Yacine Chitour
email: yacine.chitour@l2s.centralesupelec.fr

Rosane Ushirobira
email: rosane.ushirobira@inria.fr

Hassan Bouhemou

STABILIZATION FOR A PERTURBED CHAIN OF INTEGRATORS IN PRESCRIBED TIME

Keywords: Mathematics Subject Classification. 93D15, 93D21, 93D05. Key words and phrases. stabilisation, finite-time, input-to-state stability

In this paper, we consider issues relative to prescribed time stabilisation of a chain of integrators of arbitrary length, either pure (i.e., where there is no disturbance) or perturbed. In a first part, we revisit the proportional navigation feedback (PNF) approach and we show that it can be appropriately recasted within the framework of time-varying homogeneity. As a first consequence, we recover all previously obtained results on PNF with simpler arguments. We then apply sliding mode inspired feedbacks to achieve prescribed stabilisation with uniformly bounded gains. However, all these feedbacks are robust to matched uncertainties only. In a second part, we provide a feedback law yet inspired by sliding mode which not only stabilises the pure chain of integrators in prescribed time but also exhibits some robustness in the presence of measurement noise and unmatched uncertainties.

Introduction

In this paper, we consider the following problem: for n positive integer and T positive real number, the perturbed chain of integrators is the control system given by [START_REF] Aldana-Lpez | Enhancing the settling time estimation of a class of fixed-time stable systems[END_REF] ẋ(t) = J n x(t) + (d(t) + b(t)u(t)) e n , t ∈ [0, T ), x(t) ∈ R n , u(t) ∈ R

where (e i ) 1≤i≤n denotes the canonical basis of R n , J n denotes the n-th Jordan block (i.e., J n e i = e i-1 for 1 ≤ i ≤ n with the convention e 0 = 0), d(•) and b(•) denote respectively a matched uncertainty and the uncertainty on the control respectively. Moreover we assume that there exists b > 0 such that (2) b(t) ≥ b, ∀t ∈ [0, T ).

Our goal consists in designing a feedback control u that renders the system fixed-time inputto-state stable in any time T > 0 (prescribed-time stabilisation) and possibly convergent to zero (PT+ISS+C) (cf. [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF], and Definition 2 given below). Note also that one may ask robustness properties in the presence of noise measurement d 1 , for instance, if the feedback control u is static, it takes the form u = F (x + d 1 )) and unmatched uncertainty d 2 , i.e., ẋ = J n x + (d + bu)e n + d 2 .

Here, the feedback law F (x) stabilises the n-th order pure chain of integrators.

It is clearly more difficult to address the issue of prescribed time stabilisation rather than the issue of finite time stabilisation, cf [START_REF] Snchez-Torres | A class of predefined-time stable dynamical systems[END_REF][START_REF] Aldana-Lpez | Enhancing the settling time estimation of a class of fixed-time stable systems[END_REF][START_REF] Pal | Design of controllers with arbitrary convergence time[END_REF][START_REF] Wang | Prescribed-time consensus and containment control of networked multiagent systems[END_REF]. In missile guidance [START_REF] Zarchan | Tactical and strategic missile guidance[END_REF] and other applications, the issue of prescribed time stabilisation has a long history and two main approaches for solving this problem have been considered: (a) proportional navigation feedback (PNF), which is a feedback linear in the state x and with time-varying gains blowing up to infinity towards the prescribed fixed time; (b) optimal control with a terminal constraint, where the dependency of the gains with respect to the state is implicit. Stemming from the PNF design for second order chains of integrators, a general approach is proposed in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF] for n-th order perturbed chains of integrators, i.e., in [START_REF] Aldana-Lpez | Enhancing the settling time estimation of a class of fixed-time stable systems[END_REF]). The feedback law has the form u(t) = K T P 1 ( 1 T -t )(x(t)) + P 2 ( 1 T -t ), where P 1 and P 2 are polynomials (either matrix or real valued) and the vector K must be chosen in a rather involved way. The first term in the feedback is definitely of PNF type but the second one is only necessary for the convergence argument and does not appear for second order models for instance. Anyway, the controller in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF] does tend to zero as t tends to T even though the gains blow up and it exhibits excellent robustness properties in the case of matched uncertainties. However, the authors only suggest that it behaves poorly in case of measurement noise or unmatched uncertainties and also claim that all known techniques (including theirs) do not work in case of unmatched uncertainties. Finally, due to the rather complicated stability analysis as well as the involved construction of the feedback, it is not clear how to measure quantitatively the limitations of that feedback (see Section 3.2 in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF]) and to possibly improve its results.

The first part of the present paper aims at revisiting PNF design with a new perspective. We show that it can be naturally seen as a particular instance of weighted-homogeneous control systems (cf. [START_REF] Shtessel | Sliding mode control and observation[END_REF] for instance) with the usual homogeneity coefficient not anymore constant but being equal to an appropriate time-varying function. Indeed, recall that a PNF has the form [START_REF] Chitour | Lp-stabilization of integrator chains subject to input saturation using Lyapunov-based homogeneous design[END_REF] u

P N F (t) = n i=1 k i x i (t) (T -t) n-i+1 , t ∈ [0, T ),
where K = (k 1 , • • • , k n ) ∈ R n . Let r = (n -i + 1) 1≤i≤n be the weight vector and set, for λ > 0, D r λ to be the diagonal matrix made of the λ n-i+1 's. Rewriting u P N F (t) = K T D r λ(t) x(t) with λ(t) = 1 T -t suggests at once to consider the new state y(t) = D r λ(t) x(t) for which u P N F simply reduces to K T y. Now, the dynamics of y with respect to the time scale s(t) = ln( T T -t ) turns out to be given by [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF] dy ds = (D r + J n )y + (b(s)u(s) + d(s))e n , s ≥ 0, where D r is the constant diagonal matrix made of the n -i + 1's. Hence, the original problem of prescribed time stabilisation of (1) in time T > 0 has been reduced to the asymptotic stabilisation of an n-th order perturbed chain of integrators with the extra term D r y in (4) with respect to [START_REF] Aldana-Lpez | Enhancing the settling time estimation of a class of fixed-time stable systems[END_REF]. Note that, to the best of our knowledge, it seems to be the first time that one considers a timevarying homogeneity coefficient in the context of stabilisation of weighted-homogeneous systems in that general manner. Usually, the homogeneity coefficient, when non constant, is statedependent (cf. [START_REF] Praly | Generalized weighted homogeneity and state dependent time scale for linear controllable systems[END_REF] as the pioneering reference for fixed-time stabilisation of linear systems, then [START_REF] Cruz-Zavala | Second-order uniform exact sliding mode control with uniform sliding surface[END_REF] and [START_REF] Harmouche | Robust and adaptive higher order sliding mode controllers[END_REF] for instance, in the case of second order and n-th order perturbed chains of integrators respectively.) With the previous viewpoint, it is immediate to see that PNF (and its variant given in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF]) is nothing more but the stabilisation of (4) with a linear feedback. As a consequence, we recover all the results of [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF] with much simpler arguments and the limitations (as well as the advantages) of such a feedback appear in a transparent way. In particular, our convergence analysis easily reduces to the verification of a Linear Matrix Inequality (LMI), see Proposition 10 below, whose solution is essentially given in [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF]. Moreover, the fact that measurement noise and unmatched uncertainties in (1) cannot be handled with that linear feedback is obvious since the corresponding disturbances become amplified by D r λ(t) in ( 4) and one can measure explicitly their destabilising effect.

One can then turn to other types of stabilisation for (4). If the settling time of the system associated to some feedback law u = F (s, y) is infinite (i.e., the supremum over the initial conditions x 0 of the time needed to reach the origin for the trajectory of (4) fed by u = F (s, y) and starting at x 0 ), then we will unavoidably face the numerical challenge of plotting y(s(t)) = D r λ(t) x(t)) in ( 1), with D r λ(t) growing unbounded as t tends to T . Therefore, we should aim at feedback laws u = F (s, y) providing fixed-time convergence for the y variable. On the other hand, recall that the n-th order perturbed chain of integrators is the basic model for sliding mode control, cf. [START_REF] Shtessel | Sliding mode control and observation[END_REF], for which there exist plenty of efficient finite time stabilizers with eventually good robustness properties. At the heart of these stabilizers, lies the technique of weigthedhomogeneity with constant homogeneity coefficient. We will show that this technique easily extends to handle (4) and its extra linear term D r y to produce fixed-time stabilizers for (4) under the assumption that bounds on b and d in (4) are known a priori. In particular, under that assumption, this resolves in a satisfactory manner one of the issues raised in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF], namely that of avoiding a gain growing unbounded without sacrifice on the regulation accuracy in x.

The second objective of the paper consists in addressing the difficult issues of robustness with respect to measurement noise and unmatched uncertainties for prescribed time stabilisation of (1). As mentioned earlier regarding time-varying homogeneity approach, the disturbances corresponding to these perturbations become, at the best, amplified by D r λ(t) in (4). It is not clear at all how to handle (4) with disturbances growing unbounded. This is why we present in the second part of the paper a feedback design that does not involve any time-varying function λ(•). This will allow us to provide partial robustness results in case of measurement noise and unmatched disturbances on the feedback. Here, robustness must be understood in the ISS setting of Definition 4 and not anymore according to Definitions 2 and 3. Our construction is based on fixed-time stabilisation with a control on the settling time in the case of an unperturbed chain of integrators and then on the use a simple trick to extend that solution to prescribed-time stabilisation. To perform that strategy, one must get an explicit hold on several parameters. To be more precise, the fixed-time stabilisation design relies on sliding mode feedbacks with statedependent homogeneity degree. This idea was first considered in [START_REF] Harmouche | Robust and adaptive higher order sliding mode controllers[END_REF] and [START_REF] Harmouche | Stabilisation of perturbed chains of integrators using Lyapunov-based homogeneous controllers[END_REF] with a completely explicit feedback law. The latter bears a serious drawback since it is discontinuous. This defect has been removed in a subsequent work in [START_REF] Lopez-Ramirez | Fixed-time output stabilization and fixedtime estimation of a chain of integrators[END_REF], relying on an appropriate perturbation argument. However that latter solution does not bear an explicit character, which is an issue in order to estimate the settling time, and hence it requires important extra work for practical implementations. Moreover, it can be adapted only to a restricted set of perturbations.

Our feedback design for fixed-time stabilisation relies on the sliding mode feedback laws proposed by [START_REF] Hong | Finite-time stabilization and stabilizability of a class of controllable systems[END_REF] for finite-time stabilisation of an n-th order pure chain of integrators . Recall that, in that reference, it is proved that, for every homogeneity parameter κ ∈ [-1 n , 1 n ], there exists a control law u = ω H κ (x) which stabilizes ẋ = J n x + u e n and a Lyapunov function V κ for the closed-loop system satisfying Vκ ≤ -CV 2+2κ 2+κ κ , for some positive constant C, independent of κ. One of the main avantages of these feedbacks and Lyapunov functions is that they admit explicit closed forms formulas computable once the dimension n is given. In order to first obtain fixed-time stabilisation, we choose, as in [START_REF] Harmouche | Stabilisation of perturbed chains of integrators using Lyapunov-based homogeneous controllers[END_REF], a feedback law of the type u = ω H κ(x) (x), where the homogeneity parameter is a state function and, by using the smart idea of [START_REF] Lopez-Ramirez | Fixed-time output stabilization and fixedtime estimation of a chain of integrators[END_REF], we can also make x → κ(x) continuous. We finally use a standard homogeneity trick to pass from fixed-time to prescribed-time stabilisation.

The structure of the paper goes as follows. In Section 2, general stability notions and homogeneity properties are recalled. In Section 3, time-varying weighted homogeneity is considered for n-th order perturbed chains of integrators: Subsection 3.1 studies thoroughly linear timevarying homogeneous feedbacks while in Subsection 3.2, we provide sliding mode based feedbacks with uniformly bounded gains. We gather in Section 4 a new design of a sliding mode inspired feedback for which we characterise explicitly the parameters and we give some ISS properties in presence of measurement noise and unmatched disturbances. Finally we collect in an appendix the proofs of technical results used in the text.

Stability definitions

In this paper, we will consider various non autonomous differential equations ẋ = f (x, t), where x ∈ R n and f : R n × R + → R n is a vector field. When it exists, the solution of ẋ = f (x, t) for an initial condition x 0 ∈ R n is denoted by X(t, x 0 ). We recall the main stability notions needed in the paper, see [START_REF] Khalil | Nonlinear systems[END_REF]. Definition 1. Let Ω be an open neighborhood of the origin assumed to be an an equilibrium point of f . 1 At 0, the system is said to be:

(a) Lyapunov stable if for any x 0 ∈ Ω the solution X(t, x 0 ) is defined for all t ≥ 0, and for any > 0, there exists δ > 0 such that for any x 0 ∈ Ω, if x 0 ≤ δ then X(t, x 0 ) ≤ , ∀t ≥ 0.; (b) asymptotically stable if it is Lyapunov stable and for any κ > 0, > 0 there exists T (κ, ) ≥ 0 such that for any x 0 ∈ Ω, if x 0 ≤ κ then X(t, x 0 ) ≤ , ∀t ≥ T (κ, ); (c) finite-time converging from Ω if for any x 0 ∈ Ω there exists 0 ≤ T < +∞ such that X(t, x 0 ) = 0 for all t ≥ T . The function T(x 0 ) = inf {T ≥ 0 | X(t, x 0 ) = 0, ∀t ≥ T } is called the settling time for x 0 of the system; (d) finite-time stable if it is Lyapunov stable and finite-time converging from Ω. (e) fixed-time stable if it is finite-time stable and sup x0∈Ω T(x 0 ) < +∞ and the latter is referred as the settling time of the system.

Furthermore, for prescribed-time stability and robustness issues, we consider disturbances d : [0, ∞) → R p which are measurable functions where d [t0,t1) denotes the essential supremum over any time interval [t 0 , t 1 ) in [0, ∞). ) is finite. We have the following two definitions (cf. [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF] and [START_REF] Lopez-Ramirez | Fixed-time output stabilization and fixedtime estimation of a chain of integrators[END_REF]). Definition 2. A system ẋ = f (x, t, d) is prescribed-time input-to-state stable in time T (PT-ISS) if there exist functions β ∈ KL 2 , γ ∈ K and λ : [t 0 , t 0 +T ) → R * + such that λ tends to infinity as t tends to t 0 +T and, for all t ∈ [t 0 , t 0 +T ) and bounded d, Note that (P T -ISS) is a much stronger property than ISS.

If [t 0 , t 1 ) = [0, ∞), we say that d is bounded if d ∞ := d [0,∞
x(t) ≤ β ( x 0 , λ(t))+γ d [t0,t] . Definition 3. A system ẋ = f (x, t, d) is fixed-time input-to-
Remark 5. Definitions 2 and 3 have been given in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF] but with the explicit choice λ(t) = t-t0 T +t0-t . Next, basic definitions of homogeneity are collected. Definition 6.

(i) A function f : R n → R is said to be homogeneous of degree m ∈ R with respect to the weights r = (r 1 , ...,

r n ) ∈ R n >0 if for every x ∈ R n and λ ∈ R * + , f (D r λ x) = λ m f (x), where D r λ = diag (λ ri ) n i=1
defines a family of dilations. We also say that f is r-homogeneous of degree m.

(b) A vector field Φ = (f 1 , . . . , f n ) : R n → R n is said to be homogeneous of degree m ∈ R if for all 1 ≤ k ≤ n, for all x ∈ R n and λ ∈ R * + , f k (D r λ x) = λ m+r k f k (x), i.e.
, each coordinate function f k is homogeneous of degree m + r k . We also say that F is r-homogeneous of degree m. (c) Let Φ be a continuous vector field. If Φ is r-homogeneous of degree m, then the system ẋ = Φ(x), x ∈ R n is r-homogeneous of degree m.

The next lemma is important in the proof of our results in Section 4 (see for instance [START_REF] Lopez-Ramirez | Fixed-time output stabilization and fixedtime estimation of a chain of integrators[END_REF]).

Lemma 7. [START_REF] Nakamura | Smooth lyapunov functions for homogeneous differential inclusions[END_REF] Let ẋ = f (x, t) be a r-homogenous system of degree κ asymptotically stable at the origin. Then at the origin, the system is globally finite-time stable if κ < 0, globally exponentially stable if κ = 0 and globally fixed-time stable with respect to any open set containing the origin if κ > 0.

Time-varying homogeneity

Let n be a positive integer, (e i ) 1≤i≤n the canonical basis of R n and J n the n-th Jordan block, i.e. J n e i = e i-1 , 1 ≤ i ≤ n, with the convention that e 0 = 0. For λ > 0, using the notation above for D r λ (see also [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF]), set, for

r i := n -i + 1, 1 ≤ i ≤ n, (5)
D r λ = diag(λ ri ) n i=1 , D r λ J n (D r λ ) -1 = λJ n , D r λ e n = λe n .
(The second relation above simply says that the linear vector field induced by J n on R n is r-homogeneous of degree -1.)

In the literature devoted to prescribed time stabilization (see [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF] and references therein) and as clearly stated in Definitions 2 and 3, the quantity t-t0 T +t0-t is a new time scale which tends to infinity as t tends to the prescribed convergence time T . This fact suggests to consider the homogeneity parameter λ depending on the time t in such a way that, if one sets the new time to be equal to [START_REF] Filippov | Differential equations with discontinuous righthand sides[END_REF] s

: [0, T ) → R * + , s(t) = t 0 λ(ξ)dξ,
then s(t) tends to infinity as t tends to T . If x(t) denotes a trajectory of (1), it is natural to consider the change of coordinates and time given by ( 7)

y(s) = D r λ(t) x(t), ∀ t ∈ [0, T ).
In order to analyse the dynamics of y, we use y to denote the derivative of y with respect to the new time s. Using (1), ( 5) and ( 6), we obtain:

(8) λ y = ẏ = λ ∂D r λ ∂λ (D r ) -1 λ y + λ (J n y + b u e n + d e n ) .
For every µ > 0, we also have that

∂D r µ ∂µ (D r ) -1 µ = 1 µ D r , with D r := diag(r i ) 1≤i≤n .
Then ( 8) becomes

(9) y = λ λ 2 D r + J n y + (b(s) u(s) + d(s)) e n .
Here we consider the control u and both b and f as functions of the new time s.

Let a : [0, T ] → R be a non negative continuous function so that the

C 1 function A : [0, T ] → R defined by A(t) = T t a(s) ds is positive on [0, T ). Setting (10) λ(t) := 1 A(t) , t ∈ [0, T ), one gets that λ(t) λ 2 (t) = a(t), t ∈ [0, T ).
It is then immediate to see that the function λ : [0, T ) → R * + is increasing, tends to infinity as t tends to T and the time s defined in ( 6) realizes an increasing

C 1 bijection from [0, T ) to [0, ∞).
We still use a(s) to denote a(t). With this choice, ( 9) becomes

(11) y = (a(s)D r + J n ) y + b(s) u(s) + d(s) e n .
To solve the original problem of designing a feedback control u that renders the system FT-ISS-C in time T > 0, the idea consists in choosing [START_REF] Khalil | Nonlinear systems[END_REF] u = F (y(s)),

where F : R n → R is a continuous function to be chosen later.

Remark 8. An important feature in stabilisation of control systems is the fact that one usually requires the feedback law to remain bounded and ideally, to tend to zero as the state x tends to zero, even if in presence of disturbance. In the context of prescribed time stabilisation of (1), this feature is automatically guaranteed by our view point of time-varying homogeneity since the feedback law takes the form (12): bounding u(t) uniformly on [0, T ] simply reduces to bound the artificial state y(s) uniformly on R + .

Remark 9. For the stabilisation of [START_REF] Hong | Stabilization of uncertain chained form systems within finite settling time[END_REF], one can of course rely on linear feedback laws, as done in the next section (and already done in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF]) but also on sliding mode type of feedbacks which insure fixed time stabilisation (in the scale s) with robust properties, see Subsection 3.2 below.

3.1. Linear feedback. We now revisit the results obtained in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF] at the light of the timevarying homogeneity introduced in the previous section. To establish the connection with that reference, one must compares our change of variable defined in [START_REF] Gauthier | Observability and observers for nonlinear systems[END_REF] and the one considered in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF]. At once, one can see that the function µ(•) = T m+n (T -t) m+n in Eq. (31) of [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF] corresponds, up to a positive constant, to the time-varying homogeneity parameter λ(•) where a(t) is chosen as a(t) = (T -t) m-1 (with m ≥ 2 integer). In opposite to [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF], in our approach one does not have to take time derivatives of λ(•) (or equivalently of µ(•)), and hence our computations are simpler (in particular no need of Lemmas 2 and 3 in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF]).

As for the feedback control in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF], it is given by u 42) and (50) in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF]), where L 0 is a linear combination of successive derivatives of µ and the state components, L 1 contains a gain matrix K n-1 , k is a scalar gain and z is a change of variable of the n-th coordinate of the state. The above expression of the feedback u shows that this choice of feedback can be essentially reduced to a linear one (realized by the constant k and the R n-1 vector K n-1 in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF]). This is the reason why we take here F (y) = -K T y for some vector K ∈ R n to be fixed later. In that case, after replacing u in [START_REF] Hong | Stabilization of uncertain chained form systems within finite settling time[END_REF], it follows

= -1 b (d + L 0 + L 1 + kz) (cf. Eqs. (
(13) y = a(s)D r + J n -b(s) e n K T y + d(s) e n ,
that is an equation of the type [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF]. In [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF], such systems were considered (without the term a(s)D r ) and it was proven that there exists a positive constant µ > 0, a real symmetric positive definite n × n matrix S > 0 and a vector K ∈ R n such that ( 14)

y = M (s)y + f (s) e n where M (s) = a(s)D r + J n -b(s) e n K T with b(s) subject to
J n -b e n K T T S + S J n -b e n K T ≤ -µ Id n , ∀ b ≤ b ≤ b,
where Id n denotes the n × n identity matrix and S, K and µ depend on b and b.

A careful examination of the argument shows actually that one can remove the upper bound on the parameter b. We thus get the slightly stronger result, whose proof is given in Appendix, for sake of completeness.

Proposition 10. Let n ∈ N and b > 0. Then there exists a positive constant ρ > 0, a real symmetric positive definite n × n matrix S and K ∈ R n so that

(15) J n -b e n K T T S + S J n -b e n K T ≤ -ρ Id n , ∀b ≥ b.
With an obvious perturbation argument, we immediately derive the following proposition:

Proposition 11. Using the notation of Proposition 10, there exist ρ 0 , C 0 ∈ R * + , a real symmetric positive definite n × n matrix S and K ∈ R n so that

(16) aD r + J n -b e n K T T S + S aD r + J n -b e n K T ≤ -ρ 0 Id n , ∀a ∈ [-C 0 , C 0 ], b ≥ b.
We apply the above proposition to derive ISS properties of (13) and we prove the following proposition.

Proposition 12. Consider the dynamics given in [START_REF] Hong | Stabilization of uncertain chained form systems within finite settling time[END_REF] and D r η defined in [START_REF] Cruz-Zavala | Second-order uniform exact sliding mode control with uniform sliding surface[END_REF]. Then there exists K ∈ R n and η 1 > 0 such that, for η ≥ η 1 , the state feedback u = -K T D r η y provides the following estimate:

(17) |y i (s)| ≤ max(1, η n-1 ) η n-i exp(-C S ρ 1 η s) y(0) + C S η n-i+1 max r∈[0,s] |d(r)|, ∀s ≥ 0, 1 ≤ i ≤ n.
where C S and ρ 1 are positive constants only depending on the lower bound b.

Proof. Fix η > 0 and set z η = D r η y. From [START_REF] Lopez-Ramirez | Fixed-time output stabilization and fixedtime estimation of a chain of integrators[END_REF], with u = -K η y = -Kz η , one gets that z η verifies the following dynamics, after setting the time ξ := ηs, [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF] dz

η dξ = D r η y η = a(ξ/η) η D r + J n -b(ξ/η) e n K T z + d(ξ/η) e n . Set C a = sup s≥0 |a(s)| = max t∈[0,T ] |a(t)|.
One takes the Lyapunov function V (z η ) = z T η Sz η and takes its time derivative along [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF]. Then, by taking η ≥ Ca C0 and using Proposition 11, one gets

dV (z η (ξ)) dξ ≤ -ρ 0 z η (ξ) 2 + 2|z T η (ξ)Se n | max r∈[0,ξ/η] |d(r)| ≤ -ρ 0 z η (ξ) 2 + C S z T η (ξ) max r∈[0,ξ/η] |d(r)|, (19)
where C S stands for "any" constant only depends on S, i.e. on b and C a .

One deduces that there exists constants C S such that

z η (ξ) ≤ exp(-C S µξ) z η (0) + C S max r∈[0,ξ/η] |d(r)|, ∀ξ ≥ 0.
We now write the previous inequality in terms of y(s). After noticing that

η n-i+1 |y i (s)| = |(z η ) i (ξ)| ≤ z η (ξ) , 1 ≤ i ≤ n, z η (0) ≤ η max(1, η n-1 ) y(0) ,
one gets [START_REF] Shtessel | Sliding mode control and observation[END_REF].

One can rewrite the previous argument using an LMI formulation. For that purpose, one needs a result similar to Proposition 11, which involves the extra parameter η. More precisely, one easily shows the following proposition.

Proposition 13. Let n ∈ N and b ∈ R * + .
Then there exist a positive constant ρ, a real symmetric positive definite n × n-matrix S and a vector K ∈ R n such that, for every C > 0 there exists η 1 so that, the following holds true, (20)

aD r +J n -b e n K T η T S η +S η aD r +J n -b e n K T η ≤ -µ * η (D r ) 2 η , a ∈ [-C, C], η ≥ η 1 b ≥ b, where S η = D r η SD r η and K η = D r η K.
To see that, simply take K η = D r η K 1 and multiply the LMI ( 16) on the left and on the right by D r η . That yields [START_REF] Snchez-Torres | A class of predefined-time stable dynamical systems[END_REF]. Using Proposition 12 and the fact that

λ(t) n-i+1 |x i (t)| ≤ |y i (s)|, 1 ≤ i ≤ n, t ≥ 0,
we deduce PT-ISS-C for x in any time T > 0.

We gather in the following corollary our findings, which are similar to Theorem 1 in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF].

Corollary 14. Consider the dynamics given in (1). Let a : [0, T ] → R any non negative continuous function such that

T t a(ξ)d ξ > 0 for t ∈ [0, T ). Then set (21) λ(t) = 1 T t a(ξ)dξ , s(t) = t 0 λ(ξ)dξ, 0 ≤ t < T.
There exists K ∈ R n such that, for every η ≥ 1, the state feedback

(22) u = -K T D r ηλ(t) x(t), t ≥ 0, where D r
λ is defined in [START_REF] Cruz-Zavala | Second-order uniform exact sliding mode control with uniform sliding surface[END_REF], provides the following estimate, for every t ≥ 0 and

1 ≤ i ≤ n, ( 23
) |x i (t)| ≤ 1 (ηλ(t)) n-i+1 η max(1, η n-1 ) exp (-C S µη s(t)) x(0) + C S max r∈[0,t] |d(t)| .
where C S and µ are positive constants only depending on the lower bound b.

Remark 15. The case where a(t) = (T -t) m with m positive integer corresponds to [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF] and one can choose another a which goes faster to 0 as t tends to T , for instance exp(-1/T -t)/(T -t) 2 , which yields to λ(t) = exp(-1/(T -t)) and then faster rates of convergence.

One should now refer to Section 3.2 in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF] which provides the advantages and limits of such a feedback regulation. In the sequel, we only insist on what we believe are the advantages of our approach with respect to that of [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF] as well as the inherent limitations in terms of robustness of feedback strategies based on time-varying homogeneity.

Remark 16. The disturbance we consider here has a simpler expression than that of [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF], the latter being bounded by |d(t)|ψ(x), with d any measurable function on [0, ∞) and ψ ≥ 0 a known scalar-valued continuous function. To lighten the presentation, we do not consider the function ψ since the analysis in this case is entirely similar to the above by using Eq. (25) in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF].

Remark 17. Let us compare our results with those obtained in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF]. First of all, we recover at once the main result of that reference (Theorem 2 and Inequality (79)) by choosing the function a(•) appearing in the theorem to be equal to C(T -t) m where C is a positive constant and m a positive integer. We have though slightly better results since we can prescribe the rate of exponential decrease as well as the estimate on the error term modeled by f (•) thanks to the occurence of the parameter η in our findings. Indeed the choice of the function λ(•) in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF] (called µ(•) in Equation (30) of [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF]) must be specific because it relies on the fact that time derivatives of µ(•) must be expressed as polynomials in λ(•), cf. Lemmas 2 to 4 in the reference therein. Instead, using our presentation, it turns out there is more freedom in the choice of λ. More importantly, our presentation yields simpler proofs of convergence with a unique time scale for variables and everything boiling down to LMIs. Another advantage of the more transparent structure of the feedback is given in the subsequent remarks, where we are able to explain in a very explicit manner the limitations of the present feedback law, as they are suggested in the discussion 3.2 in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF], as well as in the conclusion of that reference.

Remark 18. As noticed in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF], the linear feedback defined in [START_REF] Zarchan | Tactical and strategic missile guidance[END_REF] is not suitable if it is subject to measurement noise on x(•). More precisely, this amounts to have instead of (22) a feedback ũ given by ũ(t) = -K T D r ηλ(t) x(t) + d(t) = u(t) -K T D r ηλ(t) d(t), t ≥ 0, i.e., with a disturbance d in (1) of the form ηλ(t) max(1, (ηλ) n-1 (t)) d(t) . We can only derive from Corollary 14 the following estimate, for every t ≥ 0 and 1 ≤ i ≤ n, (24)

|x i (t)| ≤ η max(1, η n-1 ) (ηλ) n-i+1 (t) exp (-C S µη s(t))) x(0) + C S max(1, (ηλ) n-1 (t)) max r∈[0,t] d(r) (ηλ) n-i (t) .
The right-hand side blows up as t tends to T , except for i = 1, with a loss of regulation accuracy (we do not have anymore convergence to zero but to an arbitrary small neighborhood).

On the other hand, by choosing η of the amplitude of λ(t) as t tends to T , we deduce at once from (24) the following corollary.

Corollary 19. With the notations of Corollary 14, assume that one feeds the dynamics given in (1) with the pertubed feedback ũ(t) = -K T D r ηλ(t) x(t) + d(t) . Then for every time T < T , there exists η > 0 such that, one gets that

(25) max t∈[0,T ] |x i (t)| ≤ η x(0) + C T ,T max t∈[0,T ] |d(t)|, ∀t ≥ 0, 1 ≤ i ≤ n,
where C T ,T is a positive constant, tending to infinity as T tends to T .

The previous result of semi-global nature has been already suggested in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF] and has been obtained in the present paper thanks to the extra parameter η. In particular, it follows the idea that in order to obtain estimates for prescribed time control in time T , one can use the previous strategy of prescribed time control in a time T > T and then use (25). This estimate is a direct result of the use of the time-varying function λ(•) but, as regards measurement noise it is definitely not satisfactory.

Remark 20. Looking back at [START_REF] Khalil | Nonlinear systems[END_REF], the most natural choice is a linear feedback and it has been (essentially) first addressed in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF] and revisited here. One can also use other feedback laws, especially those providing finite-time stability (in the scale time s). If there is measurement noise, i.e., of the type x + d, the feedback implemented in (11) will be u(s) = F (y(s) + D r λ(t(s)) d(s)) and it is likely that one can find appropriate perturbations so that the last coordinate of x will become unbounded as t tends to T . Already, in the linear case, for a double integrator for instance, it is easy to choose bounded disturbances d in the case where a(t) = 1 such that y 2 (s) has the magnitude of λ 2 (s), and then x 2 (t) has the magnitude of λ(t) as t tends to T . It is not difficult to extend that remark to any feedback law F which is differentiable at zero. Such a fact prevents to get any type of ISS results and it indicates that no property such as (P T -ISS) can hold in presence of measurement noise. This is why time-varying homogeneity based feedbacks are not, in our opinion, well-suited for prescribed-time stabilization in presence of measurement noise. One must to follow another approach and this is the purpose of the next section.

3.2.

Fixed-time feedbacks. We now consider feedback laws in (12) which will provide fixedtime stabilisation for [START_REF] Hong | Stabilization of uncertain chained form systems within finite settling time[END_REF] under the assumption of a priori knowledge on the uncertainties bounds. More precisely, we will simply show that the feedback law provided in [9, Theorem 5] does the job and we have the following.

Proposition 21. Set ε = ±. Assume that there exists α ∈ (0, 1/n), c > 0, two continuous feedback laws u ε : R n → R and two C 1 functions V ε : R n → R + , which are positive definite, r-homogeneous of degree larger than one and such that the following holds true:

(a): u ε stabilizes the n-th order pure chain of integrator ẋ = J n x + ue n in finite-time and along the trajectories of the corresponding closed-loop system, one has

(26) Vε ≤ -cV 1+εα ε ;
(b): for every x ∈ R n , the following geometric condition holds true:

(27) ∂V ε ∂x n u ε (x) ≤ 0 and u ε (x) = 0 ⇒ ∂V ε ∂x n = 0.
Define the feedback law ω 0 : R n → R as

(28) ω 0 (x) = u + (x), if V -(x) > 1, u -(x), if V -(x) ≤ 1.
Consider the dynamics [START_REF] Hong | Stabilization of uncertain chained form systems within finite settling time[END_REF] and assume that b(•) verifies (2) and d ∞ ≤ D for some positive constant D. Then the feedback law ω : R n → R defined as

(29) ω(x) = 1 b ω 0 (x) + Dsgn(ω 0 (x)) ,
where ω 0 is defined in (28) and sgn stands for the set-valued function "sign", globally stabilises (11) in fixed-time.

Here the sgn function makes the closed-loop system corresponding to [START_REF] Hong | Stabilization of uncertain chained form systems within finite settling time[END_REF] and u = ω a differential inclusion and its trajectories must be understood in the Filippov sense, cf. [START_REF] Filippov | Differential equations with discontinuous righthand sides[END_REF]. Note also that examples of feedbacks u ε and the Lyapunov functions V ε verifying Items (a) and (b) are also provided in Definition 23 and Proposition 24 given in the next section.

Proof. First of all notice that, by multiplying (11) by D r µ with µ ≥ 1 and considering the new state D r µ y(s/µ), C a = sup s≥0 |a(s)| becomes C a /µ and hence arbitrary small.

Set E := min V-(x)=1 V + (x) > 0 and consider the sets

S 1 = {x ∈ R n : V + (x) ≤ E}, S 2 = {x ∈ R n : V -(x) ≤ 1}.
By definition of E, we have that S 1 ⊂ S 2 . We claim that the closed-loop system corresponding to [START_REF] Hong | Stabilization of uncertain chained form systems within finite settling time[END_REF] and u = ω is globally fixed-time stable with respect to S 2 . For that purpose, we compute the time derivative of V + along the trajectories outside S 2 and get

V+ = a ∇V + (y), D r y + n-1 i=1 ∂V ε ∂x i x i+1 + ∂V ε ∂x n (bω + d), ≤ a ∇V + (y), D r y -cV 1+α + + ∂V ε ∂x n ( b b -1)ω 0 + sgn(ω 0 )( b b D -|d|) , ≤ c 2 V + -cV 1+α + ≤ - c 2 V 1+α + . ( 30
)
To get the above we have used Item (a), i.e.,

n-1 i=1 ∂V ε ∂x i x i+1 + + ∂V ε ∂x n ω 0 (x) ≤ -cV 1+α + ,
Item (b), and the fact that the function ∇V + (y), D r y having the same degree of r-homogeneity as V is smaller than c 2 V 1+α + outside S 2 for C a small enough. The claim is proved by using Lemma 7.

As soon as a trajectory x of the closed-loop system corresponding to [START_REF] Hong | Stabilization of uncertain chained form systems within finite settling time[END_REF] and u = ω reaches S 2 , it verifies V -(x) = 1. Morever for trajectories in S 2 , a computation entirely similar to (30) yields the differential inequality

V-≤ -c 2 V 1-α -
, which proves that any trajectory starting at V -(x) = 1 enters in S 2 , remains in it for all subsequent times and, again according to Lemma 7, converges to the origin in a uniform finite-time. That concludes the proof of Proposition 21.

Remark 22. Note that the feedback ω defined in (29) exhibits a discontinuity at V -= 1. By using the feedback law of Theorem 28, one can remove that discontinuity, if in addition, an upper bound for b is assumed to be known.

Robust prescribed-time stabilisation

In the previous section, a linear feedback u = K T y was considered but this choice faces a pernicious problem as soon as there is some noise measurement on the state. We propose in this section an alternative feedback law for prescribed-time stabilisation with ISS properties in presence of measurement noise and unmatched uncertainties. The construction of this feedback runs in two steps, the first one deals with the fixed-time stabilisation in the unperturbed case and the second addresses the ISS issue in the perturbed case.

4.1.

A special fixed-time stabilisation design. The unperturbed case associated with (1), namely

(31) ẋ = J n x + u e n ,
which is referred in the sequel as the n-th order pure chain of integrators.

To proceed, we rely on the original idea of [START_REF] Harmouche | Stabilisation of perturbed chains of integrators using Lyapunov-based homogeneous controllers[END_REF] and use the perturbation trick of [START_REF] Lopez-Ramirez | Fixed-time output stabilization and fixedtime estimation of a chain of integrators[END_REF] to provide an explicit and continuous feedback law.

We next provide the necessary material needed to describe the feedback design given in [START_REF] Harmouche | Stabilisation of perturbed chains of integrators using Lyapunov-based homogeneous controllers[END_REF]. The following construction, which is based on a backstepping procedure, has been given first in [START_REF] Hong | Finite-time stabilization and stabilizability of a class of controllable systems[END_REF] and we will modify it to handle the present situation.

Definition 23. Let j > 0, j = 1, • • • , n be positive constants. For κ ∈ -1 n , 1 n , define the weights r(κ) = (r 1 , • • • , r n ) by r j = 1 + (j -1)κ , j = 1, • • • , n. Define the feedback control law (32) u = ω H κ (x) := v n
, where the v j = v j (x) are defined inductively by:

(33) v 0 = 0, v j = -j x j βj-1 -v j-1 βj-1 r j +κ r j β j-1 ,
and where the β i 's are defined by β 0 = r 2 , (β j + 1)r j+1 = β 0 + 1 > 0, j = 1, ..., n -1.

For 1 ≤ j ≤ n, we also consider the union of the homogeneous unit spheres associated with r,

κ ∈ -1 2n , 1 2n , i.e., (34)
S j = κ∈[-1 2n , 1 2n ] x ∈ R j | |x 1 | 2 r 1 + • • • + |x n | 2 rn = 1 .
Then S j is clearly a compact subset of R j and dealing with this set constitutes the main difference with [START_REF] Hong | Finite-time stabilization and stabilizability of a class of controllable systems[END_REF].

We have then the following proposition.

Proposition 24. There exist positive constants j > 0, j = 1, • • • , n such that for every κ ∈ -1 2n , 1 2n , the feedback law u = ω H κ (x) defined in (32) stabilizes the system (31). Moreover, there exists a homogeneous C 1 -function V κ : R n → R + given by

Vκ(x) = n j=1 |xj| β j-1 +1 -|vj-1| β j-1 +1 βj-1 + 1 -vj-1 β j-1 (xj -vj-1) , (35)
which is a Lyapunov function for the closed-loop system (31) with the state feedback ω H κ , and it satisfies

(36) Vκ ≤ -CV 1+α(κ) κ , α(κ) := κ 2 + κ ,
for some positive constant C, independent of κ. Moreover, V κ is r(κ)-homogeneous of degree (2 + κ) with respect to the family of dilations D r(κ) λ λ>0 . Remark 25.

(i): The previous proposition is essentially Theorem 3.1 of [START_REF] Hong | Finite-time stabilization and stabilizability of a class of controllable systems[END_REF], except that the gains i are uniform with respect to κ ∈ -1 2n , 1 2n . The choice of 1 2n has been made because the previous proposition actually holds true for κ ∈ -

1 n , 1 n at the exception that V 1 n is not C 1 on R n . (ii):
The critical exponent 1+α(κ) appearing in (36) is larger than one if κ > 0 and smaller than one if κ < 0. (iii): For κ = 0, one gets a linear feedback and V 0 is a positive definite quadratic form, hence there exists a real symmetric positive definite n × n matrix P such that V 0 (x) = x T P x for every x ∈ R n . Finally, the time derivative of V 0 is associated with the n × n matrix L T P + P L where L is the companion matrix associated with the coefficients -l 1 , • • • , -l n on the last line. We deduce at once that L is Hurwitz since the differential inequality (36) for κ = 0 is equivalent to the LMI, A T P + P A ≤ -CP . We set Q := -(A T P + P A), which is a real symmetric positive definite n × n matrix.

Proof. The argument follows closely that of Theorem 3.1 of [START_REF] Hong | Finite-time stabilization and stabilizability of a class of controllable systems[END_REF], but we will bring some technical changes to obtain the required uniformity with respect to κ ∈ -1 2n , 1 2n . Moreover, in order to show in the next section the explicit character of our construction, we will provide quantitative estimates on the several constants involved in the construction, which are new with respect to [START_REF] Hong | Finite-time stabilization and stabilizability of a class of controllable systems[END_REF].

Let κ ∈ -1 2n , 1 2n and set, for 1

≤ j ≤ n, x (j) = (x 1 , • • • , x j ), W κ,j (x (j) ) = xj vj-1 s βj-1 -v j-1 βj-1 ds = 1 β i-1 + 1 |x j | βj-1+1 -|v j-1 | βj-1+1 -v j-1 βj-1 (x j -v j-1 ) , (37) and
(38)

V κ,0 := 0, V κ,j := W κ,j + V κ,j-1 .

It holds that V κ (x) = V κ,n (x (n) ) = n i=1 W κ,j (x (j)
). The choice of the j is made recursively at each step 1 ≤ j ≤ n by considering, as in [START_REF] Hong | Finite-time stabilization and stabilizability of a class of controllable systems[END_REF], the following expression

(39) d dt V κ,j = d dt V κ,j-1 + ∂V κ,j-1 ∂x j-1 (x j -v j-1 ) + j-1 i=1 ∂W κ,j ∂x i x i+1 + ∂W κ,j ∂x j v j ,
where d dt V κ,j-1 is used to denote the time derivative of V κ,j-1 is taken along the (j -1)th pure chain of integrators. Note that the functions ∂Vκ,j-1 ∂xj-1 and ∂Wκ,j ∂xi are continuous. We also get that, for 1 ≤ j ≤ n, one has

∂V κ,j-1 ∂x j-1 (x j -v j-1 ) = ∂W κ,j-1 ∂x j-1 (x j -v j-1 ) = x j-1 βj-2 -v j-2 βj-2 (x j -v j-1 ) (40) ∂W κ,j ∂x j v j = -j Z κ,j , Z κ,j = x j βj-1 -v j-1 βj-1 2(1+κ)/rj βj-1 . (41)
We will need the following elementary fact: for every α in a compact set of R * + and M > 0, there exists two positive constants A, B such that, for every real numbers |x|, |y| ≤ M , (42) (1,α) .

A|x -y| max(1,α) ≤ | x α -y α | ≤ B|x -y| min
We next prove by induction on 1 ≤ j ≤ n, that there exists positive real numbers

1 , • • • , n such that (43) max{ d dt V κ,j | - 1 2n ≤ κ ≤ 1 2n , x (j) ∈ S j } ≤ - l 1 2 j-1 .
By homogeneity and for j = n, one immediately gets (36) and the conclusion of the proposition.

In the rest of the argument, we use K j , M j , L j to denote positive constants depending on S j and 1 , • • • , j-1 but independent of j . For j = 1, (43) reduces to d dt V κ,1 = -l 1 and any positive l 1 does the job. For the inductive step with 2 ≤ j ≤ n, assume that 1 , • • • , j-1 have been built with the required properties, in particular we have d dt V κ,j-1 ≤ -l1 2 j-2 on S j-1 . From (40), we get

(44) ∂V κ,j-1 ∂x j-1 (x j -v j-1 ) ≤ K j |x j -v j-1 |.
For 1 ≤ i ≤ j -1, the continuous function ∂Wκ,j ∂xi is r(κ)-homogeneous of degree (2 + κ) with respect to the family of dilations D r(κ) ε ε>0 . (Actually, one restricts this homogeneity to x (j) .) Moreover, it is equal to zero if x j = v j-1 . Hence, by using repeatedly (42) and noticing that for some positive constants b, b. Then, there exists m ∈ (0, 1) and κ 0 ∈ (0, 1 2n ) such that, the undisturbed n-th order chain of integrators defined by

(51) ẋ(t) = J n x(t) + b(t)u(t), b ≥ b(t) ≥ b, t ≥ 0,
together with an adapted feedback law given by ω H κ(x) (x), with κ(•) defined in (49) is globally fixed-time stable at the origin in at most time T (m, κ 0 ) upper bounded as

(52) T (m, κ 0 ) ≤ 1 C r(m, κ 0 ) -α(κ0) α(κ 0 ) -2 ln(2m) + r(m, -κ 0 ) -α(-κ0) -α(-κ 0 ) ,
where r(m, κ 0 ) > 0 (and r(m, -κ 0 ) > 0) is the largest (smallest) number r > 0 such that B κ0 <r

(B -κ0 <r ) is contained in (contains) B 0 <1+m (B 0 <1-m
) and the constant C has been introduced in (36).

By adapted, we mean the following: strictly speaking, we must choose the feedback law ω H κ(x) (x)/b. However, we can replace n by either n /b or by b n in order to satisfy (48). Hence, with no loss of generality, we assume b = 1.

Proof. For this result, we follow the perturbative argument considered in the proof of Lemma 2 in [START_REF] Lopez-Ramirez | Fixed-time output stabilization and fixedtime estimation of a chain of integrators[END_REF]. For that purpose, the time derivative of V 0 along non trivial trajectories of System (51) closed by the feedback law given by ω H κ(x) (x) can be written as

(53) V0 = 2x T P (J n x + bω H κ(x) (x)e n ) ≤ -x T Qx + 2|x T P e n |δ(x), δ(x) := b|ω H κ(x) (x) -ω H 0 (x)|.
We have to first to show that trajectories of

(54) ẋ = J n x + ω H κ(x) (x) e n ,
are well-defined and second that trajectories starting in B 0 >1+m reach B 0 1+m in finite time, then "cross" it till reaching B 0 1-m in finite time and finally remain in B 0 <1-m for all larger times, while converging to zero in finite time.

Since the right-hand side of (54) is continuous, there exist solutions from any initial condition defined at least on a non trivial interval. Clearly, there exists R > 0 such that trajectories starting at any x 0 ∈ B κ0 >R stay in the compact set B κ0 <Vκ 0 (x0) and hence are defined for all times. Both the convergence parts of the claim follow from the arguments of [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF] and Lemma 7, where one proves the following • the closed-loop system (54) is r(κ 0 )-homogeneous of degree 2 + κ 0 in B 0 >1+m and hence converges in finite-time to B 0 1+m , • the closed-loop system (54) is r(-κ 0 )-homogeneous of degree 2 -κ 0 in B 0 <1-m and hence converges in finite-time to the origin.

For the remaining part of the argument, it amounts to show that, for m ∈ (0, 1) and κ 0 ∈ (0, 1/n) small enough, the time derivative of V 0 along trajectories of (54) is negative in B 0 1-m,1+m . To see that, it is enough to notice that the function δ defined in (53) is continuous and tends to zero if either m or κ 0 tends to zero.

It remains to provide a first quantitative estimate of the "fixed-time" part of the theorem. The time needed for the closed-loop system (54) to converge to B 0 1+m is at most equal to the time T + (m, κ 0 ) needed to converge to B κ0 <r(m,κ0) . By integrating (36), one derives that

T + (m, κ 0 ) ≤ 1 Cα(κ 0 )r(m, κ 0 ) α(κ0) .
A similar reasoning yields that the time T -(m, κ 0 ) needed to converge from B -κ0 <r(m,-κ0) to the origin verifies the following

T -(m, κ 0 ) ≤ 1 -Cα(-κ 0 )r(m, -κ 0 ) α(-κ0) .
(Recall that α(-κ 0 ) = -κ0 2-κ0 < 0.) It remains to upper bound the time T 0 (m, κ 0 ) needed to "cross" B 0 1-m,1+m . For that purpose, choose m and κ 0 small enough so that

(55) M δ (m, κ 0 ) := max x∈B 0 1-m,1+m |δ(x)| ≤ C(1 -m) 2 .
In that case, (53) becomes V ≤ -CV /2 and one gets

T 0 (m, κ 0 ) ≤ -2 ln(2m) C .
We conclude that the closed-loop system (54) is globally-fixed time stable with respect to the origin in settling time less than or equal to T (m, κ 0 ) given by

T (m, κ 0 ) := T + (m, κ 0 ) + T 0 (m, κ 0 ) + T -(m, κ 0 ).
One derives (52) and this concludes the proof of the theorem.

Remark 29. The above result is the counterpart of Lemma 2 in [START_REF] Lopez-Ramirez | Fixed-time output stabilization and fixedtime estimation of a chain of integrators[END_REF] for our feedback law ω H κ . Note that in that reference, the statements of Lemma 2 and Theorem 4 as well as the argument of Lemma 2 consider the euclidean norm x instead of B 0 1 in the definition of κ(•). As one can see from the above argument, using that norm cannot not provide the required results. However [START_REF] Lopez-Ramirez | Fixed-time output stabilization and fixedtime estimation of a chain of integrators[END_REF] does consider the correct controller in Lemma 3 and in the last section of the corresponding reference.

It remains to use a standard time rescaling technique with homogeneity (cf. [START_REF] Hong | Stabilization of uncertain chained form systems within finite settling time[END_REF] and [START_REF] Harmouche | Stabilisation of perturbed chains of integrators using Lyapunov-based homogeneous controllers[END_REF]) to pass from the result of fixed-time stability contained in Theorem 28 to a result about prescribedtime stability.

Theorem 30. Let m ∈ (0, 1), κ 0 ∈ (0, 1/n) defined in Theorem 28 and the feedback law ω H κ(x) (x) defined in (49) which renders System (31) globally fixed-time stable at the origin in settling time less than or equal to T (m, κ 0 ) defined in (52). Then, given any T > 0, the the feedback law ω H κ(D λ x) (D r λ x) renders System (31) globally fixed-time stable at the origin in settling time less than or equal to T as soon as λ ≥ T (m, κ 0 )/T . Proof. For λ > 0, one sets y(s) = D λ x(t) with the time scale s = λt. One deduces at once that y converges in finite time to the origin with a settling time upper bounded by T (m, κ 0 ) as well as x, with a settling time upper bounded by T (m, κ 0 )/λ. To guarantee that the latter is less than or equal to T , it is enough to choose λ as stated. 4.2. Explicit determination of the main parameters. In order to fully compare our controller u = ω H κ(x) , with x → κ(x) given in Definition 27 with the controller provided in [START_REF] Lopez-Ramirez | Fixed-time output stabilization and fixedtime estimation of a chain of integrators[END_REF], we must explain how to choose the parameters m ∈ (0, 1) and κ 0 ∈ 0, 1 2n introduced in Definition 27. We also have to estimate the quantities r(m, κ 0 ) and r(m, -κ 0 ) in order to get a hold on the upper bound T (m, κ 0 ) of the settling time to reach precise estimates of the rescaling factor λ appearing in Theorem 30.

For that purpose, we first need an explicit bound on the coordinates of x ∈ B κ 1-m,1+m with -1 2n ≤ κ ≤ 1 2n . This is the content of the next lemma.

Lemma 31. Let m ∈ (0, 1). Then, there exists an explicit positive constant X n (depending on m and the j 's) such that, for 0 ≤ j ≤ n, x ∈ B κ 1-m,1+m and -

1 2n ≤ κ ≤ 1 2n , |x j |, |v j | ≤ X n . Proof. Fix m ∈ (0, 1), x ∈ B κ 1-m,1+m and -1 2n ≤ κ ≤ 1 2n
. The proof of the lemma goes by induction on j, where we prove the statement with a constant X j explicitly depending on m and the j 's. This is clearly true for j = 1 since

|v 1 | = 1 |x 1 | 1+κ and |x1| 1+β 0 1+β0 ≤ V κ (x) ≤ 1 + m.
Assume that the thesis holds true for j -1 ≥ 1. One then deduces from the definition of W κ,j in (37) and the induction hypothesis that

|x j | 1+βj-1 ≤ (2 + β j-1 )X 1+βj-1 j-1 + (1 + β j-1 )X βj-1 j-1 |x j | + (1 + β j-1 )(1 + m).
Since β j-1 > 0, one deduces at once a first explicit bound for x j and then for v j by using (33).

The following lemma provides the required differences between useful quantities evaluated at any κ ∈ [-1 2n , 1 2n ] and κ = 0. For 0 ≤ j ≤ n, we introduce the notation v κ j := v j , where the latter has been defined in (33).

Lemma 32. Let m ∈ (0, 1). Then there exists explicit positive constants C 1 n , C 2 n (depending on m and the j 's) such that, (1,rn) , and (1,rn) .

(56) max{|ω H κ (x) -ω H 0 (x) | - 1 2n ≤ κ ≤ 1 2n , x ∈ B κ 1-m,1+m } ≤ C 1 n |κ| min
(57) max{|V κ (x) -V 0 (x) | - 1 2n ≤ κ ≤ 1 2n , x ∈ B κ 1-m,1+m } ≤ C 2 n |κ| min
Proof. Fix m ∈ (0, 1). We will actually prove by induction on 1 ≤ j ≤ n, that there exists an explicit positive constant C 1 j (depending on m and 1 , • • • , j ) such that, (58) max{|v κ j (x) -v 0 j (x) | - (1,rj ) , where B κ j,m is the set of x (j) ∈ R j for which 1 -m ≤ V κ,j (x (j) ) ≤ 1 + m. The result is immediate for j = 0 and hence we turn to the inductive step for 1 ≤ j ≤ n, assuming that the hypothesis holds for j -1.

1 2n ≤ κ ≤ 1 2n , x (j) ∈ B κ j,m } ≤ C 1 j |κ| min
Let -1 2n ≤ κ ≤ 1 2n and x (j) ∈ B κ j,m . Then

v κ j (x) -v 0 j (x) = -j x j βj-1 -v κ j-1 βj-1 r j+1 r j β j-1 + j (x j -v 0 j-1 ) = -j (F + G), where F = x j βj-1 -v κ j-1 βj-1 r j+1 a r j β j-1 -x j βj-1 -v 0 j-1 βj-1 r j+1 a r j β j-1 , G = x j βj-1 -v 0 j-1 βj-1 r j+1
r j β j-1 -(x j -v 0 j-1 ). By applying (42) with α = rj+1 rj βj-1 , then with α = β j-1 and A, B depending on X n obtained in Lemma 31, we get

(59) |F | ≤ B v κ j-1 βj-1 -v 0 j-1 βj-1 min(1, r j +κ r j β j-1 ) ≤ B 2 |v κ j-1 -v 0 j-1 | νj ,
where ν j := min(1, β j-1 ) min(1,

rj +κ rj βj-1 ) ≤ 1.
To bound G, we consider

M := max(|x j |, |v 0 j-1 |), ε := sign(x j v 0 j-1 ) ∈ {-1, 1}, τ := max(|x j |, |v 0 j-1 |) M ∈ [0, 1], N := 1 + εt βj-1 ,
where we have assumed with no loss of generality that M > 0.

An easy computation yields that

(60) G = (M r j+1 r j -M )N r j+1 r j β j-1 + M N r j+1 r j β j-1 -N + ε(t βj-1 -t) .
We now use the following elementary fact: for x ≥ 0 and α > 0, (1,α) .

|x α -x| ≤ |α -1| ln(x)x min
By applying that fact to (60), we deduce that there exists an explicit positive constant D j (depending on m and 1 , • • • , j-1 ) such that |G| ≤ |D j |κ|. From (59) and the previous inequality, we get that

|v κ j (x) -v 0 j (x)| ≤ l B 2 |v κ j-1 -v 0 j-1 | νj + D j |κ| .
By applying the induction hypothesis on |v κ j-1 -v 0 j-1 |, we prove the inductive step with C 1 j := l j (B 2 C νj-1 j-1 + D j ). This concludes the proof of (58). We now turn to the proof of (57). It is enough to prove the result for one single W κ,j . let -1 2n ≤ κ ≤ 1 2n and x (j) ∈ B κ j,m . On gets

W κ,j (x (j) ) -W 0,j (x (j) ) = β j-1 -1 β j-1 + 1 |v κ j-1 | βj-1+1 -|v 0 j-1 | βj-1+1 + 1 2 |x κ j | βj-1+1 -x 2 j -(|v κ j-1 | βj-1+1 -(v 0 j-1 ) 2 ) -x j v κ j-1 βj-1 -v 0 j-1 βj-1 + (v κ j-1 -v 0 j-1 ) + |v κ j-1 | βj-1+1 -(v 0 j-1 ) 2
). Following the same type of estimates used to derive (58), one gets (57).

We can now provide explicit bounds on κ 0 , for the results of the previous section to hold.

Proposition 33. Let m ∈ (0, 1). Then there is an explicit κ 0 (m) ∈ [-1 2n , -1 2n ] such that, for every κ 0 ∈ (0, κ 0 (m)), the statements of Theorem 28 and Theorem 30 hold true. Proof. To determine κ 0 (m), we rewrite (53) as follows,

V0 ≤ -CV 0 + 2 V 0 V 0 (e n )|δ(x)|.
The constant C above has been characterized in (36).

Along trajectories of System (31) closed by the feedback law given by ω H κ(x) (x) inside B 0 1-m,1+m , one gets by using 1 -m ≤ V 0 (x) ≤ 1 + m and (58) that

CV 0 ≥ C(1 -m), 2 V 0 V 0 (e n )|δ(x)| ≤ 2 √ 1 + m V 0 (e n )C 1 n κ 1-(n-1)/2n 0 .
One chooses then κ 0 (m) > 0 so that V0 ≤ -CV0 2 inside B 0 1-m,1+m , which yields that

κ 0 (m) := C(1 -m) 4 √ 1 + m V 0 (e n )C 1 n 2n n+1 .
As for Theorem 30, the only task to complete for an explicit characterization of the parameter λ appearing in the statement consists in estimating explicitly lower bounds for r(m, κ 0 ) and r(m, -κ 0 ). We provide indications for only r(m, κ 0 ). By definition, every x ∈ B κ0 <r(m,κ0) belongs to B 0 <1+m . There exists x ∈ B κ0 r(m,κ0) ∩ B 0 ≤1+m and then |r(m, κ 0

) -(1 + m)| ≤ C 2 n κ
1-(n-1)/2n 0 according to (57). One deduces immediately an explicit lower bound for r(m, κ 0 ) 4.3. ISS-type of result. In this section, we provide the second step for our partial solution of the prescribed-time stabilization of the n-th order chain of integrators in presence of disturbances. More precisely, the aim consists in stabilizing (31) with a static feedback law u = F (x), in a robust manner, i.e., with respect to measurement noise and external disturbances. The corresponding n-th order perturbed chain of integrators is given by

(61) ẋ = J n x + b(x)F (x + d 1 ) e n + d 2 ,
where d 1 ∈ R n is the measurement noise and d 2 ∈ R n the external perturbation. We set d = (d 1 , d 2 ) ∈ R 2n and we refer to it as the perturbation. Note that we are allowing unmatched uncertainties.

We now provide an ISS type of result regarding the robust properties of the perturbed system (61) stabilized with k

(x) = ω H κ(x) (x) given by (62) ẋ = J n x + bω H κ(x+d1) (x) e n + d 2 , x, d 1 , d 2 ∈ R n
, where b verifies (50). As before, we can assume with no loss of generality that b = 1. We have the following result. To prove the theorem, we are not able to exhibit an ISS-Lyapunov function but, by taking into account Theorem 28 and using the characterization of (ISS) provided by [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF]Theorem 2.1], it is enough to prove the following proposition.

Proposition 37. There exists a function F of class KL such that, for every bounded disturbances d 1 , d 2 : R + → R n and any trajectory of (62), the following holds true

(63) lim sup t→∞ Z(x(t)) ≤ F ( d 1 ∞ + d 2 ∞ ), where (64)
Z(x) = min V 0 (x), V 1+α(κ0) κ0 (x), V 1-α(κ0) -κ0 (x) , x ∈ R n .
Proof. The argument is similar to Item (S -∞) in [3, Proposition 2]. It is based on the following three inequalities, whose proofs are given in Appendix.

(i): On the open set B 0 >1+m , the time derivative Vκ0 of V κ0 along trajectories of (62) verifies almost everywhere

(65) Vκ0 ≤ - C 2 V 1+α(κ0) κ0 + F 1 ( d 1 ∞ + d 2 ∞ ),
where F 1 is a function of class KL.

(ii): On the set B 0 1-m,1+m , the time derivative V0 of V 0 along trajectories of (62) verifies almost everywhere

(66) V0 ≤ - C 2 V 0 + F 2 ( d 1 ∞ + d 2 ∞ ),
where F 2 is a function of class KL. (iii): On the open set B 0 <1-m , the time derivative V-κ0 of V -κ0 along non trivial trajectories of (62) verifies almost everywhere

(67) V-κ0 ≤ - C 2 V 1+α(-κ0) -κ0 + F 3 ( d 1 ∞ + d 2 ∞ ),
where F 3 is a function of class KL.

Let x(•) be a non trivial trajectory of (62).

Assuming that we have at hand the above inequalities. Suppose first that there exists a time t 0 ≥ 0 such that one of the following situations occurs:

(a): for every t ≥ t 0 , x(t) ∈ B 0 >1+m . By using (65), one gets that

lim sup t→∞ V 1+α(κ0) κ0 (x(t)) ≤ 2F 1 ( d 1 ∞ + d 2 ∞ ) C ; (b): for every t ≥ t 0 , x(t) ∈ B 0 1-m,1+m
. By using (66), one gets that lim sup

t→∞ V 0 (x(t)) ≤ 2F 2 ( d 1 ∞ + d 2 ∞ ) C ;
(c): for every t ≥ t 0 , x(t) ∈ B 0 <1-m . By using (67), one gets that lim sup

t→∞ V 1+α(-κ0) κ0 (x(t)) ≤ 2F 3 ( d 1 ∞ + d 2 ∞ ) C .
Let I + (I -respectively) be the set of times t such that x(t) ∈ B 0 >1+m (x(t) ∈ B 0 <1-m respectively). If such a t 0 does not exists, either I + or I -is an infinite (countable) union of disjoint non trivial intervals (s k , t k ), k ≥ 0, where lim k→∞ s k = ∞. We analyse only the case where I + = ∪ k≥0 (s k , t k ) since handling the other case is entirely similar.

Set

C V := max x∈B 0 1+m V κ0 . For k ≥ 0, consider the trajectory x(•) on [t k , s k+1 ]. Recall that V 0 (x(t k )) = V 0 (x(s k+1 )) = 1 + m by definition of t k , s k+1 . Then, there exists tk ∈ [t k , s k+1 ) such that V 0 (x( tk )) ≤ V 0 (x(s k+1 )) and V 0 (x(t ) ) ≥ 1 -m on [ tk , s k+1 ]. Integrating (66) from tk to s k+1 yields that C(1-m) 2 ≤ F 2 ( d 1 ∞ + d 2 ∞ ). Set now L := lim sup t→∞ V 1+α(κ0) κ0 (x(t)). If L ≤ C 1+α(κ0) V , then L ≤ 2C 1+α(κ0) V C(1 -m) F 2 ( d 1 ∞ + d 2 ∞ ).
Otherwise, assume that L > C 1+α(κ0) V

. Consider then the non empty set of v > C 1+α(κ0) V for which there exist two sequences t k ≤ tk < sk+1 < s k such that Clearly L is the supremum of such v's. Integrating (65) between tk and sk+1 yields at once that v ≤ 2F1( d1 ∞ + d2 ∞)

C

. We deduce at once that the content of Item (b) above holds true. By collecting all the cases, we conclude the proof of Proposition 37.

Conclusion

In this paper, we have addressed the issue of prescribed-time stabilisation of an n-chain of integrators, n ≥ 1, either pure or perturbed. We have first recasted the results obtained in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF] within the framework of time-varying homogeneity and hence provided simpler proofs. As noticed in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF], the feedback laws (linear or finite time) arising from this time-varying approach do not perform well when the n-chain of integrators is subject to perturbations (especially measurement noise), even if one stops before the prescribed settling time. We instead propose to rely on feedback laws handling fixed-time stabilisation and to apply a standard trick of timescale reparametrisation and homogeneity to render the modified stabilisers fit for prescribed-time stabilisation of an n-th order chain perturbed of integrators. We perform that strategy in two steps. The first one consists in using feedbacks similar to those of [START_REF] Harmouche | Stabilisation of perturbed chains of integrators using Lyapunov-based homogeneous controllers[END_REF] and then by relying on a nice deformation argument proposed in [START_REF] Lopez-Ramirez | Fixed-time output stabilization and fixedtime estimation of a chain of integrators[END_REF]. In a second step, we obtain an ISS type of result in the presence of measurement noise for prescribed-time stabilisation of an n-th perturbed chain of integrators. However, such an approach is meaningful if one can get an explicit hold on the various parameters involved in the above construction. This is why we devoted a section for such an objective. 6. Appendix 6.1. Proof of Proposition 10. We next prove the result for η = 1 and the argument is inspired from the proof of Lemma 4.0 of [START_REF] Gauthier | Observability and observers for nonlinear systems[END_REF], and partly given in [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF]. Given a vector K = (k 1 , • • • , k n ) T ∈ R n with positive entries, we consider the invertible n × n matrix M K defined by (68)

M K =      k 1 k 2 • • • k n 0 k 1 • • • k n-1 . . . . . . . . . . . . 0 0 • • • k 1     
.

Note that

M K e n = K, M T K e 1 = K, M K J n M -1 K = J n . The last equation comes from the fact that M K is a polynomial function of J n , namely M K = n i=1 k i J i-1 n . Multiplying the LMI ( 14) on the left and on the right by (M T K ) -1 and M -1 K respectively yields the following LMI

J n -b Ke T 1 T S 1 + S 1 J n -bKe T 1 ≤ -ρM T K M K , ∀b ≥ b.
where S 1 = (M T K ) -1 SM -1 K . Let ρ * > 0 such that ρM T K M K ≥ ρ * Id n . We are left to prove that there exists ρ * > 0, S 1 symmetric positive definite and a vector K ∈ R n so that the following LMI holds true, for ρ small enough, and hence (70). We now assume that (76) V κ0 (d 1 ) V κ0 (x) > ρ.

In that case, the conclusion follows if one can prove that there exists C 4 > 0 independent of x, d 1 such that (77)

V κ0 (x + d 1 ) ≤ C 4 V κ0 (d 1 ) + F 1 ( d 1 ),
for some KL function F 1 . Indeed, in (73), the term in parentheses becomes bounded by F 2 ( d 1 ) for some KL function F 2 and then one gets (70) after using Holder's inequality with appropriate weights.

We are then left to prove (77). For that purpose set f (s) := V κ0 (x + sd 1 ) for s ∈ [0, 1] and let s * ∈∈ [0, 1] such that f (s * ) = max s∈[0,1] f (s). We will prove (77) with f (s * ) on the left-hand side and hence the conclusion. We can therefore assume with no loss of generality that s * = 1. One obtains that f (1) -f (0) ≤ for some KL function F 3 . In the above, we have used (72), the definition of s * = 1 and for the final inequality, Holder's inequality with appropriate weights. Combining the previous inequality with (76), one concludes the argument for (77) and hence that of (70).

Theorem 34 .

 34 With the assumptions of Theorem 30, System (62) is (ISS) for any bounded d = (d 1 , d 2 ). If d 1 = 0 and d 2 is parallel to e n (matched uncertainty), then convergence occurs in fixed time. The same conclusion holds for any prescribed time T by using the feedback k µ (x) = ω H κ(D r µ x) (D r µ x) with µ > 0 depending on T . Remark 35. This result improves [13, Corollary 1] where only the property (ISpS) was obtained. Remark 36. Using k µ instead of k 1 will modify the gain functions in Definition 4 since the disturbance d = (d 1 , d 2 ) must be modified to (D r µ d 1 , D r µ d 2 /µ).

V

  1+α(κ0) κ0 (x( tk )) = V 1+α(κ0) κ0 (x(s k+1 )) = v and V 1+α(κ0) κ0 (x(t)) ≥ v, t ∈ [ tk , sk+1 ].

1 TS 1 + 1 ≤-b k 1 ẽT 1 -b L J n- 1 . 1 2+κ 0 .

 1111110 S 1 J n -b Ke T -ρ * Id n , ∀b ≥ b.For n = 1, (69) reduces to -2bkS ≤ -µ * . By taking S = 1/2 and k = 1/b we get the result with ρ * = 1.Let n be a positive integer larger than or equal to two. Set ẽ1 = (1,• • • , 0) T ∈ R n-1 and K = (k 1 , L T ) T with L ∈ R n-1 to be determined. Notice that J n -b Ke T 1 =Using the homogeneity property of V κ0 , one gets that there exists a positive constant C 3 independent of x, d 1 such that M (x, d 1 ) ≤ C 3 ρ Since ω H κ0 is bounded on B 0 1 , one gets ∇V κ (x), e n ω H κ(x+d1) (x) -ω H κ(x) (x)

| 2 +

 2 ∇V κ0 (x + sd 1 ), d 1 |ds ≤ sd 1 )(d 1 ) i ds ≤ C d 1 )|(d 1 ) i | ≤ f (1) CF 3 ( d ),

  state stable in time T and convergent to zero (PT-ISS-C) if there exist functions β, β f ∈ KL, γ ∈ K and λ : [t 0 , t 0 + T ) → R * + such that λ tends to infinity as t tends to t 0 + T and, for all t ∈ [t 0 , t 0 + T ) and bounded d, x(t, d, x 0 ) ≤ β f β ( x 0 , t -t 0 ) + γ d [t0,t] , λ(t) .

	Definition 4. A system ẋ = f (x, t, d) is input-to-state practically stable (ISpS) if, for any
	bounded disturbance d, there exist functions β ∈ KL, γ ∈ K and c > 0 such that, for all t ≥ 0
	and bounded d, x(t, d, x 0 ) ≤ β ( x 0 , t) + γ d [0,t] + c. The system is input-to-state stable
	(ISS) if c = 0.
	1 Meaning that for f (t, 0) = 0 for all t ≥ 0.

2

A function γ : R + → R + is said to belong to a class K if it is strictly increasing and continuous with γ(0) = 0.

A function α is said to belong to a class K∞ if α ∈ K and it increases to infinity. A function β : R + × R + → R + is said to belong to a class KL if for each fixed t ∈ R + , β(•, t) ∈ K∞ and if for each fixed s ∈ R + , β(s, t) -→ t→∞ 0.

This research was partially supported by the iCODE Institute, research project of the IDEX Paris-Saclay, and by the Hadamard Mathematics LabEx (LMH) through the grant number ANR-11-LABX-0056-LMH in the ''Programme des Investissements d'Avenir''.

1

v 1 , • • • , v j-1 do not depend on j , one deduces that there exists L j , M j > 0 such that, for every x (j) ∈ S j , if βj = min(1, β j-1 ), then, for every κ ∈ [-1 2n , 1 2n ] and x (j) ∈ S j ,

|Z κ,j | ≥ M j |x j -v j-1 | 2(1+κ)/rj βj . (46) (Note that we used in the above that |x j | ≤ 1 for 1 ≤ j ≤ n and x (j) ∈ S j as well as a bound on the v j obtained with an immediate inductive argument based on (33).)

Inserting ( 44), ( 45) and ( 46) in (43), one deduces that, (47)

This is possible since the right-hand side of the above inequality does not depend on j . In that case, d dt V κ,j ≤ -l1 2 j-2 . This concludes the proof of the inductive step.

Remark 26. One can notice in the above argument a difference with respect of that of [START_REF] Hong | Finite-time stabilization and stabilizability of a class of controllable systems[END_REF] which consists in introducing the constants K j , L j and M j . The latter provide an explicit choice in order to be as explicit as possible in view of numerical determination of the constants 1 , • • • , n .

We next consider a state varying homogeneity degree given next.

Definition 27. For m ∈ (0, 1) and κ 0 ∈ (0, 1 2n ), define the following continuous function κ : R n → [-κ 0 , κ 0 ] by

We also need the following notation. For κ ∈ [-1 2n , 1 2n ] and a, b non negative real numbers, let B κ a,b , B κ ≤a and B κ ≥b respectively be the subsets of R n defined respectively by

The last set corresponds to the weighted spheres associated with the positive definite functions V κ .

In the spirit of [START_REF] Lopez-Ramirez | Fixed-time output stabilization and fixedtime estimation of a chain of integrators[END_REF], we are now able to consider the feedbacks which will ultimately yield prescribed time stability, which is described next.

Theorem 28. Assume that the uncertainty b is bounded, i.e.,

For Ω ∈ R n-1 , consider the n × n matrix A Ω given by

We make the linear change of variable y = A Ω x and we require the following condition on (k 1 , L), i.e., k 1 Ω + L = 0. One gets that

This linear change of variable amounts to multiply (69) on the left by (A T Ω ) -1 and on the right by A -1

Ω and we still denote by S the matrix (A T Ω ) -1 SA -1 Ω . We now pick Ω so that J n-1 + Ωẽ T 1 is Hurwitz and there exists a positive constant µ > 0 and a real symmetric positive definite (n -1) × (n -1) matrix S n-1 > 0 such that

After choosing S = 1 0 0 S n-1

, one simply finds k 1 > 0 large enough to get the result.

Remark 38. One must notice the similarity of the argument which is essentially that of [START_REF] Gauthier | Observability and observers for nonlinear systems[END_REF] and [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF], with the corresponding one in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF]. The one given here is more transparent and also allows to use the extra degree of freedom given by η.

6.2. Proof of Equations (65), ( 66) and (67). For κ ∈ {-κ 0 , 0, κ 0 }, taking the time derivative Vκ of V κ0 along a trajectory of (62) yields the inequality

We will prove that in each region of interest, there exists KL functions

and

(71)

Once this is established, one gets the conclusion by taking

We start by proving (71). For κ ∈ {-κ 0 , 0}, the region of interest is bounded. Hence one immediately concludes by applying Cauchy-Schwartz inequality and taking an upper bound for the continuous function ∇V κ on the region of interest. For κ = κ 0 , we recall that, for 1 ≤ i ≤ n, ∇V κ (x), e i is r(κ 0 )-homogeneous of degree (2 + κ 0 ) -r i with respect to the family of dilations D r(κ0) λ λ>0 . It is therefore immediate to see that there exists a positive constant

Since every r i is positive and hence 1 -ri 2+κ0 < 1 + α(κ 0 ), one can apply an appropriately weighted Holder inequality to get (71).

We know turn to an argument for (70). We provide an argument only for κ = κ 0 since for the other cases it is similar. If κ(x + d 1 ) = κ 0 , then V 0 (x + d 1 ) ≤ 1 + m implying that x ≤ C 1 d 1 for some positive constant independent of x, d 1 . Hence one can bound the left-hand side of (70) by F 1 ( d 1 ) for some KL function F 1 , and then conclude. We now treat the case where κ(x + d 1 ) = κ 0 . Recall that ω H κ0 is r(κ 0 )-homogeneous of degree r n+1 := 1 + nκ 0 with respect to the family of dilations D r(κ0) λ λ>0 . For non zero x ∈ R n , we define the normalized vector

Then one gets on B κ0 >1+m ,

where

Moreover, we have the following result: there exists a positive constant B such that, for every

, which is an immediate consequence of (42).

Consider ρ > 0 to be fixed small later. Assume first that

The term in brackets in (75) can be written as

≤1+m and hence, since V κ0 is of class C 1 , there exists a positive constant C 2 independent of x, d 1 , such that

By using (74), we can bound the term in parentheses in (75) as follows,

. In turn, one deduces that