

Introduction

Consider a standard control system of the form ẋ(t) = f (x(t), u(t)) a.e., u(t) ∈ U,

and let V be a Lyapunov function for the system : We have V (x) ≥ 0, V (x) = 0 iff x = 0, V (x) → ∞ as x → ∞, and (for some function W ) the Infinitesimal Decrease Condition

min u∈U ∇V (x), f (x, u) ≤ -W (x) < 0, x = 0. ( 2
)
It is well-known (but true) that the existence of V implies (open-loop) Asymptotic Controllablity to the origin : for every α ∈ IR n , there is a control u(t) such that the solution x(•) of ( 1) with initial condition x(0) = α satisfies x(t) → 0 as t → ∞ (and in addition, convergence to zero takes place in a certain uniform and stable manner that we will not dwell upon here). A related and important goal in many situations is to produce a state feedback k(•) : IR n → U which stabilizes the system; i.e., such that the system ẋ = f (x, k(x)) is globally asymptotically stable. This article explores the question of how to define such a feedback law through the use of a given Lyapunov function V .

The ideal case, a well-known heuristic useful for motivational purposes, is the one in which we can find a continuous function k(x) that selects a value of u ∈ U attaining (or almost) the minimum in [START_REF] Bacciotti | Local Stabilizability of Nonlinear Control Systems[END_REF] :

∇V (x), f (x, k(x)) ≤ -W (x) ∀x = 0.
Then any solution of ẋ = f (x, k(x)) is such that d dt V (x(t)) = ∇V (x(t)), ẋ(t) ≤ -W (x) < 0, a monotonicity conclusion that, together with the growth property of V , assures that x(t) → 0 as t → ∞.

There are two fundamental difficulties with this ideal picture, and both concern regularity issues. The first is that a differentiable Lyapunov function may not exist, and the second is that even when a smooth V exists, the continuous selection k(•) does not generally exist. If we have recourse to a discontinuous feedback k(•),then the issue arises of how to interpret the discontinuous differential equation ẋ = f (x, k(x)). The primary goal of this article is to give a general answer to the problem of defining a (discontinuous) stabilizing feedback based upon a given (nondifferentiable) Lyapunov function, one for which infinitesimal decrease is known to hold only on a restricted set. The construction is described in section 1, while section 2 establishes that under mild conditions, a Lyapunov function of the type required in the previous section always exists. In the final section, we address the issue of robustness of the feedback with respect to measurement error and small perturbations of the dynamics, a particularly important issue when discontinuity is present. Some works and general references related to the results of this article include [START_REF] Bacciotti | Local Stabilizability of Nonlinear Control Systems[END_REF][START_REF] Bacciotti | Lyapunov and Lagrange stability : inverse theorems for discontinuous systems[END_REF][START_REF] Clarke | Qualitatives properties of trajectories of control systems: a survey[END_REF][START_REF] Coron | Global asymptotic stabilization for controllable systems without drift[END_REF][START_REF] Coron | A relation between continuous time-varying and discontinuous feedback stabilization[END_REF][START_REF] Coron | Feedback stabilization of nonlinear systems : Sufficients conditions and Lyapunov and input-output techniques[END_REF][START_REF] Freeman | Robust Nonlinear Control Design. State-Space and Lyapounov Techniques[END_REF][START_REF] Lakshmikantham | Practical Stability of Nonlinear Systems[END_REF][START_REF] Leitmann | One approach to the control of uncertain dynamical systems[END_REF][START_REF] Lin | A smooth converse Lyapounov theorem for robust stability[END_REF][START_REF] Sontag | A lyapunov-like characterization of asymptotic controllability[END_REF][START_REF] Subbotin | Generalized Solutions of First-Order PDEs[END_REF][START_REF] Tsinias | A lyapunov description of stability in control systems[END_REF]. We proceed now to situate our result with respect to the literature.

The possible nonexistence of continuous stabilizing feedback was brought to light in the seminal work of Sontag and Sussmann [START_REF] Sontag | Remarks on continuous feedback[END_REF] and of Brockett [START_REF] Brockett | Asymptotic stability and feedback stabilization[END_REF]. The latter who developed a necessary condition for continuous stabilizability and adduced the following example, the "non-holonomic integrator" :

   ẋ1 = u 1 ẋ2 = u 2 ẋ3 = x 1 u 2 -x 2 u 1 (u 1 , u 2 ) ∈ B =: U.
This system is globally asymptotically controllable yet fails to admit a continuous stabilizing feedback (by Brockett's condition). In considering the use of discontinuous feedback laws k(•), one could have recourse to the Filippov solution concept [START_REF] Filippov | Differential Equations with Discontinuous Righthand Sides[END_REF] : x is a solution of ẋ = f (x, k(x)) =: g(x) provided that we have ẋ ∈ δ>0 meas(Ω)=0

clco(g([x + δB] \ Ω)).

However, as shown by Ryan [START_REF] Ryan | On brockett's condition for smooth stabilizability and its necessity in a context of nonsmooth feedback[END_REF] and by Coron [START_REF] Coron | Global asymptotic stabilization for controllable systems without drift[END_REF], Brockett's condition continues to hold for this solution concept, so that the nonholonomic integrator (for example) cannot be stabilized by a discontinuous feedback in the Filippov sense.

In [START_REF] Clarke | Asymptotic Controllability Implies Feedback Stabilization[END_REF] it was shown that any globally asymptotically controllable system is stabilizable by a (possibly discontinuous) feedback if the trajectory x(•) associated to the feedback is defined in a natural way that involves discretizing the control law (closed-loop system sampling) in a manner similar to [START_REF] Krasovskii | Game-Theoretical Control Problems[END_REF]. We proceed now to describe this concept, which is the one used in this article.

Let π = {t i } i≥0 be a partition of [0, ∞), by which we mean a countable, strictly increasing sequence t i with t 0 = 0 such that t i → ∞ as i → ∞. The diameter of π, denoted diam(π), is defined as sup i≥0 (t i+1t i ). Given an initial condition x 0 , the π-trajectory x(•) corresponding to π and an arbitrary feedback law k : IR n → U is defined in a step-by-step fashion as follows. Between t 0 and t 1 , x is a classical solution of the differential equation

ẋ(t) = f (x(t), k(x 0 )), x(0) = x 0 , t 0 ≤ t ≤ t 1 .
(Of course in general we do not have uniqueness of the solution, nor is there necessarily even one solution, although the latter will be ruled out by the feedback constructed in section 1, which will preclude blow up of the solution in finite time.) We then set x 1 := x(t 1 ) and restart the system with control value k(x 1 ):

ẋ(t) = f (x(t), k(x 1 )), x(t 1 ) = x 1 , t 1 ≤ t ≤ t 2 ,
and so on in this fashion. The resulting trajectory x is a physically meaningful one that corresponds to a natural sampling procedure and piecewise constant controls; the smaller diam(π), the greater the sampling rate. Since our results are couched in term of π-trajectories, the issue of defining a solution concept for discontinuous differential equations is effectively sidestepped. Our approach will lead to precise estimates of how small the step size diam(π) must be for a prescribed stabilization tolerance to ensue, and of the resulting stabilization time, in terms of the given data.

The next major point to address concerns the nonsmoothness of the Lyapunov function V . An early and important result of Artstein [START_REF] Arstein | Stabilization with relaxed controls[END_REF] implies in particular that the nonholonomic integrator fails to admit a smooth V (see [START_REF] Clarke | Asymptotic stability and smooth Lyapunov functions[END_REF] for related results). It has been shown by Sontag [START_REF] Sontag | A lyapunov-like characterization of asymptotic controllability[END_REF], however, that globally asymptotically controllable systems always admit a continuous Lyapunov function V satisfying the following nonsmooth version of the infinitesimal decrease condition :

inf u∈U DV (x; f (x, u)) ≤ -W (x) < 0, x = 0, (3)
where the lower Dini derivate DV is defined by

DV (x; v) := lim inf t↓0 v →v V (x + tv ) -V (x) t . (4)
Among the several important ways in which the theory of nonsmooth analysis intervenes in this article is that of asserting the equivalence to (3) of another, and for our purposes more useful, form of the infinitesimal decrease condition :

inf v∈cof (x,U ) v, ζ ≤ -W (x) < 0 ∀x = 0, ∀ζ ∈ ∂ P V (x). ( 5
)
Here ∂ P V (x) refers to the proximal subdifferential of V at x (which may very well be empty); ζ belongs to ∂ P V (x) iff there exists σ and η > 0 such that

V (y) -V (x) + σ y -x 2 ≥ ζ, y -x ∀y ∈ B(x, η).
The equivalence of (3) and ( 5) is a consequence of Subbotin's Theorem (see for example [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF], our principal source for the theory of nonsmooth analysis).

The essential reason for which proximal calculus is well-suited to our approach is because of its relation to metric projection onto sets, upon which is based the "proximal aiming" method that we employ. The crux is this : when x(t i ) = x lies outside a level set S = S(c) := {V ≤ c} and admits closest point (or projection) s in S, then xs is a "proximal normal" to S at s, and for some λ > 0 we have λ(xs) ∈ ∂ P V (x). Then [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF] can be invoked at s to find a suitable value of the control u which moves the state toward S, in the sense that the Euclidian distance d S decreases at a certain positive rate ∆ :

d S (x(t)) -d S (x(t i )) ≤ -∆(t -t i ), ∀t i ≤ t ≤ t i+1 ,
provided x(t i ) is close enough to S to start with, and provided diam(π) is small enough. A sequence of such feedbacks is amalgamated in the first section to produce the stabilizing feedback k(•) that is sought.

The approach as presented in §1 requires the Lyapunov function V to be Lipschitz (in the zone under consideration). It is not known whether a globally asymptotically controllable system must admit a suitably Lipschitz V , but that is not quite the setting in which we work. Instead, Theorem 1 derives finite-time stabilizability , but to a close approximation of some inner level set S(a), as a consequence of the supposed existence of a Lipschitz Lyapunov function. In contrast, [START_REF] Clarke | Asymptotic Controllability Implies Feedback Stabilization[END_REF] obtains asymptotic stabilizability to the origin (the case S(a) = {0}), and the proof uses Moreau-Yosida inf convolution to produce a Lipschitz Lyapunov function as an intermediate step. This methodology is also employed in [START_REF] Clarke | The synthesis of universal pursuit strategies in differential games[END_REF], in a differential game setting. The direct use of a Lipschitz Lyapunov function, when it is possible, leads to a far more transparent feedback construction, and has the important consequence of yielding robustness, as we discuss presently. The fact that under mild assumptions, suitable Lipschitz functions exist leading to practical stabilization to any required tolerance, is proven in section 2.

Ledyaev and Sontag [START_REF] Yu | A lyapunov characterization of robust stabilization[END_REF] have recently proved that there is a close relationship between the issues of "how regular a Lyapunov function does the system admit" and "how robust a stabilizing feedback does the system admit". Consider for example a perturbed equation ẋ = f (x, k(x + p)), where p represents a measurement error. Full robustness of the feedback k is taken to mean that for any ǫ, there is a δ > 0 such that whenever the perturbation p(t) satisfies p(t) ≤ δ for all t, then stabilization to the ǫ-ball takes place. This seems to have been studied by Hermes [START_REF] Hermes | Resonance, stabilizing feedback controls, and regularity of viscosity solutions of Hamilton-Jacobi-Bellman equations[END_REF]. Then [START_REF] Yu | A lyapunov characterization of robust stabilization[END_REF] asserts that the system admits a fully robust stabilizing feedback iff it admits a smooth (C 1 or C ∞ ) Lyapunov function. Thus the nonholonomic integrator, which can be stabilized by a discontinuous feedback (in view of [START_REF] Clarke | Asymptotic Controllability Implies Feedback Stabilization[END_REF]), does not admit a fully robust stabilizing feedback. It appears that Hermes [START_REF] Hermes | Discontinuous vector fields and feedback control[END_REF] was the first to raise the question of robustness of discontinuous feedback with respect to measurement error.

The above concept of full robustness, unrelated as it is to the system sampling method that we employ, is not the one discussed in this article. Instead, we introduce a type of relative robustness in which we require the size of the measurement error to be limited as a function of the maximum step size δ of the underlying partition. This step size δ must still be small enough (for stabilization), but at the same time the individual steps must be big enough to preclude a possible chattering phenomenom, even in the presence of small errors. This consideration, which leads us to specify "reasonably uniform" sampling in section 3, appears to be new in this context. The term "reasonably uniform" is taken here to mean that the following hold :

δ 2 ≤ t i+1 -t i ≤ δ ∀i ≥ 0,
although it is possible to replace the factor 1 2 by any constant in (0, 1).

To conclude with the nonholonomic integrator, then, it turns out that the system does admit a stabilizing feedback to within any prescribed tolerance r, in the sense that we will have x(t) ≤ r for all t ≥ T , whenever x is a π-trajectory, where π is a reasonably uniform partition whose diameter is sufficiently small, and whenever measurement and external error do not exceed a critical level related to the diameter (or sampling rate).

A feedback construction

For a given function V : IR n -→ (-∞, ∞], we shall deal frequently with the sublevel sets S(r) defined as follows :

S(r) := {x ∈ IR n : V (r) ≤ r}.
In addition, the following sets are considered :

S(a, b) := {x ∈ IR n : a ≤ V (x) ≤ b}.
Let a and b be two given numbers with a < b. The following hypotheses are made concerning the function V and the system function f : (H3) f (x, u) is continuous on S(b) + ηB as a function of x for each u ∈ U, and ∃m > 0 such that

(H1) V is lower semicontinuous, S(b) = ∅, and for some η > 0, V is Lipschitz of rank L V on S(a, b) + ηB : |V (x) -V (y)| ≤ L V x -y ∀(x, y) ∈ S(a, b) + ηB. (
f (x, u ≤ m ∀x ∈ S(b) + ηB, ∀u ∈ U. (H4) f is Lipschitz in x of rank L f on S(a, b) + ηB : f (x, u) -f (y, u) ≤ L f x -y ∀x, y ∈ S(a, b) + ηB, ∀u ∈ U.
(H5) There exists ω > 0 such that, for every x ∈ S(a, b) + ηB, we have

inf v∈cof (x,U ) DV (x; v) ≤ -ω.
Remark 1.1 We do not require that f and V be defined except on S(b) + ηB; the Lipschitz conditions on these functions, as well as the Infinitesimal Decrease Condition (H5), are posited only on a neighborhood of S(a, b). No hypotheses are made concerning the abstract set U, nor on the nature of the dependence of f on the control variable.

Theorem 1 For any γ > 0 sufficiently small, there exist positive numbers δ, T and a feedback k : S(b) + ηB -→ U such that whenever a partition π satisfies diam(π) < δ, then any π-trajectory x(•) having x(0) ∈ S(b) + γB satisfies

x(t) ∈ S(b) + γB ∀t ≥ 0, x(t) ∈ S(a) + γB ∀t ≥ T.
Remark 1.2 Thus we almost recover the conclusion of the "ideal case" discussed in the Introduction, but in approximate terms, with a discontinuous feedback, and for a nonsmooth Lyapunov function satisfying localized hypotheses. The proof is constructive and gives estimates of γ, δ and T in terms of the given data.

Remark 1.3 Note that (H5) is a weaker hypothesis than ( 3). An important fact used in the proof is that the Infinitesimal Decrease Condition (H5) has a strictly equivalent formulation in proximal terms, as follows :

inf u∈U ζ, f (x, u) ≤ -ω ∀x ∈ S(a, b) + ηB, ∀ζ ∈ ∂ P V (x).
Here, ∂ P V is the proximal subdifferential of V [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF]. The equivalence of the formulations is nontrivial, and is a consequence of Subbotin's Theorem [9, Theorem

The proof of Theorem 1 is based upon defining a feedback control, via projections. The first two lemmas below guarantee that the projections lie in the set where the hypotheses are active. Finally, we note xs < γ, whence ys < 2γ and s lies in S(a, b) + ηB.

The next "solvability" result is central to our approach. The notation u + stands for max{u, 0}.

Lemma 1.3 For any r ∈ [a, b], for any x ∈ S(a, b), we have d(x, S(r)) ≤ m ω (V (x) -r) + .
Proof We shall invoke results (and terminology) from [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF] to give a short proof of this result, whose proof from first principles would be lengthy. We define a lower semicontinuous function g : IR n -→ [0, ∞] as follows : It follows from this that for any ǫ > 0, for any x ∈ C, for any

g(x) := (V (x) -r) + + I S(b) (x)
ζ ∈ ∂ P g(x), there exists u ∈ U such that ζ, f (x, u) ≤ -ω + ǫ.
Since f (x, u ≤ m and since ǫ > 0 is arbitrary, we derive ζ ≥ ω m . This verifies the hypothesis of the Solvability Theorem [9, Theorem 3.3.1] (applied with V = Ω = IR n ), whose conclusion is precisely the desired one since S(r) = {x : g(x) = 0}.

We now proceed to fix γ > 0 such that

γ < min δ 1 , η 2 , ω 12L f L V , (6)
and we define

β := min δ 1 , (b -a) 2 , γω 4m . ( 7
)
Let N be the first integer such that

b -N β > a ≥ b -(N + 1)β.
Note that N ≥ 1 since β < ba. We proceed to define certain sets Ω i (i = 0, 1, • • • , N + 1) that lie at the heart of our construction. For 0 ≤ i ≤ N -1, we set

Ω i := [S(b -iβ) + γB] \ [S(b -(i + 1)β) + γB];
for i = N we set

Ω N := [S(b -N β) + γB] \ [S(b -N β) + γ 4 B];
and finally, we define Ω N +1 := S(b -N β) + γ 4 B. We now gather some facts about these sets.

Lemma 1.4 (a)

The Ω i are disjoint, and

Ω i is contained in S(a, b) + γB for i ≤ N . (b) N +1 i=0 Ω i = S(b) + γB. (c) If x ∈ Ω i for some i ∈ {0, 1, • • • , N } and s ∈ proj(x, S(b -iβ)), then s ∈ S(a, b) + ηB. (d) S(b -iβ) + γ 4 B ⊂ S(b -(i + 1)β)) + γB (i = 0, 1, • • • , N -1). (e) For every i ∈ {0, 1, • • • , N }, ∀x ∈ Ω i , we have γ 4 ≤ d(x, S(b -iβ)) < γ. (f ) S(b -N β) + γ 2 B ⊂ S(a) + γB. Proof (a): That the Ω i are disjoint is evident; that they lie in S(a, b) + γB for i ≤ N
follows from Lemma 1.2 for i < N and from Lemma 1.

1 for i = N (recall that b -N β -a ≤ β ≤ δ 1 and γ < δ 2 ). (b): Evident. (c): Direct from Lemma 1.2 (i < N ) or Lemma 1.1 (i = N ). (d): Let x lie in S(b -iβ) + γ 4 B, and let s ∈ S(b -iβ) satisfy x -s < γ 4 . Then V (s) ≤ b -iβ, and if V (s) ≤ b -(i + 1)β the conclusion is immediate. Otherwise we have V (s) > b -(i + 1)β > a, so that s ∈ S(a, b). By lemma 1.3 there exists y ∈ S(b -(i + 1)β) such that s -y ≤ m ω [V (s) -b + (i + 1)β] ≤ mβ ω ≤ γ 2 ,
in view of [START_REF] Clarke | Asymptotic stability and smooth Lyapunov functions[END_REF]. Then xy ≤ xs + sy < γ 4 + γ 2 < γ, which established the desired conclusion.

y -s ≤ m ω [V (s) -a] ≤ m ω [b -N β -a] ≤ mβ ω ≤ γ 2 .
But then xy < γ, so again x ∈ S(a) + γB.

Lemma 1.5 Let x ∈ Ω i (i = 0, 1, • • • , N ), and let s ∈ proj(x, S(b -iβ)). Then there exists u ∈ U such that x -s, f (s, u) ≤ -ω 2L V x -s .
Proof By definition, xs lies in the proximal normal cone N P (x, S(biβ)).

Note that s lies in S(a, b)+ηB (by Lemma 1.2 for i < N , Lemma 1.1 for i = N ), so that V is Lipschitz of rank L V in a neighborhood of s. A basic calculus result [9, 1.11.26] yields the existence of λ > 0 such that λ(xs) ∈ ∂ L V (s), and necessarily λ xs ≤ L V . In accord with Remark 1.1, there exists u ∈ U such that λ(xs), F (s, u) ≤ -ω 2 .

The result follows.

Defining the feedback

We now define a feedback k(•) on S(b) + γB as follows. If x ∈ Ω i for some i ∈ {0, 1, • • • , N }, then we set k(x) = u, where u corresponds to x (and a projection s) as in lemma 1.5. There remain the points x in Ω N +1 to consider (see Lemma 1.4 (b)). For such x, we define k(x) to be any point in U, for example one which minimizes f (x, u) .

The remainder of the proof consists in establishing that for suitably small mesh size, any π-trajectory generated by k(•) with initial condition in S(b) + γB remains in S(b) + γB, enters S(a) + γB within a certain (uniform) time, and then remains in that set subsequently. We consider countable partitions {t j } such that t 0 = 0, lim j→∞ t j = ∞, and such that t j+1t j ≤ δ ∀j ≥ 0, where δ is any positive number satisfying

δ < min γ 4m , ω 6mL f L V , γω 48m 2 L V , 1 . (8)
For such a partition, let x 0 be any point in S(b) + γB, and let x(•) be a πtrajectory with x(0) = x 0 . We denote x(t j ) by x j , and we set ∆ := ω 60LV .

Lemma 1.6 For some t j ∈ π, suppose that

x j ∈ Ω i , i ∈ {0, 1, • • • , N }. Then x(t) ∈ S(b) + γB for t ∈ [t j , t j+1 ], and
d(x(t), S(b -iβ)) ≤ d(x j , S(b -iβ)) -∆(t -t j ) ∀t ∈ [t j , t j+1 ].
Proof We have x j ∈ S(a, b) + γB by Lemma 1.4 (a), and ẋ(t) ≤ m while x(t) lies in S(b) + ηB. Since δm < γ 4 by (8) and γ < η 2 , it follows that x(t) lies in S(a, b) + ηB for t ∈ [t j , t j+1 ], as does the point s that figures in the definition of k(x j ); this was pointed out in the proof of Lemma 1.5, where we also deduced the inequality

x j -s, f (s, k(x j )) ≤ -ω 2L V x j -s . (9)
We fix t ∈ (t j , t j+1 ) and set Ψ := x(t)-s x(t)-s .(Note that x(t) = s, since x j -s ≥ γ 4 by Lemma 1.4 (e), and since x(t)x j < δm < γ 4 .) We now observe two inequalities :

d(x(t), S(b -iβ)) ≤ x(t) -s = Ψ, x(t) -s , d(x j , S(b -iβ)) = x j -s ≥ Ψ, x j -s .
These together imply

d(x(t), S(b -iβ)) -d(x j , S(b -iβ)) ≤ Ψ, x(t) -x j = τ Ψ, f j , (10)
where we introduce the notation τ := tt j ,

x(t) = x j + τ f j , f j := 1 τ t tj f (x(r), k(x j ))dr.
We also set fj :

= f (s, k(x j )) = 1 τ t tj f (s, k(x j ))dr.
Note that f j -fj ≤ 1 τ t tj L f x(r)s dr (the Lipschitz condition applies because we are in S(a, b) + ηB)

≤ 1 τ t tj L f ( x(r) -x j + x j -s )dr ≤ L f (τ m + d(x j , S(b -iβ))).
It follows from this and (9) that we have

x j -s, f j = x j -s, fj + f j -f j ≤ -ω 2L V x j -s + L f d(x j , S(b -iβ)){τ m + d(x j , S(b -iβ))} ≤ d(x j , S(b -iβ))[ -ω 2L V + L f δm + γL f ] (since τ < δ and d(x j , S(b -iβ)) ≤ γ) ≤ d(x j , S(b -iβ))[ -ω 2L V + ω 6L V + ω 6L V ]
(we have invoked ( 8) and ( 6))

≤ - γω 24L V (since d(x j , S(b -iβ)) ≥ γ 4 by Lemma 1.4 (e)).
We shall use this bound on x js, f j to derive one on x(t)s, f j as follows:

x(t) -s, f j = x j + τ f j -s, f j ≤ x j -s, f j + τ f j 2 ≤ -γω 24L V + δm 2 ≤ -γω 48L V (11)
(in light of ( 8)).

We also have

x(t) -s = x j + τ f j -s ≤ x j -s + τ f j ≤ γ + δm < 5γ 4
(by ( 8)).

Combining this with [START_REF] Coron | Global asymptotic stabilization for controllable systems without drift[END_REF] we arrive at

Ψ, f j = x(t) -s x(t) -s , f j ≤ -γω 48L V / 5γ 4 = -∆.
Together with [START_REF] Clarke | The synthesis of universal pursuit strategies in differential games[END_REF], this gives the inequality asserted by the Lemma. Since this inequality evidently implies

d(x(t), S(b -iβ)) < γ,
it also follows that x(t) ∈ S(b) + γB.

Lemma 1.7 If x j ∈ Ω i where 0 ≤ i ≤ N , then x j+1 lies in Ω i for some i ≥ i.

Proof Since x j ∈ Ω i , we have d(x j , S(biβ)) < γ, and (by Lemma 1.6)

d(x j+1 , S(b -iβ)) < γ. Now let 1 ≤ k < j. Since S(b -iβ) ⊂ S(b -(k + 1)β), we deduce d(x j+1 , S(b -(k + 1)β)) < γ. But then x j+1 / ∈ Ω k by definition of Ω k . Since x j+1 ∈ S(b) + γB by Lemma 1.6, we must have x j+1 ∈ Ω i for some i ≥ i, in view of Lemma 1.4 (b). Lemma 1.8 If x(τ ) ∈ Ω N +1 for some τ ∈ π, then x(t) ∈ S(a) + γB ∀t ≥ τ .
Proof We know that x(τ ) lies in the interior of S(a) + γB by Lemma 1.4 (f). For t > τ , as long as d(x(t), S(b -N β)) does not attain or exceed γ 2 , then x(t) remains in S(a) + γB. Thus ẋ(t) remains bounded by m and no blow-up occurs (i.e., x(t) is well-defined). It suffices therefore to prove that the continuous function g(t) := d(x(t), S(b -N β)) does not become greater than or equal to γ 2 for some t 0 > τ . We have g(τ ) ≤ γ 4 . Note that S(b-N β)+γB lies in S(b)+γB, so that the bound ẋ ≤ m continues to apply for some time after any positive point t 0 at which g(t) becomes equal to γ 2 . Thus g(t) is locally well-defined. Yet Lemma 1.6 (for i = N ) shows that g is decreasing when g(t) lies between γ 4 and γ. The conclusion is that g(t) ≤ γ 2 ∀t > τ (or in fact that g(t) never even exceeds γ 4 ). One last lemma and the proof is complete.

Lemma 1.9 Let T := (1 + b-a β )(1 + 45γLV ω ). Then x(t) ∈ S(a) + γB ∀t ≥ T.
Proof In view of Lemma 1.8 and Lemma 1.4 (f), it suffices to prove that there is a node τ ∈ π with τ ≤ T for which x(τ ) ∈ Ω N +1 . Note that x(0) belongs to some Ω i (0 ≤ i ≤ N + 1) by Lemma 1.4 (b); if i = N + 1 we are done, so assume i ≤ N . Since δ < 1 by ( 8), there is a node τ 1 lying in the open interval (σ, σ + 1), where σ := 3γ 4∆ . By Lemma 1.7, x(τ 1 ) belongs either to Ω i or to Ω i for some i > i. In the former case, it follows that x(t) lies in Ω i for every node t ∈ π lying between 0 and τ 1 , and the inequality of Lemma 1.6 applies to give

d(x(τ 1 ), S(b -iβ)) ≤ d(x(0), S(b -iβ)) -∆τ 1 < γ - 3γ 4 = γ 4 .
However, the left side is no less than γ 4 by Lemma 1.4 (e). This contradiction shows that, in fact, x(τ 1 ) must belong to some Ω i for an index i > i. The same argument, beginning now at (τ 1 , x(τ 1 )), yields the existence of a node τ 2 ∈ π with τ 2 ≤ 2σ + 2 such that x(τ 2 ) belongs to Ω i , where i > i . Continuing in this manner, we find that (since there are at most N + 1 steps as above prior to landing in Ω N +1 ), there is a node τ ∈ π with τ ≤ (N + 1)(σ + 1) such that x(τ ) ∈ Ω N +1 . But N < b-a β implies that the T defined in the statement of the Lemma is greater than (N + 1)(σ + 1).

Construction of a Lyapunov function

We show in this section thst under reasonable assumptions, there always exist Lyapunov functions having the properties required for the feedback construction of the preceding section, and giving rise to practical feedback stabilization of arbitrarily prescribed range. While the result below seems new and the approach to proving it has some novel features, there is a familiar heuristic at work : the Lyapunov function is contructed as the value function associated with a parametrized family of optimal control problems.

The function f (x, u) describing the dynamics is supposed in this section to satisfy much the same regularity conditions as before. Specifically, we require that for any bounded subset S of IR n , there exist constants m = m(S) and

L = L(S) such that f (x, u) ≤ m ∀x ∈ S, ∀u ∈ U f (x, u) -f (y, u) ≤ L x -y ∀x, y ∈ S, ∀u ∈ U.
(As before, U is just an abstract set, and no hypotheses are made concerning the nature of the dependence of f on u.)

In addition, we require controllability to the ball of radius r 0 via relaxed trajectories. Let us now proceed to make this precise. We are given r 0 ≥ 0, and we define a multifunction Γ on IR n by Γ(x) := clco{f (x, u) : u ∈ U}.

By "trajectory" we mean an absolutely continuous function x(•) on an interval [0, T ] such that ẋ(t) ∈ Γ(x(t)) a.e. t ∈ [0, T ].

Given α ∈ IR n , we define T 0 (•) as the least time required for a trajectory to go from α to the closed ball r 0 B :

T 0 (α) := inf{T ≥ 0 : x(•) is a trajectory on [0, T ], x(0) = α, x(T ) ≤ r 0 }.
the controllability hypothesis that we make is that every α admits a trajectory steering it to r 0 B in finite time, a time which goes to zero as α approaches r 0 B. Equivalently :

(CH) T 0 (α) < ∞ ∀α ∈ IR n , and lim α ↓r0

T 0 (α) = 0.

Theorem 2 For any pair r and R with R > r > r 0 , there exist a function V and numbers a, b, γ, η such that

S(a) + γB ⊂ B(0, r) ⊂ B(0, R) ⊂ S(b) + γB,
and such that V satisfies all the hypotheses of Theorem 1; in addition S(b) is compact. Thus there is a feedback defined on a neighborhood of B(0, R) which stabilizes the state to B(0, r) in the sense of Theorem 1.

We begin by defining another multifunction Γ (more useful than Γ for being compact-valued) :

Γ(x) := clco{ v 1 + v : v ∈ Γ(x)}.
We set

T0 (α) := inf{T ≥ 0 : x(•) is a Γ -trajectory on [0, T ], x(0) = α, x(T ) ≤ r 0 }.
Evidently (or by convention) we have T0 = 0 on r 0 B. (d) There exists a positive number ǫ such that whenever α ∈ B(0, r 0 + ǫ), and whenever the Γ-trajectory x(•) has x(0) = α and x(T ) ≤ r 0 for some T ≤ T0 (α) + ǫ, then we have x ∞ ≤ r 0 + 1. We can suppose ǫ < 1, ǫ < rr 0 , and that sup{ T0 (α) : α ∈ B(0, r 0 + ǫ)} < (rr 0 ) 2 17(r 0 + 1)

. ( 12
)
Proof We omit the routine proof of (a). For (b), let α ∈ IR n be given. By assumption, there is a Γ-trajectory x on an interval [0, T ] such that x(0) = α, x(T ) = r 0 . We set T := ∈ Γ(x(τ )) a.e., so that x is a Γ-trajectory. Hence T0 (α) ≤ T < ∞.

We turn now to (c). Let α i be a sequence for which α i decreases to r 0 . Then T 0 (α i ) → 0 by assumption. Let m be such that f (x, u) ≤ m for (x, u) ∈ B(0, r 0 + 1) × U. Then, as soon as T 0 (α i ) is strictly less than 1 m , there is a Γ-trajectory x i on an interval [0, T i ] such that

x i (0) = α i , x i (T i ) ≤ r 0 , T i < 1 m , T i < T 0 (α i ) + 1 i .
It follows that x i ∞ < r 0 + 1. Now let xi be the Erdmann Transform of x i as given above. Then

T0 (α i ) ≤ Ti = Ti 0 (1 + ẋi (t) )dt ≤ (1 + m)T i < (1 + m)(T 0 (α i ) + 1 i ).
It follows that T0 (α i ) → 0, as required.

We now examine (d). If the assertion is false, there exist a sequence α i with α i ↓ r 0 and corresponding Γ-trajectories

x i with x i (0) = α i , x i (T i ) ≤ r 0 , such that T i ≤ T0 (α i ) + 1 i , x i ∞ > r 0 + 1.
Since T0 (α i ) → 0 by (c), we have T i → 0. On the other hand, there is a subinterval of [0, T i ] in which x i goes from being r 0 + 1 to r 0 , and since ẋi (t) ≤ 1 the length of that subinterval (and hence, T i ) is at least 1. This contradiction establishes the first part of (d); the rest follows immediatly by shrinking ǫ as required, in light of (c).

We proceed now to define a new multifunction F (x) whose effect is to enlarge the set Γ(x) for x < r 0 + ǫ. We set

F (x) :=    Γ(x) for x ≥ r 0 + ǫ Γ(x) + 2[ r0+ǫ-x ǫ ] B for r 0 ≤ x ≤ r 0 + ǫ Γ(x) + 2 B for x ≤ r 0
Having done this, we define a value function V (•) on IR n in terms of the trajectories of F as follows :

V (α) := inf{ T 0 x(t) dt : T ≥ 0, x(0) = α, ẋ ∈ F (x)a.e., x(T ) ≤ r 0 }.
We stress that T is a choice variable here, in this free time problem. Proof The assertions of (a) are immediate. Since F (x) is uniformly bounded, the attainment and the lower semicontinuity asserted in (b) follow from standard "compactness of trajectories" arguments; see [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF]Chapter 4] for details. The first assertion of (c) is clear, and the other one stems from Lemma 2.1 as follows.

Let α ∈ B(0, r 0 + ǫ), and let the Γ-trajectory x satisfy x(0) = α, x(T ) ≤ r 0 , and T ≤ T0 (α) + δ, for some δ ∈ (0, ǫ). Then x ∞ ≤ r 0 + 1, and we deduce

V (α) ≤ T 0 x(t) dt ≤ ( T0 (α) + δ)(r 0 + 1).
Since T0 (α) ↓ 0 as α ↓ r 0 , (c) follows. Finally we turn to (d). If α > r 0 +1+ǫ, then the time required for a trajectory to go from

x = α to x = r 0 + ǫ is at least α -r 0 -ǫ. But then V (α) ≥ (r 0 + ǫ)( α -r 0 -ǫ). This established (d).
The next step invokes Hamiltonian conditions for optimal control, and uses the lower Hamiltonian h associated with F :

h(x, p) := min{ p, v : v ∈ F (x)}. Lemma 2.3 Let ζ ∈ ∂ P V (α)
, where α > r 0 . Let x be a trajectory solving the problem that defines V (α), with associated time T . Then there exists an absolutely continuous function p on [0, T ] such that

(-ṗ - x x , ẋ) ∈ ∂ C h(x, p) a.e. t ∈ [0, T ] (13)
p(0) = ζ (14) h(x(t), p(t)) + x(t) = 0 ∀t ∈ [0, T ]. (15)
Proof By definition of ∂ P V (α), we have for some σ ≥ 0 and for all α near α,

V (α ) + σ α -α 2 -ζ, α ≥ -ζ, α .
Let x be a trajectory near x (in the L ∞ norm), put α = x (0) and α = x(0) and rearrange to derive that x (•) = x(•) solves locally the problem of minimizing

T 0 x (t) dt -ζ, x (0) + σ x (0) -x(0) 2
over the trajectories x for F satisfying x (T ) ≤ r 0 . (Here T and x (0) are free.) We apply the corollary of Theorem 3.6.1 of [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF] (in reverse time) to deduce the existence of an absolutely continuous function q on [0, T ] satisfying 16)

(-q, ẋ) ∈ ∂ C [H(x, q) -x ](x, q) a.e. t ∈ [0, T ], (
q(0) = -ζ, (17)
H(x(t), q(t)) -x(t) = 0, t ∈ [0, T ], (18)
where H(x, p) is the function -h(x, -p) and ∂ C denotes the generalized gradient. The Hamiltonian inclusion above implies

( q - x x , ẋ) ∈ ∂ C h(x, -q) a.e.
Now putting p := -q gives the conclusions the form asserted in the statement of the lemma.

Lemma 2.4 For any constant c > 0, there is a constant M c with the following property. If α ∈ S(c) and if the trajectory x on [0, T ] attains the infimum defining

V (α), then x ∞ ≤ M c , T ≤ M c .
Proof If x ∞ > c + r 0 + 1, then the time required for x to attain the value r 0 + 1 exceeds c (since ẋ ≤ 1). But then V (α) ≥ (1 + r 0 )c > c. This shows that x ∞ is bounded by c + r 0 + 1. By Lemma 2.1(c), lim α ↓r0 T0 (α) = 0. So there exists ρ > r 0 such that ∀x ∈ B(0, ρ), | T0 (x)| ≤ 1. Now for α ≥ ρ, let τ ρ denote the first time that x(t) attains the value ρ. Then V (α) ≥ ρτ ρ , whence τ ρ ≤ c ρ for α ∈ S(c). We deduce that T0 (α) ≤ τ ρ + 1 (by construction of ρ)

≤ c ρ + 1.
If α < ρ, the same bound evidently holds. It suffices now to set

M c := max{ c ρ + 1, c + r 0 + 1}. Lemma 2.5 V is locally Lipschitz on IR n .
Proof We prove first that V is locally Lipschitz on the open set {V > 0} = comp( B(0, r 0 )). Let α 0 belong to this set, take any δ > 0 such that δ < α 0 -r 0 , and any element ζ ∈ ∂ P V (α), where

α -α 0 < δ, V (α) ≤ V (α 0 ) + δ =: c. (19)
The conclusions of Lemma 2.4 are available for any trajectory solving the V (α) problem. If K is a Lipschitz constant for F on the ball B(0, M c + 1) (where M c comes from Lemma 2.4), then the Hamiltonian inclusion (13) implies

ṗ ≤ K p + 1. (20)
The condition (15) at t = T givesp(T ) ≥ -r 0 since x(T ) = r 0 , and since F (x(T )) = clcof (x(T ), U) + 2 B ⊃ B. Thus p(T ) ≤ r 0 . This, together with (20) and Gronwall's Lemma, leads to

ζ = p(0) ≤ e KT p(T ) + T 0 e K(T -s) ds ≤ r 0 e KT + e KT -1 K ≤ r 0 e KMc + e KMc -1 K ,
since T ≤ M c by Lemma 2.4. This establishes a uniform bound on elements of ∂ P V (α) whenever α satisfies [START_REF] Krasovskii | Game-Theoretical Control Problems[END_REF], which proves that V is Lipschitz on a neighborhood of α 0 [9, 1.11.11]. Thus V is locally Lipschitz on the set where it is strictly positive.

There is a neighborhood N of B(0, r 0 ) on which V is bounded above, in view of Lemma 2.2(c). The argument above therefore yields a bound L on elements of ∂ P V (α) for all α ∈ N \ B(0, r 0 ), so that V is uniformly Lipschitz of rank L on α ∈ N \ B(0, r 0 ) by [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF]Theorem 1.7.3]. Of course, V = 0 on B(0, r 0 ). That V is Lipschitz on N , and hence locally Lipschitz on IR n , now follows. We derive V (α) ≤ (r 0 + 1) T0 (α), and (from ( 12))

V (α) < (r -r 0 ) 2 17 .
Now let α ≥ r-r0 2 , and let x solve the problem defining V (α). There is an interval of length at least r-r0 4 during which x(t) ≥ r-r0 4 (since ẋ ≤ 1), whence

V (α) > (r -r 0 ) 2 16 .
The result follows.

Let x be a trajectory solving the problem defining V (α). Then, by Lemma 2.3, we have (at t = 0) :

h(α, ζ) + α = 0.
Since α > r 0 + ǫ by Lemma 2.7, we have

F (α) = Γ(α),
so that the preceding equality yields, for any δ > 0, the existence of some element v ∈ Γ(α) such that

ζ, v 1 + v ≤ -α + δ < -(r 0 + ǫ) + δ.
For δ small enough, the right side is negative, whence

ζ, v < -(r 0 + ǫ) + δ.
Given that Γ(x) := clcof (x, U), this yields the existence of u ∈ U for which

ζ, f (α, u) < -(r 0 + ǫ) + 2δ.
Since δ is arbitrarily small, (21) ensues. Since S(b) is compact, f is Lipschitz in x and bounded on S(b)+ηB, in accord with hypotheses (H3) (H4) of section 1. When the level sets are compact, (H2) always holds. The verification of this is the last property to confirm. Lemma 2.9 Hypothesis (H2) holds.

Proof If (H2) fails, then there exist sequences α i ∈ IR n , ǫ i ↓ 0, and u i ∈ B(0, 1) such that V (α i ) ≤ a + ǫ i and V (α i + u i ) > b.

Since S(b) is compact, we can suppose by passing to a subsequence that α i → α 0 . Then V (α 0 ) ≥ b > a ≥ V (α 0 ), a contradiction. The setting of Theorem 1 is established, and Theorem 2 is proved.

Robustness

We prove in this section that the feedback constructed in §1 is robust with respect to small measurement error and persistent external disturbance, in a precise sense that requires two stipulations. The first is that the measurement error must not exceed in order of magnitude the step size of the underlying discretization, a condition which appears to be rather natural. The second requirement is perhaps more surprising, and surfaces from the nature of the feedback construction. It dictates that each step be "big enough" (while continuing to be "small enough") so as to counteract the measurement error by means of the attractive effect inherent in the construction. Thus the partitions used to discretize the effect of the control are taken to be "reasonably uniform".

Our perturbed system is modeled by

ẋ = f (x, k(x + p)) + q,
where the external disturbance q : [0, ∞) -→ IR n is a bounded measurable function : q(t) ≤ E q t ≥ 0 a.e.

Given a partition π = {t i } i≥0 of [0, ∞) and the initial condition x 0 , the resulting π-trajectory of our perturbed system is defined by successively solving the differential equation

ẋ(t) = f (x(t), k(x(t i ) + p i )) + q(t), t ∈ [t i , t i+1 ],
with x(0) = x 0 . The continuous function x(t) is the real state of the system, while the sequence {x(t i ) + p i } corresponds to the inexact measurements used to select control values.

Theorem 3 The feedback k : S(b) + γB → U constructed in Theorem 1 is robust in the following sense. There exist positive numbers δ 0 , T and E q such that, for every δ ∈ (0, δ 0 ) there exists E p (δ) > 0 having the following property : for any partition π = {t i } i≥0 having

δ 2 ≤ t i+1 -t i ≤ δ i ≥ 0,
where 0 < δ < δ 0 , for any set of measurement errors {p i } i≥0 having

p i ≤ E p i ≥ 0,
for any initial condition x 0 such that x 0 + p 0 ∈ S(b) + γB, for any disturbance q having q ∞ ≤ E q , the resulting π-trajectory x satisfies

x(t i ) + p i ∈ S(b) + γB ∀i ≥ 0, x(t) ∈ S(b) + 2γB ∀t ≥ 0, x(t) ∈ S(a) + γB ∀t ≥ T.
Remarks 3.1 (a) Note that unlike T and E q , the maximum admissible measurement error E p depends on δ. Note also that (in contrast to Theorem 1) x(t) may not lie in S(b) + γB for all t, although for large it must do so.

(For example, we do not require x(0) ∈ S(b) + γB.) However, the "observed values" of the state, namely the values x(t i ) + p i (i ≥ 0), all fall in S(b) + γB the domain of definition of k.

(b) Certain other kinds of error, for example a disturbance d(•) entering into the dynamics in the form ẋ = f (x, k(x) + d), can be reduced to that of external disturbance by positing suitable continuity of f in the control variable.

(c) The maximum admissible disturbance measure E q will be seen to be proportional to ω LV . This has a natural physical meaning, as can easily be seen in the case of smooth V and a continuous feedback k(x) such that ∇V (x), f (x, k(x)) ≤ -W (x). Then we see that the perturbed system

ẋ = f (x, k(x)) + q is stabilized by k if q ∞ < W (x)
∇V (x) for every x, a bound akin to that involving ω LV .

We adapt the proof of Theorem 1, whose first five lemmas hold with no change whatever, as does the definition of k(•). Recall that γ, β and N were introduced (earlier; see (6) (7)). We now define our upper bound for δ :

δ 0 := min 6γL V ω , 1, γω 24L V (m + ω 6LV + 1) 2 , γ 20m (22)
E q := ω 6L V (23)
T := 1 + b -a β 1 + 81γL V ω (24)
(this T differs slightly from the one in Theorem 1) and we let π = {t i } i≥0 be a partition as described i the statement of Theorem 3, with corresponding measurement errors {p i } i≥0 having p i ≤ E p for some E p > 0 satisfying

E p < min{ 3γ 80 , δ, δω 432L V }. (25)
We also admit any disturbance q(•) for which q ∞ ≤ E q , and we take x 0 such that x 0 + p 0 ∈ S(b) + γB. We shall show that the corresponding π-trajectory has the required properties. We introduce the notation

x i := x(t i ), y i := x i + p i
for the actual and the measures space state values at time t i , and proceed to develop modified versions of the fourlast lemmas figuring in the proof of Theorem 1. We set ∆ := ω 108LV . Lemma 3.1 For some t j ∈ π, suppose that y

j ∈ Ω i , i ∈ {0, 1, • • • , N }. Then x(t) ∈ S(b) + 2γB ⊂ S(b) + ηB, t j ≤ t ≤ t j+1 y j ∈ S(b) + γB, y j+1 ∈ S(b) + γB, d(y j+1 , S(b -iβ)) ≤ d(y j , S(b -iβ)) -∆(t j+1 -t j ).
Proof Note that y j ∈ S(b) + γB by Lemma 1.4; it will follow from the last conclusion of the current Lemma that y j+1 ∈ S(b) + γB. Also, x jy j = p j ≤ E p , together with x(t)x j ≤ δm, yield

where s ∈ proj(y j , S(biβ)). Fix t ∈ (t j , t j+1 ) and set Ψ := x(t)s x(t)s .

Note that x(t) = s since y js ≥ γ 4 by Lemma 1.4 (e), while ≤ Ψ, x(t)y j = Ψ, x j + τ (f j + q j )y j ≤ τ Ψ, f j + q j + p j ,

where we have introduced τ := tt j , f j := 1 τ t tj f (x(r), k(y j ))dr, q j := 1 τ t tj q(r)dr.

We also set fj := f (s, k(y j )); note fj ≤ m. We have

f j -fj ≤ 1 τ t tj L f x(r) -s dr ≤ L f τ t tj
( x(r)x j + x js )dr ≤ L f (δm + x jy j + y js )

≤ L f (δm + E p + γ) ≤ 5 4 L f γ. (28)
We deduce

x(t)s, f j + q j = x j + τ (f j + q j )s, f j + q j = y jsp j , f j + q j + τ f j + q j 2 ≤ y js, f j + E p (m + E q ) + δ(m + E q ) 2 = y js, fj + y js, f j -fj + (m + E q )[δm + δE q + E p ] ≤ -ω 2L V y js + 5 4 y js L f γ + (m + E q )[δm + δE q + E p ] (where we have used ( 26) and ( 28) )

≤ d(y j , S(b -iβ)) - ω 2L V + ω 6L V + δ[m + E q + 1] 2
(by [START_REF] Clarke | Asymptotic Controllability Implies Feedback Stabilization[END_REF], and since E p < δ by ( 25))

≤ γ 4 - ω 3L V + γω 24L V = - γω 24L V
(since d(y j , S(biβ)) ≥ γ 4 by Lemma 1.4, and since δ < δ 0 defined by ( 22)). Note also that x(t)s = y jp j + τ (f j + q j )s ≤ γ + E p + δ(m + E q ) ≤ 5γ 4 + δE q < 9γ 4

(note δE q < γ because of δ < δ 0 , in view of (22) ( 23)). It follows that Ψ, f j + q j = x(t)s x(t)s , f j + q j ≤ -(γω)/(24L V ) (9γ)/4 = -ω 54L V .

Substituting into [START_REF] Sontag | Nonsmooth control-Lyapunov functions[END_REF] ≤ -∆(t j+1t j ), by [START_REF] Sontag | A lyapunov-like characterization of asymptotic controllability[END_REF], and since t j+1t j ≥ δ 2 .

Lemma 3.2 If y j ∈ Ω i , where i ≤ N , then y j+1 lies in Ω k for some k ≥ i.

Proof We know that y j+1 ∈ Ω k for some k, Then x(t) ∈ S(a) + γB ∀t ≥ T .

Proof In view of Lemma 3.3, it suffices to prove that some node τ ∈ π with τ ≤ T is such that y(τ ) ∈ Ω N +1 . The argument is identical to that used to prove Lemma 1.8, with ∆ replaced by ∆, and applied to the y i rather than the x i .

H2) ∃δ 1 ∈

 1 (0, ba) and δ 2 > 0 such that S(a + δ 1 ) + δ 2 B ⊂ S(b).

Lemma 1 . 1 Lemma 1 . 2

 1112 Let ǫ lie in [0, δ 1 ] and suppose that x is a point in the set [S(a + ǫ) + min{δ 2 , η}B] \ S(a + ǫ). Then x ∈ S(a, b), and if s ∈ proj(x, S(a + ǫ)), then s ∈ S(a, b) + ηB. Proof Since we have S(a+δ 1 )+δ 2 B contained in S(b) by hypothesis, it follows that x lies in S(b). Since x does not belong to S(a + ǫ), we deduce x ∈ S(a, b). Finally, we have sx < min{δ 2 , η} ≤ η, whence s ∈ S(a, b) + ηB. Let 0 < γ < η/2, and suppose that for some r and r with a ≤ r < r ≤ b we have x ∈ [S(r) + γB] \ [S(r ) + γB]. Then x ∈ S(a, b) + γB, and if s ∈ proj(x, S(r)), then s ∈ S(a, b) + ηB. Proof There exists y ∈ S(r) having yx < γ. Since x does not belong to S(r )+γB, we have V (y) > r necessarily. Thus y ∈ S(a, b) and x ∈ S(a.b)+γB.

  , where I S(b) (•) is the indicator function of the set S(b). At any point x in the open set C := {y : g(y) > 0} at which g is finite, we have x ∈ S(a, b), and the infinitesimal decrease condition implies that inf{Dg(x; v) : v ∈ cof (x, U)} ≤ -ω.

  (e): For i = N , this is immediate from the definition of Ω N ; for i < N , it is a consequence of (d). (f): Let x belong to S(b-N β)+ γ 2 B, and let s ∈ S(b-N β) satisfy x-s < γ 2 . If V (s) ≤ a, then x ∈ S(a) + γB. Otherwise, s belongs to S(a, b), and lemma 1.3 implies the existence of y ∈ S(a) such that

Lemma 2 . 1

 21 (a) Γ is locally Lipschitz and has nonempty convex compact values in B(0, 1).(b) T0 (α) is finite ∀α ∈ IR n .(c) lim α ↓r0 T0 (α) = 0.

T 0 ( 1 = t 0 ( 1

 0101 + ẋ(t) )dt and we define a function x on [0, T ] by x(τ ) := x(t), where t = t(τ ) is determined in [0, T ] by τ + ẋ(r) )dr (this change of variables or time scale is known as the Erdmann Transform.

Lemma 2 . 2

 22 (a) F is compact and convex-valued, uniformly bounded and locally Lipschitz.(b) V (•) is finite-valued and lower semicontinuous, and the infinum defining V (α) is attained for every α.

  (c) V (α) > 0 iff α > r 0 , and lim α ↓r0 V (α) = 0.(d) The sublevel sets S(b) := {x : V (x) ≤ b} of V are compact.

Lemma 2. 6 sup{V-r 0 2 }

 62 (α) : α ∈ B(0, r 0 + ǫ)} < inf{V (α) : α ≥ r 1 Proof Let α < r 0 + ǫ, fix δ ∈ (0, ǫ) and let the trajectory x on [0, T ] satisfy x(0) = α, x(T ) ≤ r 0 , T < T0 (α) + δ. Then by Lemma 2.1 we have x ∞ ≤ r 0 + 1 and soV (α) ≤ T 0x(t) dt ≤ ( T0 (α) + δ)(r 0 + 1).

Lemma 2 . 7 Lemma 2 . 8

 2728 There exist positive numbers a, b, η with a < b such that S(a) + ηB ⊂ B(0, r) ⊂ B(0, R) ⊂ S(b) and S(a, b) + ηB ⊂ {α : r 0 + ǫ < α }.Proof Pick a number a > 0 lying between the two quantities in the statement of Lemma 2.6. Then evidentlyS(a) ⊂ B(0, rr 0 2 ), cl(compS(a)) ⊂ comp B(0, r 0 + ǫ),whence S(a) + ηB ⊂ B(0, r) for η > 0 suitable small. It also follows that the compact set clS(a, b) is contained in the open set {α : r 0 + ǫ < α }. By shrinking η further if necessary, this will be true of clS(a, b)+ηB as well. Finally, any b suitably large will satisfy B(0, R) ⊂ S(b), since V is bounded on bounded sets. The Infinitesimal Decrease Condition (H5) of §holds on S(a, b) + ηB, with ω := -r 0ǫ. Proof As remarked in §1, it suffices to show that for any α ∈ S(a, b) + ηB, for any ζ ∈ ∂ P V (α), one has : inf{ ζ, f (α, u) : u ∈ U} ≤ -r 0ǫ.

x

  (t) ∈ S(a, b) + γB + (E p + δm)B ⊂ S(b) + 2γB, since E p + δm < γ in view of[START_REF] Leitmann | One approach to the control of uncertain dynamical systems[END_REF] and[START_REF] Sontag | A lyapunov-like characterization of asymptotic controllability[END_REF]. Since 2γ < η by[START_REF] Clarke | Asymptotic Controllability Implies Feedback Stabilization[END_REF], this gives x(t) ∈ S(b) + ηB. By Lemma 1.5 we havey js, f (s, k(y j )) ≤ -ω 2L V y js ,

y 4 ,

 4 jx(t) ≤ y jx j + x jx(t) < E p + δm < γ as already noted. We observe the relationsd(x(t), S(biβ)) ≤ x(t)s = Ψ, x(t)s , d(y j , S(biβ)) = y js ≥ Ψ, y js ,whence d(x(t), S(biβ))d(y j , S(biβ))

  leads tod(x(t), S(biβ))d(y j , S(biβ)) ≤ -2 ∆(tt j ) + E p .We obtain from thisd(y j+1 , S(biβ))d(y j , S(biβ)) ≤ y j+1x(t j+1 ) -2 ∆(tt j ) + E p ≤ -∆(t j+1t j ) + [2E p -∆(t j+1t j )]

5 .

 5 since y j+1 belongs to S(b)+γB by Lemma 3.1. Suppose that k < i. We have d(y j , S(b-iβ)) < γ by definition of Ω i , and Lemma 3.1 implies d(y j+1 , S(biβ)) < γ. But S(biβ) ⊂ S(b -(k + 1)β), so that d(y j+1 , S(b -(k + 1)β)) < γ. But then y j+1 /∈ Ω k by definition of Ω k (note that k ≤ N ). This contradiction proves the Lemma.Lemma 3.3 If τ ∈ π is such that y(τ ) ∈ Ω N +1 , then x(t) ∈ S(a) + γB ∀t ≥ τ. Proof We first establish d(y(τ ), S(b -N β)) ≤ 2γ 5 for all nodes τ ≥ τ.(29)We consider first τ = τ + 1. We have d(y(τ ), S(b -N β)) ≤ γ 4 , whenced(y(τ + 1), S(b -N β)) ≤ d(x(τ + 1), S(b -N β)) + E p ≤ d(x(τ ), S(b -N β)) + E p + mδ ≤ d(y(τ ), S(b -N β)) + 2E p + mδ < y(τ + 1), S(b -N β)) is in fact ≤ γ 4 ,then this same argument yields d(y(τ + 2), S(b -N β)) ≤ 2γ If however d(y(τ + 1), S(b -N β)) > γ 4 , then y(τ + 1) lies in Ω N by definition, and Lemma 3.1 yields again d(y(τ + 2), S(b -N β)) < 2γ 5 . Continuing in this way, we obtain (29) for all nodes τ ≥ τ . We use (29) to argue as follows : let t ≥ τ , and let τ ≥ τ be a node adjacent to t; then d(x(t), S(b -N β)) ≤ d(x(τ ), S(b -N β)) + δm ≤ d(y(τ ), S(b -N β)) + E p + δm gives x(t) ∈ S(a) + γB by Lemma 1.4 (f).


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

EXISTENCE OF LIPSCHITZ AND SEMICONCAVE CONTROL-LYAPUNOV FUNCTIONS

Keywords: 1991 Mathematics Subject Classification. 93D05, 93D20, 93B05, 34D20, 49J52, 49L25, 70K15 Asymptotic stabilizability, semiconcave Lyapunov function, nonsmooth analysis, viscosity solutions. L. Rifford

Given a locally Lipschitz control system which is globally asymptotically controllable to the origin, we construct a control-Lyapunov function for the system which is Lipschitz on bounded sets and we deduce the existence of another one which is semiconcave (and so locally Lipschitz) outside the origin. The proof relies on value functions and nonsmooth calculus.

Introduction

This paper is concerned with the stabilization problem for a standard control system of the form ẋ(t) = f (x(t), u(t)). Lyapunov-like techniques have been successfully used in many problems in control theory, such as stabilizability, asymptotic controllability and stability. Stabilization by smooth feedback has been a subject of research by many authors. Among them, Artstein provided an important contribution (see [START_REF] Artstein | Stabilization with relaxed controls[END_REF]), proving that the control system admits a smooth Lyapunov function if and only if there is a stabilizing relaxed feedback. Moreover, if the system is affine in the control, it is further the case that there exists an ordinary stabilizing feedback continuous outside the origin. In general however such a feedback fails to exist, as pointed out by Sontag and Sussmann [START_REF] Sontag | Remarks on continuous feedback[END_REF] and by Brockett [START_REF] Brockett | Asymptotic stability and feedback stabilization[END_REF] among others ( [START_REF] Ryan | On Brockett's condition for smooth stabilizability and its necessity in a context of nonsmooth feedback[END_REF], [START_REF] Clarke | Asymptotic stability and smooth Lyapunov functions[END_REF]). Consequently, the existence of a smooth Lyapunov function fails in general. This fact leads to the design of time-varying (see [START_REF] Coron | Global asymptotic stabilization for controllable systems without drift[END_REF], [START_REF] Coron | Stabilization in finite time of locally controllable systems by means of continuous time-varying feedback laws[END_REF]) or discontinuous feedbacks. The construction of the latter (see [START_REF] Clarke | Asymptotic controllability implies feedback stabilization[END_REF]) has used the existence of a locally Lipschitz control-Lyapunov function whose decrease condition is stated in terms of Dini derivates or equivalently of proximal subgradients. The first result of this article is that, under certain mild assumptions on f (a local Lipschitz condition and bounded dynamics near the origin), for Globally Asymptotically Controllable systems, such control-Lyapunov functions always exist. This fact extends the well-known result of Sontag [START_REF] Sontag | A Lyapunov-like characterization of asymptotic controllability[END_REF] and brings an affirmative answer to a conjecture that has been attributed to Sontag and Sussmann. Furthermore, the main result shows that a semiconcave control-Lyapunov function outside the origin always exists under the same assumptions. The semiconcavity is an intermediate property between Lipschitz continuity and continuous differentiability. Semiconcave functions have been used for instance to obtain uniqueness results for weak solutions of Hamilton-Jacobi equations, see [START_REF] Kružkov | Generalized solutions of Hamilton-Jacobi equations of eikonal type I[END_REF], [START_REF] Lions | Generalized solutions of Hamilton-Jacobi equations[END_REF]. More recently, attention has been focused on the differential properties of such functions, see [START_REF] Alberti | On the singularities of convex functions[END_REF], [START_REF] Ambrosio | On the propagation of singularities of semi-convex functions[END_REF]. This semiconcavity will be exploited in forthcoming work to construct stabilizing feedbacks having certain regularity properties. Some works and general references related to this article include [START_REF] Bacciotti | Local Stabilizability of Nonlinear Control Systems[END_REF][START_REF] Bacciotti | Lyapunov and Lagrange stability: inverse theorems for discontinuous systems[END_REF][START_REF] Freeman | Robust Nonlinear Control Design. State-Space and Lyapunov Techniques[END_REF][START_REF] Hermes | Resonance, stabilizing feedback controls, and regularity of viscosity solutions of Hamilton-Jacobi-Bellman equations[END_REF][START_REF] Rosier | Étude de quelques problèmes de stabilisation[END_REF].

Definitions and statements of the results

In this paper, we study systems of the general form

ẋ(t) = f (x(t), u(t)) (2.1)
where the state x(t) takes values in a Euclidian space X = R n , the control u(t) takes values in a given set U , and f satisfies the following hypotheses: Assumption 2.1. f is locally Lipschitz in x(uniformly in u). That is, for all x ∈ X, there exists V x a neighborhood of x and L x ≥ 0 such that

f (y , u) -f (y, u) ≤ L x y -y ∀y, y ∈ V x , ∀u ∈ U.
Assumption 2.2. f is bounded on the ball R B × U for all R > 0 (or equivalently, in view of the preceding assumption for some R > 0.) A special element "0" is distinguished in U , and the state x = 0 of X is an equilibrium point, i.e., f (0, 0) = 0 (No linear structure on U is used, however). The set of admissible controls is the set of measurable and locally essentially bounded functions u : R ≥0 -→ U . R ≥0 denotes nonnegative reals, B the open ball B(0, 1) := {x : x < 1} in X and B the closure of B. We now introduce our definitions and the main result. Definition 2.3. The system (2.1) is Globally Asymptotically Controllable (abbreviated GAC) if there exist a nonincreasing function M : R >0 -→ R >0 such that lim R↓0 M (R) = 0 and a function T : R >0 × R >0 -→ R ≥0 with the following property: For any 0 < r < R, for each ξ ≤ R, there exist a control u : R ≥0 -→ U and corresponding trajectory x(•) : R ≥0 -→ X such that 1) lim t→∞ x(t) = 0; 2) ∀t ≥ 0, x(t) ≤ M (R); 3) ∀t ≥ T (r, R), x(t) ≤ r. Remark 2.4. A routine argument involving continuity of trajectories with respect to initial states shows that the requirements of the above standard definition are equivalent to the following apparently weaker pair of conditions used in some references (see [START_REF] Sontag | Nonsmooth control-Lyapunov functions[END_REF], [START_REF] Sontag | General classes of control-Lyapunov functions[END_REF]):

1) For each ξ ∈ X there is a control u : R ≥0 -→ U that drives ξ asymptotically to 0; 2) for each ǫ > 0, there is a δ > 0 such that for each ξ ∈ X with ξ ≤ δ there is a control u : R ≥0 -→ U that drives ξ asymptotically to 0 and such that the corresponding trajectory x(•) satisfies x(t) ≤ ǫ for all t ≥ 0. Moreover, the authors of [START_REF] Sontag | Nonsmooth control-Lyapunov functions[END_REF], [START_REF] Sontag | General classes of control-Lyapunov functions[END_REF] add a condition on bounded controls; this one implies the assumption 2.2 by restriction on the system near the origin.

A function V : X -→ R ≥0 is positive definite if V (0) = 0 and V (x) > 0 for x = 0, and

proper if V (x) → ∞ as x → ∞.
IGD, , , Definition 2.5. A Lyapunov pair for the system (2.1) is a pair (V, W ) consisting of a continuous, positive definite, proper function V : X -→ R and a positive definite continuous function W : X -→ R, with the property that for each x ∈ X \ {0} we have

∀ζ ∈ ∂ P V (ξ), inf u∈U ζ, f (x, u) ≤ -W (x).
(2.2)

Here ∂ P V (x) refers to the proximal subdifferential of V at x (which may be empty): ζ belongs to ∂ P V (x) iff there exists σ and η > 0 such that

V (y) -V (x) + σ y -x 2 ≥ ζ, y -x ∀y ∈ x + ηB.
The condition (2.2) is in fact equivalent to another one often used in the definition of nonsmooth Lyapunov function (see [START_REF] Sontag | A Lyapunov-like characterization of asymptotic controllability[END_REF], [START_REF] Sontag | Nonsmooth control-Lyapunov functions[END_REF], [START_REF] Sontag | General classes of control-Lyapunov functions[END_REF]); this other notion is based on the notion of directional or Dini subderivate. The equivalence between these two conditions is a consequence of Subbotin's Theorem (see for example [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF], our principal source for the theory of nonsmooth analysis and [START_REF] Clarke | Feedback stabilization and Lyapunov functions[END_REF] for a discussion of the equivalence). We remark that there exists a complete calculus of proximal subdifferentials, one that extends all the theorems of the usual smooth calculus. Definition 2.6. A control-Lyapunov function (CLF) for the system (2.1) is a function V : X -→ R such that there exists a continuous positive definite W : X -→ R with the property that (V, W ) is a Lyapunov pair for (2.1).

We will say that V is a locally Lipschitz control-Lyapunov function if V is a control-Lyapunov function which is locally Lipschitz on X. We claim the following theorem. Theorem 2.7. Let (f, U ) be a control system as described above. Then under the assumptions 2.1 and 2.2, if the system is Globally Asymptotically Controllable, there exists a locally Lipschitz control-Lyapunov function. Remark 2.8. The converse is true and relatively easy if we suppose that f is continuous in u (we need this to obtain the existence of trajectories).

We now recall the definition of a semiconcave function [START_REF] Kružkov | Generalized solutions of Hamilton-Jacobi equations of eikonal type I[END_REF] in an open set Ω of X. Definition 2.9. Let u : Ω -→ R be a continuous function on Ω; it is said to be semiconcave on Ω if for any point x 0 ∈ Ω there exist ρ, C > 0 such that

u(x) + u(y) -2u x + y 2 ≤ C x -y 2 , (2.3)
for all x, y ∈ x 0 + ρB.

We shall deduce as a corollary of the preceding theorem the main result of this article. Theorem 2.10. Let (f, U ) be a control system as described above. Then under the assumptions 2.1 and 2.2, if the system is Globally Asymptotically Controllable, there exists a continuous control-Lyapunov function semiconcave on X \ {0}. Remark 2.11. The CLF is a viscosity supersolution of

sup u∈U {-f (x, u), DV } -W ≥ 0. IGD, , ,
We begin by giving some regularity results about certain value functions. Then we give the proof of Theorem 2.7. In the last section, we conclude with the proof of Theorem 2.10.

A Result on Value Functions in finite time

Throughout this section, we are given a multifunction F mapping X to the subsets of X, and we consider the differential inclusion . We need for this section two properties of F which turn out to be particulary important. Assumption 3.1. The multifunction F is locally Lipschitz with non-empty compact convex values. Assumption 3.2. For some positive constants K and M , and for all x ∈ X,

ẋ(t) ∈ F (x(t)) a.e.. ( 3
v ∈ F (x) =⇒ v ≤ K x + M
(that is called the linear growth condition). Under these two conditions, for all x 0 ∈ X, there exists a trajectory of (3.1) defined on R ≥0 such that x(0) = x 0 , and for any trajectory with initial data x 0 we have the following estimate ∀t ≥ 0, x(t) ≤ x 0 e Kt + M te Kt .

(3.2)

This inequality is an easy consequence of Gronwall's Lemma (see [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF]).

Let there be given a function L : X -→ R ≥0 and a compact set T of X satisfying Assumption 3.3. L is locally Lipschitz and for all x ∈ X, L(x) ≥ 1. Assumption 3.4. There exists δ > 0 such that ∀x ∈ T , δ B ⊂ F (x).

We proceed now to define a value function V (•) on X in terms of trajectories of F as follows:

V (x) := inf T 0 L(x(t))dt : x(0) = x, ẋ(t) ∈ F (x(t)) a.e. and x(T ) ∈ T .
(Note that T is a choice variable in this "free-time" problem.) We introduce the notation R := {x ∈ X : V (x) < +∞} , the letter R stands for reachable: the set of points where V is finite is the sets of points which can be driven to the target T in finite time. We have the following theorem. Theorem 3.5. Assume (3.1)- (3.4). Then

(i) R is open; (ii) V is locally Lipschitz in R; (iii) ∀x ∈ R \ T , ∀ζ ∈ ∂ P V (x), min v∈F (x) ζ, v ≤ -L(x). IGD, , ,
Proof. First, by the Lipschitz condition on F and (3.4), there exists 0 < r ≤ 1 such that ∀x ∈ T + r B, δ 2 B ⊂ F (x). Hence, each state x of T + r B can be driven to T by a trajectory of (3.1) in time 2 δ d(x, T ) (where d(x, T ) denotes min τ ∈T x -τ ). This proves that V is finite on T + r B, if we set m := max x∈T +r B L(x), we have

∀x ∈ T + r B, V (x) ≤ 2m δ d(x, T ). (3.3) Fix now x 0 / ∈ T such that V (x 0 ) < +∞.
By the definition of V , there exists a F -trajectory x 0 (•) and T > 0 such that x 0 (0) = x 0 , x 0 (T ) ∈ T and

T 0 L(x(s))ds ≤ V (x 0 ) + 1.
(3.4)

The estimate (3.2) gives

∀t ∈ [0, T ], x 0 (t) ≤ x 0 e KT + M T e KT . Let A := [ x 0 + M T ]e KT , let λ F the Lipschitz constant of F on the ball (A + 1) B. Fix y ∈ B(x 0 , re -λ F T ).
By the corollary 1 p121 of [START_REF] Aubin | Differential Inclusions[END_REF], there exists a F -trajectory y(•) such that y(0) = y and verifying ∀t ∈ [0, T ], y(t) -x 0 (t) ≤ e λ F T y -x 0 , and y(t

) ≤ A + 1. (3.5)
Consequently, if we set λ L the Lipschitz constant of L on the ball (A + 1) B, we obtain

T 0 L(y(s))ds ≤ T 0 L(x 0 (s))ds + T 0 [L(y(s)) -L(x 0 (s))]ds ≤ V (x 0 ) + 1 + T 0 λ L y(s) -x 0 (s) ds ≤ V (x 0 ) + 1 + T λ L e λ F T y -x 0 ≤ V (x 0 ) + 1 + T λ L r.
On the other hand, d(y(T ), T ) ≤ y(T ) -x 0 (T ) ≤ e λ F T y -x 0 ≤ r; this implies by (3.3) that

V (y(T )) ≤ 2m δ d(y(T ), T ) ≤ 2mr δ .
Consequently, we have that for all y ∈ B(x 0 , re -λ F T ),

V (y) ≤ T 0 L(y(s))ds + 2mr δ (3.6) ≤ V (x 0 ) + 1 + T λ L r + 2mr δ =: c < +∞. (3.7)
We have shown that B(x 0 , re -λ F T ) ⊂ R which gives (i). Now, let x ∈ B(x 0 , re -λ F T ), then for each positive integer n, there exists a F -trajectory x n (•) and T n x ≥ 0 such that x n (0) = x, x n (T n x ) ∈ T and

T n x 0 L(x n (s))ds ≤ V (x) + 1 n .
IGD, , , (3.6). As before, the estimate (3.2) gives for each n

Thus L ≥ 1 implies T n x ≤ V (x) + 1 n ≤ c + 1 n by
∀t ∈ [0, T n x ], x n (t) ≤ x e KT n x + M T x e KT n x ≤ [ x + M (c + 1)]e K(c+1) ≤ [ x 0 + 1 + M (c + 1)]e K(c+1) .
So we find a uniform bound for ẋn (•) on the intervals [0, T n x ] ⊂ [0, V (x)+1]. Hence, the theorem of Arzela-Ascoli and the compactness of trajectories (see [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF]) imply that there exists a trajectory x(•) with initial data x such that x(T x ) ∈ T and

V (x) = Tx 0 L(x(s))ds,
with T ≤ V (x). That means that the infimum is attained in the definition of V . We set A := [ x 0 + 1 + M (c + 1)]e K(c+1) , and λ F the Lipschitz constant of F on the ball (A + 1) B. We proceed to show that V is Lipschitz on the ball B(x 0 , r 2 e -λ F (c+1) ). Fix x, y in B(x 0 , r 2 e -λ F (c+1) ). Then there exists as above x(•) a F -trajectory and T x ≥ 0 such that x(0) = x, x(T x ) ∈ T and

V (x) = Tx 0 L(x(s))ds.
≤ V (x) + T x λ L e λ F Tx y -x ≤ V (x) + cλ L e λ F c y -x Now, V (y(T x )) ≤ 2m δ d(y(T x ), T ) ≤ 2m δ e λ F c y -x . Hence, we conclude that V (y) ≤ V (x) + cλ L + 2m δ e λ F c y -x .
Thus, because all the constants in the preceding inequality are independent of x and y, we find

|V (y) -V (x)| ≤ (c + 1)λ L + 2m δ e λ F (c+1) y -x ,
which proves (ii). We now have to prove (iii). For that, consider x ∈ R \ T , and x(•) a trajectory of (3.1) for which the infimum of the definition of V (x) is attained. Let ζ belonging to ∂ P V (x), then there exists σ and η > 0 such that V (y) -V (x) + σ y -x 2 ≥ ζ, y -x ∀y ∈ x + ηB.

IGD, , ,

By optimality of the trajectory x(•), for all t ∈ [0, T ], V (x(t)) = T t L(x(s))ds. Then, for t sufficiently small,

T t L(x(s))ds - T 0 L(x(s))ds + σ x(t) -x 2 ≥ ζ, x(t) -x , which gives - 1 t t 0 L(x(s))ds + tσ x(t) -x t 2 ≥ ζ, x(t) -x t .
We find (iii) by passing to the limit: t ↓ 0.

Remark 3.6. In [START_REF] Clarke | Feedback stabilization and Lyapunov functions[END_REF], a result of this type is proven differently by an appeal to Hamiltonian necessary conditions. Remark 3.7. The conclusions of Theorem 3.5 remain true if we weaken the assumption 3.4 to the proximal condition

min v∈F (x) ζ, v ≤ -δ ζ forall x ∈ T where ζ ∈ N P T (x)
. (This result is a consequence of proximal criteria for attainability, see [START_REF] Clarke | Qualitative properties of trajectories of control systems: a survey[END_REF], [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF].) This kind of condition added to the smooth regularity of F is used in [START_REF] Cannarsa | Convexity properties of the minimum time function[END_REF] to obtain the semiconcavity of the minimum-time function. However, these results (on the Lipschitz property or on the semiconcavity property) do not hold if we omit the linear growth condition (3.2); see for example [7, Ex.1.3,p.238] Remark 3.8. The conclusion (iii) can be strengthened to equality. The value function V is the viscosity solution of a certain Hamilton-Jacobi equation (see [START_REF] Bardi | Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations[END_REF], [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF]).

Proof of Theorem 2.7

We suppose first that we have constructed a control-Lyapunov function V which is continuous on X and locally Lipschitz on X \ {0}. Thus, there exists another continuous positive definite function W : X -→ R ≥0 such that (V, W ) is a Lyapunov pair for (2.1). We proceed to show that we can deduce the existence of a new control-Lyapunov function which is locally Lipschitz on all the space X. We set for any 0

≤ a ≤ b S V (b) := {x; V (x) ≤ b} and S V [a, b] := {x; a ≤ V (x) ≤ b},
they are compact sets of X. We proceed to construct a sequence of functions on X which will converge uniformly to our desired locally Lipschitz control-Lyapunov function. First, we set V 0 (x) := max{V (x), 1}. This function is locally Lipschitz on X, proper, positive, constant on S V (1) and verifies

∀x / ∈ S V (1), ∀ζ ∈ ∂ P V 0 (x), inf u∈U ζ, f (x, u) ≤ -W (x).
By assumption, for all n ≥ 0, V is Lipschitz on we can choose this constant smaller than 1).

S V [ 1 2 n+1 , 1 2 n ], we note K( 1 2 n+1 , 1 2
We define now a sequence inductively; suppose V n given, we set

V n+1 (x) :=      V n (x) si x / ∈ S V ( 1 2 n ) V n (x) + 1 K( 1 2 n+1 , 1 2 n ) [V (x) -1 2 n ] si x ∈ S V [ 1 2 n+1 , 1 2 n ] V n (x) - 1 2 n+1 K( 1 2 n+1 , 1 2 n ) si x ∈ S V ( 1 2 n+1 )
We have the following lemma. Lemma 4.1. For all n ≥ 1, V n is 1-Lipschitz on S V (1), proper, and constant on S V ( 1 2 n ). Moreover, V n satisfies the following properties:

∀x ∈ X, V n (x) ≥ 1 - n k=1 1 2 k K( 1 2 k , 1 2 k-1 )
, and

∀x ∈ S V ( 1 2 n-1 ) \ S V ( 1 2 n ), ∀ζ ∈ ∂ P V n (x), inf u∈U ζ, f (x, u) ≤ - W (x) K( 1 2 n , 1 2 n-1 ) . (4.1)
Proof. We're going to prove only the last assertion. The other ones are left to the reader, they are the consequence on an easy inductive proof.

Let n ≥ 1, x ∈ S V ( 1 2 n-1 ) \ S V ( 1 2
n ), and ∀ζ ∈ ∂ P V n (x). We remark that for all y not in S V ( 1 2 n ), we have

V n (y) = min V n-1 (y), V n-1 (y) + V (y) -1 2 n-1 K( 1 2 n , 1 2 n-1 )
.

For x, the minimum is attained in the second term , so

ζ ∈ ∂ P V n-1 (x) + V(x) -1 2 n-1 K( 1 2 n , 1 2 n-1 ) = ∂ P V n-1 (x) + V(x) K( 1 2 n , 1 2 n-1 )
. (4.2)

First case: n > 1. We remark now that ∀y ∈ S V ( 1 2 n-2 ),

V n-1 (x) = max C n-2 + V (x) -1 2 n-2 K( 1 2 n-1 , 1 2 n-2 ) , C n-2 - 1 2 n-1 K( 1 2 n-1 , 1 2 n-2 )
, where C n-2 is the value of V n-2 on the set S V ( 1 2 n-2 ). We deduce by (4.2) that

ζ ∈ ∂ P max V (x) K( 1 2 n-1 , 1 2 n-2 ) + A, A + V (x) K( 1 2 n , 1 2 n-1 )
.

where

A = C n-2 - 1 2 n-2 K( 1 2 n-1 , 1 2 n-2 ) et A = C n-2 - 1 2 n-1 K( 1 2 n-1 , 1 2 n-2 )
. Hence, we obtain that ζ is in the set

∂ P max 1 K( 1 2 n-1 , 1 2 n-2 ) + 1 K( 1 2 n , 1 2 n-1 ) V (x) + A, V (x) K( 1 2 n , 1 2 n-1 ) + A .
Now, by the basic calculus on the proximal subgradients, we have

ζ ∈ co 1 K( 1 2 n-1 , 1 2 n-2 ) + 1 K( 1 2 n+1 , 1 2 n ) ∂ P V (x), 1 K( 1 2 n , 1 2 n-1 ) ∂ P V (x). IGD, , ,
Then, there exists

ζ 1 et ζ 2 in ∂ P V (x) and t ∈ [0, 1] such that ζ = t 1 K( 1 2 n-1 , 1 2 n-2 ) + 1 K( 1 2 n , 1 2 n-1 ) ζ 1 + (1 -t) 1 K( 1 2 n , 1 2 n-1 ) ζ 2 . = t( 1 K( 1 2 n-1 , 1 2 n-2 ) + 1 K( 1 2 n , 1 2 n-1 ) ) + (1 -t) 1 K( 1 2 n , 1 2 n-1 ) ζ where ζ ∈ ∂ P V (x), because ∂ P V (x) is a convex set. Now, we invoke the decrease property of V , inf u∈U ζ, f (x, u) ≤ -W (x). Then inf u∈U ζ, f (x, u) ≤ - 1 K( 1 2 n , W (x) 2 n-1 )
, which gives the result. Second case: If n = 1, the proof is similar. Now, note that for each x = 0, the sequence (V n (x)) n≥0 is stationary, thus it converges. So, we can define

V(x) := lim n→∞ V n (x) -C.
where

C := 1-+∞ n=0 1 2 n+1 K( 1 2 n+1 , 1 2 n ) ∈ [0, 1]
(because the Lipschitz constants have been chosen smaller than 1). By the preceding lemma, V n is always positive and for x = 0,

V n (0) = 1 - n k=1 1 2 k K( 1 2 k , 1 2 k-1 ) → n→∞ 1 -C =: V(0).
We deduce that V(0) = 0 and then that V is positive definite. On the other hand, it is locally Lipschitz everywhere as a simple limit of Lipschitz functions (with the same constant in each compact set on X) and it verifies the decreasing property (2.2) with a continuous positive definite function W defined as follows:

W(x) := inf y∈X {w(y) + x -y }, forall x ∈ X;
where

w(x) :=      W (x) if x / ∈ S V (1) W (x) K( 1 2 n+1 , 1 2 n ) if x ∈ S V ( 1 2 n ) \ S V ( 1 2 n+1 ) 0 if x = 0.
The decrease property is an immediate consequence of (4.1).

To complete the proof of Theorem 2.7, we now have to prove the existence of a control-Lyapunov function which is continuous on X and locally Lipschitz outside the origin. We begin by defining a multifunction F , which is useful because uniformly bounded:

∀x ∈ X, F (x) := cl co f (x, u) 1 + f (x, u) ; u ∈ U . IGD, , ,
We study the differential inclusion ẋ(t) ∈ F (x(t)) a.e.

This dynamic has the same properties as the system (2.1). Proposition 4.2.

(i) F is locally Lipschitz and compact convex valued;

(ii) The system ẋ(t) ∈ F (x(t)) is GAC.

Proof. (i) We omit the proof.

(ii) Let x ∈ X with x ≤ R be given. By assumption, there is a trajectory x(•) of (2.1) on [0, ∞) which verifies the assumptions of Global Asymptotic Controllability (Definition 2.3). We set φ(t) := We deduce that if τ ≥ T (r, R)(1 + N R ), then τ ≥ φ(T (r, R) and consequently x(τ ) ≤ r.

The new differential inclusion ẋ ∈ F (x) is GAC with suitable constants M (R) and T (r, R)

:= T (r, R)(1 + N R ).
We shall use the notation M (•) and T (

constants M (R) ↓ 0 and Tβ (r, R) = T (r, R) max x∈M (R) B {β(x) -1 }. IGD, , ,
We remark that for x ≥ 2M (1), we have

Γ 0 (x) ≤ [1 + T ( 1 2 , 1) M (1) ] -1 .
Then the time used for a Γ 0 trajectory with initial condition α to reach the ball 2M (1) B is greater than [1 +

T ( 1 2 ,1)
M (1) ]M (1). Hence, V 0 (α) ≥ M (1) + T ( 12 , 1) > m 0 . Consequently, S 0 ⊂ 3M (1) B. (c) This last assertion is a consequence of Lemma 4.6.

Second step

We now define a value function with a decrease property which holds closer to the origin. We set

Γ 1 (x) :=            1 + ( x -M ( 1 2 )) T ( 1 4 , 1 2 ) M ( 1 2 ) 2 -1 F (x) for x ≥ M ( 1 2 ) F (x) for 1 2 ≤ x ≤ M ( 1 2 ) F (x) + 8[ 1 2 -x ] B for 1 4 ≤ x ≤ 1 2 F (x) + 2 B for x ≤ 1 4
We have immediatly the following result. Lemma 4.8. Γ 1 is compact convex valued and the system ẋ(t) ∈ Γ 1 (x(t)) is GAC (with possible constants M 1 (R) = M (R) ↓ 0 and T1 (r, R)).

We need an auxiliary function with the local Lipschitz property. We define for all x ∈ X: B 0 (x) := max{V 0 (y) : y ≤ x + M (1)}.

As before, the new multifunction leads to a value function R 1 associated to the set A 1 := 1 4 B. We set for all x in X:

R 1 (x) := inf T 0 L 1 (x(t))dt : x(0) = x, ẋ ∈ Γ 1 (x) a.e. and x(T ) ∈ τ 1 where L 1 (x) := 1 + max{0, x -3M (1)} B 0 (x) ρ 1 M (1) 2 and ρ 1 := m 0 /2 m 0 1 + (3M (1) -M ( 1 2 )) T ( 1 4 , 1 2 ) M ( 1 2 ) 2 + T1 ( 1 4 , 1) ≤ 1.
The theorem 3.5 gives the following lemma. Lemma 4.9.

(a) R 1 is locally Lipschitz on X; (b) ∀ x ≥ 1 2 , ∀ζ ∈ ∂ P R 1 (x), min v∈F (x) ζ, v ≤ -L 1 (x).
Proof. L 1 and Γ 1 being locally Lipschitz and the system associated to Γ 1 GAC, R 1 is finite everywhere and the theorem 3.5 proves the assertions.

As in the first step, we're going to evaluate the size of a certain level set given by R 1 . We set m R 1 := max{R 1 (y) : y ∈ 1 2 B} and S R 1 (m R 1 ) = {x : R 1 (x) ≤ m R 1 }. IGD, , , second one. In this case, x / ∈ S 0 and ζ ∈ ∂ P ( Ṽ0 + m 0 )(x) = ∂ P Ṽ0 (x). We conclude by Lemma 4.7 (c). (f) This is an easy consequence of the Lemma 4.11(b).

Third Step

We finish the construction of the sequence (V n ) n≥0 by induction on n. Assume (V k , A k , L k , R k , Γ k ) have been already defined for 1 ≤ k ≤ n with the following properties.

1) As before, we can define a new function V n+1 . We proceed as follows; for all x ∈ X, we set Γ n+1 (x) :=

1 2 k B ⊂ S k ⊂ 3M ( 1 2 k ) B; 2) For x ≥ 5M ( 1 2 k-1 ), V k (x) = V k-1 (x); 3) For 1 2 k ≤ x ≤ 5M ( 1 2 k-1 ), ∀ζ ∈ ∂ P V k (x
           1 + ( x -M ( 1 2 n+1 )) T ( 1 2 n+2 , 1 2 n+1 ) M ( 1 2 n+1 ) 2 -1 F (x) if x ≥ M ( 1 2 n+1 ) F (x) if x ∈ [ 1 2 n+1 , M ( 1 2 n+1 )] F (x) + 2 n+3 [ 1 2 n+1 -x ] B if x ∈ [ 1 2 n+2 , 1 2 n+1 ] F (x) + 2 B if x ≤ 1 2 n+2
We need as before an auxiliary function with the local Lipschitz property.

We define for all x ∈ X:

B n (x) := max{V n (y) : y ≤ x + M ( 1 2 n )}. From this multifunction, we define a value function associated to the set T n+1 := 1 2 n+2 B. We set for any x ∈ X, R n+1 (x) := inf where L n+1 (x) := 1 + max{0, x -3M ( 1 2 n )} Bn(x)

ρ n+1 M ( 1 2 n ) 2 with ρ n+1 := ρ n m n /2 m n 1 + (3M ( 1 2 n ) -M ( 1 2 n+1 )) T ( 1 2 n+2 , 1 2 n+1 ) M ( 1 2 n+1 ) 2 + Tn+1 ( 1 2 n+2 , 1 2 n ) ≤ ρ n .
The differential inclusion ẋ ∈ Γ n+1 (x)) is GAC, we denote by Tn+1 (•, •) its new constant (we saw that we can choose Mn+1 = M ). On the other hand, we set m R n+1 := max{R n+1 (y) : y ∈ 1 2 n+1 B}, and S R n+1 (m R n+1 ) := {x : R n+1 (x) ≤ m R n+1 }.

IGD, , ,

. 1 )

 1 A solution x(•) of (3.1) on the interval [a, b] is taken to mean an absolutely continuous function x : [a, b] -→ X which, together with ẋ, satisfies (3.1); such an arc will be called a F -trajectory on the interval [a, b]

By [ 4 ,

 4 Cor.1,p.121], there exists a F -trajectory y(•) such that y(0) = y and verifying ∀t ∈ [0, T x ], y(t) -x(t) ≤ e λ F Tx y -x , and y(t) ≤ A + 1. Consequently, if we set as before λ L the Lipschitz constant of L on the ball (A + 1) B, we obtain Tx 0 L(y(s))ds ≤ Tx 0 L(x(s))ds + Tx 0 λ L y(s) -x(s) ds

t 0 [ 1 = t 0 [ 1

 0101 + ẋ(s) ]ds and we define a function x on [0, ∞] by x(τ ) := x(t), where t = t(τ ) is determined in [0, ∞] by τ + ẋ(s) ]ds (this change of variables or time scale is known as the Erdmann Transform.) Then dx dτ = ẋ(t) 1 + ẋ(t) ∈ Γ(F x(τ )) a.e., so that x is a F -trajectory. But by construction, for all τ ≥ 0, x(τ ) ≤ M (R) and if τ ≥ φ(T (r, R)) then x(τ ) ≤ r. The trajectory x(•) remains in the ball M (R) B, so if N R denotes the maximum of f (x, u) for x ∈ M (R) B and u ∈ U (finite by the assumption 2.2), we have ∀t ≥ 0, φ(t) ≤ t(1 + N R ).

  ), min v∈F (x) ζ, v ≤ -ρ k ; 4) L k = 1 on the ball 3M ( 1 2 k-1 ) B; 5) ∀x ∈ R N , V k (x) = 0 ⇐⇒ x ∈ 1 2 k+1 B; 6) ∀k ∈ [1, n], m k ≤ m k-1 2 , and ρ k ≤ ρ k-1 ≤ 1 =: ρ 0 .With the following definitions:m k := max{V k (x); x ≤ 1 2 k },andS k := {x; V k (x) ≤ m k }, where ρ k are some positive constants.

T 0 L

 0 n+1 (x(t))dt : x(0) = x, ẋ ∈ Γ n+1 (x) and x(T ) ∈ T n+1

  •, •) for the constant of global asymptotic stability of F . Remark 4.3. We have in fact by a similar proof the following property. Proposition 4.4. Let β : X -→ R >0 locally Lipschitz. Then the differential inclusion

ẋ(t) ∈ β(x(t))F (x(t)) a.e. is locally Lipschitz with convex compact values and is GAC with appropriate

Acknowledgments. I would like to thank Professor Francis Clarke for his comments and his suggestions.

First Step:

We proceed to define a first multifunction Γ 0 as follows:

By construction (and by the proposition 4.4), we have immediately the following lemma. Lemma 4.5. Γ 0 is compact convex valued, locally Lipschitz, uniformly bounded (by 1) and the differential inclusion ẋ ∈ Γ 0 (x(t)) is GAC.

On the other hand, B ⊂ Γ 0 (x) for all x in 1 2 B. Hence, the theorem 3.5 can be applied with T = T 0 := 1 2 B and L = L 0 := 1. So we define the value function V 0 (x) := inf{T : x(0) = x, ẋ(t) ∈ F (x(t))a.e. and x(T ) ∈ 1 2 B}, for all x ∈ X. Lemma 4.6. V 0 est locally Lipschitz on X, positive, proper and for all x / ∈ B,

Proof. This is an easy corollary of the theorem 3.5.

We set m 0 := max{V 0 (x); x ≤ 1} and S 0 := {x; V 0 (x) ≤ m 0 }. We define a new function Ṽ0 as follows: Ṽ0 (x) := max{0, V 0 (x) -m 0 }. The first inclusion is given by the definition of Ṽ0 . However, the second one is less easy. Since the system ẋ ∈ F (x)) is GAC, for all x ∈ B there exists a F -trajectory x(•) such that [START_REF] Alberti | On the singularities of convex functions[END_REF] x(0) = α et ẋ(t) ∈ F (x(t)) p.p, (2) ∀t ≥ 0, x(t) ≤ M (1), [START_REF] Artstein | Stabilization with relaxed controls[END_REF]

Consequently, m 0 ≤ T ( 1 2 , 1). Let us consider now α ∈ X such that α ≥ 3M (1).

IGD, , ,

By the proposition 4.2, for any x ∈ 1 2 B, there exists a F -trajectory x(.) such that [START_REF] Alberti | On the singularities of convex functions[END_REF]

.

). Now, we consider an initial state α such that α ≥ 3M ( 12 ). We remark that for x ≥ 2M (1), we have

Then the time used for a Γ 1 trajectory with initial condition α to reach the ball 2M ( 12 ) B is greater than [1 +

Indeed, The proof gives the following lemma. Lemma 4.10.

Proof. (a) Let x ∈ S 0 . Indeed, there exists a Γ 0 -trajectory x(•) which relies

x to the set B with time

] we can write Γ 1 (x) ⊂ β(x)Γ 0 (x) with β(x) as follows (assuming that M ( 1 2 ≥ 1):

]

We observe that if M ( 1 2 ) < 1, we have to omit it in the definition of β. Now, an appropriate change of variables (see Proposition 4.4) provides that there exists a Γ 1 -trajectory x(•) which remains in 3M (1) B and drives x to B with a time

. Now, we can extend this trajectory until T 1 with the following property (by Lemma 4.8): ∀t ≥ T, x(t) ∈ M (1) B and x(T + T1 ( 14 , 1)) ∈ 1 4 B. In IGD, , , so doing, we have constructed a trajectory which remains in 3M (1) B ( where L 1 = 1) and reaches the set T 1 .

. We conclude by the definition of ρ 1 . (b) Let x such that x ≥ 5M [START_REF] Alberti | On the singularities of convex functions[END_REF]. By the definition of B 0 we have

On the other hand, the time required for driving from { y ≥ x -M (1)} to { y ≥ x -2M (1)} is greater than M (1) (the dynamic is bounded by 1). Consequently,

We finish this step by defining a new function V 1 as follows.

We set

We have the following lemma. Lemma 4.12. V 1 is locally Lipschitz on X. Moreover, we have,

Proof. (a) By the Lemma 4.11 (a), for any

And by the remark of (a),

First Case: The minimum is attained for the second term.

We conclude by Lemma 4.9 (b). Second Case: the minimum is attained for the first term and not for the

Proof. (a) and (b) are evident by the theorem 2.7. The assertion (c) is proved as before (Lemma 4.10). We prove now (d) and (e), we begin with (e). Let there be given x ≥ 5M ( 12 n ). By the definition of B n we have

On the other hand, the time required for driving from

We prove now (d). Let x ∈ S n . Indeed, there exists a Γ n -trajectory x(•) which relies x to the set 1 2 n B with time

An appropriate change of variables (see Proposition 4.4) provides that there exists a Γ n+1 -trajectory x(•) which remains in 3M ( 1 2 n ) B and drives x to

. Now, we can extend this trajectory until T n+1 with the following property (by Lemma 4.8

In this way, we have constructed a trajectory which remains in 3M ( 12 n ) B ( where L n+1 = 1) and reaches the set T n+1 . Consequently,

n ). We conclude by the definition of ρ n+1 .

IGD, , ,

We can now define the new function V n+1 . We set for all x ∈ X,

where Ṽn (x) := max{0, V n (x) -m n }.

As before, we consider

We have the following lemma. Lemma 4.14. V n+1 is locally Lipschitz on X. Moreover, we have

The proof is similar to the proof of Lemma 4.12. This is let to the reader.

Fourth

Step: The Function V We study the convergence of the sequence (V n ) n≥0 . For that, we need a last lemma.

Proof. We do an inductive proof. The result being already proven for n=1 (Lemma 4.12), assume that we have proved the result for n ≥ 1; we establish the property for n + 1. Let us consider 0 ≤ k ≤ n + 1 and x ∈ S k . First Case: k ≤ n.

If x /

∈ S n , then by definition of V n , V n (x) = Ṽn (x) + m n . Hence, V n+1 (x) ≤ V n (x) ≤ m k by induction. Otherwise, x ∈ S n . In this case, Ṽn (x) = 0 implies V n+1 (x) ≤ m n ≤ m k by the property on the sequence (m k ). Second Case: k = n + 1. The property follows from the definition of S n+1 .

We can now conclude. Let us consider K a compact set of X \ {0}. Then, as lim n→∞ M ( 1 2 n ) = 0, there exists a positive integer n K such that

By the second property of the sequence (

Hence, the sequence (V n (x)) n≥0 converges for all x in K and the limit is locally Lipschitz function in K (as a stationnar limit of locally Lipschitz functions). On the other hand, for any n ≥ 0, V n (0) = 0; so we can define for all x ∈ X, V (x) := lim n→∞ V n (x).

By the proof above, V is locally Lipschitz on X \ {0}, positive definite and proper (because 1)); we have to show that V is continuous at the origin. This fact is a consequence of the preceding lemma. Let us consider x p -→ p→∞ 0. We want to show that f (x p ) -→ p→∞ 0.

IGD, , ,

Let ǫ > 0. There exists n 0 ≥ 0 such that m n 0 ≤ ǫ (because m n ≤ m 0 2 n ). Thus, by the last lemma, ∀n ≥ n 0 , ∀x ∈ S n 0 , V n (x) ≤ ǫ. But for p sufficiently great (p ≥ P ), x p ∈ 1 2 n 0 B ⊂ S n 0 . We deduce that ∀n ≥ n 0 , ∀p ≥ P, V n (x p ) ≤ ǫ. By passing to the limit: ∀p ≥ P, V (x p ) ≤ ǫ, which gives the continuity on the origin. Now, we set ∀x ∈ X,

We can now define the function W by

W is a positive definite and locally Lipschitz function. The decrease condition (2.2) is the consequence of the stationarity of the sequence (V n (x)) n≥0 outside the origin and of the Lemma 4.14 (d). This completes the proof of Theorem 2.7.

Existence of a semiconcave CLF

We begin by some preliminaries on the semiconcavity. It is easy to show that any semiconcave function in Ω is locally Lipschitz. Concave functions are of course, semiconcave. Another class of semiconcave functions is that of C 1 functions with locally Lipschitz gradient. Moreover we have the two following lemmas. Lemma 5.1. Let Ψ : R -→ R be an increasing semiconcave function and u : Ω -→ R be a semiconcave function on Ω. Then Ψ • u is a semiconcave function on Ω. Lemma 5.2. Let u, v : Ω -→ R be two semiconcave functions on Ω, then the function min{u, v} is semiconcave on Ω.

A convenient way to build semiconcave approximations of a given function is provided by the method of inf-convolution, a standard tool in convex and non-smooth analysis. Let Ω be a subset of X and u a positive function in Ω. Define, for any α > 0,

(5.1)

Lemma 5.3. Let u : X -→ R be a locally Lipschitz and proper function.

Then u α is semiconcave on X (the infimum is attained in the definition of u α ) and moreover, u α u, as α → +∞, locally uniformly in X.

Proof. We leave the proof to the reader.

We can link the proximal subdifferentials of u and its inf-convolution. We have the following Lemma. (We refer to [14, Theorem.5.1,p.44] for the proof.) Lemma 5.4. Suppose that x ∈ X is such that ∂ P u α (x) is nonempty. Then there exists a point ȳ ∈ X satisfying the following:

a) The infimum in (5.1) is attained uniquely at ȳ. b) The proximal subgradient

Proof of Theorem 2.10 By Theorem 2.7, there exists a control-Lyapunov pair for the system (2.1); without loss of generality, we can suppose that the function W is 1-Lipschitz on X (othewise, we can set W (x) := inf y∈X {W (y) + x -y }). For any 0 < r < R, we define the following sets:

Let there be given an integer n ∈ N * . By the Lipschitz property of f and V , we can consider

where the constant M n is defined by

On the other hand, we note w n the minimum of W on S V [ 1 2n , 11n], and we set

We define by inf-convolution the function V αn as follows:

Lemma 5.5. Let x 0 ∈ S V (M n ). If the infimum in the definition of V αn (x 0 ) is attained at ȳ, then x 0 -ȳ ≤ min{

Proof. If the infimum is attained for ȳ, then

we find a contradiction. Hence,

On the other hand, we have found the estimate

x 0 -ȳ which implies the desired inequality by the bound on x 0 -ȳ . IGD, , , Lemma 5.6. Let x 0 ∈ S V [ 1 2n , 11n] and ζ ∈ ∂ P V αn (x 0 ), then

Proof. By the Lemmas 5.4 and 5.5, the infimum in the definition of V αn (x 0 ) is attained uniquely at a point ȳ ∈ S V (11n) which satisfies x 0 -ȳ ≤ wn 2(1+L n V L n f ) and such that ζ ∈ ∂ P V (ȳ). Thus, by the Lipschitz properties of f , V and W , we can write:

Lemma 5.7. There exists Ψ n : R ≥0 -→ R ≥0 C ∞ increasing which satisfies the following properties:

2 . Proof. The different properties lead to defining a piecewise affine function which we then regularize to render it C ∞ , giving Ψ n .

We now set Ṽn := Ψ n • V αn , this function is semiconcave on X by Lemma 5.1. The definitive Lyapunov pair (V, W) is defined for all x ∈ X by:

]. Hence, Lemma 5.7 implies that Ṽn (x 0 ) = V αn (x 0 ). On the other hand, for any p ≥ n, by construction α p ≥ α n and then V αp (x 0 ) ≥ V αn (x 0 ). The same argument as above on Ψ p leads to Ṽp (x 0 ) = V αp (x 0 ) ≥ Ṽn (x 0 ) = V αn (x 0 ). consequently, we have shown that V(x 0 ) = min 1≤p≤n Ṽp (x 0 ). Now, if the minimum in the definition of V(x 0 ) is attained for Ṽn 0 (x 0 ) (with 1 ≤ n 0 ≤ n) then ζ ∈ ∂ P V(x 0 ) =⇒ ζ ∈ ∂ P Ṽn 0 (x 0 ) = Ψ (V αn 0 (x 0 ))∂ P V αn 0 (x 0 ). We now have to show the inequality (5.5). First Case: If V (x 0 ) > 11n 0 and V αn 0 (x 0 ) ≤ 11n 0 , then there exists ȳ ∈ x 0 + B IGD, , , (because α n 0 ≥ 11n 0 ) such that V αn 0 (x 0 ) = V (ȳ) + α n 0 x 0 -ȳ 2 . Therefore, ȳ ∈ S V (11n 0 ) and x 0 ∈ S V (M n 0 ) by definition of M n 0 . By Lemma 5.5 and Lemma 5.7 (ii), we obtain V αn 0 (x 0 ) ≥ 11n 0 -1 8n 0 and Ṽn 0 (x 0 ) ≥ 11n 0 +V (x 0 ). But Ṽn (x 0 ) = V αn (x 0 ) ≤ V (x 0 ). Hence, n 0 = n, and we have the decrease property by Lemma 5.6. Second Case: If V (x 0 ) > 11n 0 and V αn 0 (x 0 ) > 11n 0 , then Lemma 5.7(ii) implies Ṽn 0 (x 0 ) ≥ 11n 0 + V (x 0 ), we conclude as in the first case. Third Case: If V (x 0 ) < 1 2n 0 , then

But we proved that Ṽn (x 0 ) = V αn (x 0 ) ≤ V (x 0 ), so the minimum is also attained for n; then we have (5.5) by Lemma 5.6. Fourth Case: If x 0 ∈ S V [ 1 2n 0 , 11n 0 ], then we conclude by Lemma 5.6 and Lemma 5.7 (iv). This last Lemma shows that the minimum in the definition of V(x) is always attained for x = 0. Therefore, the function V is semiconcave outside the origin (by Lemma 5.2). On the other hand, V is continuous on the origin (because 0 ≤ V ≤ V ) and satisfies the decrease condition by 5.5. Consequently V provides a control-Lyapunov function; which proves the Theorem 2.10.