Janusz Wolny
email: wolny@novell.ftj.agh.edu.pl

Anna Wn

Jean-Louis Verger-Gaugry

Fractal Behaviour of Diffraction Pattern of Thue-Morse Sequence

Keywords: Thue-Morse sequence, singular continuous diffraction pattern, aperiodic structures, average unit cell, structure factor, Fourier transform, automata sequences, combinatorics on words, structure of quasicrystals, fractals

The Thue-Morse sequence is an example of an aperiodic structure with a singular continuous contribution to the diffraction pattern. Many satellites observed near the main diffraction peaks have been indexed and their intensities have been numerically calculated. Details of such a calculation are presented for satellite peaks indexed by 1/3. An average-unit-cell approach was successfully used to describe probabilities of atomic displacements from the points of the reference lattice. For a certain parameterisation of such probabilities, the results of fairly simple integer calculations could be easily generalised. For some defined sets of points, the analytical expressions for diffraction intensities were found and tested numerically on a wide range of scattering vectors and for up to 2 32 atoms. c 2000

INTRODUCTION

The one-sided Thue-Morse sequence can be defined on a two-letter alphabet {0, 1}, using the following substitution rule σ : σ (0) = 0 1, σ (1) = 1 0 [1][2][START_REF] Allouche | Automata and automatic sequences[END_REF][START_REF] Queffelec | Dynamical Systems-Spectral Analysis[END_REF][START_REF] Queffelec | Spectral study of automatic and substitutive sequences[END_REF][START_REF] Luck | [END_REF][7][8][9][10][11]. Starting with 0, one obtains the sequence 0110100110010110 . . . , which is neither periodic nor quasiperiodic. For such a sequence the two figures are replaced by the corresponding two bonds (a and b). In this paper it is assumed that b = 0.1 and a = 10b = 1 and the "atoms" are placed at the bond's edges (see also [START_REF] Allouche | Automata and automatic sequences[END_REF][START_REF] Queffelec | Dynamical Systems-Spectral Analysis[END_REF][11][12][START_REF] Gähler | The diffraction pattern of self-similar tilings[END_REF]). In the diffraction pattern of such a structure one can find two components that scale differently with the number of atoms (N ). The first component is known as the atomic contribution (Bragg peaks); it has intensities that scale as N 2 . The second one is a singular continuous component, which scales fractally with the number of atoms. For the Thue-Morse sequence the numbers of the two bonds are equal, and a periodic subset made of every second position, with a periodicity constant equal to λ 0 = (a + b) = 1.1, can be selected. The corresponding wave vector is then equal to k 0 = 2π/(a + b) ≈ 5.71. Therefore, for the Thue-Morse sequence, there is an average periodic structure with some appropriate decoration of the unit cell. Occupation probabilities of atomic positions in such a unit cell are given by [14-16] P(0) = 0.5 and P(±0.1) = 0.25, which leads to the following expression for the intensities of Bragg peaks at k = n • k 0 :

I (k) N 2 = 1 4 [1 + cos(kb)] 2 . ( 1
)
Equation (1) gives correct intensities at periodic positions of Bragg peaks and, in general, describes the so-called envelope function shown as a dashed line in Fig. 1a. In the diffraction pattern of the Thue-Morse sequence there is a lot of diffuse scattering, namely the previously mentioned singular continuous part of the diffraction pattern (see Fig. 1b) [START_REF] Keane | Generalized Morse sequence[END_REF]18]. It is already known that there is a cumulation of diffuse scattering around q 0 = k 0 /3, which leads to well-defined satellite reflections around such particular positions of scattering vectors. However, one should remember that peak positions fluctuate around q 0 . This feature is also discussed in our paper. As already shown in [START_REF] Wolny | The reference lattice concept and its application to the analysis of diffraction patterns[END_REF], intensities at satellite positions scale fractally, disappearing gradually from the diffraction pattern with an increasing number of atoms. In the following we describe intensity at satellite positions given by q 0 , using the average-unit-cell approach recently developed by Wolny [START_REF] Wolny | The reference lattice concept and its application to the analysis of diffraction patterns[END_REF][START_REF] Wolny | Average unit cell of the Fibonacci chain[END_REF][START_REF] Wolny | Kwazikrysztaly i inne dwuwymiarowe uklady o zabronionej symetrii[END_REF][START_REF] Wolny | Average unit-cell approach to diffraction on Thue-Morse sequence and decorated quasicrystals[END_REF]. This allows us not only to calculate the fractal exponents for the scaling behavior of diffraction intensities with respect to the number of atoms, but also to write an analytical expression for diffraction intensities obtained for particular numbers of atoms.

AVERAGE-UNIT-CELL APPROACH

Using the concept of reference lattice [START_REF] Wolny | The reference lattice concept and its application to the analysis of diffraction patterns[END_REF][START_REF] Wolny | Kwazikrysztaly i inne dwuwymiarowe uklady o zabronionej symetrii[END_REF], we can write the structure factor for monoatomic structure as

F(k) = N f λ/2 -λ/2 P k (u) exp(iku) du, (2)
where P k (u) is the probability distribution of atomic displacements from the points of the reference lattice. This distribution depends on the scattering vector k = 2π/λ; and for all higher harmonics (i.e., m • k, where m ∈ Z) we have

P mk (u) = n∈Z P k u + n λ m (3)
and

F(mk) = N f +λ /2 -λ /2 P mk (u ) • e imku • du = N f +λ/2 -λ/2 P k (u) • e imku • du. ( 4
)
This means that the probability distribution P k (u), |u| ≤ λ/2, describes an average unit cell for the scattering vector k and all its higher harmonics. 1) connects all maxima of the main diffraction peaks. (b) Intensities of satellite reflections are well described by envelope functions given by (10). Envelope curves have been calculated as a function of k for two different satellite positions: q = +k 0 /3 (dotted line) and q = -k 0 /3 (dashed line).

It has already been shown in [START_REF] Wolny | Average unit cell of the Fibonacci chain[END_REF] that for a modulated structure the expression for the structure factor of the mth satellite of any nth main reflection is

F(nk, mq) = N f u 1 -u 1 v 1 -v 1 P k,q (u, v) exp[i(nku + mqv)] du dv, (5)
for m, n ∈ Z, u 1 = π/k, v 1 = π/q, while the probability distribution P k,q (u, v) describes an average unit cell for a series of main reflections and their satellites.

THUE-MORSE SEQUENCE

For the considered Thue-Morse sequence there is a periodic superstructure consisting of every second atom at the lattice positions x n = n(a + b), n ∈ Z. The unit cell of this superstructure is decorated by two atoms, one located at position 0 and the other occupying aperiodically the two possible positions at distances a or b from the previous one. For an infinite cluster, occupation probabilities are equal (P a = P b = 1/4), which leads to the expression for the Bragg reflections given by (1). For any finite structure the difference of these two probabilities is bounded,

|P a -P b | ≤ 1 N , ( 6
)
where N is the number of atoms in the structure; the proof can be found in [START_REF] Verger-Gaugry | Generalized Meyer sets and Thue-Morse quasicrystals with toric internal space[END_REF].

Satellite Reflections at q = k 0 /3
Using two reference lattices, one with periodicity λ 0 = (a + b) = 1.1 (corresponding to the scattering vector k 0 of the main reflections) and the other with periodicity equal to 3λ 0 (which corresponds to the scattering vector q = k 0 /3 describing positions of an appropriate satellite), one gets the probability distribution shown in Fig. 2. The probability distribution in such an average unit cell is non-zero for nine positions only; and, for the number of atoms given by N = 6n, n ∈ Z + , the distribution values are related as follows:

P 1 = P 4 = P 7 = 1/6
and P 2 + P 3 = P 5 + P 6 = P 8 + P 9 = 1/6.

This relation is a consequence of the fact that for each period λ 0 there is an extra atom placed at distance a or b from the super-lattice position (see Table I). Using (7) and Fig. 2b, a set of three parameters i (i = 1, 2, 3) can be defined, such that

1 = (P 2 -P 3 )N , 2 =(P 6 -P 5 )N, 3 =(P 9 -P 8 )N (8)

TABLE I Occupation Probabilities of Nine Different Atomic

Positions in 3(a + b) Supercell and

Position in a 3(a + b) n supercell P n 1 0 1 / 6 2 b P 2 + P 3 = 1 / 6 3 a 4 a + b 1 / 6 5 ( a + b ) + b P 5 + P 6 = 1 / 6 6 ( a + b ) + a 7 2 ( a + b ) 1 / 6 8 2 ( a + b ) + b P 8 + P 9 = 1 / 6 9 2 ( a + b ) + a
P 2 = 1 12 + 1 2N , P 3 = 1 12 - 1 2N P 5 = 1 12 - 2 2N , P 6 = 1 12 + 2 2N ( 9
)
P 8 = 1 12 - 3 2N , P 9 = 1 12 + 3 2N .
The structure factor (i.e., the Fourier transform of ( 9)) is then given by

F(k + q) = F 1 (k + q) + F 2 (k + q), ( 10
)
where

F 1 (k + q) = N 6 [1 + 2 cos(q(a + b)) + cos((k + q)b) + cos(kb -qa) + cos(kb + q(a + 2b))] F 2 (k + q) = 1 2 [exp(i(k + q)b) -exp(i(qa -kb))] ( 11
)
i 2 sin(kb + q(a + 2b))

- 3 2 [exp(i(kb -qa)) -exp(-i(k + q)b)].
These equations describe correctly the structure factor for each scattering vector given by

k n,m = nk 0 + mq 0 n, m ∈ Z; k 0 = 2π a + b ; q 0 = k 0 3 , ( 12
)
where nk 0 is the position of the main Bragg peak and mq 0 corresponds to the position of the appropriate mth satellite. The first part of the structure factor (F 1 ) does not depend on the number of atoms; for q = 0, it leads to expression (1) for the diffraction intensity of Bragg reflections. Parameters i (shown in Fig. 3) depend on the number of atoms. Figure 3 atoms or more precisely when N is equal to 6 • 4 n (n = 1, 2, . . .), the following relation is fulfilled:

3 = 2 = 1 2 . ( 13
)
In a double logarithmic scale the plots versus the number of atoms are linear (Fig. 4), with the linear coefficient equal to α = log 4 (3) ≈ 0.7925. The exact value of the coefficient has been obtained for numerical integer calculations performed for a number of atoms equal to 6 • 4 1 = 24 and 6 • 4 2 = 96. For those numbers, 1 is equal to 4 or 12, which corresponds to probabilities P 2 = 4/24 or 14/96 and P 3 = 0 or 2/96, respectively. The plots of multiplied by a factor of N -α versus the number of atoms (Fig. 5 These relations have been checked in our "computer experiment." and i • N -α (i.e., the product with the number of atoms to the power of -α) is a periodic function of log N , with a periodicity constant equal to log 4. This allows us to plot i • N -α versus ξ , where

ξ ≡ log 4 N mod 1, (17)
which is shown in Fig. 6a. For each delta, a common curve is obtained, which is independent of the number of atoms. Actually, the curves look rather complicated, and a more general expression cannot be given yet. However, we have found the exact values for some special points. The two sets of points are as follows:

Set I:

N = 24 m∈Z + β m • 4 m , where β m ∈ {0, 1}, (18) 1
= 4 m∈Z + β m • 3 m , 3 = 2 = 1 2 .
Some examples of such numbers are (N , 1 ) ∈ {(24, 4), (96, 12), (120, 16), (384, 36) . . .}. Set II:

N = 12 m∈Z + β m • 4 m ,
where

β m ∈ {0, 1}, (19) 1
= 2 m∈Z + β m • 3 m , 2 = 1 , 3 =0.
Some examples of such numbers are (N , 1 ) ∈ {(12, 2), (48, 6), (60, 8), (192, 18) . . .}.

such differences decrease rapidly to zero as N -1 . The first 10 points in this figure are marked with open circles. For N above 100 one can hardly see any difference from the previous behaviour, with the points almost lined up along the common curve. One can conclude that for a large enough number of atoms (i.e., for more than a few hundred atoms), there are almost no differences between the diffraction pattern obtained for the number of atoms equal to the neighbouring numbers given by even and those given by odd multiples of 6.

DIFFRACTION ANALYSIS

Using formulas ( 10) and ( 11), one can calculate intensities of diffraction peaks for the scattering vectors given by ( 12)

I = F 2 real + F 2 imag , ( 20
)
where F real and F imag are the real and imaginary parts of the structure factor, respectively. The formulas can be essentially reduced for certain points described by ( 18) and ( 19). The results are as follows:

Set I:

F real (q) = N 6 [1 + 2 cos(q(a + b)) + cos(qb) + cos(qa) + cos(q(a + 2b))] + 3 4 1 [cos(qb) -cos(qa)] ( 21
)
F imag (q) = 1 4
[sin(qb)sin(qa) -2 sin(q(a + 2b))],

where 1 is given by (18). This is the envelope function plotted in Fig. 8a; it correctly describes intensities at any position of the main peaks and their satellites given by (12).

Set II:

F real (q) = N 6 [1 + 2 cos(q(a + b)) + cos(qb) + cos(qa) + cos(q(a + 2b))] + 1 2 1 [cos(qb) -cos(qa)] ( 22
)
F imag (q) = 1 2 [sin(qb) -sin(qa) -2 sin(q(a + 2b))],
where 1 is given by [START_REF] Wolny | Average unit cell of the Fibonacci chain[END_REF]. The appropriate plot of the envelope function is shown in Fig. 8b. The two envelope functions in Figs. 8a and 8b have been calculated using two different formulas (( 21) and ( 22)). There is only a small difference between those envelopes, as they correctly describe the intensities of the main reflections and their satellites. The scaling exponent for the diffraction intensities of satellite reflections is 2α = 2 log 4 3, which is very close to the one obtained previously [START_REF] Wolny | The reference lattice concept and its application to the analysis of diffraction patterns[END_REF].

In Figs. 8a and 8b one can see small differences between the envelope function and the maximum intensities of the appropriate satellites because the maximum intensity does not correspond to the exact position of the satellite with index equal to 1/3. The small shift from such a particular position is shown in Fig. 9. It oscillates and goes to zero for the number of atoms increasing to infinity. However, it should be stressed that for any scattering vector given by ( 12), the envelope function gives exactly the same value as the Fourier transform. A special test has been also performed for N = 144, when the shift of peak position is almost zero (Fig. 9). For this particular number, the envelope function fits the intensities of the satellites almost perfectly (Fig. 8c). It should be noted that Figs. 1b and8c show the same diffraction pattern (i.e., for N = 144) but the diffraction peaks are connected by FIG. 8-Continued different envelope functions given by (11). In Fig. 1b, the two different curves are plotted as a function of k (11), for q = ±k 0 /3 respectively; and in Fig. 8c the envelope curve is plotted as a function of q (11), for k = 0.

For satellites indexed by 1/5 and 1/7, similar properties have been observed [START_REF] Wolny | Average unit-cell approach to diffraction on Thue-Morse sequence and decorated quasicrystals[END_REF]. Fractal exponents have been determined to be equal to α = log 16 5 for index equal to 1/5 and α = log 64 7 for index equal to 1/7. Properly defined parameters can be expressed similarly to (18) and ( 19), but the coefficients must be changed. For example, when the satellite index FIG. 9. Shift of peak position with respect to the perfect position of the satellite indexed by 4 + 1/3, versus the number of atoms in cluster.

is equal to 1/5, the value of the base of the power expansion for N in ( 18) and ( 19) should be changed from 4 to 4 2 , and for 1 , from 3 to 5.

CONCLUSIONS

In this paper, the average-unit-cell approach has been used for the diffraction analysis of singular continuous diffraction patterns of the Thue-Morse sequence. For satellite reflections with indices equal to 1/3 of the main Bragg peaks, an average unit cell was constructed in the space defined by two variables, u and v. The probability distribution obtained is nonzero at only nine positions (Fig. 2a), which has allowed us to calculate envelope functions for the main reflections as well as for the satellites. We noticed that probability distributions depend on three parameters i (i = 1, 2, 3) defined by (8). For such 's, a general expression for the structure factor has been obtained (11). The parameters depend on the number of atoms, scaling as N α [α = log 4 3]. For numbers of atoms equal to even multiples of 6, formula ( 16) holds, and only two 's are independent. As a result of our computer experiment, we also found that for some specially defined numbers of atoms (i.e., for Sets I and II), relations ( 18) and ( 19) hold respectively, and one parameter is enough to describe intensities of the main reflections and their satellites. It has been shown that for double logarithmic plots, 's change periodically with the number of atoms, with the periodicity constant equal to log(4), along a line with a slope equal to some fractal exponent. After appropriate scaling of the 's (i.e., multiplication by the factor N -α ) and after introduction of a new variable ξ (17), a common curve (Fig. 6a) is obtained for all numbers of atoms. A similar curve is also obtained for all odd multiples of 6 as a limit of the number of atoms going to infinity. This saw-like curve is rather complicated and analytical expressions have been suggested for special sets of numbers only (i.e., Set I given by (18) and Set II by [START_REF] Wolny | Average unit cell of the Fibonacci chain[END_REF]). For those particular numbers, exact formulas for diffraction intensities have been written as [START_REF] Wolny | Average unit-cell approach to diffraction on Thue-Morse sequence and decorated quasicrystals[END_REF] and ( 22).

All the formulas in this paper have been tested numerically using integer calculations for the number of atoms limited by 2 32 . Exact integer calculations and a wide range of N allow us to formulate more general conclusions. However, strict mathematical proofs (in progress) are still required. Some results are already available and will be published separately [START_REF] Verger-Gaugry | Generalized Meyer sets and Thue-Morse quasicrystals with toric internal space[END_REF][START_REF] Verger-Gaugry | Mathematical quasicrystals with toric internal spaces, diffraction and Thue-Morse sequence[END_REF].

Similar results have been obtained for all prime numbers less than or equal to 19, except for 17, where the fractal exponent has been approximated as equal to 0.633 ± 0.0002, which corresponds to the already known value α = {log 17 + 2 log(4 + √ 17)}/16 log 2. More details of calculations will be published elsewhere. The next singular number is expected for m = 41. However, the numerical calculations require calculations for integer numbers more than 2 32 and are in progress. This paper is an example of a simple "computer experiment." In such an experiment, calculations are performed for integer numbers only, and the strictly valid results can be easily generalised and checked for arbitrarilry large numbers. Any appropriate hypothesis can then be proven for arbitrarily large numbers, and its validity extrapolated for the number of points to infinity. Results of such computer experiments can serve as a substitute for an appropriate mathematical proof.

FIG. 1 .

 1 FIG. 1. Diffraction pattern of Thue-Morse sequence. (a) Envelope function given by (1) connects all maxima of the main diffraction peaks. (b) Intensities of satellite reflections are well described by envelope functions given by(10). Envelope curves have been calculated as a function of k for two different satellite positions: q = +k 0 /3 (dotted line) and q = -k 0 /3 (dashed line).

FIG. 2 .

 2 FIG. 2. (a)Positions of non-zero probability distributions of atomic displacements are marked in twoparameter space (u, v). (b) Probability distribution versus v for the number of atoms equal to 72. Peaks marked as P 1 , P 4 , and P 7 are equal to 1/6 and correspond to periodical positions of the superstructure. Other probabilities scale with the number of atoms and approach 1/12 for infinite cluster.

FIG. 3 .FIG. 4 .

 34 FIG. 3. Plots of i /N , i = 1, 2, 3 versus log N . Open circles mark points for which N = 6 • 4 n and 2 =

3 = 2

 32 ) are almost constant (within the range of some fluctuations) in the logarithmic scale of N ; therefore, i ∝ N α , where α = log 4 (3). (14) More precisely, for N = 6 • 4 n (n = 1, 2, 3, . . .), = 1 /2.

FIG. 5 .

 5 FIG. 5. Parameters i (i = 1, 2) multiplied by a factor of N -α versus logarithm of the number of atoms. Periodic plots with periodicity constant equal to log 4 can be found.

FIG. 6 .

 6 FIG. 6. Plots of i • N -α (i = 1, 2, 3) versus ξ = [log 4 N mod 1] for the number of atoms equal to even (a) and odd (b) multiples of 6. The first ten odd multiples are marked as open circles.

FIG. 8 .

 8 FIG. 8. Diffraction patterns of the Thue-Morse sequence and corresponding envelope functions obtained for different numbers of atoms: (a) N = 96 is the second point of Set I (18) and the envelope function was calculated according to (21); (b) N = 60 is the third point of Set II (19) and the envelope function was calculated according to (22); (c) N = 144 is the number of atoms for which satellites are placed almost at perfect positions (see Fig. 9); in this case, a general expression (11) has been used for the envelope function calculation.

= 1 /2.

ACKNOWLEDGMENTS

Discussions with J. P. Gazeau and L. Pytlik were very helpful. Financial support from the Polish Committee for Scientific Research under Grant 2 P03B 041 16 is also acknowledged.

Those two sets give an infinite number of different points in Fig. 6a. More details for Set II can be found in Fig. 7. Those points can be easily ordered in a sequence as follows:

Series 1: ξ = log 4 [START_REF] Allouche | Automata and automatic sequences[END_REF]

Number of Points Equal to N

For all numbers of atoms equal to odd multiples of 6, the positions of points for each i (Fig. 6b) depart from the common curves shown in Fig. 6a for even multiples. However,


Youness Braidiz
email: youness.braidiz@inria.fr

Andrey Polyakov
email: andrey.polyakov@inria.fr

Denis Efimov
email: denis.efimov@inria.fr

Wilfrid Perruquetti
email: wilfrid.perruquetti@ec-lille.fr

On finite/fixed-time stability analysis based on sup-and sub-homogeneous extensions

Keywords: Finite/Fixed-time stability, Asymptotic stability, Nonlinear systems

Fixed-time and finite-time stabilities of non-linear systems are investigated using the introduced notions of sub-and sup-homogeneity. These concepts allow the systems to be analyzed using the homogeneity, even if they do not admit homogeneous approximations. Finite-time and fixed-time stability properties can be established using homogeneity degree of sub-and sup-homogeneous extensions.

Introduction

Finite-time stability (FTS) and stabilization is rather popular research direction in the last decades (see, e.g., [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][START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF][START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF][START_REF] Braidiz | On finite-time stability of homogeneous systems with multiplicative bounded function[END_REF][START_REF] Braidiz | On finite-time stability of subhomogeneous differential inclusions[END_REF][START_REF] Braidiz | On fixed-time stability of a class of nonlinear time-varying systems[END_REF]). Lyapunov function method is the main tool for FTS analysis. Two possible approaches for its application are known today. The first one is to find a strict Lyapunov function, which derivative satisfies a very specific estimate [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]. Design of such a Lyapunov function is a nontrivial task even in two dimensional case [START_REF] Polyakov | Robust finite-time stabilization and observation of a planar system revisited[END_REF]. The second option is to consider the so-called homogeneous approximation [START_REF] Andrieu | Homogeneous approximation, recursive observer and output feedback[END_REF] of the system provided that this approximation exists. Negative homogeneity degree and an asymptotic stability of the homogeneous approximation at zero implies FTS of the original system [START_REF] Andrieu | Homogeneous approximation, recursive observer and output feedback[END_REF]. The second approach is much more simple and useful for control systems analysis and design. Indeed, the equations describing the approximated dynamics are more elementary and the homogeneity of this system can be utilized for the asymptotic stability analysis [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF]. The aim of this paper is to study a possibility to use the same homogeneity-based methodology for systems, which do not admit homogeneous approximations.

By definition, the homogeneity is a dilation symmetry. The notion of a homogeneous function was first introduced by Leonhard Euler in 18th century. Nowadays, the homogeneity is used in systems analysis and control design. Many generalizations of this concept can be found for ordinary differential equations [START_REF] Zubov | Methods of AM Lyapunov and their Application[END_REF][START_REF] Khomenyuk | Systems of ordinary differential equations with generalized homogeneous righthand sides[END_REF][START_REF] Hermes | Nilpotent approximations of control systems and distributions[END_REF][START_REF] Kawski | Geometric homogeneity and stabilization[END_REF], time-delay systems [START_REF] Efimov | Weighted homogeneity for time-delay systems: Finite-time and independent of delay stability[END_REF], discrete-time systems [START_REF] Sanchez | On homogeneity of discretetime systems: stability and convergence rates[END_REF] and partial differential equations [START_REF] Polyakov | On homogeneous finite-time control for linear evolution equation in hilbert space[END_REF]. The homogeneity is utilized for FTS and fixed-time stability (FxTS) analysis of nonlinear systems [START_REF] Nakamura | Stabilization of nonholonomic systems using homogeneous finite-time control technique[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF], [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF]. Local homogeneity and homogeneous approximations [START_REF] Andrieu | Homogeneous approximation, recursive observer and output feedback[END_REF] allow the mentioned analysis to be applied for a larger class of systems. Homogeneous differential inclusions (DIs) were studied as well [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF][START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF][START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF]. In this paper we introduce a new property of DIs, which is called as sup/sub-homogeneity. Sup/sub-homogeneity is a certain relaxation of the usual homogeneity (dilation symmetry) of DIs. A system may not admit a homogeneous approximation being, at the same time, sup-or sub-homogeneous in the proposed sense. Below we show that the usual homogeneity-based arguments can be utilized for FTS/FxTS analysis in this case. This work proves the existence of a homogeneous Lyapunov function for a globally asymptotically stable (GAS) sup/sub-homogeneous DI. This result is a generalization of the converse Lyapunov theorem for homogeneous systems [START_REF] Rosier | Homogenous Lyapunov function for homogenous coninuous vector field[END_REF], [START_REF] Bernuau | On an extension of homogeneity notion for differential inclusions[END_REF], [START_REF] Bernuau | Robustness and Stability of Nonlinear Systems : a homogeneous point of view[END_REF], [START_REF] Bernuau | Homogeneity of differential inclusions[END_REF], [START_REF] Bernuau | Robustness of homogeneous and locally homogeneous differential inclusions[END_REF]. The property of weak FTS/FxTS is also investigated for the considered class of DIs.

To summarize, the main contribution of this work is a new simple approach to analyze FTS or FxTS of nonlinear systems, which do not admit local homogeneous approximations. The principal novelty is the concept of sub/sup -homogeneous extension, which can be constructed for any nonlinear system (in contrast to homogeneous approximation).

The paper is organized as follows. After introducing notations and definitions of asymptotic, finitetime, fixed-time stability and homogeneity in Section 2, we investigate the FTS of sub-homogeneous DIs (in Subsection 3.1) and FxTS of sup-homogeneous DIs (in Subsection 3.2). In Subsection 3.3 we construct examples of sup-and sub-homogeneous extensions, which are used next for the FTS and FxTS analysis of some classes of nonlinear systems. In Section 4, to illustrate the obtained results, an example of finite-time control design is considered.

Preliminaries

Stability of differential equations and inclusions

Let us consider a system:

   ẋ(t) = f (x(t)), t > 0, x(0) = x 0 ∈ R n (1)
where x(t) ∈ R n is the system state and f : R n → R n is a continuous vector field, f (0) = 0. This system has (probably non-unique) solutions being continuously differentiable functions satisfying (1).

Below we deal only with the so-called uniform stability analysis of differential equations and inclusions (see e.g. [START_REF] Clarke | Asymptotic stability and smooth Lyapunov functions[END_REF][START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF] and [START_REF] Polyakov | Stability notions and Lyapunov functions for sliding mode control systems[END_REF] for more details), so we omit the word "uniform" for shortness.

Definition 2.1. The origin of the system (1) is globally

• Lyapunov stable, if ∃α ∈ K ∞ such that x(t) ≤ α( x 0 ), ∀t ≥ 0,
• asymptotically stable, if there exists β ∈ KL such that x(t) ≤ β( x 0 , t), ∀t ≥ 0,

• FTS, if it is globally Lyapunov stable and there exists a locally bounded settling-time function

T : R n → [0, +∞) such that x(t) = 0, ∀t ≥ T (x 0 ),
• FxTS, if it is globally FTS and the settling-time function T is uniformly bounded,

• nearly FxTS if it is globally Lyapunov stable and any neighborhood of the origin is globally fixed-time attractive, i.e., for any nonempty neighborhood M ⊂ R n of the origin there exists a positive number 0 < T M < +∞ such that x(t) ∈ M, ∀t ≥ T M , for any x 0 ∈ R n and any solution t → x(t) of the system (1), where the classes of functions K ∞ and KL are defined in the usual way (see e.g. [START_REF] Khalil | Nonlinear systems, 3rd. New Jewsey[END_REF]).

Notice that usually the asymptotic stability is a combination of two properties: Lyapunov stability and the asymptotic convergence lim t→+∞ x(t) → 0. However, in the case of autonomous system the latter is equivalent to the existence of the function β ∈ KL satisfying Definition 2.1 (see [START_REF] Clarke | Asymptotic stability and smooth Lyapunov functions[END_REF], [START_REF] Khalil | Nonlinear systems, 3rd. New Jewsey[END_REF] for more details). More details about FTS and FxTS stability and illustrative examples can be found in the survey [START_REF] Polyakov | Stability notions and Lyapunov functions for sliding mode control systems[END_REF].

Let us consider the following differential inclusion

ẋ(t) ∈ F (x(t)), ∀t ≥ 0, (2)
where F : R n ⇒ R n is a set-valued mapping.

By definition, a solution of ( 2) is an absolutely continuous function satisfying (2) almost everywhere.

Such solutions exist, for example, if F is nonempty-valued, compact-valued, convex-valued and uppersemi-continuous [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF]. This condition is referred below as the standard assumption. In this paper, we deal also with DI, which do not satisfy the standard assumption. Such DIs are also studied in the literature (see, e.g., [START_REF] Tolstonogov | Differential Inclusions in Banach Spaces[END_REF], [START_REF] Polovinkin | Multi-valued Mappings and Differential Inclusions[END_REF]).

For each Ω ⊆ R n , we denote the set S(Ω) ⊂ C(R + , R n ) of all forward complete solutions1 t → x(t) of (2) satisfying x(0) ∈ Ω. If Ω is a singleton {x 0 }, we write S(x 0 ). We denote S = S(R n ) as the set of all forward complete solutions. The next definition concerns GAS and FTS/FxTS properties of DIs (see, e.g., [START_REF] Clarke | Asymptotic stability and smooth Lyapunov functions[END_REF], [START_REF] Polyakov | Stability notions and Lyapunov functions for sliding mode control systems[END_REF] for more details).

Definition 2.2. The origin of the system (2) is

• strongly asymptotically stable (resp., FTS/FxTS) if all solutions of (2) satisfy the asymptotic stability (resp., FTS/FxTS) property given in Definition 2.1;

• weakly asymptotically stable (resp., FTS/FxTS) if for any x 0 ∈ R n there exists a non-empty subset of solutions Ŝβ (x 0 ) ⊂ S(x 0 ) such that all solutions from the set Ŝβ = The strong GAS means that ∃β ∈ KL such that x(t) ≤ β( x 0 , t) for all t ≥ 0, all x 0 ∈ R n and all x ∈ S(x 0 ). In the case of the weak GAS, the latter holds for solutions from a nonempty set Ŝβ , which may depend on the function β ∈ KL (see the example above). That is why β is indicated in the notation Ŝβ .

Homogeneity

In control and systems theory, homogeneity simplifies qualitative analysis of nonlinear systems. Being a dilation symmetry, it allows a local properties (e.g., stability) of dynamical systems to be extended globally. The homogeneity can be introduced using various types of dilations [START_REF] Zubov | Methods of AM Lyapunov and their Application[END_REF][START_REF] Khomenyuk | Systems of ordinary differential equations with generalized homogeneous righthand sides[END_REF][START_REF] Kawski | Geometric homogeneity and stabilization[END_REF]. In this paper we deal with the so-called linear dilation [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF] in R n given by

d(s) = e G d s = +∞ i=0 s i G i d i!
where G d ∈ R n×n is an anti-Hurwitz2 matrix called as the generator of the dilation group [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]. The definitions of d-homogeneous functions, single-valued and multi-valued vector fields can be found in [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF], [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF].

Definition 2.3. A multi-valued vector field F : R n ⇒ R n (a function h : R n → R) is said to be d-homogeneous of degree ν ∈ R if F (d(s)x) = e νs d(s)F (x) (resp. h(d(s)x) = e νs h(x)) for all s ∈ R, x ∈ R n .
The homogeneous ball of the radius r > 0 is denoted as

B d (r) = {x ∈ R n : d(-ln(r))x ≤ 1} . For any r > 0 the set B d (r) is a compact in R n and d(s)B d (r) = B d (e s r), ∀s ∈ R.
For more details about geometric structures induced by linear dilations we refer the reader to [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF]Chapter 6]. For n = 1, the dilation d can be chosen as follows: d(s) = e s .

Main Results

Sub-homogeneous DI

In this section, we define sub-homogeneous DI and we investigate its FTS.

Definition 3.1. A set-valued mapping F : R n ⇒ R n is said to be d-sub-homogeneous of degree ν ∈ R, if for all x ∈ R n and all s ≥ 0 we have

e νs d(s)F (x) ⊆ F (d(s)x),
where d is a linear dilation in R n .

Obviously, the sub-homogeneity implies the inclusion F (d(s)x) ⊆ e νs d(s)F (x), ∀s ≤ 0, ∀x ∈ R n but not the equality as in the conventional case. Notice that d(s) has different properties for positive and negative values the group parameter s ∈ R. Indeed, d(s) is a strong contraction d(s) < 1 for s < 0 and a strong expansion d(s) > 1 for s > 0 provided that the matrix norm d(s) is properly defined (see, e.g., [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF] for more details). Therefore, the sub-homogeneity introduced above (the sup-homogeneity introduced below) is a symmetry with respect to a semi-group of contractions/expansions.

Below we show that such DIs may appear as extensions of some dynamical systems, which do not have homogeneous approximation. The sub-homogeneity simplifies the finite-time stability analysis in the latter case.

The following proposition ensures a symmetry of solutions of sub-homogeneous DI (2) The next corollary shows that the latter assumption is fulfilled under certain conditions.

Proposition 3.1. Let F be d-sub-homogeneous of degree ν ∈ R. If t → x(t)
Corollary 3.1. Let F be d-sub-homogeneous of degree ν ∈ R and Ŝβ ⊂ S be a set of weakly GAS solutions of (2) with some β ∈ KL. If

∃β d ∈ KL : β d (ρ,t) ≥ sup s≥0 d(-s) β d(s) ρ, e -νs t , (3)
for all t ≥ 0 and all ρ ≥ 0, then there exists Ŝβ d ⊂ S satisfying Assumption 3.1 such that Ŝβ ⊂ Ŝβ d .

Proof of Corollary 3.1: If

Ŝβ d (x 0 ) := s≥0 x ∈ C(R + , R n ) : x(t) = x(e νs t), t ≥ 0, x ∈ Ŝβ (d(-s)x 0 ) ,
where Ŝβ is the set of weakly GAS solutions with β ∈ KL, then Ŝβ ⊂ Ŝβ By construction, for any x 0 ∈ R n the inclusion x ∈ Ŝβ d (x 0 ) means that for any s ≥ 0 there exists

x ∈ Ŝβ (d(s)x 0 ) such that x(t) = d(-s)x(e -νs t), x(0) = d(s)x 0 .
Hence, we derive

x(t) = d(-s)x(e -νs t) = d(-s) • x(e -νs t) ≤ d(-s) β( d(s) • x 0 , e -νs t) ≤ β d ( x 0 , t),
for all t ≥ 0, all x 0 ∈ R n and all x ∈ Ŝβ d .

Taking into account that any function β ∈ KL admits the estimate [33, Lemma 8]

β(ρ, t) ≤ σ 1 (ρ)σ 2 (e -t ), for some σ 1 , σ 2 ∈ K ∞ ,
the condition (3) can be represented as follows

∃σ d ∈ K ∞ : σ d (ρ) ≥ sup s≥0 d(-s) σ 1 ( d(s) ρ),
provided that ν ≤ 0. Indeed, since e -e -νs t ≤ e -t , ∀s ≥ 0, ∀ν ≤ 0, ∀t ≥ 0 then we can select

β d (ρ, t) = σ d (ρ)σ 2 (e -t ), ρ, t ≥ 0.
Theorem 3.1. Let F be d-sub-homogeneous of degree ν ∈ R and satisfy the standard assumption. Let 120 p be an arbitrary natural number. The origin of (2) is GAS if and only if there exists a pair (V, W ) of continuous functions such that 1) V ∈ C p (R n , R + ), V is positive definite and d-homogeneous with degree k > 0 such that the matrix

pG d -kI n is Hurwitz; 2) W ∈ C ∞ (R n \ {0}, R + ), W is positive definite and d-homogeneous with degree k + ν; 125 3) max h∈F (x) ∇V (x), h ≤ -W (x) for all x = 0,
where

∇V (x) = ∂V (x) ∂x1 , • • • , ∂V (x)
∂xn stands for the gradient of the function V : R n → R at point

x ∈ R n and ∇V (x), h = n i=1 ∂V (x)
∂xi h defines the directional derivative of a continuously differentiable function V in the direction h ∈ R n .

V 0 and W 0 the claimed result is proven in [START_REF] Clarke | Lyapunov functions and feedback in nonlinear control[END_REF], i.e., there exists a pair (V 0 , W 0 ) of continuous functions (see [START_REF] Clarke | Lyapunov functions and feedback in nonlinear control[END_REF]), such that:

1) V 0 ∈ C ∞ (R n , R + ), V 0 is positive definite; 2) W 0 ∈ C ∞ (R n \ {0}, R + ), W 0 is positive definite; 3) max h∈F (x) ∇V 0 (x), h ≤ -W 0 (x) for all x ∈ R n \{0}.
135 Since V 0 (0) = 0, V 0 (x) → +∞ as x → +∞ and d is a dilation then there exist γ > 0 and

0 < v 1 < v 2 such that V 0 (d(s)x) ≤ v 1 for any x : x = 1 and ∀s ≤ 0, V 0 (d(s)x) ≥ v 2 for any x : x = 1 and ∀s ≥ γ.
Following Rosier's theorem [START_REF] Rosier | Homogenous Lyapunov function for homogenous coninuous vector field[END_REF] to build a homogeneous Lyapunov function V we consider a C ∞ function

a : [0, +∞) → [0, 1] such that a(t) =    0, if t ≤ v 1 , 1, if t ≥ v 2 ,
and ȧ(t) > 0 for all t ∈ (v 1 , v 2 ). Take k > max {-ν, 0} then the function

x → V (x) := R e -ks a • V 0 (d(s)x)ds is well defined, continuous on R n and C ∞ on R n \{0}. Moreover, V is d-homogeneous V (d(τ )x) =
e kτ V (x), and for any x : x = 1 and any h ∈ F (x) we have

∇V (x), h = R e -ks ȧ (V 0 (d(s)x)) ∇V 0 (d(s)x), d(s)h ds. and max h∈F (x) ∇V (x), h = R e -ks ȧ (V 0 (d(s)x)) max h∈F (x)
∇V 0 (d(s)x), d(s)h ds using sub-homogeneity of F , one gets

max h∈F (x) ∇V (x), h ≤ γ 0 e -ks ȧ (V 0 (d(s)x)) max h∈e -νs d(-s)F (d(s)x) ∇V 0 (d(s)x), d(s)h ds = γ 0 e -(k+ν)s ȧ (V 0 (d(s)x)) max h∈F (d(s)x) ∇V 0 (d(s)x), h ds ≤ - γ 0 e -(k+ν)s ȧ (V 0 (d(s)x)) W 0 (d(s)x)ds = -W (x) ≤ 0.
These inequalities are derived by using the sub-homogeneity of F. W is a d-homogeneous positive definite function with the degree (ν + k) > 0. This proves that V is a d-homogeneous Lyapunov function for [START_REF] Bernuau | Robustness and Stability of Nonlinear Systems : a homogeneous point of view[END_REF]. Now, let k > max{-ν, 0} be such that the matrix pG d -kI n is Hurwitz. Then for s ∈ R and

x : x = 1 we have

d p d(d(s)x) p V (d(s)x) = e ks d(-ps) d p dx p V (x). To guarantee that V ∈ C p (R n , R + ) it is enough to show lim s→-∞ d p V (d(s)x) d(d(s)x) p = lim s→-∞ e ks d(-ps) d p dx p V (x) = 0.
This is true, because the matrix pG d -kI n is Hurwitz and e ks d(-ps) = e s(kIn-pG d ) .

We use the same argument to prove that W ∈ C ∞ (R n \ {0}, R + ).

The latter result generalizes the Rosier's theorem [START_REF] Rosier | Homogenous Lyapunov function for homogenous coninuous vector field[END_REF], [START_REF] Bernuau | On an extension of homogeneity notion for differential inclusions[END_REF], [START_REF] Bernuau | Robustness and Stability of Nonlinear Systems : a homogeneous point of view[END_REF], [START_REF] Bernuau | Homogeneity of differential inclusions[END_REF] to sub-homogeneous DI.

τ R = sup x∈ Ŝβ (B d (2R)) inf {T > 0 : x(t) ∈ B d (R), ∀t ≥ T } .
Let us denote

ρ(R) = sup z∈B d (2R) z and ρ(R) = inf z∈R n \B d (R) z .
Notice that ρ(R) > 0, ρ(R) > 0 for any R > 0 and

x(t) ≤ β( x(0) , t) ≤ β(ρ(R), t), ∀t ≥ 0 for any x ∈ Ŝβ (B d (2R))
, where monotonicity of β ∈ KL with respect to the first argument is taken into account. Since β(ρ(R), t) monotonically tends to 0 as t → +∞ then there exists a finite number T β R > 0 such that β(ρ(R), t) ≤ ρ(R), ∀t ≥ T β R . This means τ R < +∞ for any R > 0. On the other hand, we derive

τ R/2 = sup x∈ Ŝβ (B d (R)) inf {T > 0 : x(t) ∈ B d (R/2), ∀t ≥ T } = sup x∈ Ŝβ (B d (R)) inf {T > 0 : d(ln(2))x(t) ∈ B d (R), ∀t ≥ T } = 2 ν sup x∈ Ŝβ (B d (R)) inf    2 -ν T > 0 : d(ln(2))x(2 ν t) ∈ B d (R), ∀t ≥ 2 -ν T    ≤ 2 ν sup y∈ Ŝβ (d(ln(2))B d (R)) inf T > 0 : y(t) ∈ B d (R), ∀t ≥ T = 2 ν sup y∈ Ŝβ (B d (2R)) inf T > 0 : y(t) ∈ B d (R), ∀t ≥ T = 2 ν τ R ,
where y(t) = d(ln(2))x(2 ν t). Hence, the convergence time of x ∈ Ŝβ (B d (2R)) to zero admits the estimate

τ R +∞ i=1 2 νi < +∞ provided that ν < 0.
Theorem 3.2 generalizes the results from [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF], [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF] about FTS of homogeneous DI with negative degree.

Proposition 3.1 implies that Assumption 3.1 is fulfilled for the set of all solutions of (2) provided that ( 2) is strongly GAS.

Corollary 3.2. Let F be d-sub-homogeneous of degree ν < 0. If (2) is strongly GAS then it is strongly globally FTS.

Sup-homogeneous DI

Definition 3.2. The set-valued mapping

F : R n ⇒ R n is said to be d-sup-homogeneous of degree ν ∈ R if it satisfies e νs d(s)F (x) ⊆ F (d(s)x), ∀s ≤ 0, ∀x ∈ R n ,
where d is a linear dilation in R n .

The following proposition shows the symmetry of solutions to the differential inclusion (2) provided that F satisfies the sup-homogeneity property.

∃β d ∈ KL : β d (ρ, t) ≥ sup s≤0 d(-s) β d(s) ρ, e -νs t , (4)
for all ∀t ≥ 0 and all ρ ≥ 0, where β ∈ KL corresponds to Ŝβ .

Taking into account that any function β ∈ KL admits the estimate [33, Lemma 8]

β(ρ, t)) ≤ σ 1 (ρ)σ 2 (e -t ), for some σ 1 , σ 2 ∈ K ∞ ,
the condition (4) can be represented as follows

∃σ d ∈ K ∞ : σ d (ρ) ≥ sup s≤0 d(-s) σ 1 ( d(s) ρ),
provided that ν ≥ 0. Indeed, since e -e -νs t ≤ e -t , ∀s ≤ 0, ∀ν ≥ 0, ∀t ≥ 0 then we can select

β d (ρ, t) = σ d (ρ)σ 2 e -t , ρ, t ≥ 0.
The following theorem shows the existence of a homogeneous Lyapunov function for an asymptotically stable sup-homogeneous differential inclusion.

Theorem 3.3. Let F be d-sup-homogeneous of degree ν ∈ R and satisfy the standard assumption. Let p be an arbitrary natural number. The origin of (2) is GAS if and only if there exists a pair (V, W ) of continuous functions 1) V ∈ C p (R n , R + ), V is positive definite and d-homogeneous of a degree k > 0 such that the matrix

pG d -kI n is Hurwitz; 2) W ∈ C ∞ (R n \ {0}, R + )
, W is positive definite and d-homogeneous of the degree k + ν;

3) max

h∈F (x) ∇V (x), h ≤ -W (x) for all x ∈ R n \{0}.
Proof of Theorem 3.3: We follow the arguments of the proof of Theorem 3.1 and we construct the so-called cut-off function a. For any constant γ > 0, by the definition of the dilation d and since

V 0 (0) = 0 and V 0 (x) → +∞ as x → +∞, there exist two numbers 0 < v 1 < v 2 , such that

V 0 (d(s)x) ≤ v 1 , for x = 1 and ∀s ≤ -γ, V 0 (d(s)x) ≥ v 2 , for x = 1 and ∀s ≥ 0. (5)
This is the difference with the proof of Theorem 3.1 (the property (5) holds for a different interval of s). The cut-off function a will have the same form that we defined in the proof of Theorem 3.1, then for inf {T > 0 :

k > max {-ν, 0} The Lyapunov function V is given by V (x) = R e -ks a • V 0 (d(s)x)ds and the function W (x) = R e -(k+ν)s ȧ (V 0 (d(s)x)) W 0 (d(s)x)ds.
x(t) ∈ B d (R), ∀t ≥ T } .
If ν > 0 then repeating the proof of Theorem 3.2 we derive τ 2 i R ≤ 2 -iν τ R and the time of convergence of any solution from Ŝβ to a d-homogeneous ball of the radius R > 0 is finite

+∞ i=0 τ 2 i R ≤ τ R +∞ i=0 2 -iν = 2 ν τ R 2 ν -1 < +∞
independently of the initial value x(0).

Corollary 3.3. Let F be d-sub-homogeneous of degree ν > 0. If (2)
is strongly GAS then it is strongly globally nearly FxTS.

FTS/FxTS of non-homogeneous systems

In this section, we consider the dynamical system [START_REF] Andrieu | Homogeneous approximation, recursive observer and output feedback[END_REF]. The goal of this section is to study the FTS and FxTS of globally asymptotically stable nonlinear systems which are not homogeneous and do not allow a homogeneous approximations at the zero or/and at infinity. To present a homogeneity-based analysis we construct sub-or sup-homogeneous extension of the function f . Homogeneous extensions were introduced in [START_REF] Polyakov | On homogeneous finite-time control for linear evolution equation in hilbert space[END_REF] to simplify a stability analysis of non-homogeneous PDEs. In this paper, we deal with finite-dimensional models and prove that, under certain conditions on f , the GAS nonlinear system (1) is globally FTS or globally nearly FxTS, according to the degree of the sub/sup-homogeneous extension.

Sub-homogeneous extension

Studying the finite-time convergence of a dynamical system by construction of a Lyapunov function is not easy and sometimes it is quite impossible. To overpass this difficulty, we consider the DI (2) with

F (x) = s≤0 e -νs d(-s)f (d(s)x) , (6)
where d is a linear dilation and ν ∈ R. Indeed, the set valued mapping F is d-sub-homogeneous with the degree of homogeneity ν ∈ R. The set F (x) is nonempty for every x ∈ R n . Since, by construction, f (x) ∈ F (x) for all x ∈ R n then any solution of ( 1) is a solution of ( 2) with the right-hand side (6) and Corollary 3.2 immediately implies Corollary 3.4. Let f be continuous and the system (2) with F given by (6) for some ν < 0 is GAS then (1) is globally FTS.

Consider the following example: f (x) = -x 1/3 (2 -cos 1 |x| . The function f does not have a homogeneous approximation at the origin, but it has a sub-homogeneous extension, which is given by

F (x) = -x 1/3 s≤0 2 -cos 1 |e s x| = -x 1/3 [1, 3].
Where d(s) = e s and ν = -2/3.

The following corollary deals with the FTS of the dynamical system (1).

Corollary 3.5. If (1) is GAS with β ∈ KL satisfying the condition (3) for some ν < 0 then it is globally FTS.

Proof of Corollary 3.5: The set of solutions of ( 1) is a subset Ŝβ of weakly asymptotically stable solutions of (2) with F given [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF]. Using Corollary 3.1 we conclude that all conditions of Theorem 3.2 are fulfilled and the origin of ( 1) is globally FTS.

Sup-homogeneous extension

In this section we define the sup-homogeneous extension of a vector field f as follows

F (x) = s≥0 e -νs d(-s)f (d(s)x) , (7)
where d is a linear dilation and ν ∈ R. x(T )

x 2 ν +∞ τ (R) 2 +ν∞ -1 F sub (x) F sup (x) x 0 • B R • Figure 1: Fixed-time stable solution.
Example: Let x → f (x) given by f (x) = x(3-cos (|x|)) . The function f does not have a homogeneous approximation at infinity, but it has a homogeneous extension, which is defined by

F (x) = x s≥0 {(3 -cos (|e s x|))} = x[2, 4],
where d(s) = e s and ν = 0.

Similarly to the previous section, the following condition of the nearly FxTS of the system (1) can be derived.

Corollary 3.6. Let f be continuous and the system (2) with F given by (7) for some ν > 0 is GAS then (1) is globally nearly FxTS.

Corollary 3.7. If (1) is GAS with β ∈ KL satisfying the condition (4) then it is globally nearly FxTS.

Combining the above results we derive the condition of global FxTS.

Corollary 3.8. If (1) is GAS with β ∈ KL satisfying both (3) and (4) with some degrees ν -< 0 and ν + > 0, respectively, then (1) is globally FxTS.

Proof of Corollary 3.8: Using Corollary 3.7 we deduce that the system (1) is nearly fixed-time stable and all trajectories arrive to a ball B R in a fixed time. In addition, Corollary 3.5 implies that the system (1) is FTS for all x 0 ∈ B R . These two facts imply that the state of the system (1) converges to the origin in a fixed time (see Fig. 1).

Example

Let us consider the system

   ẋ1 = x 2 , ẋ2 = -k 1 (x 1 ) -k 2 (x 2 ), (8)
where x 1 (t), x 2 (t) ∈ R are components of the state vector x = (x 1 , x 2 ) and k

i ∈ C(R, R), k i (ρ)ρ > 0, ∀ρ ∈ R, i = 1, 2.
Let us denote by f the vector field, which defines the right-hand side of the considered system. It is easy to see that f is continuous on R 2 . The system (8) can be interpreted as a mechanical model with the total energy (i.e., the sum of kinetic and potential energy) given by

U (x) = x1 0 k 1 (σ)dσ + x 2 2
2 .

The mechanical system is dissipative since

∇U (x), f (x) = -k 2 (x 2 )x 2 < 0 for x 2 = 0.
Applying LaSalle invariance principle (Barbashin-Krasovski Theorem) we conclude that the origin of the considered system is GAS. Its FTS and nearly FxTS can be studied using the sub/sup-homogeneous extensions. Taking Let the functions k 1 , k 2 and the degree ν be such that e -(1+2ν)s k 1 (e s x 1 ) = k 1 (x 1 ), ∀s, x 1 ∈ R. Then ∇U (x), g(s, x) = -e -(1+2ν)s x 2 k 2 (e (1+ν)s x 2 ) < 0 and take sub-homogeneous extension

d(s) = diag{e s , e (1+ν)
F (x) = co s≤0 {g(s, x)} , (9)
where coA denotes the closed convex hull of A ⊂ R n . Then, we have

sup h∈F (x) ∇U (x), h ≤ 0, x 2 = 0.
If the corresponding number ν is negative and

0 < lim inf s→-∞ e -(1+2ν)s x 2 k 2 (e (1+ν)s x 2 ) < lim sup s→-∞ e -(1+2ν)s x 2 k 2 (e (1+ν)s x 2 ) < +∞, ∀x 2 = 0,
then, obviously, F is nonempty-, compact-, convex-valued and upper-semi-continuous. Moreover, the sub-homogeneous extension (2),( 9) is strongly GAS due to LaSalle invariance principle (see Theorem 14 [START_REF] Desch | Meagre functions and asymptotic behaviour of dynamical systems[END_REF] for an extension of the LaSalle invariance principle to DI). Hence, from Corollary (3.6) we conclude that ( 8) is globally FTS. For example, if

k 1 (x 1 ) = -x 1 3 1 , k 2 (x 2 ) = -x 2 |x 2 | (1 + |x 2 |(1 + cos(1/x 2 )) + 0.5 sin(1/x 2 )) ,
the system (8) does not have an asymptotically stable d-homogeneous approximation at 0, but the system is globally FTS since the conditions given above are fulfilled for ν = -1/3.

Similarly if ν > 0 and

0 < lim inf s→+∞ e -(1+2ν)s x 2 k 2 (e (1+ν)s x 2 ) < lim sup s→+∞ e -(1+2ν)s x 2 k 2 (e (1+ν)s x 2 ) < +∞, ∀x 2 = 0,
then the sup-homogeneous extension

F (x) = co s≥0 {g(s, x)} (10)
is nonempty-, compact-, convex-valued and upper-semi-continuous. Moreover, (2),( 10) is GAS, which implies the globally nearly FxTS of the origin for the system (8). For example, if k 1 (x 1 ) = -x 3 1 , k 2 (x 2 ) = -x 2 |x 2 |(1 + 0.5 sin(x 2 )).

Notice also that a strict Lyapunov function V (x) = U γ (x) + εx 1 x 2 with ε, γ > 0 can be utilized for FTS (nearly FxTS) analysis of [START_REF] Braidiz | On finite-time stability of subhomogeneous differential inclusions[END_REF]. However, in this case, to prove FTS we need to show V (x) ≤ -cV 1-α (x) with α ∈ (0, 1), c > 0 at least close to the origin (to infinity with α > 1, respectively).

The corresponding derivations are much more complicated (see e.g. [START_REF] Polyakov | Robust finite-time stabilization and observation of a planar system revisited[END_REF]) than the given above sup/subhomogeneous extension-based analysis.

Conclusion and Discussion

The notions of sup-and sub-homogeneity are introduced. It is shown that GAS of sub/suphomogeneous DIs satisfying the standard assumptions are characterized by a homogeneous Lyapunov function. The strong and weak FTS/FxTS of sub/sup-homogeneous GAS DIs are investigated. Sub/suphomogeneous extensions for non-homogeneous systems are constructed.

The suggested example illustrates how the introduced notions allow us to use the conventional homogeneity-based FTS analysis for a larger class of systems. The second order system considered in the example was shown to be GAS by using the classical non-strict Lyapunov function and the LaSalle invariance principle. However, this Lyapunov function cannot be used for FTS/FxTS analysis of the system. The homogeneity-based tools can be utilized in this case (see, e.g., [START_REF] Andrieu | Homogeneous approximation, recursive observer and output feedback[END_REF]) provided that the system admits GAS homogeneous approximations. We showed that the latter assumption can be relaxed to the existence of GAS sup-or sub-homogeneous extensions of the considered system. In the studied example, the GAS of the corresponding extensions has been easily proven using the same non-strict Lyapunov function. Hence, we deduced the FTS and nearly FxTS only regarding the signs of the homogeneity degrees of the extensions. The presented example showed the usefulness of the proposed notions for simplification of FTS/FxTs analysis of some non-homogeneous systems, which do not have homogeneous approximations at zero or at infinity. Further development of this concept seems to be a promising direction of the research for finite and infinite-dimensional systems. The sup/subhomogeneity-based robustness (input-to-state stability analysis) can be mentioned as an interesting problem for future investigation.

  d and the properties mentioned in Assumption 3.1 are fulfilled by construction and Ŝβ d ⊂ S due to Proposition 3.1. Let us prove that Ŝβ d is a set of weakly GAS solutions with β d ∈ KL.

Theorem 3 . 2 .

 32 If the set Ŝβ ⊂ S of weakly GAS solutions satisfies Assumption 3.1 for ν < 0 then Ŝβ is a set of weakly globally FTS solutions. Proof of Theorem 3.2: Let Ŝβ be the set of solutions of (2) satisfying Assumption 3.1. 140 Let R > 0 be an arbitrary real number and let us define a time τ R for weakly GAS solutions from Ŝβ which start from the set B d (2R) and converge into the set B d (R):

Proposition 3 . 2 . 2 : 3 . 2 .

 32232 Let F be d-sup-homogeneous of degree ν ∈ R. If x(•) is a solution of (2), then for any s ≤ 0 the function t → d(s)x(e νs t) is a solution of (2).Proof of Proposition 3.Proof of proposition uses the sup-homogeneity property and follows the same steps as the proof of Proposition 3.1.Let us introduce the assumptionAssumption The property in Assumption 3.1 is fulfilled for s ≤ 0 .Similarly to Corollary 3.1, it can be shown that Assumption 3.2 is fulfilled if there exists a set Ŝβ ⊂ S of weakly stable solutions and

Theorem 3 . 4 .

 34 If Assumption 3.2 is fulfilled for ν > 0 then (2) is weakly globally nearly FxTS. Proof of Theorem 3.4: Let Ŝβ be the set of solutions of satisfying Assumption 3.2. For a given R > 0 and let us define τ R = sup x∈ Ŝβ (B d (2R))

  s }, s ∈ R, and ν > -1 we derive g(s, x) := e -νs d(-s)f (d(s)x) = x2 -e -(1+2ν)s (k1(e s x1)+k2(e (1+ν)s x2)) .

A solution of (2) said to be forward complete if it is defined on [0, +∞) and it does not blow up in a finite time.

A matrix G ∈ R n×n is anti-Hurwitz if -G is Hurwitz.

This work was partially supported by ANR 15 CE23 0007 (Project Finite4SoS), by the Government of Russian Federation (Grant 08-08) and the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898.