Ana Belén

Galán López

Ariane Burke

; Sandrine Costamagno

Reconstructing habitat type and mobility patterns of Rangifer tarandus during the Late Pleistocene in Southwestern France: an ecomorphological study

WHAT WE EXPECT TO FIND

Modern ethological data indicate that reindeer herds adopt different mobility strategies that correlate with habitat type and topography. Long-distance migrations tend to take place in tundra and steppe habitats, while animals living in or near woodlands tend to migrate over much smaller distances (Schaefer et al. 2000;[START_REF] Weinstock | Late Pleistocene reindeer populations in Middle and Western Europe, an osteometrical study of Rangifer tarandus[END_REF][START_REF] Thomas | COSEWIC assessment and update status report on the Woodland Caribou Rangifer tarandus caribou in Canada[END_REF][START_REF] Courtois | Significante of caribou (Rangifer tarandus) ecotypes from a molecular genetics viewpoint[END_REF][START_REF] Goddard | Boreal caribou in Northeastern British Columbia[END_REF]. The mobility patterns of prehistoric reindeer, therefore, should be predictable if palaeoenvironmental reconstructions allow us to identify whether or not they lived in more open or more wooded environments. Palaeolithic environment does not offer a current analogue and in consequence it is not possible to infer reindeer mobility patterns from palaeoenvironmental reconstructions. On the other hand, it has been proven that an animal´s environment influences its' mobility patterns which, in turn, affect bone density and limb bone morphometry as has been demonstrated in several species, e.g., African bovids, cervids or wild horses [START_REF] Plummer | Hominid paleoecology at Olduvai Gorge, Tanzania, as indicated by antelope remains[END_REF]Bishop, 1994, DeGusta and[START_REF] Degusta | A method for inferring paleohabitats from the functional morphology of bovid astragali[END_REF][START_REF] Bignon | Geometric morphometrics and the population diversity of Late Glacial horses in Western Europe (Equus caballus arcelini): phylogeographic and anthropological implications[END_REF][START_REF] Van Asperen | Ecomorphological adaptations to climate and substrate in late Middle Pleistocene caballoid horses[END_REF][START_REF] Curran | Expanding ecomorphological methods: geometric morphometric analysis of Cervidae post-crania[END_REF] including humans [START_REF] Stock | Long bone robusticity and subsistence behaviour among Later Stone Age foragers of the forest and fynbos biomes of South Africa[END_REF].

Thus, we expect that the postcranial morphology of tundra dwelling and forest dwelling reindeer will reflect differences in adaptation, specifically their migratory behaviour.

The analysis of the ecological preferences of extinct fauna can contribute to reconstructing the paleobiology of Magdalenian hominids in South-Western France. Through the analysis of habitat preferences, we have the potential to advance in our understanding of the ecological context of hominid adaptation and evolution [START_REF] Bishop | Suid paleoecology and habitat preferences at African Pliocene and Pleistocene hominid localities[END_REF].

SAMPLE AND METHODS

Rangifer tarandus metapodia and phalanges from woodland (non-migratory) and tundra (migratory) settings will be use to create a referential framework to study ecomorphological differences between animals from different biomes and with different mobility patterns. To date we have collected 15 metatarsals, 14 metacarpals and 35 phalanges from non migratory ecotypes, and 31 metatarsals, 37 metacarpals and 50 phalanges from migratory settings.

The impact of habitat type and mobility on bone density and morphology of reindeer living in different habitats will be studied using Computer Tomography (CT), a non-invasive technique. This technique will be used to obtain detailed images of the cross-sectional area (virtual cross-sections) and visualise the internal structure of the bone, assessing its robusticity at five locations along the diaphysis for metapodia (20%, 35%, 50%, 65% and 80%) and one (50%) for phalanges.

Once cross-sectional images are obtained, they will be analysed using Geometric morphometrics (GMM) and Cross-Sectional Geometry (CSG).

GMM will be used to quantify the size and overall shape of the sections. With quantified geometric information about shape differences we can examine ecomorphological patterns and test morphological differences between populations in a highly effective, reproducible, accurate and statistically powerful way [START_REF] Viscosi | Leaf Morphology, Taxonomy and Geometric Morphometrics: A Simplified Protocol for Beginners[END_REF].

INTRODUCTION

The link between seasonal game movements and the season of occupation of human territories during the Magdalenian period (18.000-14.000 cal BP) in South-Western France remains an open question. The EMORPH project proposes an actualistic approach to reconstructing migratory patterns of Palaeolithic reindeer as a means of studying how their migration patterns affected human hunting strategies and socio-economic decisions. This research will allow us to better understand the behaviour of Magdalenian people and identify the precise role of reindeer in their economy.

An animal´s habitat preferences and mobility patterns hypothetically affect their bone density and limb bone morphometry. The goals of this project are twofold: 1) test this ecomorphological hypothesis for a given taxon (Rangifer tarandus), establishing a reference sample that includes animals from different habitat types and mobility patterns, and; 2) apply the knowledge gained towards an analysis of archaeological populations. This project will quantify the link between habitat type and mobility and bone density and morphology using Computer Tomography (CT) and geometric morphometry (GMM). The study will focus on reindeer because it is a key prey species in many prehistoric contexts in both Europe and North America. Modern reference samples will be collected in North America, where Rangifer tarandus (caribou, aka reindeer) herds still exist in woodland (nonmigratory) and tundra (migratory) habitats, making the creation of a referential framework possible.

Once the relationship between habitat, mobility and bone structure has been quantified, the information collected will be applied to faunal assemblages from Upper Palaeolithic archaeological sites in Southwestern France and used to reconstruct prey mobility. Prey mobility, in particular the migration of key species such as reindeer, is thought to influence the seasonal mobility of human predators and affect their hunting strategies. There is a longstanding debate as to the migratory behaviour of reindeer in Western Europe during the Late Pleistocene and its influence on Magdalenian populations. This project will enable us to test whether Magdalenian people were hunting animals which migrated over long or short distances. This information will be used to explore the impact of prey mobility on human mobility, hunting strategies and potentially technological choices.

Finally, the overarching goal of the project is to improve our knowledge about our past and the results are expected to have a significant impact on our understanding of the Magdalenian period.

Furthermore, being aware that it is not always is possible to have access to a CT scanner, linear measurements are also included in this project.

GMM analysis of the resulting images will be obtained using ImageJ [START_REF] Schindelin | Fiji: an open-source platform for biological-image analysis[END_REF], to set landmarks and obtain geometric coordinates. Then, the data collected will be transported, rotated and scaled through a Generalised Procrustes Analysis (GPA) in MorphoJ [START_REF] Klingenberg | MorphoJ: an integrated software package for geometric morphometrics[END_REF]. Next, a PCA analysis of landmark coordinates will be conducted to examine changes in shape and morphological differences and a Multivariate Analysis of Covariance (MANCOVA) will be used to statistically test for differences in mean shape as a function of ecological predictors such as vegetation type, substrate and mobility patterns.

Cross-Sectional Analysis will be carried out in ImageJ [START_REF] Schindelin | Fiji: an open-source platform for biological-image analysis[END_REF]) using the plugin BoneJ to obtain cross sectional properties (standardized prior to analysis). Then, in order to quantify and test differences between groups, t-tests will be applied in R (R Development Core Team, 2008).

Finally, linear measurements will be scaled prior to analysis and a Principal Component Analysis (PCA) will be carried out in R. Once the most effective variables are identified, a logistic regression (LR) will be performed to determine which variables discriminate between migratory and non-migratory settings.

The referential framework created from contemporary reindeer will be put into practice during the second phase of the project, using archaeological material from Southwestern France, allowing us to address on-going debates about prey mobility and its impact on human mobility patterns during the Upper Palaeolithic.

The calculation of Cross-Sectional Geometry (CSG) properties will allow us to interpret skeletal morphology within the context of the mechanical environment (Ruff, 2008) and identify how these properties might reflect mobility patterns [START_REF] Demes | Long bone cross-sectional dimensions, locomotor adaptations and body size in prosimian primates[END_REF].

CT scanner 20% 35% 50% 65% 80%

EMORPHPROJECT

ACKNOWLEGMENTS

This project has received funding from the European Union´s Horizon 2020 research and innovation programme under the Grant Agreement 794925.


Francisco Lopez-Ramirez

Denis Efimov

Andrey Polyakov

Wilfrid Perruquetti

Conditions for Fixed-Time Stability and Stabilization of Continuous Autonomous Systems

This work presents Lyapunov analysis conditions for fixed-time stability, a property where all the system's trajectories converge exactly to zero in a finite amount of time that is independent of the system's initial condition. Necessary and sufficient conditions for fixed-time stability without taking into account the regularity of the settling-time function are presented first. Next, a characterization for fixed-time stability with continuous settling-time function is introduced. A particular form of the characterizing functions follows, it allows to establish more constructive conditions and in order to obtain a converse result, the concept of complete fixed-time stability is introduced. A set of academic examples and an example of allocation of mobile agents illustrate the given concepts. Finally, a sufficient condition for fixed-time stabilization of nonlinear affine systems is obtained.

Introduction

In the theory of dynamical systems, the paramount Lyapunov's direct method allows to determine the stability of a system via the existence of a positive definite function V with a negative derivative, derived from the system equations [START_REF] Khalil | Nonlinear Systems[END_REF]. Converse results obtained by Massera and Kurzweil [START_REF] Khalil | Nonlinear Systems[END_REF] show that whenever a system is stable, there exists such a function V , called Lyapunov function. Further results show that a similar technique can be used to determine rates of convergence e.g. asymptotic or exponential stability [START_REF] Hafstein | A constructive converse Lyapunov theorem on exponential stability[END_REF]. These rates of convergence guarantee that an equilibrium point will be reached in an infinite amount of time. In contrast, finite-time stability, described early in the works of Roxin [START_REF] Roxin | On finite stability in control systems[END_REF] and Zubov [START_REF] Zubov | Methods of A. M. Lyapunov and their application[END_REF], provides a faster than asymptotic rate where the system's trajectories converge exactly to zero in a finite amount of time. The definition of this type of stability implies the existence of a function, often called settling-time function, that determines, given an initial condition, the minimum time at which the system's trajectories have converged exactly to zero and remain at zero for all future time. A counterpart of finite-time stable systems is that, as shown in [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF], a finite-time stable system cannot be Lipschitz continuous at the origin, so that uniqueness of solutions in the backward time is lost. Another aspect to take into account is the regularity of the settlingtime function since it might be discontinuous with respect to the initial conditions. The classical work of Bhat and Bernstein [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF] provides an insightful study of finite-time stable systems and provides a sufficient condition for finite-time stability with continuous settling-time function using a Lyapunov-like formulation, i.e. a positive definite function that satisfies a differential inequality. It was later given in [START_REF] Moulay | Finite time stability and stabilization of a class of continuous systems[END_REF] necessary and sufficient conditions for finite-time stability for both cases of settling-time function (continuous and discontinuous) and in [START_REF] Moulay | Finite time stability conditions for non-autonomous continuous systems[END_REF] analogous results for non-autonomous systems were given.

Yet another type of stability, reported in works such as [START_REF] Cruz-Zavala | Uniform robust exact differentiator[END_REF][START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF][START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF][START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF][START_REF] Jiménez-Rodríguez | On optimal predefined-time stabilization[END_REF], occurs when the settling-time function of a finite-time stable system is bounded on the whole domain by a finite value. This means that all the system's trajectories will converge exactly to zero before a specified, known in advance time regardless of the initial conditions. Being a particular case of finite-time stability, this type of stability, known as fixedtime stability, inherits many of the results derived for finite-time stable systems. Although some sufficient conditions for fixed-time stability can be found in [START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF][START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF][START_REF] Polyakov | Finite-time and fixed-time stabilization: implicit Lyapunov function approach[END_REF][START_REF] Lu | A note on finite-time and fixed-time stability[END_REF] and references therein, so far, necessary and sufficient conditions that take into account the regularity of the settling-time function haven't been presented. The aim of this article is to provide these conditions along with a characterization of the fixed-time stability property using Lyapunov theory. The theoretical tools employed to derive these conditions are founded in the works [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF][START_REF] Moulay | Finite time stability and stabilization of a class of continuous systems[END_REF] as well as in classic results on direct and converse Lyapunov theorems [START_REF] Krasovskiȋ | Stability of motion. Applications of Lyapunov's second method to differential systems and equations with delay[END_REF]. The main challenge consists in the analysis of the system's behavior far away from the origin since fixed-time stability, contrary to finite-time stability, is a property that preserves the finiteness of the settling-time over the whole domain of definition, even if this domain is unbounded.

In addition, based on Sontag's universal formula [START_REF] Sontag | A "universal" construction of Artstein's theorem on nonlinear stabilization[END_REF], we will present a sufficient condition for fixed-time stabilization of continuous autonomous systems that are affine in the control.

Preliminaries

Notation

R denotes the set of real numbers while

R + = {x ∈ R : x > 0} and R ≥0 = {x ∈ R : x ≥ 0}. a class-K function is a continuous, strictly increasing function γ : R ≥0 → R ≥0 with γ(0) = 0 while a class-K ∞ function is a class K function γ such that γ(s) → ∞ as s → ∞.
throughout the article, the domain Γ ⊆ R n will denote an open connected set containing the origin and ∂Γ will denote its boundary.

B n denotes the unit open ball in R n . a continuous function V : Γ → R ≥0 is said to be radially unbounded on Γ if V (x) → +∞ as x → ∂Γ. If Γ is unbounded then, in addition, V (x) → +∞ as x → +∞. a function V : Γ → R is said to be C 0 (or to belong to the C 0 class), if it is continuous, C 1 if it is continuously differentiable and if the derivative V (n) (x)
exists and it is continuous for all x ∈ Γ and for any integer n, then it is said to be C ∞ .

The Lie derivative of

V : Γ → R along g : R n → R n is represented by L g V : R n → R, L g V = ∇V (x), f (x) .
: [a, b) → R, b > a, is the function D + g : [a, b) → R given by D + g(t) = lim sup h→0 + 1 h [g(t + h) -g(t)], t ∈ [a, b). If g is differentiable at t, then D + g(t)
is the ordinary derivative of g at t.

Theoretical Background

Consider the following autonomous system

ẋ = f (x), x ∈ Γ, (1)
where f : Γ → R n is a continuous function and f (0) = 0. Let us assume that f is such that (1) has the properties of existence and uniqueness of solutions in forward time outside the origin. Then Φ(t, x) denotes the solution to system (1) starting from x ∈ Γ at t = 0. Definition 1 ( [START_REF] Bacciotti | Liapunov functions and stability in control theory[END_REF][START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF]). The origin of the system (1) is said to be

• Lyapunov stable if for any x 0 ∈ Γ the solution Φ(t, x 0 ) is defined for all t ≥ 0, and for any > 0 there is δ > 0 such that for any x 0 ∈ Γ, if x 0 ≤ δ then Φ(t, x 0 ) ≤ for all t ≥ 0;

• asymptotically stable if it is Lyapunov stable and Φ(t, x 0 ) → 0 as t → +∞ for any x 0 ∈ Γ;

• finite-time stable if it is Lyapunov stable and finite-time converging from Γ, i.e. for any x 0 ∈ Γ there exists 0 ≤ T < +∞ such that Φ(t, x 0 ) = 0 for all t ≥ T . The function T (x 0 ) = inf{T ≥ 0 : Φ(t, x 0 ) = 0 ∀t ≥ T } is called the settling-time function of system (1);

• fixed-time stable if it is finite-time stable and sup x0∈Γ T (x 0 ) < +∞.

The set Γ is called the domain of attraction. If Γ = R n , then the corresponding properties become global.

Lyapunov Functions

For a continuous function V : Γ → R, the upper-right Dini derivative of V along the solutions of ( 1) is given by V

(x) := D + (V • Φ)(0, x), (2)
if V is continuously differentiable on Γ\{0}, then

V (x) = d(V • Φ) dt (0, x) = ∂V ∂x f (x), x ∈ Γ\{0}. ( 3
)
Definition 2 ([15]). A function V : Γ → R ≥0 is called a strict Lyapunov function for system (1) if it fulfills the following properties:

L1 Positive Definiteness. V (x) > 0 for all x ∈ Γ \ {0} and V (0) = 0.
L2 V is radially unbounded on Γ.

L3 V is continuous on Γ and V (x) < 0 for each x ∈ Γ\{0}.

Theorem 1 ([15, Theorems 5.2 and 5.3]). The origin of (1) is asymptotically stable on Γ if and only if there exists a strict Lyapunov function for (1).

The next result is a Corollary of Theorem 1 and will be used later in the development of the proofs of some results. Remark that it is stated in a converse form and that instead of a differential inequality, an equality is obtained.

Corollary 1. Suppose that the origin of system (1) is asymptotically stable on Γ. Then there exist a strict Lyapunov function Ṽ : Γ → R ≥0 for (1) and a continuous positive definite function W : Γ → R ≥0 that satisfy

M1 V (x) = -W (x) ∀x ∈ Γ. M2 α 1 ( Ṽ (x)) ≤ W (x) ≤ α 2 ( Ṽ (x)) ∀x ∈ Γ, for some α 1 , α 2 ∈ K ∞ .
Proof. Following Theorem 1, there exists a continuous positive definite and radially unbounded function V such that

W * (x) := - ∂V ∂x f (x) > 0 ∀x ∈ Γ\{0}.
Let us propose the function Ṽ (x) =

V (x) 0 ξ(p)dp, where ξ ∈ K ∞ is defined later on. Note that from its definition, Ṽ is continuous and radially unbounded (the integral of the radially unbounded function ξ over V is itself radially unbounded). Then

V (x) = ξ(V (x)) V (x) = -ξ(V (x))W * (x), ξ(s) := s 1 if s ≤ 1 max{1, w(1) w(s) } if s > 1 and w(s) := inf V (x)=s W * (x)
By defining W (x) := ξ(V (x))W * (x) we obtain M1 and it becomes clear that W is continuous, W (0) = 0 and W (x) > 0 for all x ∈ Γ\{0} such that M2 is satisfied with the above selected ξ and the class-K ∞ functions

α 1 (s) = s s + 1 inf x∈Γ: Ṽ (x)≥s W (x), α 2 (s) = s + sup x∈Γ: Ṽ (x)≤s W (x).
Since Ṽ satisfies all conditions of Definition 2, it constitutes a strict Lyapunov function for system (1).

Relevant Theorems and Lemmas on Finite-Time Stability for Autonomous Systems

The following theorem gives necessary and sufficient conditions for finite-time stability of autonomous systems. Theorem 2 ([6]). Let us consider system (1) with uniqueness of solutions in forward time outside the origin. The following properties are equivalent:

1) the origin of system (1) is finite-time stable on Γ.

2) there exists a class-C ∞ strict Lyapunov function V : Γ → R ≥0 for system (1), satisfying for all x ∈ Γ 0

V (x) ds V (Φ(θ x (s), x)) < +∞, (4)
where the map s θx → t fulfills the identity s = V (Φ(θ x (s), x)).

Moreover if 1) or 2) are verified, all smooth Lyapunov functions V : Γ → R ≥0 for the system (1) satisfy for all x ∈ Γ

T (x) = 0 V (x) ds V (Φ(θx(s), x)) < +∞.
(

) 5
The following theorem, introduced initially in [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF] provides a particular differential inequality from which finite-time stability with continuous settling-time function can be asserted. Theorem 3 ([6, Proposition 12], [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF]). For the system (1), the following properties are equivalent:

1. the origin is finite-time stable on Γ with a continuous settling-time function.

2. there exist real numbers c > 0, α ∈ (0, 1) and a strict Lyapunov function

V : Γ → R ≥0 satisfying V (x) ≤ -cV (x) α ∀x ∈ Γ.
The next lemma states relevant properties of the solutions of finite-time stable systems and is used later in the article. Lemma 1 ([5, Proposition 2.4]). Suppose that the origin of system (1) is finite-time stable on Γ. Let T : Γ → R ≥0 be the settling-time function. Then the following statements hold:

i. If x ∈ Γ and t ∈ R ≥0 , then T (Φ(t, x)) = max{T (x) -t, 0}. (6)
ii. T is continuous on Γ if and only if T is continuous at 0.

Since fixed-time stability implies finite-time stability, this lemma remains valid for the fixed-time case.

Necessary and Sufficient Conditions For Fixed-Time Stability

Let us introduce the following example that illustrates the behavior of a fixed-time stable system.

Example 1. Consider the scalar system

ẋ = -x 1 2 -x 3 2 . ( 7
)
The trajectories of this dynamics, for any initial condition x 0 ∈ R and any t ≥ 0, can be obtained by direct integration and are given by It is clear from the system's solutions that for any t ≥ 2 arctan

Φ(t, x 0 ) = tan[arctan(|x 0 | 1 2 ) -1 2 t] 2 sign(x 0 ) if 0 ≤ t ≤ T (x 0 ) 0 if t > T (x 0 ) .
(|x 0 | 1 2 ), Φ(t, x 0 ) = 0 so that the settling-time function is T (x 0 ) = 2 arctan(|x 0 | 1 2
). Moreover, this function is bounded on R, indeed sup x0∈R T (x 0 ) = π so that all trajectories, regardless of where they start, converge exactly to zero in t ≤ π seconds (see Figure 1). System ( 7) possesses yet another property: for all initial conditions x 0 = 0 there exist well-defined solutions in the backward time. This property will be called complete fixed-time stability and it is defined as follows:

Definition 3. The system (1) is called complete fixed-time stable, if it is fixed-time stable, and for x 0 ∈ Γ \ {0} there exist unique solutions in the backward time denoted as Φ(-t, x 0 ) for t ∈ [0, τ (x 0 )), where

τ : Γ \ {0} → R ≥0 is the escape-time function such that lim sup t→τ (x0) Φ(-t, x 0 ) δΓ = 0 ∀x 0 ∈ Γ \ {0}. Note that by definition, if T (x 0 ) ≤ T max for all x 0 ∈ Γ, then τ (x 0 ) ≤ T max also for all x 0 ∈ Γ \ {0}, since T (Φ(t, x 0 )) + τ (Φ(t, x 0 )) = const
for all t ≥ 0 such that Φ(t, x 0 ) = 0. Therefore, by Lemma 1:

τ (Φ(-t, x 0 )) = max{0, τ (x 0 ) -t}, then, provided that τ is continuous, ∂τ (Φ(-t, x 0 )) ∂t = -1 for all t ∈ [0, τ (x 0 )) and all x 0 ∈ Γ \ {0}.
Analogously to Theorem 2 we first present a necessary and sufficient condition for an autonomous system to be fixed-time stable. Theorem 4. Consider system (1). The following properties are equivalent 1) The origin is fixed-time stable on Γ.

2) There exists a strict Lyapunov function V for system [START_REF] Khalil | Nonlinear Systems[END_REF]

satisfying for all x ∈ Γ sup x∈Γ 0 V (x) ds V (Φ(θ x (s), x)) < +∞, ( 8
)
where the map s θx → t fulfills the identity s = V (Φ(θ x (s), x)).

Proof. 1) ⇒ 2). If the system (1) is fixed-time stable, then its settling-time function is such that T * := sup x∈Γ T (x) < +∞ and Φ(t, x) = 0 for all t ≥ T * and for all x ∈ Γ. Since fixed-time stability implies asymptotic stability, according to Theorem 1, there exists a strict Lyapunov function V for (1) and therefore there exists a well defined mapping [0, T (x)) → (0, V (x)], t → V (Φ(t, x)) strictly decreasing and differentiable for all t ∈ [0, T (x)). Hence, for any x ∈ Γ, there exists a differentiable inverse mapping (0, V (x)] → [0, T (x)), s θx → t, also decreasing that satisfies for all s ∈ (0, V (x)]

θ x (s) = 1 V (Φ(θ x (s), x))
.

The change of variables s = V (Φ(t, x)) and the fact that V (Φ(T (x), x)) = 0 ∀x ∈ Γ lead to

T (x) = T (x) 0 dt = 0 V (x) θ x (s)ds = 0 V (x) ds V (Φ(θx(s), x)) . (9)
Then we have that

+ ∞ > sup x∈Γ T (x) = sup x∈Γ 0 V (x) ds V (Φ(θ x (s), x)) . ( 10
)
for all x ∈ Γ and the conclusion readily follows.

2) ⇒ 1). According to Theorem 1, because there exists a strict Lyapunov function V for system (1), its origin is asymptotically stable and therefore the inverse mapping θ x exists. The equation ( 10) implies, furthermore, that the origin is fixed-time stable.

Even though the conditions ( 4) and ( 8) are in general difficult to verify (they require explicit knowledge of the system's trajectories), they can be used to discard finite-time or fixed-time stability of certain systems. The next example, borrowed from [START_REF] Moulay | Finite time stability and stabilization of a class of continuous systems[END_REF], shows that linear systems cannot be finite-time stable.

Example 2. Consider the Cauchy problem

ẋ = -x, x 0 = 1, (11)
and the Lyapunov function candidate V (x) = x 2 . Then Φ(t, x 0 ) = e -t , V (Φ(t), x 0 ) = e -2t , θ(s) = -1 2 ln(s) and Φ(θ(s), x 0 ) = √ s lead to V (Φ(θ(s), x 0 ) = -2s, where s > 0. Substituting in (5) yields

T (1) = 1 0 ds 2s = +∞
and according to Theorem 2, the system (11) is not finite-time stable.

Example 3. The scalar system ẋ = -x µ , µ ∈ (0, 1) has, for any initial condition x 0 ∈ R, the explicit solution

Φ(t, x 0 ) = (|x 0 | 1-µ -(1 -µ)t) 1 1-µ sign(x 0 ) if 0 ≤ t ≤ T (x 0 ) 0 if t ≥ T (x 0 ) .
Although in this simple example it is possible to derive the settling-time function T from the general solution, as we did in Example 1, and immediately check that it is not globally bounded, here we will propose the

C 1 candidate function V (x) = |x| 1-µ , which yields Φ(θ x0 (s), x 0 ) = s 1 1-µ sign(x 0 ), s > 0 and V (Φ(θ x0 (s), x 0 )) = -(1 -µ). Then we have that 0 V (x0) ds V (Φ(θ x0 (s), x 0 )) = 1 1 -µ |x0| 1-µ 0 ds = 1 1 -µ |x 0 | 1-µ (12)
and it becomes clear that (4) is satisfied while (8) is not. Hence, the system ẋ = -x µ is finite-time stable but not fixed-time stable.

Note that in Theorem 4 nothing is said about the regularity of the settling-time function, therefore T might be discontinuous.

Sufficient Conditions for Fixed-Time Stability With Continuous Settling-Time Function

The next set of results presents more constructive conditions for fixed-time stability, they will no longer require explicit knowledge of the system's solutions and in the sufficiency case, they will guarantee continuity of the settling-time function.

Theorem 5. Suppose that there exists a continuously differentiable strict Lyapunov function V : Γ → R ≥0 for system (1) such that S1 there exists a continuous positive definite function r : R ≥0 → R ≥0 that satisfies

sup x∈Γ V (x) 0 dz r(z) < +∞; S2 the inequality V (x) ≤ -r(V (x)) holds for all x ∈ Γ.
Then the origin of (1) is fixed-time stable with continuous settling-time function T : Γ → R ≥0 and

T (x) ≤ sup x∈Γ V (x) 0 dz r(z) ∀x ∈ Γ. (13)
Proof. Let us define the inverse mapping (0, V (x)] → [0, T (x)) as s ϕx → t, where T : Γ → R ≥0 is the settlingtime function i.e. T (x) := inf{T ≥ 0 : V (Φ(T, x)) = 0}. Since V is, by assumption, differentiable on x ∈ Γ and satisfies S2, ϕ x (s) is strictly decreasing and differentiable with

ϕ x (s) = 1 V (Φ(ϕ x (s), x))
.

Then we have that for all x ∈ Γ

T (x) = T (x) 0 dt = - 0 V (x) ds -V (Φ(ϕx(s), x))
.

From condition S2 we have that -

V (x) ≥ r(V (x)) ∀x ∈ Γ so that - 0 V (x) ds -V (Φ(ϕx(s), x)) ≤ V (x) 0 ds r(V (Φ(ϕx(s), x))) ≤ sup x∈Γ V (x) 0 ds r(V (Φ(ϕx(s), x))) = sup x∈Γ V (x) 0 ds r(s) < +∞,
where in this last step, the condition S1 was used. Therefore, sup x∈Γ T (x) < +∞. Equivalently, the origin of ( 1) is fixed-time stable. Taking any x k ∈ Γ\{0} converging to zero, we have that

T (x k ) ≤ V (x k ) 0 ds r(s)
, by continuity of V and since r is assumed to be positive definite we obtain lim

x k →0 V (x k ) 0 ds r(s) = 0,
therefore T is continuous at the origin and due to Lemma 1.ii, T is continuous on Γ and, from the analysis above, bounded by [START_REF] Polyakov | Finite-time and fixed-time stabilization: implicit Lyapunov function approach[END_REF].

As can be seen from this theorem, fixed-time stability with continuous T can be completely characterized by the pair of functions (V, r) satisfying conditions S1 and S2. The following corollary gives more insight about some of the forms that the function r might take.

Corollary 2. Suppose there exists a strict Lyapunov function V : Γ → R ≥0 for system (1) such that

V (x) ≤ -c 1 V (x) α -c 2 V (x) β , ∀x ∈ Γ, ( 14
)
with c 1 , c 2 > 0, α ∈ [0, 1) and β > 1. Then the origin of (1) is fixed-time stable with continuous settling-time function T (x) satisfying

T (x) ≤ 1 c 1 (1 -α) + 1 c 2 (β -1)
.

Proof. Although this result was proven in [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF], here we will use the characterization proposed in Theorem 5.

Let us consider the ball D := {x ∈ R n : V (x) ≤ 1} and denote T D := inf{x ∈ Γ\D, T ≥ 0 : V (Φ(T, x)) ≤ 1}, and T 0 := inf{x ∈ D, T ≥ 0 : V (Φ(T, x)) = 0} as the time functions that provide the time that takes to any trajectory outside D to arrive to D, and the time that takes to any trajectory in D to arrive to zero, respectively. Then we have that for all x ∈ Γ\D, V (x) ≤ -r 1 (V (x)), where r 1 (z) = c 2 z β , and therefore

T D (x) ≤ sup x∈R n V (x) 1 dz r1(z) = +∞ 1 dz c2z β = z 1-β c2(β -1) 1 +∞ = 1 c2(β -1) < +∞.
For all x ∈ D, we have that V (x) ≤ -r 2 (V (x)), where r 2 (z) = c 1 z α and therefore

T 0 (x) ≤ 1 0 dz r 2 (z) = 1 0 dz c 1 z α = z 1-α c 1 (1 -α) 1 0 = 1 c 1 (1 -α) < +∞.
Hence, S1 and S2 are satisfied with r(z) = r 1 (z) + r 2 (z) and

T (x) = T D + T 0 ≤ 1 c 1 (1 -α) + 1 c 2 (β -1) < +∞ ∀x ∈ Γ.
Example 4 (Two dimensional systems). Consider the systems

Σ 1 : ẋ1 = -x 1 γ + x 2 ẋ2 = -x 2 γ -x 1 , Σ 2 : ẋ1 = -x 1 γ -x 3 1 + x 2 ẋ2 = -x 2 γ -x 3 2 -x 1 , x ∈ R 2 , γ ∈ (0, 1)
and the Lyapunov function candidate

V (x) = 1 2 (x 2 1 + x 2 2 ). For Σ 1 we have V (x) Σ1 = -(|x 1 | γ+1 + |x 2 | γ+1 ) ≤ -V (x) γ+1 2 , while for Σ 2 V (x) Σ2 = -(x 4 1 + x 4 2 ) -(|x 1 | γ+1 + |x 2 | γ+1 ) ≤ -V (x) γ+1 2 -V (x) γ+1 ,
where γ+1 2 < 1 and 1 + γ > 1. Then, according to Corollary 2, Σ 1 is finite-time stable while Σ 2 is fixed-time stable with T (x) ≤ 1 1-γ + 1 γ . Figure 2 shows the trajectories of Σ 1 and Σ 2 under small initial conditions (left) and with slightly larger ones (right). The time t represents the instant at which x ≤ 10 -3 . It is possible to see that the settling time of Σ 1 increases significantly with larger initial conditions, that of Σ 2 remains in a close vicinity.

Necessary Conditions for Fixed-Time Stability.

The first theorem of this section presents a necessary condition for fixed-time stability using a similar characterization as the one employed in Theorem 5. Theorem 6. Consider system (1) and suppose that the origin is fixed-time stable on Γ. Then there exist a strict Lyapunov function V and a class-K ∞ function q that satisfies N1 Proof. Since fixed-time stability implies asymptotic stability we know, from Corollary 1, that there exist a strict Lyapunov function V and a continuous positive definite function W that satisfy the M conditions. Then we have that for all x ∈ Γ: I there exist a decreasing differentiable mapping [0, T (x)] → (0, V (x 0 )], t → V (Φ(t, x)), with T (x) = inf{T ≥ 0 : V (Φ(T, x)) = 0} and its corresponding inverse mapping (0, V (x)] → [0, T (x)], s ϕx → t, also decreasing and differentiable such that ϕ x (s) = 1/ V (Φ(ϕ x (s), x)). II Since V (x) and W (x) := V (x) satisfy M, there exists some q ∈ K ∞ such that -q(V (x)) ≤ V (x) ∀x ∈ Γ and N2 is satisfied. III Then

sup x∈Γ V (x) 0 dz q(z) < +∞; N2 -q(V (x)) ≤ V (x) ∀x ∈ Γ.
T (x) = T (x) 0 dt = - 0 V (x) ds -V (Φ(ϕ x (s), x))
.

From II, -V (x) ≤ q(V (x)) and therefore

- 0 V (x) ds -V (Φ(ϕx(s), x)) ≥ V (x) 0 ds q(V (Φ(ϕx(s), x))) . Hence +∞ > sup x∈Γ T (x) ≥ sup x∈Γ V (x) 0 ds q(s)
and N1 is fulfilled.

It is difficult to obtain a converse result to Corollary 2 without adding some assumptions about the behavior of V when x → ∂Γ. Nonetheless, for the case of complete fixed-time stability, the next necessary and sufficient condition can be obtained.

Theorem 7. The system (1) is complete fixed-time stable on Γ with continuous settling-time T and escapetime τ functions if and only if there exists a strict Lyapunov function V : Γ → R ≥0 and some constants c 1 > 0, c 2 > 0, α > 0 and β > 0 such that

∂V (x) ∂x f (x) ≤ -c 1 V α α+1 (x) -c 2 V β+1 β (x)
for all x ∈ Γ \ {0}.

Proof. The system is fixed-time stable, and in complement to the maximal time of convergence to the origin:

T max = lim sup x δΓ →0 T (x),
let us introduce also into consideration the minimal time of convergence from the boundary of Γ:

T min = lim inf x δΓ →0 T (x),
which is well defined due to continuity of T and T min ≤ T max . Obviously, T (0) = 0 and T (x 0 ) > 0 for all x 0 ∈ Γ \ {0}, and

T (Φ(t, x 0 )) = max{0, T (x 0 ) -t}, then ∂T (Φ(t, x 0 )) ∂t = -1
for all t ∈ [0, T (x 0 )) and all x 0 ∈ Γ \ {0}.

For any a > A = Tmax Tmin ≥ 1 introduce the set

Ω a = {x ∈ Γ : T (x 0 ) ≤ 1 a T max },
which is then a closed set containing the origin, Ω a ⊂ Γ. Define a Lyapunov function candidate

V 1 : Ω a → R ≥0 as V 1 (x) = ρ1 α+1 T α+1 (x)
for some α > 0 and ρ 1 > 0, then

∂V 1 (x) ∂x f (x) = -ρ 1 T α (x) = -ρ 1 α + 1 ρ 1 V 1 (x) α α+1
for any x ∈ Ω a \ {0}. For further analysis assume that ρ 1 is taken small enough to guarantee that V 1 : Ω a → [0, 1). Now consider an open set Ξ = Γ -Ω 2a . On this set consider the system (1) in the backward time:

ẋ(t) = -f (x(t)), t ≥ 0,
then for any x 0 ∈ Ξ there is a unique solution Φ(-t, x 0 ) (to denote a solution of this system we use the solutions of (1) with negative argument since we assumed that they exist due to complete fixed-time stability of ( 1)) that is defined on some finite interval of time for t ∈ [0, τ (x 0 )), and

lim t→τ (x0) Φ(-t, x 0 ) δΓ = 0. Again, τ (Φ(-t, x 0 )) = max{0, τ (x 0 ) -t},
then for a continuous function τ :

∂τ (Φ(-t, x 0 )) ∂t = -1 for all t ∈ [0, τ (x 0 )) and all x 0 ∈ Ξ. Define another Lyapunov function candidate V 2 : Ξ → R ≥0 as V 2 (x) = ρ2 β τ -β (x)
for some β > 0 and ρ 2 > 0, then

∂V 1 (x) ∂x f (x) = -ρ 2 τ -β-1 (x) = -ρ 2 β ρ 2 V 2 (x) β+1 β
for any x ∈ Ξ. We can also select ρ 2 sufficiently large to ensure that V 2 : Ξ → (1, +∞).

It is rest to unite these functions on the set Ξ ∩ Ω a = {x ∈ Γ :

1 2a T max < T (x) ≤ 1 a T max } to obtain a Lyapunov function V : Γ → R ≥0 : V (x) =      V 1 (x) x ∈ Ω 2a V 2 (x) x / ∈ Ω a λ(x)V 1 (x) + (1 -λ(x))V 2 (x) x ∈ Ξ ∩ Ω a , where λ(x) = 2(1 - a T max T (x)).
By definition,

V (0) = 0, V (x) > 0 ∀x ∈ Γ \ {0}, lim x δΓ →0 V (x) = +∞.
Finally,

∂V (x) ∂x f (x) = -                    ρ 1 α+1 ρ1 V (x) α α+1 x ∈ Ω 2a \ {0} ρ 2 β ρ2 V (x) β+1 β x / ∈ Ω a 2a Tmax (V 2 (x) -V 1 (x)) + ρ 2 (1 -λ(x)) β ρ2 V 2 (x) β+1 β +ρ 1 λ(x) α+1 ρ1 V 1 (x) α α+1 x ∈ Ξ ∩ Ω a .
Then the condition to ensure is that

V 2 (x) > V 1 (x) for x ∈ Ξ ∩ Ω a , which is satisfied due to selection of the coefficients ρ 1 , ρ 2 (i.e. V 1 : Ω a → [0, 1) and V 2 : Ξ → (1, +∞)).
Finally, for any α > 0 and β > 0, it is straightforward to check the existence of c 1 > 0 and

c 2 > 0 such that ∂V (x) ∂x f (x) ≤ -c 1 V α α+1 (x) -c 2 V β+1 β (x)
for all x ∈ Γ \ {0} as needed. Now assume that the last inequality is satisfied, and let us show that the system (1) is complete fixed-time stable. Obviously, it is fixed-time stable with a continuous at the origin settling-time function by Corollary 2. Consider the system behavior in the backward time, then we obtain that

- ∂V (x) ∂x f (x) ≥ c 2 V β+1 β (x),
from which we conclude that the escape-time function is upper bounded by a continuous function, and hence it is also continuous.

Mobile Agents Allocation

In order to illustrate the most important features of fixed-time stability, let us consider the problem of equidistant allocation of n mobile agents along a line segment. The one dimensional position of each agent is given by x i (t) ∈ R, i = 1, 2, ..., n and x 0 and x n+1 denote fixed endpoints on the segment where the agents will align equidistantly. The dynamical model of each agent is given by the integrator system

ẋi = v i , i = 1, 2, ..., n (15)
where x = [x 1 , x 2 , ..., x n ] T is the state vector of the multi-agent system and v i ∈ R is a continuous scalar function called allocation algorithm. It was shown in [START_REF] Petrikevich | Linear algorithms for control of geometrical allocation of agents in multi-agent systems[END_REF] that the allocation algorithm

v i = 1 2 (x i-1 -x i ) + 1 2 (x i+1 -x i ), ( 16
)
drives the states of system (15) from any initial condition to the equilibrium point

x * i ∈ R n , x * = x 0 + i n+1 (x n+1 -x 0 )
exponentially. The equilibrium points x * i represent the state where all the agents are aligned sequentially and equidistantly along the interval [x 0 , x n+1 ]. Note that the allocation algorithm given above only uses the information of the current agent x i and of the neighbor agents x i-1 and x i+1 . The extension of system ( 15)-( 16) to the 2D case is achieved by combining independently the systems where x, y ∈ R n are the vectors of horizontal and vertical positions, respectively. Then, each agent position (x i (t), y i (t)) → (x * i , y * i ) as t → ∞, where x * i is as above and y * i = y 0 + i n+1 (y n+1 -y 0 ). This corresponds to the allocation of the agents along the line segment that joins the points (x 0 , y 0 ) and (x n+1 , y n+1 ). Figure 3 shows a numerical simulation of ( 17) over a time interval t ∈ [0, 40] seconds, with agents' initial position close to the target line segment.

ẋi = 1 2 (x i-1 -x i ) + 1 2 (x i+1 -x i ), ẏi = 1 2 (y i-1 -y i ) + 1 2 (y i+1 -y i ), (17)
Consider now the allocation algorithm

v i = φ 1 2 (x i-1 -x i ) + 1 2 (x i+1 -x i ) (18)
where φ(s

) := c 1 s 1-1 µ + c 2 s 1+ 1 µ , c 1 , c 2 > 0, µ > 1.
In [START_REF] Parsegov | Nonlinear fixed-time control protocol for uniform allocation of agents on a segment[END_REF], it was shown using Corollary 2, that the system (15)-( 18) is fixed-time stable and that its settling-time function satisfies the estimate

T (x) ≤ πµn 1 4µ 2 λ√ c 1 c 2 , λ = 2 sin 2 π 2n+1 .
The extension of system [START_REF] Krasovskiȋ | Stability of motion. Applications of Lyapunov's second method to differential systems and equations with delay[END_REF] with the allocation algorithm [START_REF] Petrikevich | Linear algorithms for control of geometrical allocation of agents in multi-agent systems[END_REF] to the 2D case is given by ẋi

= φ( 1 2 (x i-1 -x i ) + 1 2 (x i+1 -x i )), ẏi = φ( 1 2 (y i-1 -y i ) + 1 2 (y i+1 -y i )). (19)
Figure 4 presents a comparison between ( 17) and ( 19) with larger initial conditions than in Figure 3 and the same time interval t ∈ [0, 40]. It is possible to see in the inner plots that the fixed-time algorithm [START_REF] Petrikevich | Linear algorithms for control of geometrical allocation of agents in multi-agent systems[END_REF] manages to drive the agents to their target positions (Fig 4 .b) whereas the exponential algorithm ( 16) falls short, i.e. the given time interval is not long enough to drive the agents to the equilibrium point. Moreover, the calculation of the settling-time estimate gives T (x) ≤ 26.21, which means that the agents will converge exactly to their target positions in less than 26.21 seconds, irrespectively of their initial position. Figure 4.c shows a plot over time of the norm of the distance between the agents' position and the target points, i.e. e i (t) = (x i (t) -x * i ) 2 + (y i (t) -y * i ) 2 of each allocation algorithm e exp and e f x . It is noticeable that the fixed-time algorithm exhibits a much faster decay rate.

Fixed-Time Stabilization of Continuous Affine Systems

In this section, we will give a sufficient condition for fixed-time stabilization following the same structure of well known results on asymptotic stabilization of continuous autonomous systems. 17) and [START_REF] Parsegov | Nonlinear fixed-time control protocol for uniform allocation of agents on a segment[END_REF], respectively. c) depicts a plot of the norm of the distance between the initial points and the target points of each system, denoted eexp and e f x , respectively.

Consider the following affine in the input u system:

ẋ = f 0 (x) + m i=1 f i (x)u i , x ∈ R n and u ∈ R m , (20)
where f 0 (0) = 0, f i is continuous for all 0 ≤ i ≤ m and such that (20) has uniqueness of solutions in forward time. Its closed-loop representation is given by

ẋ = f 0 (x) + m i=1 f i (x)u i (x), x ∈ R n . (21)
Let us recall the definition of stabilization and propose a definition of fixed-time stabilization. In the latter, we will only consider fixed-time stabilization with continuous settling-time functions.

Definition 4. The control system [START_REF] Isidori | Nonlinear control systems. Communications and Control Engineering Series[END_REF] is stabilizable (respectively fixed-time stabilizable) if there exists a C 0 feedback control law u : Γ → R m such that:

1. u(0) = 0;

2. the origin of the system (21) is asymptotically stable (respectively fixed-time stable with a continuous settling-time function).

Such a feedback law u(x) is called a stabilizer (respectively fixed-time stabilizer ) for system [START_REF] Isidori | Nonlinear control systems. Communications and Control Engineering Series[END_REF].

A radially unbounded, positive definite,

C 1 function V : Γ → R ≥0 is a control Lyapunov function (CLF) for the system (20) if for all x ∈ Γ\{0}, inf u∈R m (a(x) + B(x), u ) < 0, ( 22
)
where a(x) = L f0 V (x), B(x) = (B 1 (x), . . . , B m (x)) with B i (x) = L fi V (x) for 1 ≤ i ≤ m. Such a control Lyapunov function satisfies the small control property (SCP) if for each > 0, there exists τ > 0 such that, if x ∈ τ B n , then there exists some u ∈ B m such that

a(x) + B(x), u < 0. ( 23
)
The small control property is equivalent to lim sup

x →0 a(x) B(x) ≤ 0 (24) and this limit may very well be -∞. For a CLF, a(x) < 0 whenever B(x) = 0, and one may think that this is a necessary condition for asymptotic stabilization. However, as shown in [START_REF] Sontag | A "universal" construction of Artstein's theorem on nonlinear stabilization[END_REF], it is also a sufficient one. Indeed, E. Sontag shows that if a radially unbounded and positive definite C 1 function V satisfies b(x) = 0 ⇒ a(x) < 0 where b(x) = B(x) 2 , then the feedback law u = w(x),

w i (x) :=      - a(x) + a(x) 2 + b(x) 2 b(x) B i (x) if x ∈ Γ\{0} 0 if x = 0 , (25)
known as Sontag's universal formula, is a stabilizer for [START_REF] Isidori | Nonlinear control systems. Communications and Control Engineering Series[END_REF]. It was also shown in [START_REF] Sontag | A "universal" construction of Artstein's theorem on nonlinear stabilization[END_REF] that this feedback law is continuous on Γ\{0} and that if V further satisfies the SCP, then w is continuous on Γ (see, for instance, [START_REF] Isidori | Nonlinear control systems. Communications and Control Engineering Series[END_REF]Chapter 9]).

The following theorem presents an analogous formulation for fixed-time stabilization and in order to prove it, we will provide a continuous fixed-time stabilizer, akin to Sontag's universal formula.

Theorem 8. For the system [START_REF] Isidori | Nonlinear control systems. Communications and Control Engineering Series[END_REF] there is a continuous fixed-time stabilizer u = v(x) if there exists a radially unbounded and positive definite C 1 control Lyapunov function V : Γ → R ≥0 that satisfies the SCP and

a(x) 2 + b(x) 2 ρ(x) 2 ≥ (c 1 V (x) α + c 2 V (x) β ) 2 ∀x ∈ Γ, ( 26
)
for some c 1 , c 2 > 0, α ∈ (0, 1) and β > 1, where ρ : Γ → R is a continuous function on Γ\{0} such that

lim sup x→0 |ρ(x)| b(x) = 0. (27)
Proof. Let us introduce, for brevity in the notation, the function ϕ(s) := c 1 s α + c 2 s β and define the feedback law

v i (x) :=      - a(x) + a(x) 2 + b(x) 2 ρ(x) b(x) B i (x) if x ∈ Γ\{0} 0 if x = 0 , (28)
where ρ(s) := 1 + ρ 2 (x) and 1 ≤ i ≤ m. I The derivative of V along the trajectories of ( 20), (28) is given by

V (x) = ∇V (x), f 0 (x) + m i=1 f i (x)v i (x) = -a(x) 2 + b(x) 2 ρ(x) < 0 ∀x ∈ Γ\{0}. (29)
Thus the control (28) is a stabilizer for (20) and V is a strict Lyapunov function for the closed-loop system (21). II Since ρ(x) ≥ 1 for all x ∈ Γ, using the change of variables Bi (

x) = B i (x) 4 ρ(x), b(x) = b(x) ρ(x) in (28) we obtain v i (x) =        -4 1 + ρ 2 (x) a(x) + a(x) 2 + b(x) 2 b(x) Bi (x) if x ∈ Γ\{0} 0 if x = 0
, so that Sontag's universal formula is recovered multiplied by a gain 4 1 + ρ 2 (x), which is continuous for all x ∈ Γ\{0}. In addition, this observation also implies that v(x) = 0 whenever b(x) = 0. Therefore, for further analysis assume that b(x) = 0. Since by assumption V satisfies the SCP, f i : Γ → R n for 0 ≤ i ≤ m are continuous and dV dx (0) = 0, then for any > 0 there exists τ > 0 such that b(x) ≤ and

|a(x)| ≤ b(x) ≤ 2
for all 0 < x ≤ τ . Reducing τ and due to the restrictions imposed on the product ρ(x) b(x) we also have:

|ρ(x)| b(x) ≤ for all 0 < x ≤ τ . Recall that |v(x)| = |a(x) + a(x) 2 + b(x) 2 ρ(x)| b(x) ,
then if a(x) > 0 we obtain:

|v(x)| ≤ 2a(x) + b(x) ρ(x) b(x) ≤ 2 a(x) b(x) + b(x)(1 + |ρ(x)|) ≤ 4 . For a(x) ≤ 0, a(x) + a(x) 2 + b(x) 2 ρ(x) ≤ b(x) ρ(x) and |v(x)| ≤ b(x) ρ(x) ≤ b(x)(1 + |ρ(x)|) ≤ 2 .
Hence, due to an arbitrary selection of , ( 28) is a continuous feedback stabilizer for [START_REF] Isidori | Nonlinear control systems. Communications and Control Engineering Series[END_REF]. III Now let us show that the control (28) is a fixed-time stabilizer for [START_REF] Isidori | Nonlinear control systems. Communications and Control Engineering Series[END_REF]. We have from (29) that

V (x) = - a(x) ϕ(V (x)) 2 + (1 + ρ 2 (x)) b(x) ϕ(V (x)) 2 1 2 ϕ(V (x)) ≤ - a(x) 2 + ρ(x) 2 b(x) 2 ϕ(V (x)) 2 1 2 ϕ(V (x)) ≤ -ϕ(V (x)),
which implies fixed-time stability for all x ∈ Γ and from Corollary 2, (28) is moreover a continuous fixed-time stabilizer for [START_REF] Isidori | Nonlinear control systems. Communications and Control Engineering Series[END_REF].

Example 5. Consider the following system

ẋ1 = x 1 + x 2 + x 1 µ u ẋ2 = x 2 + x 2 µ u (30)
where µ > 0. For u = 0 the system (30) becomes a linear unstable system and our aim is to design a fixedtime stabilizer for it. Consider the

C 1 control Lyapunov function candidate V (x) = 1 η+1 |x 1 | η+1 + 1 η+1 |x 2 | η+1 , with η > 0. We have that a(x) = |x 1 | η+1 + x 2 x 1 η + |x 2 | η+1 , B(x) = |x 1 | η+µ + |x 2 | η+µ , b(x) = (|x 1 | η+µ + |x 2 | η+µ ) 2 .
Note that b(x) = 0 only at x = 0 and that a(0

) = 0, such that b(x) = 0 ⇒ a(x) = 0. Selecting ρ(x) 2 := max 0, ϕ(V (x)) 2 -a(x) 2 b(x) 2 , where ϕ(s) := s η + s η+1 yields a(x) 2 + b(x) 2 ρ(x) 2 ≤ (V (x) η + V (x) η+1 ) 2 ,
such that (26) is satisfied. To verify that the SCP is satisfied we investigate the behavior of provided that 0 < η + µ < 2 and that µ < η. Hence, the SCP property holds. In order to check the condition (27), assume that the expression on the right hand side is greater that zero, then we obtain

a(x) |B(x)| = |x 1 | η+1 + x 2 x 1 η + |x 2 | η+1 |x 1 | η+µ + |x 2 | η+µ . a) b) c)
|ρ(x)| b(x) = V (x) η + V (x) η+1 |x 1 | η+µ + |x 2 | η+µ 2 - a(x) |x 1 | η+µ + |x 2 | η+µ 2 .
Similarly to the calculation above, it can be easily shown that lim sup x→0 a(x) B(x) 2 = 0 under similar restrictions on µ and lim sup x→0 ϕ(V ) B(x) 2 = 0, provided that µ < η 2 . Finally, from the analysis above, it is straightforward to verify that inf u∈R (a(x) + B(x)u) < 0 for all x ∈ R 2 \{0} and therefore V (x) is a CLF.

Thus, all the conditions of Theorem 8 are satisfied and gathering all the estimates we conclude that for µ ∈ (0, 1), η + µ < 2, µ < η and µ < η 2 the control law (28) is a continuous fixed-time stabilizer for (30).

Figure 5.a shows the norm of the state of the closed loop system (30) with x 0 = (25, 50), ν = 1 2 and µ = 1 5 in semi-logarithmic scale; Figure 5.b depicts the control signal v(t). It can be seen that stability is achieved in a finite amount of time and that the control signal is indeed continuous. Figure 5.c shows the norm of the state of system (30) for four different initial conditions. The plot reveals the expected low sensitivity of the settling-time with respect to initial conditions.

Conclusions

Complete necessary and sufficient conditions for fixed-time stability of continuous autonomous systems have been presented. A characterization of this property using a simple Lyapunov dissipation inequality has been proposed and it allows, in the sufficiency case, to rule out the case of discontinuous settling-time function. It is worth noticing that in the sufficiency case, the characterizing function r is continuous and positive, and this assumption is enough to assert fixed-time stability with continuous T . In the necessary case, however, the characterizing function q is not only continuous and positive definite but also increasing and unbounded i.e. a class-K ∞ function and no assumptions on the regularity of T were made. A particular, more constructive and previously studied form of the characterizing function r has shown to be consistent with the framework here presented.

The concept of complete fixed-time stability has been introduced and a necessary and sufficient condition for this property has been obtained. Finally, a sufficient condition for fixed-time stabilization of continuous affine systems, analogous to previous results on asymptotic stabilization, has also been obtained. It is left as an open problems, to find a necessary condition for fixed-time stability with continuous settling-time function and to find a necessary condition for fixed-time stabilization of affine systems.

  The brackets • ν simplify the notation | • | ν sign(•), where | • | denotes the absolute value. The upper-right Dini derivative of a function g

Figure 1 :

 1 Figure 1: Solutions of system (7) for four different initial conditions.

Figure 2 :

 2 Figure 2: Trajectories of Σ 1 and Σ 2 for γ = 1 2 with initial conditions x 0 = (-1, 2) (left) and x 0 = (-5, 10) (right).

Figure 3 :

 3 Figure 3: Position trajectories (x i (t), y i (t)) of the multi-agent system (17) with n = 6, initial positions x(0) = [241, 207, 232, 239, 241, 215], y(0) = [233, 244, 243, 217, 210, 211] and a target line segment given by the points (x 0 , y 0 ) = (208, 227) and (x 7 , y 7 ) = (236, 230). The symbol • denotes the initial position of each agent and × denotes its target position along the line segment depicted in bold black. The number beside each initial point corresponds to the agent's index i.

Figure 4 :

 4 Figure 4: Comparison of systems (17) and (19) with initial conditions x(0) = 208, 241 + d, 207 -d, 232, 239 + d, 241 + d, 215 -d, 236, y = [227, 233 + d, 244 + d, 243 + d, 217, 210 -d, 211 + d, 230], d = 180 and parameters c 1 = c 2 = 1, µ = 1.1 for the allocation algorithm[START_REF] Petrikevich | Linear algorithms for control of geometrical allocation of agents in multi-agent systems[END_REF]. a) and b) shows the trajectories (x i (t), y i (t)) of systems (17) and[START_REF] Parsegov | Nonlinear fixed-time control protocol for uniform allocation of agents on a segment[END_REF], respectively. c) depicts a plot of the norm of the distance between the initial points and the target points of each system, denoted eexp and e f x , respectively.

Figure 5 :x 2 x 1 η |x 1 | 2 2|x 1 |

 51121 Figure 5: State norm x in semi-logarithmic scale (a) and the control signal v(t) (b) of the system (30) in closed loop with x 0 = (25, 50) , ν = 1 2 and µ = 1 5 . The norm of the state in closed loop for four different initial conditions is depicted in c.

Acknowledgements

This work was partially supported by the ANR Project Finite4SoS (ANR 15 CE23 0007), the Ministry of Education and Science of the Russian Federation (Grant 14.Z50.31.0031) and the Government of Russia (Grant 08-08).