Andrey Polyakov

Sliding Mode Control Design Using Canonical Homogeneous Norm

The problem of sliding mode control design for nonlinear plant is studied. Necessary and sufficient conditions of quadratic-like stability (stabilizability) for nonlinear homogeneous (control) system are obtained. Sufficient conditions of robust stability/stabilizability are deduced. The results are supported with academic examples of sliding mode control design.

Introduction

Symmetry is one of well-known properties of physical systems [START_REF] Gross | The role of symmetry in fundamental physics[END_REF]. A certain form of symmetry studied in systems and control theory is known as homogeneity [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Hermes | Nilpotent approximations of control systems and distributions[END_REF], [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF], [START_REF] Rosier | Etude de quelques problemes de stabilization[END_REF], [START_REF] Sepulchre | Homogeneous Lyapunov Functions and Necessary Conditions for Stabilization[END_REF]. The standard homogeneity introduced by L. Euler in 17th century is the symmetry of a mathematical object f (e.g. function, vector field, operator, etc) with respect to the uniform dilation of the argument x → λx, namely, f (λx) = λf (x), λ > 0. Type of homogeneity is basically identified by the corresponding dilation. For example, in [START_REF] Zubov | Methods of A.M. Lyapunov and Their Applications[END_REF], [START_REF] Hahn | Stability of Motion[END_REF], [START_REF] Andreini | Global stabilizability of homogenenous vector fields of odd degree[END_REF] the uniform dilation is utilized and the papers [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Coron | Adding an integrator for the stabilization problem[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] deal with the so-called weighted dilation. Nonlinear homogeneous differential equations/inclusions form an important class of control systems [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF], [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF], [START_REF] Perruquetti | Finite-time observers: application to secure communication[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF], [START_REF] Efimov | Oscillations conditions in homogenous systems[END_REF], [START_REF] Polyakov | Robust stabilization of mimo systems in finite/fixed time[END_REF]. They appear as local approximations [START_REF] Hermes | Nilpotent approximations of control systems and distributions[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF] or set-valued extensions [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] of nonlinear systems and include models of process control [START_REF] Zimenko | A note on delay robustness for homogeneous systems with negative degree[END_REF], nonholonomic systems [START_REF] Pomet | Time-varying exponential stabilization of nonholonomic systems in power form[END_REF], mechanical models with frictions [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF], etc.

The generalized homogeneity (to be studied in this paper) was introduced originally in [START_REF] Polyakov | On homogeneous distributed parameters equations[END_REF] for infinite dimensional models such as partial differential equations (PDEs). It considers a strongly continuous group of linear bounded operators generated by a possibly unbounded linear operator as a dilation in a Banach space. A lot of well-known PDEs are homogeneous in generalized sense, e.g. heat, wave, Navier-Stocks, Saint-Venant, Korteweg-de Vries, fast diffusion equations. This paper deals with the finite-dimensional models of generalized homogeneous systems represented by ordinary differential equations (ODEs). It 1 is worth stressing that all standard and weighted homogeneous systems are particular cases of the generalized homogeneous ones. Geometric homogeneity [START_REF] Khomenuk | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Rosier | Etude de quelques problemes de stabilization[END_REF], [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF] is a more general type of symmetry allowing dilations to be nonlinear.

Stabilization of nonlinear plant is a classical control problem that is hard to solve constructively in general case, so particular solutions are still of the interest [START_REF] Karafyllis | Stability and Stabilization of Nonlinear Systems[END_REF], [START_REF] Poznyak | Attractive Ellipsoids in Robust Control[END_REF]. Stability and stabilizability problems were studied for both standard [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Andreini | Global stabilizability of homogenenous vector fields of odd degree[END_REF] and weighted homogeneous [START_REF] Coron | Adding an integrator for the stabilization problem[END_REF], [START_REF] Hermes | Homogeneous feedback controls for homogeneous systems[END_REF], [START_REF] Praly | Generalized weighted homogeneity and state dependent time scale for linear controllable systems[END_REF], [START_REF] Sepulchre | Homogeneous Lyapunov Functions and Necessary Conditions for Stabilization[END_REF], [START_REF] Sepulchre | Stabilizability does not imply homogeneous stabilizability for controllable homogeneous systems[END_REF], [START_REF] Grune | Homogeneous state feedback stabilization of homogeneous systems[END_REF], [START_REF] Nakamura | Homogeneous stabilization for input-affine homogeneous systems[END_REF] systems which are the most popular today [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], [START_REF] Perruquetti | Finite-time observers: application to secure communication[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF], [START_REF] Polyakov | Robust stabilization of mimo systems in finite/fixed time[END_REF].

In the case of negative homogeneity degree the stability/stabilizability immediately implies finite-time stability/stabilizability [START_REF] Zubov | Methods of A.M. Lyapunov and Their Applications[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], [START_REF] Moulay | Finite-time stability of differential inclusions[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], that is usually required for sliding mode control [START_REF] Utkin | Sliding Modes in Control Optimization[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]. One of goals of this paper is to develop a universal sliding mode-like control design algorithm for an essentially nonlinear generalized homogeneous system. This paper deals with quadratic stability (stabilizability) problem for nonlinear generalized homogeneous (control) systems. Quadratic stabilizability is useful property for control design, since the control tuning procedures in this case can be formalized as Linear Matrix Inequalities(LMIs) [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] and/or semidefinite programming problems [START_REF] Nesterov | Interior Point Polynomial Algorithms in Convex Programming[END_REF], which are supported by MATLAB and other software.

First of all, we show that any generalized homogeneous system is diffeomorphic to a standard homogeneous one. Next, a change of coordinate, which transforms asymptotically stable generalized homogeneous system to a quadratically stable one is presented. In both cases the so-called canonical homogeneous norm [START_REF] Polyakov | On finite-time stabilization of evolution equations: A homogeneous approach[END_REF] is utilized for construction of the corresponding coordinate transformation. Combination of these two results allows the necessary and sufficient stability condition to be presented in terms of existence of the quadratic-like Lyapunov function V (x) = x Ξ(x)P Ξ(x)x, where P = P 0 and the nonsingular matrix Ξ is constant along any ray from the origin (i.e. Ξ(e s x) = Ξ(x) for x = 0, s ∈ R) and ∂Ξ(x) ∂xi x = 0 for x = 0. The properties of matrix Ξ motivate a conjecture about existence of quadratic Lyapunov function (Ξ = const), which fails in the general case. However, some examples show that the obtained stability condition may simplify finite-time stability/stabilizability analysis as well as sliding mode control design in some particular cases. As example, a class of high order sliding mode control allowing quadratic Lyapunov function is presented and an algorithm for selection of its parameters is given as LMIs.

Notation: R is the field of real numbers; R + = {x ∈ R : x > 0}; e i = (0,..., 1,..., 0) is the identity vector in R n ; • denotes a norm in R n and

A A = sup x∈R n Ax x and A A = inf x∈R n Ax x if A ∈ R n×n ;
C n (X, Y ) is the set of continuously differentiable (at least up to the order n) maps X → Y , where X, Y are open subsets of finite dimensional spaces; ∂ ∂x = ∂ ∂x1 , ∂ ∂x2 , ..., ∂ ∂xn ; I n ∈ R n×n is the identity matrix; 0 denotes zero element, e.g. 0 ∈ R n is the zero vector but 0 ∈ R n×n is the zero matrix; diag{λ 1 ,.., λ n } 2 -diagonal matrix with elements λ i ; the order relation P 0 means positive definiteness of the symmetric matrix P ∈ R n×n ; λ max (P ) and λ min (P ) denote maximal and minimal eigenvalues of the symmetric matrix P ∈ R n×n ; (λ) denotes the real part of the complex number λ; the notation P 1 2 means that P

1 2 = M such that P = M 2 .

Problem Statement

It is well known [START_REF] Utkin | Sliding Modes in Control Optimization[END_REF] that sliding mode control design procedure consists of two steps. First, we select a sliding surface σ(q) = 0, where q ∈ R r is the state vector of a plant and σ : R r → R n is assumed to be sufficiently smooth. Next, we have to design a control that guarantees finite-time reaching of the surface σ(q) = 0 and sliding on it.

Let us assume that the plant is modeled by a nonlinear ODE and the sliding surface is selected such that the differential equation describing the evolution of the sliding variable σ(t) = σ(q(t)) admits the representation

σ(t) = g(σ(t), u(t)) + ω(t, q(t), u(t)), (1)
where

t ∈ R + is time variable, σ(t) ∈ R n is the vector of sliding variables at the time instant t, u(t) ∈ R m is the control input, g ∈ C(R n+m , R n ), g(0, 0) = 0
and the function ω : R r+m+1 → R n is assumed to be unknown, but bounded in a certain way:

ω(t, q, u) ≤ ω σ(q) u ,
where ω : R + → R + is a continuous function. More constructive restrictions to ω are given below (see, Section 6). Therefore, sliding mode control design can interpreted as robust finite-time stabilization of the origin of the reduced-order system [START_REF] Gross | The role of symmetry in fundamental physics[END_REF]. In this paper we assume that only the vector of sliding variables σ can be measured and utilized for control purposes. We deal (mainly) with the problem of a dynamical feedback control design

u(t) = k(σ(t), u(t)), k ∈ C(R n+m \{0}, R m ). (2)
Finally, we restrict ourself with the class of the so-called generalized homogeneous nonlinear systems (see, [START_REF] Polyakov | Quadratic-like stabilizability of homogeneous systems[END_REF] and the next section for more details).

The main goal of the paper is to derive a necessary and sufficient condition of finite-time stabilizability (with ω = 0) and a sufficient condition of robust finite-time stabilizability (for ω = 0) of the system (1) by means of a homogeneous dynamical feedback [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF]. In particular, we are interested in a (more or less) constructive stabilizability criteria. Obviously, under assumption

f = g k ∈ C(R n+m \{0}, R n+m ), (3)
the nominal closed-loop system (1), (2) may have discontinuity only at the origin similarly to the unit control [START_REF] Utkin | Sliding Modes in Control Optimization[END_REF] or "quasi-continuous" control [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF]. It is also worth stressing that sliding mode control method frequently uses dynamical feedbacks as a part of the control law (see, e.g. super-twisting algorithm [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF]).

3 Generalized Homogeneity

Dilation Group

Homogeneity is a sort of symmetry of an object (e.g. function or vector field) with respect to a group of transformations. The corresponding group is usually called group of dilations (or simply dilation). The generalized homogeneity [START_REF] Polyakov | On homogeneous distributed parameters equations[END_REF] deals with the groups of linear transformations (linear dilations).

Definition 1 ([20]) A map d : R → R n×n is called dilation in R n if it satisfies • Group property: d(0) = I n and d(t+s) = d(t)d(s) = d(s)d(t), t,s ∈ R; • Continuity property: d is a continuous map, i.e. ∀t > 0, ∀ε > 0, ∃δ > 0 : |s-t| < δ ⇒ d(s)-d(t) A ≤ ε;
• Limit property: lim

s i G i d i! , s ∈ R. (4)
The most popular dilations in R n (see [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[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], [START_REF] Perruquetti | Finite-time observers: application to secure communication[END_REF], [START_REF] Polyakov | Robust stabilization of mimo systems in finite/fixed time[END_REF] )

• uniform (or standard) dilation (L. Euler 17th century) :

d 1 (s) = e s I n , s ∈ R,
• weighted dilation (Zubov 1958, [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF]): Obviously, monotonicity of dilation may depend on a norm • . For example, the dilation d(s) = e s cos(s) sin(s) -sin(s) cos(s)

d 2 (s) =
with

G d = 1 1 -1 1
is monotone on R 2 equipped with the weighted norm x P =

√ x P x if P = 1 1/ √ 2 1/ √ 2 1
> 0 and it is non-monotone if, for example, P =

1 3/4 3/4 1 > 0.
In the latter case, the curve {d(s)x : s ∈ R} may cross the unit sphere S in two different points.

In other words, monotonicity of dilation means that the linear maps d(s) : R n → R n are strong contractions if s < 0. Since [d(s)] -1 = d(-s) then it is expectable that these maps are strong expansions for s > 0.

Theorem 1 The next four conditions are equivalent 1) the dilation d is monotone;

2) d(s) A > 1 for s > 0;

3) the continuous function d(•)x : R → R + is strictly increasing for any fixed x ∈ S; 4) for any x ∈ R n \{0} there exists a unique pair (s 0 , x 0 ) ∈ R × S such that x = d(s 0 )x 0 .

Proof. 1)⇒2) For any x ∈ S we have 1

= x = d(-s)d(s)x ≤ d(-s) A d(s)x . Hence, 1 ≤ inf x∈S d(-s) A d(s)x = d(-s) A inf x∈S d(s)x = d(-s) A d(s) A for any s ∈ R. The derived inequality gives d(s) A ≥ 1 d(-s) A . If the dilation is monotone then d(-s) A < 1 for s > 0. So, we conclude d(s) A > 1 if s > 0.
2)⇒3) By definition of • A one has d(s)y ≥ d(s) A y for any y ∈ R n and any s ∈ R. If s 1 < s 2 then d(s 2 -s 1 ) A > 1 and for x = 0 we derive

d(s 1 )x -d(s 2 )x = d(s 1 )x -d(s 2 -s 1 )d(s 1 )x ≤ (1 -d(s 2 -s 1 ) A ) d(s 1 )x < 0. The implication 3)⇒4) is straightforward since d(0)x = x . 4)⇒1) If x ∈ S then d(s)x /
∈ S for all s = 0. Indeed, otherwise the pair (s 0 , x 0 ) ∈ R × S such that x 0 = d(s)x ∈ S is not unique. Hence, the limit property of the dilation (see, Definition 1) implies d(s)x < 1 for all s < 0 and all x ∈ S.

Theorem 1 also proves that the functions d(•) A : R → R + and d(•) A : R → R + are also continuous and strictly increasing.

Definition 3 ([34])

The dilation d is said to be strictly monotone on R n if ∃β > 0 such that d(s) A ≤ e βs for s ≤ 0.

The dilation d considered in the above example is strictly monotone on R 2 equipped with the conventional Euclidian norm.

Theorem 2 Let d be a dilation in R n then 1) all eigenvalues λ i of the matrix G d are placed in the right complex halfplane, i.e. (λ i ) > 0, i = 1, 2, ..., n;

2) there exists a matrix P ∈ R n×n such that

P G d + G d P 0, P = P 0; (5)
3) the dilation d is strictly monotone with respect to the weighted Euclidean norm • = •, • induced by the inner product x, z = x P z with P satisfying (5):

e αs ≤ d(s) A ≤ d(s) A ≤ e βs if s ≤ 0, e βs ≤ d(s) A ≤ d(s) A ≤ e αs if s ≥ 0, where α = 1 2 λ max P 1 2 G d P -1 2 + P -1 2 G d P 1 2 and β = 1 2 λ min P 1 2 G d P -1 2 + P -1 2 G d P 1 2 . Proof. 1) Since d ds d(s) = G d d(s), d(0) = I then d(s) = e sG d = +∞ i=0 s i G i d i!
is the fundamental matrix of the linear system ODEs with the matrix G d . The limit property of the dilation implies that for all x = 0 one has e sG d x → +∞ as s → +∞. This means that all eigenvalues λ i of the matrix G d are placed in the right complex half-plane, i.e. the matrix -G d is Hurwitz. Indeed, otherwise there exists an eigenvalue λ * : (λ * ) ≤ 0 and there exists a non-trivial vector x λ i * belonging to an eigensubspace of λ * such that e sG d x λ * is bounded for any s ≥ 0. In particular, if λ * is a real eigenvalue of G d and x λ * = 0 is a corresponding eigenvector then G d x λ * = λ * x λ * and

e sG d x λ * = +∞ i=0 s i G i d i! x λ * = +∞ i=0 s i λ i * i! x λ * = e sλ * x λ * .
Hence, if λ * ≤ 0 then e sG d x λ * = e sλ * x λ * ≤ x λ * for all s ≥ 0. This contradicts the limit property of the dilation. Similar constructions can be provided for the case of a complex conjugate pair of eigenvalues.

2) If -G d is a Hurwitz matrix then for any positive definite matrix Q = Q ∈ R n×n the Lyapunov equation P (-G d ) + (-G d )P = -Q has a unique positive definite solution P = P ∈ R n×n (see, e.g. [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]Chapter 1]). Hence, P G d + G d P = Q 0, i.e. the linear matrix inequality ( 5) is always feasible provided that G d is a generator of the dilation.

3) Recall that for any symmetric matrix M = M ∈ R n×n one has

λ min (M )x x ≤ x M x ≤ λ max (M )x x,
but if, in addition, M 0 then N M N 0 for any nonsingular matrix N ∈ R n×n (see, e.g. [START_REF] Poznyak | Deterministic Technique[END_REF]). Finally, taking into account

d ds d(s) = d ds e sG d = G d e sG d = e sG d G d for x ∈ R n we derive d ds d(s)x 2 = d ds x d(s) P d(s)x = x d(s) P G d + G d P d(s)x = x d(s) P 1 2 P 1 2 G d P -1 2 + P -1 2 G d P 1 2 P 1 2 d(s)x ≥ 2β x d(s) P d(s)x = 2β d(s)x 2 .
Similarly we derive d ds d(s)x 2 ≤ 2α d(s)x 2 . Note that since P 0 then

P 1 2 = P 1 2
0 (see, e.g [START_REF] Poznyak | Deterministic Technique[END_REF]), but the matrix inequality (5) implies

P 1 2 G d P -1 2 + P -1 2 G d P 1 2
0 and 0 < β ≤ α. Therefore, the obtained differential inequalities imply

e 2βs x 2 = e 2βs d(0)x 2 ≤ d(s)x 2 ≤ e 2αs d(0)x 2 = e 2αs x 2 if s ≥ 0 and e 2αs x 2 = e 2αs d(0)x 2 ≤ d(s)x 2 ≤ e 2βs d(0)x 2 = e 2βs x 2 if s ≤ 0.
or, equivalently, for x = 0 one has

e βs ≤ d(s)x x ≤ e αs if s ≥ 0 and e αs ≤ d(s)x x ≤ e βs if s ≥ 0
Hence, taking the supremum (resp. infinum) on x = 0 in the latter inequalities we conclude that the inequalities for matrix norms hold.

The latter theorem proves that any dilation d is strictly monotone if R n is equipped with the weighted Euclidian norm x =

√

x P x provided that the matrix P > 0 satisfies (5).

Canonical Homogeneous Norm

The dilation d introduces a sort of norm topology in R n by means of the socalled "homogeneous norm" [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF], [START_REF] Efimov | Oscillations conditions in homogenous systems[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF].

Definition 4 A continuous function p : R n → R + is said to be d-homogeneous norm if p(x) → 0 as x → 0 and p(d(s)x) = e s p(x) > 0 for x ∈ R n \{0} and s ∈ R.
For monotone dilations the canonical homogeneous norm • d : R n → R + is defined as

x d = e sx where s x ∈ R such that d(-s x )x = 1. (6)
In [START_REF] Polyakov | On finite-time stabilization of evolution equations: A homogeneous approach[END_REF] such a homogeneous norm was called canonical since it is induced by the canonical norm

• in R n , x d = 1 if x = 1 and, moreover, • d = • if d(s) = e s I n . Obviously, d(-ln x d )x = 1 and d(ln x d ) A ≤ x = d(ln x d )d(-ln x d )x ≤ d(ln x d ) A . Proposition 1 If d is a strictly monotone dilation then • x 1 β d -x 2 β d ≤ x 1 -x 2 for x i ∈ R n : x i ≥ 1, i = 1, 2
, where β is given in Definition 3;

• the homogeneous norm • d is Lipschitz continuous outside the origin;

• if the norm • is smooth outside the origin then the homogeneous norm

• d is also smooth outside the origin, d d(-s)x ds < 0 if s ∈ R, x ∈ R n \{0} and ∂ x d ∂x = x d ∂ z ∂z | z=d(-s)x ∂ z ∂z | z=d(-s)x G d d(-s)x s=ln x d (7) Proof.
Since for

x i ∈ R n we have x i d = e si : d(-s i )x i = 1 then 1 = d(-s 1 )x 1 = d(-s 1 )(x 1 -x 2 ) + d(s 2 -s 1 )d(-s 2 )x 2 ≤ d(-s 1 ) A (x 1 - x 2 ) + d(s 2 -s 1 ) A . Due to strict monotonicity of the dilation d we have d(-s 1 ) A ≤ e -βs1 and d(s 2 -s 1 ) A ≤ e β(s2-s1) if 0 ≤ s 2 ≤ s 1 . For 1 ≤ x 2 d ≤ x 1 d we have 0 ≤ s 2 < s 1 and, consequently, 1 ≤ e -βs1 x 1 -x 2 + e βs2-βs1 or equivalently, e βs1 -e βs2 = x 1 β d -x 2 β d ≤ x 1 -x 2 .

Generalized Homogeneous Functions and Vectors Fields

Vector fields, which are symmetric (homogeneous) in a certain sense with respect to dilation d, have a lot of properties useful for control design and state estimation of both linear and nonlinear plants as well as for analysis of convergence rates [START_REF] Rosier | Etude de quelques problemes de stabilization[END_REF], [START_REF] Praly | Generalized weighted homogeneity and state dependent time scale for linear controllable systems[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], [START_REF] Perruquetti | Finite-time observers: application to secure communication[END_REF].

Definition 5 A vector field f : R n → R n (resp. 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), ∀x ∈ R n \{0}, ∀s ∈ R. ( 8
) (resp. h(d(s)x) = e νs h(x), ∀x ∈ R n \{0}, ∀s ∈ R. ) (9)
Example 1 Let us consider the dilation

d(s) = e s 1 0 0 0 cos(s) sin(s) 0 -sin(s) cos(s)
that is strictly monotone with respect to the Euclidean norm x = √

x T x and

G d = 1 0 0 0 1 1 0 -1 1
. The vector field f : R 3 → R 3 defined as

f (x) = x 2 2 +x 2 3 x 2 1 (cos(ln |x1|)+sin(ln |x1|)) x 2 1 (cos(ln |x1|)-sin(ln |x1|))
satisfies (8) with ν = 1 and the function h : R 3 → R given by h

= x 3 1 +(x 2 2 +x 2 3 ) 3 2
satisfies (9) with ν = 3, respectively.

Example 2 (Linear homogeneous vector field of arbitrary degree) . The vector field may have different degrees of homogeneity dependently of the dilation group. Indeed, the linear vector field f :

R n → R n , f (x) = Ax, x ∈ R n defined by a chain of integrators A = 0 In-1 0 0 ∈ R n×n is d-homogeneous of degree µ ∈ [-1, 1] with d(s) = diag{e (n+(i-1)µ)s } n i=1 . Remark 1 In the general case, if G d ∈ R n×n is a generator of the dilation, i.e. d(s) = e G d s , s ∈ R, then a linear map f : R n → R n , f (x) = Ax, x ∈ R n , A ∈ R n×n is d-homogeneous of degree ν ∈ R if AG d = (νI n + G d )A. (10)
Indeed, in our case the identity (8

) becomes Ae G d s = e (νI+G d )s A. Since e G d s = +∞ i=0 G i d s i
i! then the homogeneity identity, obviously, holds provided that (10) is fulfilled.

Let F d (R n ) (resp. H d (R n )) be the set of vector fields R n → R n (resp. functions R n → R) satisfying the identity (8) (resp. (9)), which are contin- uous on R n \{0}. Let deg F d (f ) (resp. deg H d (h)) denote the homogeneity degree of f ∈ F d (R n ) (resp. h ∈ H d (R n )). Remark 2 Note that if n = 1 then the generator of the dilation G d ∈ R + is a positive scalar and d(s) = e sG d ∈ R + for any s ∈ R. Hence, obviously, F d (R) = H d (R) but deg F d (•) + G d = deg H d (•) if n = 1.
In other words, homogeneous vector fields/function R → R belongs to both sets F d (R) and H d (R), but its degree can be defined differently (i.e. according to (8) or (9)) dependently of the context (e.g. if f defines the right-hand side of ODE then the formula (8) is utilized).

The homogeneity allows local properties (e.g. smoothness) of vector fields (functions) to be extended globally [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Hermes | Nilpotent approximations of control systems and distributions[END_REF]. Below we present two useful corollaries to Definition 5.

Corollary 1 The vector field

f ∈ F d (R n ) is Lipschitz continuous (smooth) on R n \{0}
if and only if it satisfies Lipschitz condition (it is smooth) on the unit sphere S provided that d is strictly monotone on R n equipped with a (smooth on

R n \{0}) norm • .
Proof. The necessity is straightforward since any locally Lipschitz function on a compact set satisfies Lipschitz condition on it. Let us prove sufficiency. Let

x i ∈ R n \{0}, i = 1, 2 then x i = d(ln x i d )z i for some z i ∈ S, f (x 1 ) -f (x 2 ) = f (d(ln x 1 d )z 1 ) -f (d(ln x 2 d )z 2 ) = x 1 ν d d(ln x 1 d )f (z 1 ) -x 2 ν d d(ln x 2 d )f (z 2 ) = x 1 ν d d(ln x 1 d )(f (z 1 ) -f (z 2 )) + ( x 1 ν d d(ln x 1 d ) -x 2 ν d(ln x 1 d ))f (z 2 ) + x 2 ν (d(ln x 1 d ) -d(ln x 2 d ))f (z 2 ). If L > 0 is a Lipschitz constant of f on S then f (x 1 ) -f (x 2 ) ≤ L x 1 ν d d(ln x 1 d ) z 1 -z 2 + d(ln x 1 d )f (z 2 ) ( x 1 ν d -x 2 ν d ) + f (z 2 ) x 2 ν d(ln x 1 d ) -d(ln x 2 d ) A . Since d(s 1 ) -d(s 2 ) = G d s1 s2 d(s)

ds and the function d(•)

A is strictly monotone increasing then d(ln If a function (or a vector field) is smooth then homogeneity is inherited by its derivatives in a certain way.

x 1 d )-d(ln x 2 d )) A ≤ M G d |ln x 1 d -ln x 2 d |, where M = max{ d(ln x 1 d ) A , d(ln x 2 d ) A }.
Corollary 2 If h ∈ H d (R n ) ∩ C 1 (R n \{0}, R) then e deg H d (h)s ∂h(x) ∂x = ∂h(z) ∂z z=d(s)x d(s), (11) ∂h
(x) ∂x G d x = deg H d (h) h(x), (12)
for x ∈ R n \{0} and s ∈ R.

Proof. The formula (11) can be obtained using the definition of the (Frechét) derivative, which coincides with ∂h ∂x if h is smooth. Namely, lim

∆ →0 |h(x+∆)-h(x)-∂h(x)
∂x ∆| ∆ = 0 and lim

∆ →0 h(d(s)x+∆)-h(d(s)x)- ∂h(z) ∂z | z=d(s)x ∆ ∆ = 0, where ∆ ∈ R n . Since h ∈ H d then h(d(s)x+∆)-h(d(s)x)- ∂h(z) ∂z | z=d(s)x ∆ ∆ = e νs h(x+ ∆)-h(x)-e -νs ∂h(z) ∂z | z=d(s)x d(s) ∆ d(s) ∆ ≤ e νs d(s) A × h(x+ ∆)-h(x)-e -νs ∂h(z) ∂z | z=d(s)x d(s) ∆ ∆ , where ν = deg H d (h) and ∆ = d(-s)∆ such that ∆ → 0 implies ∆ → 0.
Therefore the identity (11) holds.

To prove [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF] let us consider the canonical homogeneous norm • d induced by a weighted Euclidean norm x =

√

x P x, x ∈ R n \{0}, P = P > 0 satisfying [START_REF] Rosier | Etude de quelques problemes de stabilization[END_REF]. In this case, due to [START_REF] Zubov | Methods of A.M. Lyapunov and Their Applications[END_REF]

we have ∂ x d ∂x x > 0 if x = 0. From h(x) = x ν d h(d(-ln x d )x) we derive ∂h(x) ∂x = νh(d(-ln x d )x) x ν-1 d ∂ x d ∂x + x ν d ∂h(d(-ln x d )x) ∂x = νh(d(-ln x d )x) x ν-1 d ∂ x d ∂x + x ν d ∂h(z) ∂z z=d(-ln x d )x ∂ ∂x (d(-ln x d )x) = νh(d(-ln x d )x) x ν-1 d ∂ x d ∂x + x ν d ∂h(z) ∂z z=d(-ln x d )x × d(-ln x d ) -G d d(-ln x d )x x d ∂ x d ∂x . Therefore, for x = 1 we have ∂h(x) ∂x G d x ∂ x d ∂x = νh(x) ∂ x d ∂x .
Hence, multiplying by x we obtain that (12) holds for x = 1. Since • new = γ • with γ > 0 is again a weighted Euclidean norm in R n satisfying (5) then the obtained identity holds on R n \{0}.

Quadratic Stability of Homogeneous Systems

Homogeneity may simplify the analysis of differential equations, e.g. to prove existence and uniqueness of solution on R n \{0} it is sufficient to prove that the right hand side is Lipschitz continuous (or differentiable) on a sphere (see, Corollary 2. The most important property of d-homogeneous systems is scalability of the solutions [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF], [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], [START_REF] Moulay | Finite-time stability of differential inclusions[END_REF], [START_REF] Polyakov | On homogeneous distributed parameters equations[END_REF].

Theorem 3 If ϕ ξ0 : [0, T ) → R n is a solution to ξ = f (ξ), f ∈ F d (R n ) ( 13
)
with the initial condition ξ(0) = ξ 0 ∈ R n then ϕ d(s)ξ0 : [0, e -νs T ) → R n defined as ϕ d(s)ξ0 (t) := d(s)ϕ ξ0 (te νs ) with s ∈ R is a solution to (13) with the initial condition ξ(0

) = d(s)ξ 0 , where ν = deg F d (f ).
Proof.

The scheme of the proof is standard (see [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF]) for any type of homogeneity. Since

d dt ϕ ξ0 (t) = f (ϕ ξ0 (t)) then d(s) d dt ϕ ξ0 (t) = d dt d(s)ϕ ξ0 (t) = d(s)f (ϕ ξ0 (t)) = e -νs f (d(s)ϕ ξ0 (t)).
Making the change of time t = e νs t new we complete the proof.

This theorem has a lot of corollaries, which are very useful for qualitative analysis of homogeneous systems. For example, local stability always implies the global one or the existence of strictly invariant (in forward time) compact set implies asymptotic stability [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF], [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF], [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], etc. Moreover, using the classical result [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF] about existence of homogeneous Lyapunov function for asymptotically stable homogeneous system we can prove topological equivalence of such system to a quadratically stable one.

Theorem 4 The next five claims are equivalent

1) The origin of the system (13) is asymptotically stable.

2) There exists a Lyapunov function

V ∈ H d (R n )∩ C ∞ (R n );
3) The origin of the system

ż = z 1+deg F d (f ) (In-G d )z zP z P G d z + I n f z z (14)
is asymptotically stable, where z = √ z P z with P satisfying (5).

4)

For any matrix P ∈ R n×n satisfying (5) there exists

Ψ ∈ F d (R n ) ∩ C ∞ (R n \{0}), deg F d (Ψ) = 0 such that Ψ is diffemorphism on R n \{0}, homeomorphism on R n , Ψ(0) = 0 and ∂(Ψ (ξ)P Ψ(ξ)) ∂ξ f (ξ) < 0 if Ψ (ξ)P Ψ(ξ) = 1. ( 15
)
Moreover,

Ψ d ∈ H d (R n ) ∩ C ∞ (R n \{0}
) is Lyapunov function to the system [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF], where • d is the canonical homogeneous norm induced by ξ = ξ P ξ.

5)

For any matrix P ∈ R n×n satisfying (5) there exists a map

Ξ ∈ C ∞ (R n \{0}, R n×n )
such that det(Ξ(z)) = 0, ∂Ξ(z) ∂zi z = 0, Ξ(e s z) = Ξ(z)

for z = (z 1 , ..., z n ) ∈ R n \{0}, s ∈ R, i = 1, ..., n and z Ξ (z)P Ξ(z) (In-G d )zz P z P G d z +I n f z √ z P z < 0. ( 16
)
Proof.

Without loss of generality we may assume that f is continuous at zero and f (0) = 0, since asymptotic stability of ( 14) is equivalent to asymptotic stability of the system ξ = f (ξ) := ξ

-deg F d (f ) d f (ξ)
, which is always continuous at the origin.

1) ⇔ 2) We use the scheme developed in [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF]. The Converse Lyapunov Theorem (see, e.g. [START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF]) implies that there exists a smooth Lyapunov function V : R n → R + . Let the smooth function a : R → R + be defined as a(ρ) = e

1 1-ρ if ρ > 0 and a(ρ) = 0 if ρ ≤ 1. Obviously, a (ρ) > 0 if ρ > 1. Then the function V h : R n → R + defined as V h (ξ) = +∞ -∞ e -s a(V (d(s)ξ))

ds is d-homogeneous

Lyapunov function to the system [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF]. Indeed, it is well-defined (due to cut-off function a), smooth, positive definite and radially unbounded. Finally,

V h is d-homogeneous V h (d(q)ξ) = +∞ -∞ e -s a(V (d(-s + q)ξ))ds = e q V h (ξ) and Vh (ξ) = +∞ -∞ e -s a (V (d(-s)ξ)) ∂V (z) ∂z z=d(s)ξ d(s)f (ξ)ds < 0 since f ∈ F(R n ) : ∂V (z) ∂z z=d(s)ξ d(s)f (ξ) = 1 e νs ∂V (z) ∂z f (z) z=d(s)ξ < 0.
1) ⇔ 3) Since P satisfies (5) then the dilation d is strictly monotone on R n equipped with the norm ξ = ξ T P ξ. The change of coordinates [START_REF] Perruquetti | Finite-time observers: application to secure communication[END_REF].

z = ξ d d(-ln ξ d )ξ gives z = ξ d and ξ = d(ln z ) z z , ż = (I n -G d )d(-ln ξ d )ξ d ξ d dt + ξ d d(-ln ξ d )f (ξ) = ξ d (I n -G d )d(-ln ξ d )ξ ξ d (-ln ξ d )P d(-ln ξ d )f (ξ) ξ d (-ln ξ d )P G d d(-ln ξ d )ξ + + ξ d d(-ln ξ d )f (ξ) = ξ d (I-G d )d(-ln ξ d )ξξ d (-ln ξ d )P ξ d (-ln ξ d )P G d d(-ln ξ d )ξ +I n d(-ln ξ d )f (ξ). Taking into account f ∈ F d (R n ) we derive
4) ⇒ 2) Since Ψ(d(s)ξ) = d(s)Ψ(ξ) then Ψ(0) = 0 due to continuity at zero. Note also that Ψ(ξ) = 0 for ξ = 0, otherwise (i.e. ∃ξ * = 0 : Ψ(ξ * ) = 0), due to homogeneity we derive that Ψ(ξ) = 0 on a smooth curve {d(s)ξ * , s ∈ R}, which starts at the origin goes to ∞. The latter contradicts the assumption that Ψ is diffeomorphism (continuously differentiable invertible map with continuously differentiable inverse) on R n \{0}. Since deg

Ψ(ξ) =1 < 0 implies ∂ Ψ(ξ) d ∂ξ f (ξ) Ψ(ξ) =1 < 0. Applying homogeneity we derive ∂ Ψ(ξ) d ∂ξ f (ξ) < 0 if ξ ∈ R n \{0}, i.
e. Ψ d is a Lyapunov function to the system (13).

2) ⇒ 4) Since the origin of the system (13) is asymptotically stable then there exists a smooth d-homogeneous Lyapunov function Ṽ : R n → R + of degree µ > 0. The function V = Ṽ 1/µ is also Lyapunov function to [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF] that is d-homogeneous of degree 1, continuous at the origin and smooth outside the origin. Let us consider the map Ψ : R n → R n defined as [START_REF] Zubov | Methods of A.M. Lyapunov and Their Applications[END_REF] we derive [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF]. The map Ψ is bijection. The inverse map Ψ -1 : R n → R n is given by

Ψ(ξ) = d ln V (ξ) ξ d ξ. Since V (ξ) ξ d ∈ H d (R n ), deg H d V (ξ) ξ d = 0 then lim inf ξ →0 V (ξ) ξ d = inf ξ∈S V (ξ) ξ d > 0, so Ψ is continuous at 0 and Ψ(0) = 0. Obviously, Ψ(d(s)ξ) = d(s)Ψ(ξ) and Ψ(ξ) d = V (ξ). Using
Ψ -1 (x) = d -ln V (x) x d x,
where x ∈ R n . Indeed, Ψ -1 (Ψ(ξ)) = ξ and Ψ(Ψ -1 (x)) = x for all ξ, x ∈ R n . Since Ψ and Ψ -1 are continuous at the origin and smooth outside the origin then Ψ is diffeomorphism on R n \{0} and homeomorphism on R n .

3) ⇒ 5) The system ( 14) is homogeneous of degree deg F d (f ) with respect to the dilation d 0 (s) = e s with G d0 = I n . Note that P 0 G d0 + G d0 P 0 = 2P 0 0 holds for an arbitrary symmetric positive definite matrix P 0 . Taking into account z d0 = z = z P 0 z we use the claim 4) to obtain the homogeneous Lyapunov function V defined as V (z) = Ψ 0 (z) 2 d0 = Ψ 0 (z)P 0 Ψ 0 (z) for z ∈ R n . Since Ψ 0 is diffeomorphism then det ∂Ψ0(z) ∂z = 0 for z ∈ R n \{0}. Using [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF] we derive ∂Ψ0(z) ∂z z = Ψ 0 (z), ∂V ∂z = 2z Ξ (z)P 0 Ξ(z) and ∂Ξ(z) ∂zi z = 0, where Ξ(z) = ∂Ψ0(z) ∂z . Finally, Ξ(e s z) = Ξ(z) for z ∈ R n \{0} and s ∈ R due to [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF] and deg

F d 0 (Ψ 0 ) = 0. 5) ⇒ 3) Let us consider the function V (z) = z Ξ (z)P Ξ(z)z, V ∈ C(R n , R)∩ C ∞ (R n \{0}, R). Since ∂Ξ(z)
∂zi z = 0 then ∂V (z) ∂z = 2z Ξ (z)P Ξ(z) and the condition ( 16) implies that V is the Lyapunov function to [START_REF] Perruquetti | Finite-time observers: application to secure communication[END_REF]. This theorem proves three important facts:

• Any generalized homogeneous system is diffeomorphic to a standard homogeneous one.

• Any asymptotically stable generalized homogeneous system is diffeomorphic to a quadratically stable one. Indeed, making the change of variables z = Ψ(ξ) we derive ż = f (z), where f (z) =

∂Ψ(ξ) ∂ξ f (ξ) ξ=Ψ -1 (z)
, but the criterion [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF] implies that z P ż < 0 if z P z = 1, so the homogeneous norm • d is the Lyapunov function to the latter system. Finally, the change of variable x = z d d(ln z d )z gives z d = x , so the transformed system ẋ = f (x) is quadratically stable.

• The formula [START_REF] Efimov | Oscillations conditions in homogenous systems[END_REF] gives a more or less constructive stability criterion.

Since Ξ(z) = Ξ(e s z) for z ∈ R n \{0}, s ∈ R then the map Ξ is constant along any straight line {e s z : s ∈ R} if z = 0. In addition, the property ∂Ξ(z) ∂zi z = 0 motivates the conjecture that a quadratic Lyapunov function (Ξ ≡ const) always exists for asymptotically stable standard homogeneous system. Such assumption was also studied in [START_REF] Andreini | Global stabilizability of homogenenous vector fields of odd degree[END_REF]. However, in the view of [START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF]Proposition 5.2] this conjecture seems to be wrong. Therefore, the condition

z Q 0 (In-G d )zz P z P G d z +I n f z √ z P z < 0 if z = 0, P G d + G d P > 0, P > 0, Q 0 > 0 (17)
derived from ( 16) is just sufficient stability condition. For Q 0 = P (or, equivalently, Ξ = I n ) we derive the more simplified stability con-

dition z T P f z √ z P z < 0 .
Recall [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], if the standard (or weighted) homogeneous system ( 13) is asymptotically stable and deg F d (f ) < 0 then it is globally uniformly finite-time stable, i.e there exists a locally bounded function T : R n → R such that ϕ x0 (t) = 0, ∀t > T (x 0 ) and ∀x 0 ∈ R n . Theorem 4 together with the next proposition give the necessary and sufficient condition of the uniform finite-time stability of the system (13).

Proposition 2 An asymptotically stable d-homogeneous system (13) is uniformly finite-time stable if and only if deg

F d (f ) < 0.

Proof.

Recall that in the case of homogeneous system local stability is equivalent to global one. Sufficiency is well known (see, e.g. [START_REF] Zubov | Methods of A.M. Lyapunov and Their Applications[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], [START_REF] Polyakov | On homogeneous distributed parameters equations[END_REF]). Necessity. Suppose the contrary, i.e. deg F d (f ) ≥ 0 and there exists a locally bounded function T : R n \{0} → R + such that ϕ ξ0 (t) = 0 for all t ≥ T (ξ 0 ). Uniform finite-time stability of ( 13) implies (see, [START_REF] Moulay | Finite-time stability of differential inclusions[END_REF]) existence of a Lyapunov function such that T (ξ 0 ) ≤ CV ν (ξ 0 ) where ν > 0 and C > 0 are some constants. Hence, T is continuous at 0 and T (0) = 0. On the other hand Theorem 3 implies that T (d(s

)ξ 0 ) = e -deg F d (f )s T (ξ 0 ) for any ξ 0 ∈ R n . If deg F d (f ) > 0 then
the latter identity immediately implies contradiction to the local boundedness of the function T . If deg F d (f ) = 0 then continuity of T at zero and T (0) = 0 imply the identity T (ξ 0 ) = 0, ∀ξ 0 ∈ R n , that is also impossible.

Example: Quadratically Stable High Order Sliding Mode Algorithm

In this section we demonstrate how the obtained results can be utilized for the so-called high order sliding mode control [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF] design. The canonical homogeneous norm is utilized below as an implicit Lyapunov function [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit lyapunov function approach[END_REF].

Let us consider the system

σ = Aσ + Bu(σ), A =       0 1 0 ... 0 0 0 1 ... 0 ... ... ... ... 0 0 0 ... 1 0 0 0 ... 0       , B =       0 0 ... 0 1       , (18)
u(σ) = n i=1 k i σ i N (σ) n-i+1 , σ = (σ 1 , σ 2 , ..., σ n ) ∈ R n , (19)
and the weighted dilation

d(s) = e G d s with G d =       n 0 ... 0 0 0 n -1 ... 0 0 ... ... ... ... 0 0 ... 2 0 0 0 ... 0 1      
, where N : R n → R is a continuous positive definite function that is d-homogeneous of degree 1, i.e. N (σ) > 0 for σ = 0 and

N (d(s)σ) = e s N (σ), σ ∈ R n , s ∈ R.
In view of Definition 4 the function N is simply an arbitrary homogeneous norm.

Obviously, the control law can be rewritten in the form

u(σ) = Kd (-ln N (σ)) σ, where K = k 1 k 2 ... k n-1 k n ∈ R n is a vector of control parameters. The considered feedback control function u : R n → R is d-homogeneous of degree 0, i.e. u ∈ H d (R n ) and deg H (u) = 0. Indeed, u(d(s)σ) = Kd(-ln N (d(s)σ))d(s)σ = Kd(-ln(e s N (σ)))d(s)σ = Kd(-s -ln N (σ))d(s)σ = Kd(-ln N (σ))d(-s)d(s)σ = u(σ), σ ∈ R n \{0}, s ∈ R.
Therefore, u is a quasi-continuous homogeneous sliding mode control [START_REF] Levant | Quasi-continuous high-order sliding-mode controllers[END_REF]. Indeed, it is homogeneous of degree 0, continuous outside the origin and globally bounded:

Bu(σ) ≤ BK A d(-ln N (σ))σ ≤ BK A d -ln N (σ) σ d A d(-ln σ d )σ = BK A d -ln N (σ) σ d A ≤ C < +∞, σ ∈ R n ,
since the function χ : R n → R, χ(σ) = N (σ) σ d is homogeneous of degree 0, and, consequently, globally bounded. Similarly, for f (σ) = Aσ + Bu(σ) we derive

f (d(s)σ) = Ad(s)σ + Bu(d(s)) = e -s d(s)Aσ + Bu(σ) = e -s d(s)(Aσ + Bu(σ)) = e -s d(s)f (σ), σ ∈ R n \{0}, s ∈ R,
i.e. the closed loop system has negative homogeneity degree: deg

F d (f ) = -1.
Let us find some sufficient conditions (to the vector K and to the function N ) allowing the finite-time stability of the closed-loop system. Selecting Q 0 = P in [START_REF] Polyakov | Robust stabilization of mimo systems in finite/fixed time[END_REF] we derive the following sufficient stability condition

z P f (z) < 0, f (z) = 1 z Az + BKd(-ln Ñ (z))z , for z ∈ R n \{0}, (20)
where Ñ (z) = N z z , z = √ z P z with P satisfying (5). Since N is a continuous and positive definite function then 0 < min

ξ∈R n :ξ P ξ=1 N (ξ) ≤ Ñ (z) ≤ max ξ∈R n :ξ P ξ=1 N (ξ) < +∞
and there exist numbers δ min and δ max such that

0 < δ min ≤ Ñ (z) ≤ δ max . (21)
Obviously, if N (σ) = σ d is the canonical homogeneous norm then Ñ (z) = 1, z ∈ R n \{0} and the stability condition becomes z T P (A + BK)z < 0, z ∈ R n \{0}, i.e. K can be selected as a gain matrix of a linear stabilizing feedback. Since the pair {A, B} is controllable then appropriate K and P 0 can always be found (see, e.g. [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]) to guarantee the required inequality, while the parameter γ can be selected sufficiently large to fulfill [START_REF] Rosier | Etude de quelques problemes de stabilization[END_REF]. Moreover, Theorem 4 implies that • d is the Lyapunov function to the closed-loop system.

Therefore, the usage of the canonical homogeneous norm allows the very simple rule to be applied in order to select the parameters of control. However,

• d is defined implicitly by means of the equation ( 6), and for simplicity of practical realization it is worth to know if • d can be replaced with some explicitly defined homogeneous norm N (see also [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit lyapunov function approach[END_REF], [START_REF] Zimenko | Stabilization of chain of integrators with arbitrary in finite-time[END_REF]).

Proposition 3 Let for some 0 < δ min ≤ δ max the system of LMIs

       AX +XA +BY +Y B -X ln δmax δmin BY Xd ln δmax δmin ln δmax δmin Y T B T -(G -1 d X +XG -1 d ) 0 d ln δmax δmin X 0 -X        ≺ 0, X 0, ( 22
) be feasible with respect to X ∈ R n×n , Y ∈ R 1×n . If a function N : R n \{0} → R is continuous, positive definite, N ∈ H d (R n ), deg H d (N ) = 1
and it satisfies the inequalities [START_REF] Khomenuk | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF] with P = X -1 and 0 < δ min ≤ δ max then the closed-loop system [START_REF] Zimenko | A note on delay robustness for homogeneous systems with negative degree[END_REF], [START_REF] Pomet | Time-varying exponential stabilization of nonholonomic systems in power form[END_REF] with

K = Y X -1 d(ln δ max )
is globally uniformly finite-time stable and the canonical homogeneous norm • d is its Lyapunov function.

Proof. First of all, the LMI [START_REF] Karafyllis | Stability and Stabilization of Nonlinear Systems[END_REF] implies that G -1 d X + XG -1 d 0 or, equivalently, P G -1 d + G -1 d P 0 and G d P + P G d 0. Obviously, if

z T P (A + BKd(-ln δ)) z < 0, z ∈ R n \{0}, ∀δ ∈ [δ min , 1].
then the stability condition (20) is fulfilled. For an arbitrary fixed z ∈ R n \{0} let us consider the smooth function q : [δ min , 1] → R defined as q(δ) = z T P (A + BKd(-ln δ)) z.

To estimate its derivative q (δ) we use the classical inequality 2y v ≤ y y + v v, that, obviously, holds for all y, v ∈ R n and, in particular, for 4)) then we have

y = (G -1 d P + P G -1 d ) -1 2 d(-ln δ max )K B z and v = (G -1 d P + P G -1 d ) 1 2 G d d -ln δ δmax z. Since d ds d(s) = G d d(s) = d(s)G d for s ∈ R (see (
-q (δ) = 1 δ z T P BKG d d(-ln δ)z = 1 δ y v ≤ z P BKd(-ln δmax)(G -1 d P +P G -1 d ) -1 d(-ln δmax)K B P z+z d -ln δ δmax (P G d +G d P )d -ln δ δmax z 2δ = 1 δ z T P BKd(-ln δmax)(G -1 d P +P G -1 d ) -1 d(-ln δmax)K B P z-d dδ z d -ln δ δmax P d -ln δ δmax z 2 .
Finally, we derive q(δ) = q(δ max ) - [START_REF] Polyakov | On homogeneous distributed parameters equations[END_REF]. Using Schur Complement it can be easily shown that the latter matrix inequality is equivalent [START_REF] Karafyllis | Stability and Stabilization of Nonlinear Systems[END_REF] with X = P -1 and Y = Kd(-ln δ max )P -1 .

The system of LMIs ( 22) is always feasible provided that δ min and δ max are sufficiently close to each other. Indeed, ln(δ max /δ min ) → 0 and d(ln(δ max /δ min )) → I n as δ max /δ min → 1, so the LMI [START_REF] Karafyllis | Stability and Stabilization of Nonlinear Systems[END_REF] becomes feasible.

A possible explicit selection of N is given by

N (σ) = φ (σ)Qφ(σ) 1 2n
with φ(σ) = |σ1| r 1 sign(σ1)

...

|σn| rn sign(σn) , r i = n n-i+1 , i = 1, 2, ..., n, (23)
where Q ∈ R n×n is a positive definite matrix.

To illustrate the proposition 3 we consider the system ( 18), ( 19), [START_REF] Poznyak | Attractive Ellipsoids in Robust Control[END_REF] for n = 2. Selecting δ min = 1, δ max = 1.4 we solve the LMI [START_REF] Karafyllis | Stability and Stabilization of Nonlinear Systems[END_REF]:

P =
1.1214 0.4247 0.4247 0.2327 and K = -10.8695 -6.2376 .

One can be checked numerically (e.g. using sufficiently dense grid on the unit sphere {z ∈ R n : z P z = 1}) that the homogeneous norm [START_REF] Poznyak | Attractive Ellipsoids in Robust Control[END_REF] with [START_REF] Zimenko | A note on delay robustness for homogeneous systems with negative degree[END_REF] with the static homogeneous feedback [START_REF] Pomet | Time-varying exponential stabilization of nonholonomic systems in power form[END_REF], [START_REF] Poznyak | Attractive Ellipsoids in Robust Control[END_REF]. satisfies 1 ≤ N (z) ≤ 1.4 for z P z = 1, i.e. the condition [START_REF] Khomenuk | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF] holds. The results of numerical simulation for the considered controller are depicted at Figures 2 and3. The system has been discretized by means of the explicit Euler method with the step size 0.002.

Q = 1.
Taking into account Remark 1 the similar control design scheme can be developed for an arbitrary matrix A provided that the pair {A, B} is controllable.

Stabilization of Homogeneous Control Systems via Dynamical Feedback in the Disturbance-Free Case

In this section we study the problem of stabilizability of the homogeneous system (1) in the disturbance free case (ω = 0), i.e. σ(t) = g(σ(t), u(t)), by means of the homogeneous dynamical feedback (2):

u(t) = k(σ(t), u(t)).
Based on the scheme of universal stabilizing control design given in [START_REF] Sontag | A 'universal' construction of artstein's theorem on nonlinear stabilization[END_REF] we derive the following corollary of Theorem 4. The origin of the system (1) with ω = 0 is globally asymptotically stabilizable by means of the homogeneous dynamical feedback (2) with

g k ∈ F d (R n+m )
if and only if there exist a number γ ≥ 0, a symmetric matrix P ∈ R (n+m)×(n+m) satisfying (5) and a map

Ξ ∈ C ∞ (R n+m \{0}, R (n+m)×(n+m) ): det(Ξ(z)) = 0, ∂Ξ(z) ∂zi z = 0, Ξ(e s z) = Ξ(z) for z = (z 1 , .., z n+m ) ∈ R n+m \{0}, s ∈ R, i = 1, ..., n + m such that a(z) < γ b (z)b(z) for z ∈ S, (24)
where

a(z) = z W (z) f (z), b (z) = z W (z) 0 Im , W (z) = Ξ (z)P Ξ(z) (In+m-G d )z zP z P G d z +I n+m
and S is the unit sphere in R n+m with z = √ z P z. Moreover, the corresponding stabilizing homogeneous feedback law can be designed as follows

k(ξ) = ξ deg F d ( f ) d d u (ln ξ d )k 0 (d(-ln ξ d )ξ) with ξ = σ u , (25)
where

k 0 (•) = -a(•)+ √ a 2 (•)+(b (•)b(•)) 2 b (•)b(•) b(•) if b(•) =0 0 if b(•)=0 (26)
and • d is the canonical homogeneous norm induced by ξ = ξ P ξ.

Proof. Sufficiency. Let us show that the function k 0 is continuous on S. In-

deed, k 0 (z) = a(z)+ √ a 2 (z)+(b (z)b(z)) 2 √ b (z)b(z) b(z) √ b (z)b(z)
, the first fraction is continuous and equals zero if b(z) = 0 (see, [START_REF] Sontag | A 'universal' construction of artstein's theorem on nonlinear stabilization[END_REF]), but the norm of the second fraction is globally bounded and continuous for z ∈ R n : b(z) = 0. Let us consider the closedloop system (1), ( 2), ( 25): ξ = f (ξ), where f = ( g k ) ∈ F d (R n ). Under this notation, the stability condition ( 16) becomes a(z)+b (z)k 0 (z) < 0. The latter hods for all z ∈ S. Indeed, the inequality [START_REF] Hermes | Homogeneous feedback controls for homogeneous systems[END_REF] implies that a

(z) < 0 if b(z) = 0, but for b(z) = 0 we have a(z) + b (z)k 0 (z) = a(z) -b (z) a(z)+ √ a 2 (z)+(b (z)b(z)) 2 b (z)b(z) b(z) = -a 2 (z) + (b (z)b(z)) 2 < 0.
Necessity. Let us assume that there exists a map k ∈ C(R n+m \{0}, R m ) such that the closed-loop system (1), ( 2) is globally asymptotically stable and f

= g k ∈ F d (R n+m ). Let Ξ ∈ C ∞ (R n+m \{0}, R (
n+m)×(n+m) ) be derived according to Theorem 4 such that (16) holds. To show that in this case (24) also holds for some γ ≥ 0, we rewrite the inequality (16) as a(z) < -b (z) k(z).

Since k ∈ C(R n+m \{0}, R m ) then γ = max z∈S k (z) k(z) < +∞ and using Cauchy-Schwarz inequality we derive

-b (z) k(z) ≤ k (z) k(z) b (z)b(z) ≤ γ b (z)b(z).
For Ξ(z) = I n the inequality [START_REF] Hermes | Homogeneous feedback controls for homogeneous systems[END_REF] gives the following sufficient condition of quadratic stabilizability:

z P f z √ z P z ≤ γ z P 0 0 0 Im P z, z = 0 P G d + G d P > 0, P > 0. ( 27
)
Taking into account Proposition 2 we derive criterion of finite-time stabilizability of the system (1) by means of the feedback (2) in the case (ω = 0).

Example 3 (Quadratic Dynamical Stabilization in a Finite Time) Let us consider the stabilization problem for the system

σ1 = |σ 1 | 1 3 u + σ 2 , σ2 = u, (28)
where σ 1 , σ 2 , u ∈ R. The vector field f : R 3 → R 3 defined by

f (ξ) =   |ξ 1 | 1 3 ξ 3 + ξ 2 ξ 3 0   , ξ = (ξ 1 , ξ 2 , ξ 3 ) = (σ 1 , σ 2 , u) is d-homogeneous of
A(z) =    0 1 |z1| √ z P z 1 3 0 0 1 0 0 0    .
Let X ∈ R 3×3 and γ 0 > 0 satisfy the linear matrix inequalities

XA i + A i X -2γe 3 e 3 < 0, XG d + G d X > 0, X > 0, A i = 0 1 i 0 0 1 0 0 0 , i = 0, 1, e 3 = (0, 0, 1) ∈ R 3 . Since A(z) = α(z)A 1 + (1 - α(z))A 0 , where α(z) = |z1| √ z P z
where k 0 is given by (26) with a(z) = z P f (z) and b(z) = p 13 z 1 +p 23 z 2 +p 33 z 3 .

For example, the appropriate matrix P obtained using the above LMIs is

P =   1

Remark 3

The canonical homogeneous norm is defined implicitly by the formula [START_REF] Sepulchre | Homogeneous Lyapunov Functions and Necessary Conditions for Stabilization[END_REF]. Therefore, to implement the designed control law either the canonical homogeneous norm must be calculated numerically on-line (like in [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit lyapunov function approach[END_REF], [START_REF] Polyakov | Robust stabilization of mimo systems in finite/fixed time[END_REF]) or it must be approximated with some explicitly defined homogeneous norm N (for example, using the formula [START_REF] Poznyak | Attractive Ellipsoids in Robust Control[END_REF] or the results of [START_REF] Zimenko | Stabilization of chain of integrators with arbitrary in finite-time[END_REF]). Possibility of such a replacement can studied using robustness properties of the control (see, the next section). The construction of an appropriate approximation N of the canonical homogeneous norm • , in the general case, is a non-trivial theoretical problem needed to be studied in future.

Using Cauchy-Schwartz inequality we derive Hence, denoting s = -ln z and using [START_REF] Utkin | Sliding Modes in Control Optimization[END_REF] we complete the proof. Notice that in some particular cases the robust stability condition [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF], that is less conservative than [START_REF] Utkin | Sliding Modes in Control Optimization[END_REF], can also be easily checked. , e 2 = (0, 1, 0) ∈ R 3 .

provided that z P f (z) < 0.

Discussions and Conclusions

In the paper it is proved that any asymptotically stable generalized homogeneous system is equivalent (homeomorphic on R n and diffeomorphic on R n \{0}) to a quadratically stable one. The necessary and sufficient condition of quadraticlike stability (stabilizability) of homogeneous (control) system is derived. This condition may simplify, in some particular cases (see, Section 4.1 and Examples 3, 4) the design of finite-time stabilizing feedback laws (in particular, high order sliding mode control) and formulate an algorithm for control parameters tuning in terms of LMIs. Finally, robust stability/stabilizability conditions are derived as consequences of the obtained results. According to presented examples the obtained stability/stabilizability conditions can be useful for sliding mode control design in the case of negative homogeneity degree. In general case, an appropriate computational procedure is needed to be developed for quadratic-like Lyapunov function design (as well as for the canonical homogeneous norm calculation/estimation and control application) based on (16) (resp. ( 6) and ( 25)). This issue is considered as an open important problem for future research.

  )x = 0 and lim s→+∞ d(s)x = +∞ uniformly on the unit sphere S := {x ∈ R n : x = 1}. The dilation d is a continuous group of invertible linear maps d(s) ∈ R n×n , d(-s) = [d(s)] -1 . The matrix G d ∈ R n×n defined as G d = lim s→0 d(s)-I s is known (see, e.g. [37, Ch. 1]) as the generator of the group d. It satisfies the following properties d ds d(s) = G d d(s) = d(s)G d and d(s) = e G d s := +∞ i=0

e r 1

 1 s 0 ... 0 0 e r 2 s ... 0 ... ... ... ... 0 0 ... e rns , s ∈ R and r i > 0, i = 1, ..., n satisfy Definition 1 with G d1 = I n and G d2 = diag{r i }, respectively. Schematically the difference between uniform, weighted and generalized dilations is depicted at Figure 1.

Figure 1 :

 1 Figure 1: Illustration of uniform d 1 , weighted d 2 and generalized d 3 dilations

  Lipschitz continuity follows from the proven inequality, the identity d(s)x d = e s x d and monotonicity of the dilation. The existence of the unique function s : R n → R such that d(-s(x))x = 1 has been proven in Theorem 1. Since the dilation is strictly monotone then d ds d(-s)x < 0 on S (and, on R n \{0}) for all s ∈ R (see, Theorem 1). Since the norm • is smooth outside the origin then d ds d(-s)x = -∂ z ∂z z=d(-s)x G d d(-s)x. Taking into account ∂ ∂x x d = e s ∂s ∂x s=ln x d the formula (7) is derived by means of Implicit Function Theorem [39] implying ∂s ∂x = -∂ d(-s)x ∂x ∂ d(-s)x ∂s for the function s : R n → R implicitly defined by the equality d(-s)x = 1.

  Since the homogeneous norm is Lipschitz continuous (see Proposition 1) on R n \{0} but power and logarithm functions are Lipschitz continuous outside zero then f is Lipschitz continuous outside the origin. (Differentiability of homogeneous vector field (function) f on R n \{0} can be proven by means of the formula (7), the identity d ds d(s) = G d d(s) and the representation f (x) = x ν d d(ln x d )f (z) with z = d(ln x )x ∈ S.)

  F d (Ψ) = 0 then Ψ(d(s)ξ) d = d(s)Ψ(ξ) d = e s Ψ(ξ) d and the function Ψ(•) d is d-homogeneous of degree 1, radially unbounded, continuous at the origin and continuously differentiable outside the origin. Due to (7) the inequality ∂ Ψ(ξ) ∂ξ f (ξ)

q

  (s)ds ≤ z T P (A + BKd(-ln δ max )) z + ln δmax δ min z T P BKd(-ln δmax)(G -1 d P +P G -1 d ) -1 d(ln δmax)K B P z-z P z+z d -ln δmin δmax P d -ln δmin δmax z 2 for all δ ∈ [δ min , δ max ]. In other words, the matrix inequality P (A + BKd(-ln δ max ))+(A + BKd(-ln δ max )) P +d -ln δmin δmax P d -ln δmin δmax -P + ln δmax δmin P BKd(-ln δ max )(G -1 d P + P G -1 d ) -1 d(ln δ max )K B P ≺ 0 implies

Figure 2 :

 2 Figure2: Evolution of the state of the system[START_REF] Zimenko | A note on delay robustness for homogeneous systems with negative degree[END_REF] with the static homogeneous feedback[START_REF] Pomet | Time-varying exponential stabilization of nonholonomic systems in power form[END_REF],[START_REF] Poznyak | Attractive Ellipsoids in Robust Control[END_REF].

Figure 3 : 0 ∈

 30 Figure3: Evolution of the control signal of the system[START_REF] Zimenko | A note on delay robustness for homogeneous systems with negative degree[END_REF] with the static homogeneous feedback[START_REF] Pomet | Time-varying exponential stabilization of nonholonomic systems in power form[END_REF],[START_REF] Poznyak | Attractive Ellipsoids in Robust Control[END_REF].

degree - 1

 1 with respect to the dilation d in R 3 generated by G d = diag{3, 2, 1}. Hence, the sufficient stabilizability condition (27) becomes z P A(z)z < γ √ z P z z P e 3 e 3 P z, P > 0, P G d + G d P > 0, where z = (z 1 , z 2 , z 3 ) ∈ R 3 and

1 3

 1 and α(z) ∈ [0, 1] provided that p 11 = 1, then for P = P /(p 11 ), P = X -1 the stabilizability condition (27) holds. So, the considered system can be stabilized in a finite time (due to deg d ( f ) = -1) to zero by means of the d-homogeneous dynamical feedback u = k 0 (d(-ln ξ d )ξ) ,

z. ( 32 )

 32 T P d(ln z d )ω ≤ √ z P z ω d (-ln z )P d(ln z )ω. and the sufficient condition of robust stability becomes ω d (-ln z )P d(ln z )ω < -z deg F d (f )-1 z T P f z z Denote φ(s) = ω d (s)P d(s)ω. Obviously, from Theorem 2 we derive 2βφ(s) ≤ d ds φ(s) ≤ 2αφ(s) ⇒ φ(s) ≤ e 2αs φ(0) for s ≥ 0, φ(s) ≤ e 2βs φ(0) for s ≤ 0.

Example 4 (z e 2 P e 2

 42 Robust Quadratic Finite-time Stabilization) Let us study the robustness of the control system designed in Example 3 with the disturbance function ω 0 is assumed to be bounded as follows |ω 0 | ≤ δ ξ d , ξ = (x 1 , x 2 , u) . Taking into account ξ d = z and d(s) = diag{e 3s , e 2s , e s } we derive ω d (-ln z )P d(ln z )ω = |ω 0 | e 2 d (-ln z )P d(ln z )e 2 = |ω 0 | z 2 e 2 P e 2 ≤ δ Since deg F d (f 0 ) = -1 (see, Example 3) then the condition (32) condition of the robust finite-time stability is δ ≤ -sup x =1 x P f (x) √ e 2 P e2

Remark 4

 4 The conditions (30) -(32) can be utilized for the robust global asymptotic stability analysis of an arbitrary systemσ(t) = f (σ(t)) + w(t, σ(t)),where f : R n → R is a d-homogeneous vector field and w : R × R n → R n is a disturbance. The well-known feature of the sliding mode systems is possible rejection of some bounded exogenous disturbances. The corresponding condition can be derived from (32) restricted asω d (-ln z )P d(ln z )ω ≤ δ z deg F d (f ) d where δ = -sup z =1 z P f (z).The latter inequality holds, for example, if G d = diag{n, n-1, ..., 1}, deg F d (f ) = -1 and ω = (0, 0, ..., 0, ωn ) ∈ R n with |ω n | ≤ δ √ e n P en .

Robust Stabilization via Dynamical Feedback

In this section we study the problem of robust stabilizability of the homogeneous system (1):

by means of the homogeneous dynamical feedback (2):

Necessary condition is obviously given by Theorem 5. Sufficient condition asks for an additional restriction to the uncertain term ω.

Corollary 3 Let conditions of Theorem 5 hold and

where

Then the closed-loop system (1), ( 2) is globally asymptotically stable.

Proof. Making the change of variables z = ξ d d(ln ξ d )ξ in the closedloop system (1), (2) we derive (see, Proof of Theorem ( 4))

Next, using Theorem 4, conditions of Theorem 5 and the inequality (30) we complete the proof. More constructive (but also more conservative) sufficient condition can be derived under an assumption on quadratic stabilizability of the nominal system.

Corollary 4 Let conditions of Theorem 5 hold for Ξ = I n and

where

α, β are defined in Theorem 2. Then the closed-loop system (1), ( 2) is globally asymptotically stable.

Proof. For Ξ = I n the inequality (30) is equivalent to


Article paru in S. Schwandner-Sievers and B. J. Fischer (eds.), Albanian Identities: Myth and History. Londres, Hurst and Co. (2002), p. 190-200.

CULTURE AND THE REINVENTION OF MYTH IN A BORDER AREA

Gilles de Rapper

Long-term ethnographic fieldwork allows the ethnographer to become familiar with what is called 'presentation of the self', i.e., with what people say and do when they meet other people and have to create an image of themselves. A part of the presentation of the self has to do with collective identity, that is, with the presentation of the self as a member of a community, either national, local or religious, whose existence is justified on historical grounds. It is thus possible to have an idea of the impact of the myths, created and used in the nation's claim for existence, on society, especially on local communities which are not directly involved in the process of myth-making. Here we shall examine what happens to those myths in the southern Albanian district of Devoll, on the border with Greece. The argument is that although national myths are widely spread in this peripheral area, through school and media, they are shaped to a specific form by the international border and the way it influences social organisation and local perceptions of the self and the other.

National myths and their local forms

Nationwide myths created and transmitted by political and intellectual elites differ from local ones in so far as they are based on wider knowledge of society and history, while local myths are usually rooted in local society and history, even when they pertain to the whole nation. People tend to see the nation according to the organisation and contradictions of their own local society. It is thus impossible to talk of 'local myths' on one side and of 'national myths' on the other: [191] both are national in that they offer an explanation of the existence and characteristics of the nation. Moreover, constant interactions exist between the two levels, through books, school and newspapers.

As usual with national identity [START_REF] Andras | Les manques de la nation. Sur quelques propriétés de la "patrie" et de la "nation" en Hongrie contemporaine[END_REF] and with myth in general [START_REF] Claude | Anthropologie structurale deux[END_REF], the same myths can be heard from different people and in different contexts in quite different forms. This is the case with four national myths that appear in our local society in complete contradiction with their official form, i.e. with the variant which is transmitted through schools, research institutions, state media and national politics. These four myths have to do with religion, independence, traditions and national unity.

It is common in Albania to say that all Albanians, whether Christian or Muslim, are brothers, and that their only religion is their common Albanian nationality. The dogma of national unity as against religious differentiation is at the core of the most widely-spread Albanian national rhetoric1 . However, this rhetoric is challenged when local society is underpinned by, and conceptualised in terms of, religious differentiation. This is the case in mixed areas, where Muslims and Christians live in separate villages (or in separate neighbourhoods), and both have strong identities as religious communities -as in Devoll. In this specific context, religion cannot consist of just being Albanian. On the contrary, people are very well aware of their belonging to a specific religious community, and national identity is rarely thought of outside the basic opposition between Muslims and Christians. For instance, both Muslims and Christians claim a Christian origin for the whole nation, and Christians enjoy a higher prestige as representatives of the 'real' or 'authentic' Albanian nation. Although many declare that religion does not matter, the way people talk and behave inside local society clearly shows that they have to be Christian or Muslim in order to be Albanian, and that it is better to be Christian than Muslim. Independence and isolation is another basis of national rhetoric, since it strengthens the feeling of the nation's uniqueness and [192] autonomous existence. However, when local society is shaped by a long tradition of emigration abroad, the official variant is again challenged. This is the case in the Devoll and, to a certain extent, in southeastern Albania, whence many people migrated to America during the first half of the twentieth century2 . Today the area claims a cultural superiority over all other Albanian regions due to its long and continuous links with the West. Its relative wealth and the better education of its inhabitants are expressly connected to the influence of emigration to Western countries. Remote and isolated areas are never seen as the cradle and sanctuary of authentic Albanian values, but rather as backward, poor, and dangerous places. Contact with the outside world (through knowledge of foreign languages, travel or emigration) is explicitly sought out and acts as a source of social prestige.

In the same way, local people do not seem to value tradition as a marker of national identity. On the contrary, loyalty to traditions such as ritual hospitality or manliness and heroism is often stigmatised as fanaticism, i.e. as an extremist behaviour that has a negative influence on the image people give of themselves. People contrast what they see as modern and educated behaviour with what they call fanaticism, and this is called 'culture' (kulturë).

When they feel they are not acting 'heroically' (e.g. because they are afraid of every authority), they typically reply 'We are not brave, but we have got culture' (ne s'jemi trimë, por jemi me kulturë). Heroism and traditions are used here not to assert national identity, but rather, by being denigrated, to give local society a distinctive image of modernity.

The difference between the North and the South in Albania -the distinction between Ghegs and Tosks -is usually seen as historic, linguistic and ethnic. However, in local society the words geg and tosk are never heard and the relevant distinction is simply between "North" and "South". There is no actual territorial border between North and South: the difference lies on distinctive representations of the 'self' and the 'other' rather than on geography or society. This is obvious in the conception of space. From the local point of view, the North is sometimes seen as an entirely mountainous area inhabited by backward and dirty shepherds. With no roads or contact with the outside, the North is a closed space. Local society, said to be representative of the South, is then characterised by its agriculture and its links with the world. However, at different times, due to its own mountains, local society pretends [193] to be higher, in terms of both altitude and moral value. The North is then seen as a composite of lowlands and valleys open to bad influences and filled with a filthy atmosphere. The difference between North and South thus appears as a local and contextual realisation of underlying symbolic oppositions between high and low, open and closed, dirty and clean, and is not historicised or ethnicised as a distinctive feature of the nation3 .

Once again, these variations should not be understood as an opposition between a local conception and a national one. In fact, when asked openly, people usually tend to answer by the official form of those myths, which is known nation-wide. The local forms appear in everyday and informal conversation as a reaction to events or to the behaviour of other people. The difference is thus rather between a constructed and idealised form of national identity, and a second form, rooted in everyday life and in local social relationships. Both can be seen as discourses of a society on itself, but their production and destination are different.

Moreover, in local conceptions, all these myths have to do with the concept of 'culture'. The opposition between Muslims and Christians is conceived in terms of the cultural superiority of the latter, as with the opposition of North versus South, tradition versus modernity, and isolation versus links with the outside world. The concept of culture thus appears as a key to the understanding of these myths and of their impact on society.

The concept of culture

First of all, it is important to note that the concept of culture is both a popular category which is not consciously elaborated and an over-elaborated concept in Western philosophy and social science that has had considerable influence on the Albanian national movement. Among local populations people talk about culture and are able to apply the two categories of 'with culture' (me kulturë) and 'without culture' (pa kulturë) to both individuals and communities, but no one is able to give an extensive definition of what culture is. The dictionary definition, on the other hand, does not fit with the local conception. The definition that follows is thus based on my own [194] elaboration of various applications of the word in everyday situations.

Culture can be defined by a series of four components on the one hand, and by its functioning on the other. The four components are as follows:

Language. 'To have culture' means first to speak one's own language correctly, i.e. in the standard form rather than by using a local dialect. It also means speaking at least one of the most valued foreign languages (English, French, Italian). For instance, during the electoral campaign of 1996, all the candidates of the Devoll were presented in meetings as speaking several foreign languages. Almost certainly most of them were far from actually speaking so many different languages, but the very fact that they were said to speak a foreign language has such a high cultural value that no one in the audience would think of questioning this ability. Someone who is said to speak one or more than one foreign languages 'has culture' and thus cannot be a cheater.

We are very close here to the Gellnerian model of the nation, with its need for a unified national language. Culture here means 'national culture' as opposed to the remains of local diversity, and people from the Devoll feel proud that the way they speak is very close to standard Albanian.

Knowledge. Any kind of knowledge is valued, especially if it has been learned at school or university, or by travelling abroad. The villages which are said to be 'with culture' are those where the first Albanian schools opened early in the twentieth century, as well as those from which people emigrated to the West. Once again, the insistence on schools and education reminds us of the Gellnerian model of the nation.

The outside world. 'Culture' also means contacts with the outside world through emigration or travel, or even through television or personal contacts with visiting foreigners. Once again, some foreign countries are more valued than others, and it is better, in terms of 'culture', to have family in the United States rather than in Greece, or to watch French rather than Turkish television.

Technology. 'Culture' has also a material dimension, related to the modernisation of village life. Villages with paved streets, water, electricity, and two-storied stone houses 'have more culture' than those that have a more rural aspect. Inside the house the use of a table and chairs instead of sofër (the low round table) and rugs is said to be a sign of 'higher culture'. Today TV sets, refrigerators, video recorders and cameras tend to be considered both as signs of culture (they connect [195] people to the outside world and to modernity) and as a source of social prestige. Indeed, because culture is not spread equally among the population, it also brings prestige to the individuals who are credited 'with culture'. With these two last points, 'culture' appears as a form of modernity, but more precisely as a way of learning modernity from abroad, by way of imitation.

As can be seen, the local conception of culture is not so far from the definition used in social anthropology, but it differs, first, by the importance of relations with the outside world, and, second, by its normative dimension. Clearly, from the local point of view there is only one form of 'culture', or at least one good form of it. Some people have culture, others do not, and those who have acquired it first are superior. This also means that culture is not attached to a particular group or community, but can be transmitted from one group to another. For instance, it is usually assumed that the West has more culture, and thus people in Albania should learn from foreigners how to behave and live. Culture is also a historic process, which is not given at one time to everyone. History is, so to speak, the history of how culture comes to villages and individuals all over the country.

If the local conception of culture appears to be in contradiction with the anthropological one due to its historic and normative character, it is nonetheless very close to another classical definition of it in Western tradition. Two traditions can indeed be distinguished in the definition of culture in Western thought. The first one dates back to the 18th century and the Enlightenment. Culture is seen as a distinctive feature of all human societies and is opposed to nature; it is to be understood as 'high culture' and is related to an unbroken tradition from classical antiquity. In that sense, it is considered superior: 'culture' is a kind of behaviour, knowledge and judgement to which everyone should aspire in order to be a human being and not a 'barbarian'. The second and contrary perspective relates to the 19th century and Romanticism. Culture is no longer universal and normative, but plural and peculiar: there are different cultures, each the achievement of one particular people or social, national, or religious group. Culture is thus defined as the integrated beliefs, practices and social forms that give the group its coherence and specificity4 .

The fact that the local Albanian conception is closer to the classical universalist definition does not mean that Albania was more [196] influenced by the French Enlightenment than by German Romanticism. It should rather make us question the anthropological and sociological basis of these two traditions. Both can indeed be seen as different ways of talking about identity and especially of tracing the border between the group and the outside. The relativist definition relies on the idea that each group receives its characteristics from nature, or at least from a given and unquestionable order from which all borders were drawn from the beginning. This conception will be called 'primordialist', as 'it focuses on gender and generation, kinship, ethnicity and race, for constructing and reinforcing the boundary between inside and outside' 5 . The universalist definition, on the other hand, ranks the members of the group and the outsiders on a scale of 'culture', i.e. according to their relative proximity with an absolute value, 'culture'. It will be called 'sacred' as it relates 'the collectivity to an unchanging and eternal realm of the sacred and the sublime -be it defined as God or Reason, Progress or Rationality.'6 Both conceptions must be understood as ideal types: they are not exclusive and can be found together, though not of equal importance, in the same national identity.

The concept of culture appears to be central to the construction of the Albanian national identity for two reasons: first, the tracing of the border between 'we' and 'the other' relies on the idea that culture should be the distinctive attribute of insiders while outsiders (and especially neighbours) as seen as barbarians, and second, culture is present in all symbolic oppositions on which national myths are constructed. Culture is thus not only one myth among many. As a structuring principle of local society and as a frame for every discourse of society on itself, the concept of culture appears to be at the basis of all the national myths and to constitute the myth par excellence.

Once again, this definition of culture should not be seen as a local form as opposed to a nation or state-centred form. It is rather the local interpretation of a concept that has been widely used and spread through Albanian history, from the beginning of the national movement until today. Indeed, apart from political independence, all national movements in South-Eastern Europe also aimed at the modernisation of society through contact with Western culture and technology, creation of a national unified language, and organisation of a national [197] educational system. However, we can look at the way culture has come to be so central in the definition of collective identities in this border area.

Culture vs. non-culture

First, some of the internal divisions of local society are conceived in terms of culture. This is mainly the case for the opposition between Muslims and Christians, which forms one of the main features of social organisation in this area. Christians form a minority in this area, as they do in Albania as a whole. Although they do not live in a separate territory, they are not entirely mixed with Muslims. They live in different villages and, in the few instances where they live in the same village, they still live in separate neighbourhoods. Moreover, a spatial differentiation exists between the two communities: Christians live in the mountains, Muslims in the lowlands (see table ): As elsewhere in the Mediterranean, the mountains are strongly associated with backwardness, poverty and violence, as stated earlier regarding the difference between North and South. Lowlands, on the contrary, are associated with agriculture and hence wealth and the possibility of sustaining a larger population. Muslim villages are always bigger than Christian mountain villages. This spatial difference thus relates to a social differentiation which has to do with occupation (Muslims are peasants, Christians are craftsmen) and prosperity (life is easier in the plain than in the mountains). It also has a historical dimension; because of Ottoman political and administrative organisation, the ownership of agricultural lands would be easier for Muslims than for Christians, who would be driven up into the mountains which were of no special interest to the Muslims. The association of Muslims with land and prosperity is thus a result of their privileged relationship with the state, and of the authority they exercise over Christians.

The situation described above belongs to an idealised and partly imagined past, as both emigration and communism have engendered [198] deep social and economic changes. Emigration to the West, mainly to the United States, until the Second World War was more a Christian phenomenon in the Devoll partly because of their more desperate economic situation. Mostly, however, the émigrés would maintain strong links with their village of origin, either coming back every few years or sending money both for the family and the village as a community, and to working for improved conditions for all villagers by funding road works, mills or water adduction. By thus remaining members of their original community, they would contribute to social change, introducing new habits and artefacts still classified as 'modern', which in local conceptions cannot be expressed other than as being a part of 'culture'. The role of Communism was similar since by fighting traditional land ownership and religious practice, it was a strong factor in modernisation. The development of schools was also responsible for the spreading of the idea of 'culture.' As noted earlier, villagers are still proud of the long existence of schools in their villages and of the number of children they send to high school and university: it is a sign of their high level of culture compared to neighbouring villages. Moreover, as most religious practices and beliefs were attacked by Communism, the way people identify with their religious communities had more to do with culture than with religion as such, which was also partly the case earlier, since religious communities in Ottoman society were also social groups, acting for the socialisation of their members. The opposition between religious communities is not concerned with theological differences. Nothing is done to attack the beliefs and practices of members of the other community. Rather, the way religion is experienced is partly syncretistic, with popular religion also of great importance. What is involved in the relations between the two communities is rather the construction of an image of the other, which is used to mirror one's own identity. For instance, Muslims present themselves as 'loyal' and 'generous', whereas Christians are 'disloyal' and 'selfish'. Christians on the other hand see themselves as 'intelligent' and 'thrifty' and Muslims as 'stupid' and overly 'liberal'. These crossed representations are important as they trace borders inside local society and are at the basis of most feelings of identity.

However, although each community tries to give an image of culture and are prompt to stigmatise its absence in the behaviour of members of the other community, there is a general acknowledgement of the higher culture of Christians. This is best expressed by the Muslims themselves claiming Christian origin and presenting their conversion to Islam as accidental. Christians are also generally credited with better [199] education and closer links with the outside world. Through emigration and education they are given credit for most of the social changes of the last decades, from clothes (Christians dress allafranga) to wedding ritual and food habits. Here lies the most important difference between the local conception of the nation and the national one. Both nationalist thinkers of the 19th century and communists tried to place 'culture' (meaning modernity and progress) above religion and to use it as a way to minimise religious differentiation in the new nation. However, in local conceptions, culture is a part of religion or, in other words, religious affiliation is access to culture, or to a higher level of culture. Thus the two communities are not equal in this (even if they are explicitly declared equal where personal salvation is concerned): one of them gives its members better access to culture. For historical and social reasons, dating mostly from the end of the 19th century and the 20th century, Christians as a community are given this privilege in the Devoll, in the same way as they and other minorities such as the Jews and Armenians were in other parts of the Ottoman Empire.

The cultural differentiation between Christians and Muslims probably started in the last days of the Ottoman Empire. It was given a strong emphasis during Communism, as culture became one of the symbols of the country's modernisation. The cultural difference between Christians and Muslims today results from a more recent phenomenon: the opening of the border with Greece and a large emigration to that country. In this border area Greece appears as the nearest example of a modern and prosperous state, and although the Greeks are usually criticised by both Muslims and Christians for being 'disloyal' and 'selfish', the Greek way of life is idealised as the embodiment of culture. Here Christians are again privileged due to the identification of all Christians with Greeks, which is common in this part of the Balkans.

Local myths of origin leave no room for the Illyrians. These myths are essentially summed up in statements such as 'Muslims descend from Christians' or 'our ancestors were Christians'. What local people look for is not evidence of single autochthonous nation, for which purpose the Illyrians can be cited, but rather the confirmation that they belong to the realm of culture. Similarly, the way local people talk about the difference between North and South has nothing to do with the nation's historical genesis. Local people use an imagined North as a mirror for their own weaknesses in such a way as to contrast North and South as 'without culture' (poor, backward, violent, traditional) [200] and 'with culture' (rich, developed, well-educated, modern). Once again, they get confirmation that they are on the good side, while others are not. This chapter has sought to explain what happens to national myths once they have been created and spread inside a society. There is no fundamental opposition between national myths as expressed by the centre of the state or the elites on one hand and by local population on the other. However, local myths are shaped by specific conditions and interests which are not homogenous throughout the country. Such combinations and transformations are common to most systems of myths. Indeed the material used in these myths, such as symbolic juxtapositions, the concept of culture and representations of space, is common to the greater part of Albanian national mythology. In this particular case national myths are mediated by the concept of culture which is itself the main means by which local people talk about themselves and others. This concept takes its meaning mainly from the specific social organisation of local society. We are thus reminded that the study of myths and representations should always be articulated in conjunction with the study of social structure.

Table 1 .

 1 -Space and religion in the 43 villages of the Devoll

		Lowlands	Mountains
	Orthodox Christians	3 villages	6 villages
	Muslims	25 villages	9 villages

Repetition of such statements can be both a source of amusement and exasperation among visiting foreigners. See for instance Maspero 1997.

Emigration to America is itself inscribed in a context of mobility which was first limited to the Ottoman Empire(Istanbul, Egypt)

On the local forms of national identity in the Devoll, see[START_REF] De Rapper | La Frontière albanaise. Famille, société et identité collective en Albanie du sud[END_REF]

Kuper, 1994, p.

[START_REF] Gellner | Nations and Nationalism[END_REF], ch. 7. 5 Eisenstadt & Giesen, 1995, p. 77.

Ibid. p. 82.