Andrey Polyakov
email: andrey.polyakov@inria.fr

Homogeneous Lyapunov Functions for Homogeneous Infinite Dimensional Systems with Unbounded Nonlinear Operators

Keywords: stability of distributed parameter systems, homogeneous systems, semigroup and operator theory

The existence of a locally Lipschitz continuous homogeneous Lyapunov function is proven for a class of asymptotically stable homogeneous infinite dimensional systems with unbounded nonlinear operators.

Introduction

Homogeneity is a dilation symmetry known since 18th century, when Leonhard Euler studied functions x → f (x) which are symmetric with respect to the uniform dilation x → λx of argument: f (λx) = λ ν f (x), ∀λ > 0, ∀x, where ν is a real number. Such functions were called homogeneous and the number ν was referred as the homogeneity degree. It seems that a generalized homogeneity (the symmetry with respect to a non-uniform dilation) was originally studied by Vladimir Zubov in [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], where the homogeneity was shown to be useful for analysis of nonlinear finite-dimensional dynamical systems (see also [START_REF] Hermes | Nilpotent approximations of control systems and distributions[END_REF], [START_REF] Kawski | Homogeneous stabilizing feedback laws[END_REF]. [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF]). Indeed, it simplifies stability and robustness analysis of control systems (see e.g. [START_REF] Kawski | Families of dilations and asymptotic stability[END_REF], [START_REF] Ryan | Universal stabilization of a class of nonlinear systems with homogeneous vector fields[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF]) as well as non-linear controllers/observers design (see [START_REF] Kawski | Homogeneous stabilizing feedback laws[END_REF], [START_REF] Coron | Adding an integrator for the stabilization problem[END_REF], [START_REF] Grüne | Homogeneous state feedback stabilization of homogeneous systems[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF], [START_REF] Lopez-Ramirez | Finitetime and fixed-time observer design: Implicit Lyapunov function approach[END_REF]). Homogeneity degree specifies a convergence rate of any asymptotically stable homogeneous system (see e.g. [START_REF] Nakamura | Smooth Lyapunov functions for homogeneous differential inclusions[END_REF]). For example, in [START_REF] Zubov | Methods of A.M. Lyapunov and Their Applications[END_REF] it is shown that homogeneous systems can be finite-time stable [START_REF] Bhat | Finite time stability of continuous autonomous systems[END_REF] provided that the homogeneity degree satisfies certain restriction. Some applications of the finite-dimensional homogeneity in the theory of differential operators and Lie algebras can be found in [START_REF] Folland | Subelliptic estimates and function spaces on nilpotent Lie groups[END_REF], [START_REF] Fischer | Quantization on Nilpotent Lie Groups[END_REF]. Elements of the homogeneity theory of infinite dimensional dynamical systems are introduced in [START_REF] Polyakov | On homogeneous distributed parameters equations[END_REF], [START_REF] Polyakov | On homogeneous finite-time control for linear evolution equation in hilbert space[END_REF], [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF], where it is shown that many infinite dimensional nonlinear models of mathematical physics are homogeneous in the generalized sense (e.g. Saint-Venant, Navier-Stokes, Burgers and KdV equations).

The existence of homogeneous Lyapunov function for a stable homogeneous ordinary differential equation (ODE) is proven by Vladimir Zubov in 1958 (see [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF]) and refined by Lionel Rosier in 1992 (see [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF]). The key difference between Zubov's and Rosier's theorems is the regularity of the designed homogeneous Lyapunov function. V. Zubov proved only the existence of a continuous homogeneous Lyapunov function, while L. Rosier constructed a smooth one. The regularity (at least local Lipschitz continuity) of the Lyapunov function is important for robustness (Input-to-State Stability=ISS) analysis (see e.g. [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF], [START_REF] Karafyllis | Input-to-State Stability for PDEs[END_REF], [START_REF] Mironchenko | Input-to-state stability of infinite dimensional systems: Recent results and open questions[END_REF]). Indeed, the smooth homogeneous Lyapunov function allows rather simple robustness analysis of a homogeneous ODE to be developed [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF], [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF]. Namely, any asymptotically stable homogeneous system is ISS with respect homogeneously involved perturbations.

In [START_REF] Polyakov | Input-to-state stability of homogeneous infinite dimensional systems with locally lipschitz nonlinearities[END_REF] the existence of a locally Lipschitz continuous Lyapunov function was proven for homogeneous infinite dimensional systems with locally Lipschitz nonlinearities. The mentioned class of systems is restrictive. The aim of this paper is to extend the latter result to a class of evolution equations with unbounded nonlinear operators such as Burgers or KdV equations. The well-posedness as well as stability analysis of such systems is more tricky since solutions may exist only in a linear subspace of the infinite dimensional state space. This papers shows that the methodology of homogeneous Lyapunov function design introduced in [START_REF] Polyakov | Input-to-state stability of homogeneous infinite dimensional systems with locally lipschitz nonlinearities[END_REF] can be successfully extended to the considered class of systems. As an example, the nearly fixed-time stability of a viscous Burgers equation is analyzed by means of the homogeneous Lyapunov function.

Notation. R is the field of real numbers; R + = [0, +∞); x • y = n i=1 x i y i denotes the dot product of x = (x 1 , ..., x n ) , y = (y 1 , ..., y n ) ; • B denotes a norm in a real Banach space B; •, • H denotes an inner product in a real Hilbert space H; 0 is the zero element of a Banach space; I (resp. O) denotes the identity (resp. the zero) operator; L(B 1 , B 2 ) denotes the space of linear bounded operators B 1 → B 2 with the norm A B1,B2 = sup

x =0 Ax B 2 x B 1
; if B 1 = B 2 = B we write shortly A B ; S B is the unit sphere in B; B B (r) is the ball in B of the radius r > 0 centered at 0; for r > 1 the set K B (r) ⊂ B is defined as follows K B (r) := {x ∈ B : 1/r < x B < r}; D(A) denotes the domain of an operator A; C([t 1 , t 2 ], B) is the space of continuous functions x : [t 1 , t 2 ] → B with the uniform norm x C = max t∈[t1,t2] x(t) with -∞ < t 1 < t 2 < +∞; f 1 • f 2 and f 1 f 2 denote a composition of operators f 1 and f 2 ; C ∞ c (Ω, R m ) is a set of infinitely smooth functions R n → R m with compact supports in an open connected set Ω ⊂ R n with a sufficiently smooth boundary (or

Ω = R n ); L 2 (Ω, R m ) is the Lebesgue space with the inner product u, v L 2 (Ω,R m ) = Ω u • v and the norm u L 2 (Ω,R m ) = u, u L 2 (Ω,R m ) , where u, v ∈ L 2 (Ω, R m ); ∇ = ( ∂ ∂z1 , ..., ∂ ∂zn ) , where z = (z 1 , ..., z n ) ∈ R n ; ∇ 1 := ∇ and ∇ i+1 := ∇ • ∇ i , i = 1, 2, ...; ∆ = ∇ 2 is the Laplace opera- tor; H p (Ω, R m ) is the Sobolev space with the inner product u, v H p (Ω,R m ) = +∞; K is a set of strictly increasing continuous functions σ : [0, +∞) → [0, +∞) such that σ(0) = 0; K ∞ := {σ ∈ K : σ(s) → +∞

≤ or

a.e ∈ ) means that an identity (resp. inequality or inclusion) holds almost everywhere; ẋ(t) = lim h→0 x(t+h)-x(t) h is a time derivative of the function x : R → B, where the limit is understood in the strong topology of B;

D + v(t) = lim sup h→0 + v(t+h)-v(t) h
denotes the right-hand upper Dini derivative of the function v : R → R;

D + V (x; g) = lim sup h→0 + V (x+hg)-V (x) h
denotes the right-hand upper directional derivative of the functional

V : B → R in the direc- tion g ∈ B; DF (x) ∈ L(B 1 , B 2 ) denotes the Fréchet derivative of F : B 1 → B 2 .

Model Description and Basic Assumptions

Let us consider the nonlinear system

ẋ = Ax + f (x), t > 0, x(0) = x 0 (1)
where x(t) ∈ X is a system state at the time instant t , X is a real Banach space, x 0 ∈ X is an initial state, A : D(A) ⊂ X → X is a linear (possibly unbounded) closed densely defined operator which generates a strongly continuous semigroup Φ of linear bounded operators on X, f : D(f ) ⊂ X → X is a non-linear (possibly unbounded) closed densely defined operator such that f (0) = 0 and D(f ) is a linear subspace dense in X.

The non-linear evolution equations are well-studied in the literature (see, for example, [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF][START_REF] Engel | One-Parameter Semigroups for Linear Evolution Equations[END_REF]), where the notion of solution is introduced using the theory of evolution semigroups. Definition 1. A continuous function x : [0, T ) → X is said to be a mild solution of (1) if f (x(•)) ∈ L 1 ((0, T ), X) and

x(t) = Φ(t)x 0 + t 0 Φ(t -s)f (x(s))ds, ∀t ∈ [0, T ).
If this mild solution satisfies (1) for (almost) all t ∈ (0, T ) then x is called classical (strong) solution of (1).

The above integral is understood in the sense of Bochner (see e.g. [START_REF] Driver | Analysis Tools with Applications[END_REF], page 187). The existence of the zero solution for the system (1) follows from f (0) = 0. Definition 2. A closed densely defined non-linear operator f : D(f ) ⊂ X → X is said to be M -regular if there exists a linear closed operator M : D(f ) ⊂ X → X with a bounded inverse M -1 : X → D(f ) such that the nonlinear mapping x → f (M -1 x) is locally Lipschitz continuous in x ∈ X\{0}. More precisely, for any r > 0 there exists L r > 0 such that

f (M -1 x 1 ) -f (M -1 x 2 ) X ≤ L r x 1 -x 2 X ( 2
)
for all x i ∈ K X (r) and i = 1, 2.

For example, if f (x) = g(Qx, x), where Q : D(Q) ⊂ X → X is a closed densely defined linear operator and g : X × X → X is locally Lipschitz continuous, then f is M -regular with M = Q -λI provided that λ ∈ R belongs to the resolvent set of Q. For a well-posedness of (1), we assume that f admits some "M -regularization" consistent with A.

Assumption 1. Let f be M -regular, M commutes with Φ:

Φ(t)M x 0 = M Φ(t)
x 0 for all t ≥ 0 and all x 0 ∈ D(f ),

the linear operator M Φ(t) : X → X is bounded for any t > 0 and there exists a continuous function ω

: (0, +∞) → R + such that M Φ(t) X ≤ ω(t) and t 0 ω(σ)dσ < +∞, ∀t ∈ (0, +∞). If D(f ) = X (or A = 0) then M = I (resp. Φ(t) = I, ∀t ≥ 0)
and Assumption 1 simply asks the regularity of f on X\{0} as in [START_REF] Polyakov | Input-to-state stability of homogeneous infinite dimensional systems with locally lipschitz nonlinearities[END_REF]. If A is a generator of an analytic semigroup Φ then Assumption 1 is fulfilled for an operator M = A α being a fractional power of the operator A (see, e.g., [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF], page 195). The function ω has the form ω(t) = t -α C, α ∈ (0, 1), C ≥ 1 in this case.

Since M is a closed linear operator then the linear space Y = D(f ) with the norm y Y = y + M y is a Banach space as well. Assumption 1 guarantees that for any x 0 ∈ Y\{0} the evolution equation ( 1) has a unique mild solution x x0 which depends continuously on the initial condition (see Appendix). In this paper we study the so-called generalized homogeneous evolution equations [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF]. The aim of the paper is to proof that the generalized homogeneous system (1) is globally uniformly asymptotically stable in Y if and only if it admits a locally Lipschitz continuous homogeneous Lyapunov function V : Y → R.

Homogeneous Systems

Dilations in Banach Spaces

For more details about theory of dilations in finite-dimensional and infinitedimensional spaces we refer the reader to [START_REF] Husch | Topological Characterization of The Dilation and The Translation in Frechet Spaces[END_REF], [START_REF] Kawski | Families of dilations and asymptotic stability[END_REF], [START_REF] Polyakov | On homogeneous distributed parameters equations[END_REF]. In this paper we deal only with linear dilations [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF]. Any continuous linear dilation in R n is given by d

(s) = e sGd = ∞ i=0 s i G i d i! ,
where G d ∈ R n×n is an anti-Hurwitz matrix. Nonlinear dilations in R n are studied in [START_REF] Khomenuk | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Kawski | Families of dilations and asymptotic stability[END_REF], [START_REF] Rosier | Etude de quelques problèmes de stabilization[END_REF]

(d(s)x)(z) = e αs x(e βs z), s ∈ R, x ∈ H p (R n , R m ), z ∈ R n , (3)
where α, β ∈ R are constant parameters. Since making the change of the variable in the Lebesgue integral we derive

d(s)x 2 H p = p i=0 ∇ i d(s)x 2 L2 = p i=0 e βis d(s)∇ i x 2 L2 = p i=0 e (2α-βn+2βi)s ∇ i x 2 L2 ,
then d is a strictly monotone dilation in H p (R n , R m ) provided that α > β(0.5n/2i), i = 0, 1, ..., p. Recall that L 2 = H 0 . For more details, about linear dilations in function spaces we refer the reader to [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF]Chapter 6], where in particular the strong continuity of the group d is proven.

Definition 6. A set D ⊆ B is a d-homogeneous cone in B if d(s)u ∈ D, ∀u ∈ D, ∀s ∈ R.
Notice that D becomes the conventional positive cone in B provided that d is the uniform dilation d(s) = e s I, s ∈ R.

Being a strongly continuous group of linear bounded operators, the linear dilation always has an infinitesimal generator [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF] being closed densely defined linear operator

G d : D(G d ) ⊂ B → B given by G d u = lim s→0 d(s)u-u s . Example 2.
The generator G d of the dilation considered in Example 1 is given by (see, for example, [START_REF] Polyakov | On homogeneous finite-time control for linear evolution equation in hilbert space[END_REF] or [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF], Lemma 6.4)

(G d x)(z) = αx(z) + β(z • ∇)x(z), z ∈ R n , x ∈ D(G d ) ⊂ H p (R n , R m ), ( 4
)
where z • ∇ = z 1 ∂ ∂z1 + z 2 ∂ ∂z2 + ... + z n ∂ ∂zn , the domain D(G d ) is a completion of C ∞
c with respect to the norm x H p + G d x H p . All derivatives are understood in the weak sense.

The canonical homogeneous norm

The dilation introduces an alternative norm topology in B using the so-called canonical homogeneous norm. Definition 7 ( [START_REF] Polyakov | On homogeneous finite-time control for linear evolution equation in hilbert space[END_REF]). The functional • d : B → R + given by 0 d = 0 and

u d = e su , where s u ∈ R : d(-s u )u B = 1, u = 0 (
≥ β > 0, C ≥ 1 : 1 C u α d ≤ u B ≤ u β d , u ∈ B d (1) and u β d ≤ u B ≤ C u α d , u ∈ B\B d (1). Moreover, there exist σ, σ ∈ K ∞ : σ( u B ) ≤ u d ≤ σ( u B ), ∀u ∈ B.
In [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF]Theorem 7.1] it is also proven that • d is a norm in a Banach space B homeomorphic to B. This justifies the name "norm" for the functional • d .

Example 3. Since the strongly continuous strictly monotone dilation (3) in

H p (R n , R m ) satisfies d(s)x 2 H p = p i=0 e 2s(α-βn/2+βi) ∇ i x 2 L 2 , x ∈ H p (R n , R m )
then the canonical homogeneous norm in H p is defined as x d,H p = 1/V , where V > 0 is a unique real positive root of the following fractional polynomial equation

1 = p i=0 V 2(α-βn/2+βi) a i ,
where

a i = ∇ i x L 2 , x = 0, α > βn/2 -βi, i = 0, 1..., p.
Since the right-hand side of the latter equation is continuously differentiable in a i , V ∈ (0, +∞) then, by the implicit function theorem, the function (a 0 , ..., a p ) → V defined implicitly by this equation is continuously differentiable in a i ∈ (0, +∞) as well. In other words, the canonical homogeneous norm

x d,H 1 is a continuously differentiable function of ∇ i x L 2 , i = 0, 1, ..., p for x ∈ H p (R n , R m )\{0}.
The canonical homogeneous norm can be utilized as a Lyapunov function candidate for some homogeneous systems. The differentiability of the homogeneous norm is important for the corresponding analysis. In the case of a Hilbert space H, the canonical homogeneous norm is always Frechét differentiable at least on the domain of the generator G d .

Lemma 1 ( [START_REF] Polyakov | On homogeneous finite-time control for linear evolution equation in hilbert space[END_REF] and [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF], Lemma 7.4). Let d be a strongly continuous strictly monotone dilation group in a Hilbert space H then the homogeneous norm

• d is differentiable on D(G d )\{0} and the Fréchet derivative of • d at u ∈ D(G d )\{0} is given by (D u d ) (•) = d(-ln u d ) • , d(-ln u d )u G d d(-ln u d )u, d(-ln u d )u u d . (6)
Example 4. Let us consider again the strongly continuous strictly monotone dilation

(3) in H p (R n , R m ) with α > βn/2 -βi, i = 0, 1, ..., p. From Example 3 we conclude the Fréchet differentiability of • d,H p on H p (R n , R m )\{0}. Notice that for v ∈ H 1 (R n , R m
) using integration by parts we derive

G d v, v H 1 = αv + β(z • ∇)v, v L2 + ∇ (αv + (βz • ∇)v) , ∇v L2 = (α -βn/2) v, v L2 + (α + β(1 -n/2)) ∇v, ∇v L2
and the Fréchet derivative of

• d,H 1 at the point x ∈ H 1 (R n , R m ) can be computed as follows D x d,H 1 (q) = d(-ln x d,H 1 )q, v H 1 (α-βn/2) v,v L 2 +(α+β(1-n/2)) ∇v,∇v L 2 x d,H 1 , q ∈ H 1 (R n , R m ), (7) where
v := d -ln x d,H 1 x.

Homogeneous operators

Homogeneous functionals and operators on a Banach space B (see, e.g., [START_REF] Polyakov | On homogeneous distributed parameters equations[END_REF]) are defined similarly to homogeneous functions and vector fields in R n (see e.g. [START_REF] Kawski | Families of dilations and asymptotic stability[END_REF], [START_REF] Grüne | Homogeneous state feedback stabilization of homogeneous systems[END_REF]) taking into account their possible unboundedness.

Definition 8. An operator f : D(f ) ⊂ B → B (a functional h : D(h) ⊂ B → R) is said to be d-homogeneous of a degree ν ∈ R if the domain D(f ) (resp. D(h)) is a d-homogeneous cone and e νs d(s)f (u) = f (d(s)u), ∀s ∈ R, ∀u ∈ D(f ), (resp. h(d(s)u) = e νs h(u), ∀s ∈ R, ∀u ∈ D(h)) ( 8
)
where d is a group of linear operators in B.

We say that an evolution equation (inclusion) is d-homogeneous of a degree µ ∈ R if its right-hand side is a d-homogeneous operator of the degree µ.

Example 5 ([22], Example 7.6).

The Laplace operator ∆ :

H 2 (R n , R) ⊂ L 2 (R n , R) → L 2 (R n , R) is d-homogeneous of the degree 2β with respect to the dilation d introduced in the Example 1, where ∆x = n i=1 ∂ 2 x ∂z 2 i , z i ∈ R, x ∈ H 2 (R n , R
) and the derivatives are understood in the weak sense.

Homogeneity allows local properties of nonlinear operators (such as regularity) to be extend globally [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF]Chapter 7].

Symmetry of solutions of homogeneous systems

A semigroup generated by a closed densely defined linear homogeneous operator in X is homogeneous as well [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF]Lemma 8.1].

Theorem 2 ( [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF], Theorem 8.1). Let d be a group of linear bounded invertible operators on X. Let a linear closed densely defined operator A : D(A) ⊂ X → X generate a strongly continuous semigroup Φ of linear bounded operators on X and f : D

(f ) ⊂ X → X. Let A and f be d-homogeneous operators of a degree µ ∈ R. If x : [0, T ) → X is a mild solution of (1) and x(t) a.e.
∈ D(f ) then for any s ∈ R the function x s : [0, e -µs T ) → X given by x s (t) := d(s)x(e µs t), t ∈ [0, e -µs T ) is a mild solution of the evolution equation (1) and x s (t)

a.e.

∈ D(f ).

The latter theorem proves the symmetry of solutions of (1) with A and f being d-homogeneous operators.:

x d(s)x0 (t) = d(s)x x0 (e µs t), s ∈ R, t ≥ 0, (9)
where x z denotes a solution of (1) with the initial data x(0) = z. This symmetry expands globally any local property of solutions. For example, if the origin of (1) is locally stable then, from (9) and the limit property of d, we immediately derive global stability. Similarly, the existence of solutions for small initial data implies the existence of solutions for large initial data and so on.

Example 6. The system ẋ = ∆x -(x • ∇)x, t ≥ 0, x ∈ L 2 (R n , R n ) is d- homogeneous of the degree 2 with respect to the dilation (d(s)x)(z) = e s x(e s z), z ∈ R n , s ∈ R, where ∆ : H 2 (R n , R n ) ⊂ L 2 (R n , R n ) → L 2 (R n , R n ) is the Laplace operator and f : H 1 (R n , R n ) ⊂ L 2 (R n , R n ) → L 2 (R n , R n ) given by f (x) = -(x • ∇)
x is a non-linear M -regular operator with M = (-∆) 1/2+n/8 being the fractional Laplacian. The M -regularity of f follows from the inequalities

∇M -1 v L 4 ≤ C (-∆) n/8 ∇(-∆) -1/2-n/8 v L 2 ≤ C v L 2 , ∀v ∈ L 2
being a consequence of Corollary 6.11 from [21, Chapter 1] and the Hardy-Littlewood-Sobolev fractional integration theorem (see, e.g., [30, Page 119]) yielding

(-∆) -n/8 y L 4 ≤ C y L 2 , ∀y ∈ L 2 . Assumption 1 is fulfilled for n ∈ {1, 2, 3}.
By Theorem 2, mild solutions of the considered system satisfy (9).

Homogeneous Lyapunov function theorem (Main result)

Solutions of ( 1) with the M -regular operator f are well-defined on Y\{0} (see Appendix). It is reasonable to analyze the stability and robustness of the system (1) in the norm topology of Y. The origin of the system (1) is said to be

• globally uniformly Lyapunov stable in Y if there exists ε ∈ K ∞ such that x(t) Y ≤ ε( x(t 0 ) Y ), ∀t ≥ t 0 (10)
for any mild solution x : R + → Y of (1) and any t 0 ≥ 0;

• globally uniformly asymptotically stable in Y if it is globally uniformly Lyapunov stable in Y and ∀ε > 0, ∀R > ε and ∀t 0 ≥ 0, there exists

T = T (R, ε) > 0 such that x(t 0 ) Y ≤ R implies x x0 (t) Y ≤ ε for all t > t 0 + T .
Recall that the homogeneity degree specifies convergence rates of stable homogeneous systems (see e.g. [START_REF] Nakamura | Smooth Lyapunov functions for homogeneous differential inclusions[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF] for ODE models and [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF]Theorem 8.6] for evolution systems in Banach spaces). For instance, any uniformly asymptotically stable d-homogeneous system of negative degree is globally uniformly finite-time stable (i.e. the state x = 0 is reached by any trajectory of the system is a finite instant of time), but any uniformly asymptotically stable dhomogeneous system of positive degree is globally uniformly nearly fixed-time stable (i.e. for any neighborhood U ⊂ Y of 0 there exists an instant of time T U ∈ (0, +∞) such that x(t) ∈ U, ∀t ≥ T U independently of initial conditions).

Theorem 3. Let d be a strictly monotone strongly continuous dilation in X and in Y, the evolution equation (1) be d-homogeneous of a degree µ ∈ R, the operators A, f satisfy Assumption 1 and m > 0 be an arbitrary real number. The origin of (1) is uniformly asymptotically stable in Y if and only if there exists a continuous positive definite functional V : Y → R such that 1) V is d-homogeneous of the degree m and locally Lipschitz continuous on Y\{0};

2) there exist k, k > 0 satisfying

k x m d,Y ≤ V (x) ≤ k x m d,Y , ∀x ∈ Y, (11)
where • d,Y is the canonical homogeneous norm induced by • Y ;

3) for any mild solution x x0 of (1) with x(0) = x 0 ∈ Y the inequality

D + V (x x0 (t))≤ -x x0 (t) µ+m d,Y , t > 0 ( 12
)
holds as long as x x0 (t) = 0.

Formally, the latter theorem needs a knowledge of trajectories of (1) to check the condition [START_REF] Hermes | Nilpotent approximations of control systems and distributions[END_REF]. In practice, it is important to analyze the asymptotic stability using the right-hand side of the evolution system only. This can be done under additional restriction on X. The proofs of the above theorem and the following corollary are given in Appendix.

Corollary 1. Theorem 3 remains true even when the condition 3) is replaced with

3') D + V (x; Ax + f (x)) ≤ -x µ+m d,Y , ∀x ∈ M -1 D(A), provided that X is a reflexive Banach space and M -1 D(A) is a d-homogeneous cone dense in Y.
This corollary is applicable, in particular, if X is a Hilbert space.

Example 7. Let us consider the nonlinear equation

ẋ = ∂ 2 x ∂z 2 + x ∂x ∂z - 1 3 x 4 L 2 x, t > 0, x(0) = x 0 , (13)
where x(t) ∈ L 2 (R, R). This equation admits the representation (1)

with A = ∂ 2 ∂z 2 , X = L 2 (R, R), D(A) = H 2 (R, R), f (x) = x ∂x ∂z -1 3 x 4 L 2 x and D(f ) = Y = H 1 (R, R). Notice that f is unbounded but M -regular nonlinear operator with M = ∂ ∂z + I and (M -1 y)(z) = z -∞ e -(z-s) y(s)ds, y ∈ X, z ∈ R, respectively. Indeed, the inequality (2), obviously, follows from ∂ ∂z M -1 < ∞ and M -1 y L ∞ ≤ sup z∈R z -∞ e -(z-s) |y(s)|ds ≤ y L 2 sup z∈R z -∞ e -2(z-s) ds = y L 2 √ 2
, ∀y ∈ X.

The considered system cannot be studied using the results given in [START_REF] Polyakov | Input-to-state stability of homogeneous infinite dimensional systems with locally lipschitz nonlinearities[END_REF]. The equation (13) satisfies Assumption 1 with

(Φ(t)y)(z) = 1 √ 4πt +∞ ∞ e -(z-s) 2 4t
y(s)ds, so it has a unique mild solution for any x 0 ∈ Y\{0}.

The system (13) is d-homogeneous of the degree µ = 2 with respect to the dilation (d(s)x)(z) = e s x(e s z), z ∈ R, x ∈ X. The dilation d is strictly monotone and strongly continuous in X and Y (see Example 1).

Let us show that the system (13) is globally uniformly asymptotically stable and the canonical homogeneous norm • d,Y induced by the norm

• Y = • H 1 is a d-homogeneous Lyapunov function of the system. Indeed, for v ∈ M -1 D(A) = H 3 (R, R)\{0} we have v, Av + f (v) H 1 = v, ∂ 2 v ∂z 2 + v ∂v ∂z -1 3 v 4 L 2 v L 2 + ∂v ∂z , ∂ ∂z ∂ 2 v ∂z 2 + v ∂v ∂z -1 3 v 4 L 2 v L 2 = -∂v ∂z 2 L 2 -1 3 v 6 L 2 -∂ 2 v ∂z 2 2 L 2 -∂ 2 v ∂z 2 , v ∂v ∂z -1 3 v 4 L 2 ∂v ∂z 2 L 2 = -∂v ∂z 2 L 2 -1 3 v 6 L 2- 1 2 ∂ 2 v ∂z 2 2 L 2 -1 2
∂ 2 v ∂z 2 + v ∂v ∂z 2 L 2 + 1 2 v ∂v ∂z 2 L 2 -1 3 v 4 L 2 ∂v ∂z 2 L 2 = -∂v ∂z 2 L 2 -1 3 v 6 L 2- 1 2 ∂ 2 v ∂z 2 2 L 2 -1 2
∂ 2 v ∂z 2 + v ∂v ∂z 2 L 2 -1 6 v 3 , ∂ 2 v ∂z 2 -1 3 v 4 L 2 ∂v ∂z 2 L 2 = -∂v ∂z 2 L 2 -1 3 v 6 L 2 -5 12
∂ 2 v ∂z 2 2 L 2 -1 2
∂ 2 v ∂z 2 +v ∂v ∂z 2 L 2 -1 12 v 3 + ∂ 2 v ∂z 2 2 L2 + 1 12 v 6 L6 -1 3 v 4 L 2 ∂v ∂z 2 L 2 .
Using the Gagliardo-Nirenberg-Sobolev inequality (see [START_REF] Dolbeault | One-dimensional gagliardo-nirenberg-sobolev inequalities: remarks on duality and flows[END_REF]) we derive v 6

L 6 ≤ π 2 v 4 L 2 ∂v ∂z 2 L 2 and conclude v, Av + f (v) H 1 ≤ -∂v ∂z 2 L 2 -1 3 v 6 L 2 .
The latter means that • H1 is a Lyapunov function of the system, but it is not d-homogeneous. Using d-homogeneity of operators A, f and the formula (7), for any x ∈ M -1 D(A) we obtain

D x d,H 1 (Ax + f (x)) = 2 d(-ln x d,H 1 )(Ax+f (x)), v H 1 v 2 L 2 + ∂v ∂z 2 L 2 x d,H 1 = 2 Av+f (v), v H 1 v 2 L 2 + ∂v ∂z 2 L 2 u 3 d,H 1 ≤ 2 - ∂v ∂z 2 L 2 - 1 3 v 6 L 2 v 2 L 2 + ∂v ∂z 2 L 2 x 3 d,H 1 where v = d(-ln x d,H 1 )x. Taking into account v H 1 = 1 we conclude that D x d,H 1 (Ax + f (x)) ≤ - 2 3 x µ+1 d,H 1
and the functional V : Y → R given by V (x) = 3 2 x d,H 1 satisfies Corollary 1. Notice also that the considered system is nearly fixed-time stable in the view of [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF]Theorem 8.6].

Conclusion

The main contributions of the paper are Theorem 3 and Corollary 1, which provide a characterization (in terms of a locally Lipshcitz continuous Lyapunov function) of the uniform asymptotic stability of homogeneous infinite dimensional systems with unbounded nonlinear operators in Banach and reflexive Banach spaces, respectively. The existence of the regular Lyapunov function allows rather simple ISS analysis of homogeneous systems to be developed [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF], [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF], [START_REF] Polyakov | Input-to-state stability of homogeneous infinite dimensional systems with locally lipschitz nonlinearities[END_REF]. Namely, uniform asymptotic stability of unperturbed homogeneous system implies its ISS with respect to homogeneously involved perturbations. The expansion of this result to homogeneous infinite dimensional systems with unbounded nonlinear operators is an interesting problem for the future investigation.

Φ of linear bounded operators on X and f : D(f ) ⊂ X → X satisfy Assumption 1. Let δ > 1 be an arbitrary real number and

f δ (x) = a δ ( x Y )f (x), x ∈ Y, q ∈ V where a δ ∈ C ∞ c (R + , R) such that a δ (s) = 0 for s / ∈ (1/(2δ), 2δ) and a δ (s) = 1 for s ∈ [1/δ, δ]. Then for any x 0 ∈ Y the initial value problem ẋ = Ax + f δ (x), t ≥ 0 x(0) = x 0 ∈ Y ( 14
)
has a unique mild solution x δ : [0, +∞) → Y, which is continuous in Y and coincides with the mild solution x x0 of (1) as long as x x0 (t) ∈ K Y (δ). Moreover, for any x 0 ∈ Y\{0} the system (1) has a mild solution x x0 , which is uniquely defined as long as 0 < x x0 (t) Y < +∞.

Proof.

For

x 1 ∈ X, x 2 ∈ X we have M -1 x 1 , M -1 x 2 ∈ Y and f δ (M -1 x 1 ) -f δ (M -1 x 2 ) X = a δ ( M -1 x 1 Y )f (M -1 x 1 ) -a δ ( M -1 x 2 Y )f (M -1 x 2 ) X ≤ f (M -1 x 1 ) X |a δ ( M -1 x 1 Y )-a δ ( M -1 x 2 Y )|+ |a δ ( M -1 x 2 Y )| f (M -1 x 1 )-f (M -1 x 2 ) X = f (M -1 x 1 ) • |a δ (θ)| • M -1 x 1 Y -M -1 x 2 Y + |a δ ( M -1 x 2 Y )| • f (M -1 x 1 ) -f (M -1 x 2 ) X
where the mean value theorem is utilized for θ ∈ [min

M -1 x i Y , max M -1 x i Y ], i = 1, 2. Notice that M -1 x i ∈ K Y (2δ) implies x i ∈ K X (2ξδ), where ξ = 1+ M -1 X < +∞. Local Lipschitz continuity of f • M -1 yields sup x∈K X (2ξδ) f (M -1 x) < +∞ and ∃ Lδ > 0 such that f (M -1 x 1 ) -f (M -1 x 2 ) X ≤ Lδ x 1 -x 2 X ∀x 1 , x 2 ∈ K X (2ξδ). Since a δ ∈ C ∞ c then sup s∈R |a δ (s)| < +∞. Notice also M -1 x 2 / ∈ K Y (2δ) ⇒ a δ ( M -1 x 2 Y ) = 0. Therefore, taking into account M -1 x 1 Y -M -1 x 2 Y ≤ M -1 (x 1 -x 2 ) Y ≤ ( M -1 + 1) x 1 -x 2 X , we conclude that f δ •M -1 satisfy the Lipschitz condition on X with the Lipschitz constant L δ := Lδ + ( M -1 + 1) sup θ∈R |a δ (θ)| sup x∈K(2ξδ) f (M -1 x) < +∞.
Given x 0 ∈ Y let us consider the metric space

Z = z ∈ C([0, T ], X) : z(0) = M x 0 , sup t∈[0,T ] z(t) X < +∞ ,
with T > 0 and the metric ρ(z 1 , z 2 ) = sup t∈[0,T ] z 1 (t) -z 2 (t) . Let the operator F be defined on Z as follows

(F z)(t) = Φ(t)M x 0 + t 0 M Φ(t -τ )f δ (M -1 z(τ ))dτ. The function τ → f δ (M -1 z(τ )) is Bochner integrable in X if and only if τ → f δ (M -1 z(τ )) X is Lebesgue integrable. Since z ∈ C([0, T ], X) then M - regularity of f implies f (M -1 x(•)) ∈ L 1 ((0, T ), X) and F : Z → Z. Indeed, one has (F z)(t) X ≤ Φ(t) X • M x 0 X + t 0 M Φ(t -τ ) X • ( f δ (M -1 z(τ )) -f δ (0) X dτ ≤ k M x 0 X + L δ t 0 ω(t -τ ) z(τ ) X dτ where k = sup t∈[0,T ] Φ(t) X < +∞ and ω(t -τ ) ≥ M Φ(t -τ ) X (
(F z)(t) X < +∞ for any z ∈ C([0, T ], X). Let us show that the operator F : Z → Z has the unique fixed point F (z * ) = z * with z * ∈ Z. For any z 1 , z 2 ∈ Z we have F (z 1 )(t) -F (z 2 )(t) X ≤ t 0 ω(t -τ ) f δ (M -1 z 1 (τ ), q(τ )) -f δ (M -1 z 2 (τ ), q(τ )) X dτ ≤ t 0 ω(σ)dσ sup t∈[0,T ] z 1 (τ )) -z 2 (τ )) X ≤ L δ,q T 0 ω(σ)dσ ρ(z 1 , z 2 ).
Since T 0 ω(σ)dσ → 0 as T → 0 then for a sufficiently small T > 0 we have η = L δ,q T 0 ω(σ)dσ < 1 and

F k (z 1 )(t) -F k (z 2 )(t) X ≤ η k ρ(x 1 , x 2 ), k = 1, 2, . . . .
Using the contraction principle (Banach Fixed Point Theorem) we deduce the existence of a unique fixed point z * of the operator F in Z, which corresponds to a mild solution x δ := M -1 z * of the nonlinear system [START_REF] Karafyllis | Input-to-State Stability for PDEs[END_REF]. Indeed, using Assumption 1 we derive

x δ (t) = M -1 z * (t) = M -1 Φ(t)M x 0 + M -1 t 0 M Φ(t -τ )f δ (M -1 z * (τ ), q(τ ))dτ = Φ(t)x 0 + t 0 Φ(t -τ )f δ (x δ (τ ), q(τ ))dτ. Notice also x δ (t) = M -1 z * (t) ∈ Y for all t ∈ [0, T ].
The solution x δ can be prolonged for all t ≥ 0. Suppose the converse, i.e exists t 1 ∈ R + such that x δ (t) Y → +∞ as t → t 1 . In this case, M -1 z * (t) Y → +∞ as t → t 1 and there exists t ∈ (0, t 1 ) such that M -1 z * (t) Y ≥ 2δ for all t ∈ (t , t 1 ). The latter implies that f δ (M -1 z * (t), q(t)) = 0 for all t ∈ (t , t 1 ),

z * (t) = Φ(t -t )M x δ (t ) and x δ (t) = Φ(t -t )x δ (t ), ∀t ≥ [t , t 1 ).
Since Φ is a strongly continuous semigroup of linear bounded operators on B then the inequality t 1 -t < +∞ implies sup s∈[0,t1-t ] Φ(s) X < +∞. Using assumption 1 we derive

x δ (t) Y = Φ(t -t )x δ (t ) X + M Φ(t -t )x δ (t ) X = Φ(t -t )x δ (t ) X + Φ(t -t )M x δ (t ) X ≤ Φ(t -t ) X x δ (t ) Y and lim sup t→t1 x δ (t) Y < +∞.
If x 0 ∈ K Y (δ) then, obviously, the mild solution x δ is the unique mild solution of (1) as long as

x δ (t) = M -1 z * (t) ∈ K Y (δ). Since x δ (t + h) -x(t) Y = M -1 (z * (t+h)-z * (t)) X + z * (t+h)-z * (t) X ,
t ≥ t 0 , h > 0 then the continuity of z in X implies the continuity of x δ in Y. Tending δ → +∞ we complete the proof.

Corollary 2 (Continuous dependence of initial conditions). If all conditions of Lemma 2 are fulfilled then for any T > 0 there exists L δ,T > 0 such that for any x 1 , x 2 ∈ Y one has

x δ x1 (t) -x δ x2 (t) Y ≤ L δ,T x 1 -x 2 Y , ∀t ∈ [0, T ] where x δ xi (t) is a mild solution of (14) with x(t 0 ) = x i , i = 1, 2.
Proof. Since

x δ x1 (t) -x δ x2 (t) X ≤ Φ(t) X • x 1 -x 2 X + t 0 Φ(t -τ ) X • f δ (x δ x1 (τ )) -f δ (x δ x2 (τ )) X dτ
then taking into account Lipschitz continuity of f δ • M -1 we derive

x δ x1 (t) -x δ x2 (t) X ≤ k x 1 -x 2 X + kL δ t 0 M x δ x1 (τ ) -M x δ x2 (τ ) X dτ,
where k := sup s∈[0,T ] Φ(s) X < +∞. On the other hand, we have

M x δ xi (t) = z * xi (t) = Φ(t -t 0 )M x 0 + t 0 M Φ(t -τ )f δ (M -1 z * xi (τ ))dτ, i = 1, 2,
where z * xi is defined in the proof of Lemma 2. Hence,

M x δ x1 (t) -M x δ x2 (t) X ≤ h M x 1 -M x 2 X + t 0 M Φ(t -τ ) X • f δ (M -1 z * x1 (τ )) -f δ (M -1 z * x2 (τ )) X dτ ≤ k M x 1 -M x 2 X + L δ t 0 ω(t -τ ) • M -1 z * x1 (τ ) -M -1 z * x2 (τ ) X dτ ≤ k M x 1 -M x 2 X + L δ t 0 ω(t -τ ) • x δ x1 (τ ) -x δ x2 (τ ) X dτ,
where ω(t-τ ) ≥ M Φ(t-τ ) X (see Assumption 1). Combining the two obtained estimates we derive

x δ x1 (t) -x δ x2 (t) Y ≤ k x 1 -x 2 Y + L δ t 0 (k + ω(t -τ )) x δ x1 (τ ) -x δ x2 (τ ) Y dτ.
Hence, using the Grönwall-Bellman inequality we derive

x δ x1 (t) -x δ x2 (t) Y ≤ ke L δ t 0 (k+ω(t-τ ))dτ x 1 -x 2 Y
for all t ∈ [0, T ]. Taking L δ,T := ke L δ T 0 (k+ω(t-τ ))dτ we complete the proof.

Corollary 3. Let all conditions of Lemma 2 be fulfilled and

x 0 ∈ M -1 D(A).
Then the mild solution x δ x0 of ( 14) is 1) locally Lipschitz continuous as a function [0, +∞) → X;

2) locally uniformly continuous as a function [0, +∞) → Y;

3) a strong solution of (14), x(t)

a.e. ∈ M -1 D(A) and M ẋδ (t) a.e. = M Ax δ (t) + M f (x δ (t)), t > 0,
provided that X is a reflexive Banach space.

Proof. Let us show x 0 ∈ M -1 D(A) implies x 0 ∈ D(A) provided that Assumption 1 holds. Indeed, x 0 ∈ M -1 D(A) implies M x 0 ∈ D(A) and x 0 ∈ Y (since M -1 : B → Y). In this case, we have lim

h→0 + Φ(h)M x 0 -M x 0 h = AM x 0 .
On the other hand, Assumption 1 implies

lim h→0 + Φ(h)M x 0 -M x 0 h = lim h→0 + M Φ(h)x 0 -x 0 h .
Since M by assumption is a closed linear operator and Y is its domain then the existence of the above limits yields

Φ(h)x0-x0 h → y * ∈ Y as h → 0.
Since A is a closed linear operator as well then the existence of lim Lemma 2 implies C(T ) < +∞ for any x 0 ∈ Y. Since

h→0 + Φ(h)x0-x0 h ∈ X guarantees x 0 ∈ D(A) and M AM -1 z = Az, ∀z ∈ D(A).
f δ (x δ x0 (τ )) X = f δ (M -1 M x δ x0 (τ )) -f δ (0) X ≤ L δ M x δ x0 (τ ) X then x δ x0 (t+h)-x δ x0 (t) X ≤ Φ(t+h)x 0 -Φ(t)x 0 X + t+h t Φ(t-τ )f δ (x δ x0 (τ )) X dτ ≤ t+h t Φ(τ ) X • Ax 0 X dτ + t+h t Φ(s) f δ (x δ x0 (τ )) X dτ ≤ hk( Ax 0 X +L δ C(T )) for all h ∈ [0, T ],
where k = sup s∈[0,T ] Φ(s) and the Lipschitz constant L δ is defined in the proof of Lemma 2.

2) Now let us prove the uniform continuity of the solution in Y. We have

M x δ x0 (t + h) -M x δ x0 (t) X ≤ Φ(t + h)M x 0 -Φ(t)M x 0 X + t+h t ω(t -τ ) f δ (x δ x0 (τ )) X dτ ≤ t+h t Φ(τ ) X • AM x 0 X dτ + L δ t+h t ω(t -τ ) M x δ x0 (τ ) X ≤ hk AM x 0 X + L δ C(T ) h 0 ω(s)ds
for all h ∈ [0, T ], where ω(s) ≥ M Φ(s) X (see Assumption 1). Combining the estimate obtained in the case 1), we derive

x δ x0 (t + h) -x δ x0 (t) Y ≤ h(k Ax 0 Y + L δ C(T )k) + L δ C(T ) h 0 ω(s)ds.
Taking into account h 0 ω(s)ds → 0 as h → 0, we complete the proof. 3) Let x δ : [t 0 , t 0 + T ] be a mild solution with x 0 ∈ M -1 D(A). Recall (see the proof of Lemma 2) that x δ (t) = M z(t), where t ≥ t 0 and

z(t) = Φ(t)M x 0 + t 0 M Φ(t -s)f δ (M -1 z(s))ds.
Let us show that z : [p, T ] → X satisfies a Lipschitz condition for each p ∈ (0, T ). Indeed, since

z(t + h) -z(t) = (Φ(t + h) -Φ(t))M x 0 + t+h 0 M Φ(t + h -s)f δ (M -1 z(s))ds- t t0 M Φ(t -s)f δ (M -1 z(s))ds = t+h t Φ(s)AM x 0 + h 0 M Φ(t+h-s)f δ (M -1 z(s))ds+ t 0 M Φ(t-s)(f δ (M -1 z(s+h))-f δ (M -1 z(s)))ds. Notice that t+h 0 M Φ(t + h -s)f δ (M -1 z(s))ds X ≤ h 0 ω(t+σ)• f δ (M -1 z(h-σ)) X dσ ≤ hC(T )ω(T, p), where ω(T, p) := max s∈[p,T ] ω(s).
Hence, using Lipschitz continuity of f δ we derive

z(t + h) -z(t) X ≤ (k AM x 0 X + C(T )ω(T, p)) h+ L δ t 0 ω(t -s) z(s + h) -z(s) X ds.
Applying the Grönwall-Bellman inequality we conclude = Ax δ (t) + f δ (x δ (t)), t > 0. Since X is a reflexive Banach space then it satisfies the Radon-Nykodym property, i.e. locally Lipschitz continuity of z implies its differentiability almost everywhere. Since M is a closed linear operator then, taking into account M x δ (t) = z(t), we derive ż(t)

z(t + h) -z(t) X ≤ (k AM x 0 X + C(T )ω(T, p)) e L δ t 0 ω(t-s)ds h, i.e. z : [p, T ] → X satisfies Lipschitz condition for each p ∈ (0, T ). Since z → f δ (M -1 z) is Lipschitz continuous and f δ (x δ (t)) = f δ (M -1 z(t)), t ∈ [0, T ] then the function t → f (x δ (t
a.e. = M ẋδ (t) a.e. = M Ax δ (t) + M f (x δ (t), 0), t ≥ 0 and ż(t) a.e. = M AM -1 z(t) + M f (M -1 z(t), 0), t ≥ 0, i.e. z is a strong solution of the latter equation. Assumption 1 implies that Φ(t)v := M Φ(t)M -1 v = Φ(t)v, ∀t ≥ 0, ∀v ∈ X,
and Φ is a strongly continuous semigroup of linear bounded operators on X. Since the operator M AM -1 : D(A) ⊂ X → X is, obviously, an infinitesimal generator of Φ and Φ = Φ then M AM -1 = A due to the uniqueness of an infinitesimal generator. Using [21, Chapter 4, Theorem 2.9] we derive z(t) a.e. ∈ D(A) and, consequently,

x δ (t) = M -1 z(t) a.e. ∈ M -1 D(A). The proof is complete.
Corollary 4. Let all conditions of Lemma 2 be fulfilled. Let the origin of (1) be globally uniformly Lyapunov stable in Y. Let r > 1 be an arbitrary real number and

δ > max ε(r), 1 ε -1 (1/r) ,
where ε ∈ K ∞ is given by (10) and ε -1 is the inverse function to ε. Then 1) for any x 0 ∈ Y : 1/r ≤ x 0 Y ≤ r we have

x δ x0 (t) Y ≤ ε( x 0 Y ) ≤ δ, ∀t ≥ 0 2) if ∃t 1 ≥ 0 such that x δ (t 1 ) Y ≤ 1/δ then x δ x0 (t) Y ≤ 1/r, ∀t ≥ t 1 ,
where x δ x0 is the unique mild solution of (14). Proof. 1) Notice that the the inequality (10) and x x0 (0) = x 0 yield ε(r) ≥ r and 1/δ < 1/r < r < δ. Since x δ x0 coincide with the solution x x0 of (1) as long as

x δ x0 (t) ∈ K Y (δ) then the inequality (10) implies x δ x0 (t) Y ≤ ε( x 0 Y ) ≤ δ for all t ≥ 0 and any x 0 : x 0 Y ≤ [1/r, r]. 2) Suppose the contrary, i.e. ∃t 2 > t 1 such that x δ x0 (t 2 ) > 1/r and x δ x0 (t) > 1/δ for t ∈ (t 1 , t 2 ].
In this case, we have x δ x0 (t) = x x0 (t) for all t ∈ (t 1 , t 2 ] and using the inequality (10) we conclude

x δ x0 (t) Y ≤ ε( x δ x0 (t 1 ) Y ) = ε(1/δ) ≤ ε(ε -1 (1/r)) = 1/r
for all t ≥ t 1 .

The proof of Theorem 3

Notice that the existence of a d-homogeneous Lyapunov functional V : Y → R of the degree m > 0 satisfying [START_REF] Grüne | Homogeneous state feedback stabilization of homogeneous systems[END_REF], ( 12) is equivalent to the existence of a d-homogeneous Lyapunov functional Ṽ : Y → R of the degree 1 satisfying

c x d,Y ≤ Ṽ (x) ≤ c x d,Y , ∀x ∈ Y, c, c > 0 (15)
and

D + Ṽ (x x0 (t))≤ -x x0 (t) µ+1 d,Y , t > 0, (16)
respectively. Indeed, selecting

Ṽ = mk 1-1/m V 1/m for m ∈ (0, 1) (resp. Ṽ = mk 1-1/m V 1/m for m > 1 )
we easily derive the mentioned inequalities with c = mk and c = mk

1/m k 1-1/m (resp. c = mk 1-1/m k 1/m and c = mk).
This means that without loss of generality we can fix m = 1 for the proofs of Theorem 3 and Corollary 1. Sufficiency. Let δ > 1 be an arbitrary number. Then by Lemma 2 the solution x δ x0 of the system (14) with f δ (x) = a δ ( x Y )f (x). coincides with the solution x x0 of (1) as long as x δ x0 (t) ∈ K Y (δ). Using the inequality [START_REF] Hermes | Nilpotent approximations of control systems and distributions[END_REF] we conclude that the function t → V (x δ x0 (t)) is monotone decreasing and

V (x δ x0 (t))
a.e.

= D + V (x δ x0 (t)) ≤ -x δ x0 (t) µ+1 d .
Hence, we conclude

V (x δ x0 (t)) ≤ V (x 0 ) - t 0 x δ x0 (t) µ+1 d ds ≤ V (x 0 ) - 1 k t 0 V µ+1 (x δ x0 (t))ds
as long as x δ x0 (t) ∈ K Y (δ). Tending δ → +∞ we conclude that for any x 0 ∈ Y\{0} the function t → V (x x0 (t)) is strictly decreasing and

V (x x0 (t)) ≤ V (x 0 ) - 1 k t 0 V µ+1 (x x0 (t))ds
as long as 0 < x x0 (t) Y < +∞. Hence, for any r > 0 we derive

V (x x0 (t)) ≤ r, ∀t ≥ max{0, k(V -µ (x 0 ) -r -µ )}.
The latter implies that mild solutions of (1) converges to zero uniformly on the initial data, i.e. for any R > r > 0 there exists T = T ( R, r) such that x x0 (t) ≤ r for all x 0 ∈ B Y (r) and for all t ≥ T . Moreover, by assumption, for all x 0 ∈ Y\{0} we have

k x x0 (t) d,Y ≤ V (x x0 (t)) ≤ V (x 0 ) ≤ k x 0 d,Y and x x0 (t) Y ≤ ε( x 0 ) := σ -1 k k σ( x 0 Y ) (17)
as long as x x0 (t) = 0, where σ -1 is the inverse function to σ (see Theorem 1). Since σ, σ ∈ K ∞ then ε ∈ K ∞ . Let us prove that if x x0 (t 0 ) = 0 then x x0 (t) = 0 for all t ≥ t 0 . Suppose the converse, i.e. ∃t 1 > t 0 : := x(t 1 ) Y > 0. Then due to continuity of t →

x(t) Y there exists t 2 ∈ (t 0 , t 1 ) such that x(t 2 ) Y = δ := min{ /2, ε -1 ( /2)} and x(t) Y > δ for all t ∈ (t 2 , t 1 ], where ε -1 ∈ K ∞ is the inverse function to ε.

In this case, from the inequality (17) we derive

x x0 (t) Y ≤ ε( x(t 2 ) Y ) = ε(δ) ≤ ε(ε -1 ( /2)) = /2
for all t ∈ (t 2 , t 1 ]. This contradicts to the supposition x x0 (t 1 ) Y = . Therefore, the inequality ( 17) holds for all t ≥ 0 and any mild solution of (1) in Y.

Necessity. We design the Lyapunov function in two steps: first, we construct a function V r : Y → R + such that t → V r (x 0 (t)) is strictly decreasing as long as x x0 (t) ∈ K Y (r) for some r > 1; next, using V r we generate a d-homogeneous Lyapunov function V : B → R + by means of an integral transformation inspired by [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF]. I. Let s 0 > 0, r 0 := sup which, for x 0 ∈ K Y (δ), coincides with the unique solution x x0 of (1) as long as x δ x0 (t) ∈ K Y (δ). Moreover, by Corollary 2, for any T > 0 there exists L δ,T > 0 such that

x δ x1 (t) -x δ x2 (t) Y ≤ L δ,T x 1 -x 2 Y , for all t ∈ [0, T ].
Since the origin of ( 1) is globally uniformly asymptotically stable then there exists T r > 0 such that

x 0 Y < r ⇒ x x0 (t) Y ≤ 1/r, ∀t ≥ T r .
In the view of Lemma 2 and Corollary 4 we have x δ x0 (t) = x x0 (t) as long as x x0 (t) Y ≥ 1/δ provided that x 0 ∈ K Y (r). Indeed, the global uniform Lyapunov stability of (1) and the selection of the parameter δ > r imply that x δ x0 (t) Y ≤ δ for all t ≥ 0 and all x 0 ∈ K Y (r). Moreover, if there exists t 1 > 0 such that x δ x0 (t 1 ) Y ≤ 1/δ (resp.

x x0 (t 1 ) Y ≤ 1/δ) then x δ x0 (t) Y ≤ 1/r (resp. x x0 (t) Y ≤ 1/r
) for all t ≥ t 1 . In this case, for the functional V 0 : Y → R + given by

V 0 (x 0 ) := sup t≥0 x x0 (t) Y we have V 0 (x 0 ) = sup t∈[0,Tr]
x δ x0 (t) Y for all x 0 ∈ K(r). Hence, using the triangle inequality we derive

|V 0 (x 1 )-V 0 (x 2 )| ≤ sup t∈[0,Tr] x δ x1 (t)-x δ x2 (t) Y ≤ L δ,Tr x 1 -x 2 Y , ∀x 1 , x 2 ∈ K Y (r).
The latter means that the functional V 0 satisfy the Lipschitz condition on K Y (r).

The global uniform Lyapunov stability of (1) implies that x Y ≤ V 0 (x) ≤ ε( x Y ), ∀x ∈ Y, where ε ∈ K is given by [START_REF] Folland | Subelliptic estimates and function spaces on nilpotent Lie groups[END_REF]. Moreover, the function t → V 0 (x x0 (t)) is decreasing as long as x x0 (t) ∈ K Y (r). Let us consider the functional

V 1 (x 0 ) := +∞ 0 ã( x x0 (s) Y ) x x0 (s) Y ds, where ã ∈ C ∞ c is such that ã(ρ) = 0 for s / ∈ (1/r, r), 0 < ã(ρ) ≤ 1 for ρ ∈ (1/r, r) and ã(ρ) = 1 for ρ ∈ [1/r 0 , r 0 ]. Notice also ã( x x0 (s) Y ) x x0 (s) Y = 0 if x x0 (s) / ∈ K Y (r).
Repeating the above considerations we conclude that

V 1 (x 0 ) = Tr 0 ã( x x0 (s) Y ) x x0 (s) Y ds = Tr 0 ã( x δ x0 (s) Y ) x δ x0 (s) Y ds for all x 0 ∈ K Y (r). Since x → ã( x Y )
x Y satisfy the Lipschitz condition on Y with some Lipschitz constant L > 0 then

|V 1 (x 1 ) -V 1 (x 2 )| ≤ Tr 0 ã( x δ x1 (s) Y ) x δ x1 (s) Y -ã( x δ x2 (s) Y ) x δ x2 (s) Y ds ≤ Tr 0 |ã( x δ x1 (s) Y ) x δ x1 (s) Y -ã( x δ x2 (s) Y ) x δ x2 (s) Y |ds ≤ L Tr 0 x δ x1 (s) -x δ x2 (s) Y ds ≤ T r L δ,Tr L x 1 -x 2 Y , ∀x 1 , x 2 ∈ K Y (r),
i.e. V 1 also satisfies the Lipschitz condition on K Y (r). Moreover, if h > 0 and t ≥ 0 are such that

x x0 (t + θ) ∈ K(r), ∀θ ∈ [0, h] then x x0 (t + θ) = x δ xx 0 (t) (θ), ∀θ ∈ [0, h] and V 1 (x x0 (t + h)) -V 1 (x x0 (t)) = Tr 0 ã( x δ x(t+h) (s) Y ) x δ x(t+h) (s) Y ds - Tr 0 ã( x δ x(t) (s) Y ) x δ x(t) (s) Y ds = Tr 0 ã( x δ x(t+h) (s) Y ) x δ x(t+h) (s) Y ds - h 0 ã( x δ x(t) (s) Y ) x δ x(t) (s) Y ds- Tr+h h ã( x δ x(t) (s) Y ) x δ x(t) (s) Y ds = Tr 0 ã( x δ x(t+h) (s) Y ) x δ x(t+h) (s) Y ds - h 0 ã( x δ x(t) (s) Y ) x δ x(t) (s) Y ds- Tr 0 ã( x δ x(t) (s + h) Y ) x δ x(t) (s + h) Y ds = - h 0 ã( x δ x(t) (s) Y ) x δ x(t) (s) Y ds = - h 0 ã( x x(t) (s) Y ) x x(t) (s) Y ds = -ã( x x0 (t + θ h ) Y ) x x0 (t + θ h ) Y h, θ h ∈ [0, h],
where the mean value theorem for integrals is utilized on the last step. The latter means that lim sup

h→0 + V1(xx 0 (t+h))-V1(xx 0 (t)) h ≤ -ã( x x0 (t) Y ) x x0 (t) Y as long as x x0 (t) ∈ K Y (r). Notice that 0 ≤ V 1 (x 0 ) ≤ T r ε( x 0 Y ).
In this case, the functional V r : B → R + given by

V r (x) := V 0 (x) + V 1 (x)
satisfies the Lipschitz condition on K Y (r) and

x Y ≤ V r (x) ≤ (1 + T r )ε( x Y )
for all x ∈ Y. Moreover, for any

x 0 ∈ K Y (r) the function t → V r (x x0 (
: Y → R + given by V (x) := +∞ -∞ e s-q â(V r (d(-s)x))ds
where â ∈ C ∞ is an increasing function such that

• â(ρ) = 1 for ρ ≥ (1 + T r )ε(r);

• â(ρ) = 0 for ρ ≤ 1/r;

• â (ρ) > 0 for 1/r < ρ < (1 + T r )ε(r)

• min 1/r0≤ρ≤(1+Tr)ε(r0) â (ρ) > 0 and q := ln min{e (1+µ)s 0 ,e -(1+µ)s 0 } min 1/r 0 ≤ρ≤(1+Tr )ε(r 0 ) â (ρ) r0

.

For example, the function â can be selected as follows

â(ρ) = e - ρmax-ρ min ρ-ρ min e -ρmax-ρ min ρmax-ρ +e for ρ ∈ (ρ min , ρ max ), where ρ min = 1/r, ρ max = (1 + T r )ε(r). By construction, the functional V : Y → R + is d-homogeneous of the degree 1. Moreover, V (x) ≤ +∞ -∞ e s-q â((1 + T r )ε( d(-s)x) Y ))ds ≤ s(x) -∞ e s-q â((1 + T r )ε( d(-s)x) Y ))ds ≤ e -q e s(x)
where s(x) := ln

x d,Y σ(ε -1 ( 1 r(1+Tr ) ))
and σ ∈ K ∞ is defined in Theorem 1. Indeed, if s ≥ s(x) then from Theorem 1 we derive

σ ( d(-s)x Y ) ≤ d(-s)x d,Y = e -s x d,Y ≤ e -s(x) x d,Y = σ ε -1 1 r(1+Tr) and taking into account σ, ε ∈ K ∞ we conclude d(-s)x) Y ≤ ε -1 1 r(1+Tr)
or, equivalently, (1+T r )ε( d(-s)x) Y ) ≤ 1/r for all s ≥ s(x). The latter means V r (d(-s)x) ≤ 1/r and â(V r (d(-s)x))) = 0 for all s ≥ s(x).

Similarly, we derive

V (x) ≥ +∞ -∞ e s-q â( d(-s)x) Y )ds ≥ s(x)
-∞ e s-q ds = e -q e s(x) , where s(x) := ln

x d,Y σ((1+Tr)ε(r)) . Indeed, if s ≤ s(x) then from Theorem 1 we have σ( d(-s)x Y ) ≥ d(-s)x d,Y = e -s x d ≥ e -sx x d,Y = σ((1 + T r )ε(r)). Since σ ∈ K ∞ then d(-s)x Y ≥ (1 + T r )ε(r) and â( d(-s)x Y )) = 1 for all s ≤ s(x)
. Therefore, we have proven that the inequality (11) holds for k := e -q σ ((1 + T r )ε(r))

, k := e -q σ ε -1

1 r(1+Tr) . If x(t + θ) ∈ K Y (r 0 ) for all θ ∈ [0, h] then d(-s)x(t + θ) ∈ K Y (r) for all θ ∈ [0, h] and for all s ∈ [-s 0 , s 0 ]. Indeed, d(-s)x(t + θ) Y ≤ d(-s) Y • x(t + θ) Y ≤ r 2 0 = r and 1/r 0 ≤ x(t + θ) Y = d(s) Y • d(-s)x(t + θ) Y .
The latter means that

1/r ≤ V r (d(-s)x x0 (t + θ)) ≤ (1 + T r )ε(r) for all s ∈ [-s 0 , s 0 ] and all θ ∈ [0, h]. Using V r (d(-s)x x0 (t + h)) -V r (d(-s)x x0 (t)) ≤ -W r (x d(-s)x0 (e µs t + θ h ))e µs h = -W r (d(-s)x x0 (t + e -µs θ h ))e µs h we derive V (x x0 (t + h)) -V (x x0 (t)) = +∞ -∞ e s-q [â(V r (d(-s)x x0 (t + h))) -â(V r (d(-s)x x0 (t)))]ds = s0 -s0 e s-q [â(V r (d(-s)x x0 (t + h))) -â(V r (d(-s)x x0 (t)))]ds = s0 -s0 e s-q â (θ)[V r (d(-s)x x0 (t + h)) -V r (d(-s)x x0 (t))]ds ≤ -h s0 -s0
e s-q â (θ)W r (d(-s)x x0 (t + e -µs θ h ))e µs ds,

where the mean value theorem is utilized with

θ ∈ [V r (d(-s)x x0 (t + h)), V r (d(-s)x x0 (t))].
Therefore, the function t → V (x x0 (t)) is strictly decreasing and lim sup = M Ax(t) + M f (x(t)), t ≥ 0, as long as 0 < x(t) Y < +∞. Since V (x(t+h))-V (x(t))) h = V (x(t+h))-V (x(t)+h(Ax(t)+f (x(t))) h + V (x(t)+h(Ax(t)+f (x(t)))-V (x(t)) a.e.

h→0 + V (xx 0 (t+h))-V (xx 0 (t)) h ≤ - +∞ -∞ e (1+µ)s-q â (V r (d(-s)x x0 (t)))W r ( d(-s)x x0 (t) Y )ds = -W (x x0 (
= D + V (x(t), Ax(t) + f (x(t)), ∀x 0 ∈ M -1 D(A)\{0}.

Sufficiency. The condition 3') imply the fulfillment of the condition 3) of Theorem 3 for all x 0 ∈ M -1 D(A). This means that all corresponding solutions uniformly converge to 0 and satisfy the inequality [START_REF] Folland | Subelliptic estimates and function spaces on nilpotent Lie groups[END_REF]. Since M -1 D(A) in dense in Y then the continuous dependence of solutions on initial conditions implies the same properties for all solutions with x 0 ∈ Y\{0}. This guarantees the uniform asymptotic stability of the system (1) in Y.

Necessity. In the proof of Theorem 3 we design a Lyapunov function satisfying 1), 2) and D + V (x(t)) ≤ -x(t) 1+µ d,Y , as long as x(t) ∈ Y\{0}. Let t j → 0, t j > 0 as j → +∞ be an arbitrary a sequence of time instances such that x(t j ) ∈ M -1 D(A). Since any mild solution is continuous function of time in Y then for any h > 0 there exists i : x(t i ) -x 0 Y < h 2 and x(t i + h)x(h) Y < h 2 . Obviously, i → +∞ as h → 0. Let us consider The proof is complete.

  as s → +∞}; the symbol a.e = (resp. a.e

Definition 3 .

 3 A mapping d : R → L(B, B) is said to be a group of linear dilations (or simply dilation) in a Banach space B if 1) (Group property) d(0) = I, d(t+s) = d(t)d(s), t, s ∈ R; 2) (Limit property) lim s→-∞ d(s)u B = 0 and lim s→+∞ d(s)u B = +∞ for any u = 0. Obviously, d is an one-parameter group of linear bounded invertible operators and d(-s) = (d(s)) -1 , ∀s ∈ R. The limit property specifies groups being dilations in B. Definition 4. A dilation d is strongly continuous if s → d(s)u is continuous for any u ∈ B.

  see the assumption 1). Taking into account integrability of ω on [0, T ] we derive sup t∈[0,T ]

1 )

 1 Recall (see e.g.[START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF], page 5) that if z ∈ D(A) then AΦ(s)z = Φ(s)Az andΦ(s + h)z -Φ(s)z = As+h s Φ(τ )zdτ = s+h s Φ(τ )Azdτ for all s ≥ 0 and all h > 0. Let us denote C(T ) := ess sup s∈[0,T ] M x δ x0 (s) X , where T > 0.

1 .

 1 Let δ > r and a δ ∈ C ∞ c be defined as in Corollary 4. Then for any x 0 ∈ Y the system (14) has the unique mild solution x δ x0 : [0, +∞) → Y

  t)), as long as x x0 (t) ∈ K Y (r 0 ), whereW (x) := +∞ -∞ e (1+µ)s-q â (V r (d(-s)x))W r ( d(-s)x Y )ds.The functional W is, obviously, nonnegative and d-homogeneous of the degreeµ + 1. If x Y = 1 then d(-s)x ∈ K Y (r 0 ) and W r ( d(-s)x Y ) = d(s)x Y for all s ∈ [-s 0 , s 0 ]. Using the mean value theorem for integrals we deriveW (x) ≥ s0 -s0 e (1+µ)s-q â (V r (d(-s)x)) d(-s)x Y ds ≥ 1 r0 e (1+µ)s * â (V r (d(-s * )x)),wheres * ∈ [-s 0 , s 0 ]. Since d(-s * )x ∈ K Y (r 0 ) then 1 r < 1 r 0 ≤ V r (d(-s * )x) ≤ (1 + T r )ε(r 0 ) < (1 + T r )ε(r).Hence, we derive infx Y =1 W (x) ≥ min{e (1+µ)s 0 ,e -(1+µ)s 0 } min 1/r 0 ≤ρ≤(1+Tr )ε(r 0 ) d-homogeneity of W we conclude W (x) = W (d(ln x d,Y )d(-ln x d,Y )x) = x 1+µ d,Y W (d(ln x d,Y )d(-ln x d,Y )x) ≤ x 1+µ d,Y. Therefore, we have proven that lim suph→0 + V (xx 0 (t+h))-V (xx 0 (t)) h ≤ -x x0 (t) 1+µ d,Y as long as x x0 (t) ∈ K Y (r 0 ). If x x0 (t) ∈ d(τ )K Y (r 0 ), where τ ∈ R, then d(-τ )x x0 (t) ∈ K Y (r 0 ). Using homogeneity of V we derive V (x x0 (t)) = V (d(-τ )d(τ )x x0 (t)) = e -τ V (d(τ )x x0 (t)).Since by Theorem 2 we have x d(τ )x0 (e -µτ t) = d(τ )x x0 (t) then the function t → V (x x0 (t)) is strictly decreasing as well and lim suph→0 + V (xx 0 (t+h))-V (xx 0 (t)) h = e -τ s lim sup h→0 + V (x d(τ )x 0 (e -µτ (t+h)))-V (x d(τ )x 0 (e -µτ t)) h ≤ -e -τ (1+µ) x d(τ )x0 (e -µτ t) 1+µ d = -d(-τ )x d(τ )x0 (e -µτ t) 1+µ d = -x x0 (t) 1+µ d as long as x x0 (t) ∈ d(τ )K Y (r 0 ), where τ ∈ R is an arbitrary real number. Taking into account τ ∈R d(τ )K Y (r 0 ) = Y\{0} we complete the proof.5.3. The proof of Corollary 1Since X is a reflexive Banach space then by Corollary 3 (case 3), for any x 0 ∈ M -1 D(A)\{0} we have a unique mild solution x : [0, T ) → M -1 D(A) of (1) satisfying ẋ(t) a.e.= Ax(t) + f (x(t)) and M ẋ(t) a.e.

h

  and V is locally Lipschitz continuous in Y then, taking into account|V (x(t+h))-V (x(t)+h(Ax(t)+f (x(t)))| h ≤ L x(t+h)-x(t)-h(Ax(t)+f (x(t)) Y h = L x(t+h)-x(t) h -ẋ(t)Y → 0 as h → 0 for amost all t ≥ 0, we deriveD + V (x(t))

V.≤≤

  (x0+h(Ax0+f (x0))-V (x0) h = V (x0+h(Ax0+f (x0))-V (x0+h(Ax(ti)+f (x(ti)) h + V (x0+h(Ax(ti)+f (x(ti)))-V (x(ti)+h(Ax(ti)+f (x(ti)) h + V (x(ti)+h(Ax(ti)+f (x(ti)))-V (x(ti+h)) h + V (x(ti+h))-V (x(h)) h + V (x(h))-V (x0) hThe third term in the above sum tends to zero as h → 0 + (see above). Since the operators A and M are closed thenx(t i ) → x 0 , x(t i ) ∈ M -1 D(A) implies M Ax(t i ) → M Ax 0 , Ax(t i ) → Ax 0 and |V (x0+h(Ax0+f (x0))-V (x0+h(Ax(ti)+f (x(ti))| h Ax 0 -Ax(t i ) Y + f (x 0 ) -f (x(t i )) Y → 0 as h → 0.Using the local Lipschitz continuity of V we derive|V (x0+h(Ax(ti)+f (x(ti)))-V (x(ti)+h(Ax(ti)+f (x(ti))| h ≤ L x0-x(ti) Y h Lh → 0 as h → 0.Similarly, we obtain|V (x(ti+h))-V (x(h))| h ≤ Lx 0 x0-x(ti) Y h ≤ Lh → 0 as h → 0.Therefore, ∀x 0 ∈ M -1 D(A) we have lim suph→0 + V (x0+h(Ax0+f (x0))-V (x0) h = lim sup h→0 + V (x(h))-V (x0) h ≤ -x 0 µ+1 d,Y .

  . Examples of linear dilations in Banach spaces are considered below. Let us consider the one-parameter group of linear invertible operators in the Sobolev space H p (R n , R m ) given by

	Definition 5. A dilation d is strictly monotone in B if ∃β > 0 : d(s) B ≤
	e βs , ∀s ≤ 0.
	Monotonicity of the dilation implies that the function s → d(s)u B is mono-
	tone, u ∈ B.
	Example 1.

  Obviously, d(s)u d = e s u d and u d = -u d for ∀u ∈ B and ∀s ∈ R. Moreover, u d = r ⇔ u ∈ S d (r), r > 0. Notice that • d = • B provided that d is the uniform (standard) dilation d(s) = e s I, s ∈ R.

	5)
	is called the canonical homogeneous norm in B, where d is a strictly monotone
	dilation in B.
	Theorem 1 ([24] and [22], Lemmas 7.1, 7.2). If d is a strongly continuous
	strictly monotone dilation then • d is single-valued, positive definite, locally
	Lipschitz continuous on B\{0} and there exist α

  Y ) as long as x x0 (t) ∈ K(r), where W r (ρ) := ã(ρ)ρ with ρ ∈ R + . Due to homogeneity of the system (1) the solutions x x0 are symmetric (see Theorem 2).V r (x d(-s)x0 (e µs (t + h))) -V r (x d(-s)x0 (e µs t)) ≤ -ã( x d(-s)x0 (e µs t + θ h ) Y ) x d(-s)x0 (e µs t + θ h ) Y e µs h,where θ h ∈ [0, e µs h], and µ ∈ R is the homogeneity degree of the system (1), i.e. the function t → V r (d(-s)x x0 (t)) is strictly decreasing

	Hence, if d(-s)x x0 (t + θ) ∈ K Y (r) for all θ ∈ [0, h] then using the identity
	d(-s)x x0 (t) = x d(-s)x0 (e µs t), ∀s ∈ R we derive
		V r (d(-s)x x0 (t + h)) -V r (d(-s)x x0 (t)) =
	lim sup h→0 +	Vr(d(-s)xx 0 (t+h))-Vr(d(-s)xx 0 (t)) h
			t)) is
	strictly decreasing,
		lim sup h→0 +	Vr(xx 0 (t+h))-Vr(xx 0 (t)) h

≤ -W r ( x x0 (t) ≤ -ã( d(-s)x x0 (t) Y ) d(-s)x x0 (t) Y e µs as long as d(-s)x x0 (t) ∈ K Y (r).

II. Inspired by

[START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF]

let us consider the functional V


Ali El Amine
email: ali.elamine@orange.com

Valery Guillet
email: valery.guillet@orange.com

Device-Free People Counting Using 5 GHz Wi-Fi Radar in Indoor Environment with Deep Learning

Keywords: Wi-Fi sensing, Doppler radar, presence detection, people counting, deep learning

People counting plays an important role in many people-centric applications including crowd control, traffic management and smart home energy management. With the advancements in wireless sensing, it is now possible to intelligently sense the presence of people with wireless signals. Yet, a lot of challenges arise when Wi-Fi solutions are used for counting humans due to the uncertainty of the states in the environment. In this paper, we propose a novel 3D-Convolutional Neural Network (3D-CNN) architecture able to extract features from range-Doppler images to count the number of people present in an indoor environment by detecting their movements. We generate the range-Doppler images from a Celeno Wi-Fi pulse Doppler radar that uses the 5 GHz frequency band. To the best of our knowledge, this work is the first to count people based on a Wi-Fi Doppler radar. Our experimental results show that our deep learning model is able to estimate the number of people for up to four with an average accuracy of 89%.

I. INTRODUCTION

Wireless sensing has received great attention in recent years. A lot of research efforts have revealed the sensing ability to recognize human activity, human identification, localization and beyond [START_REF] Khalili | Wi-fi sensing: Applications and challenges[END_REF]. In particular, Wi-Fi environmental sensing plays an important role in the areas of health care, smart home, security and virtual games [START_REF] Oscar | A survey on human activity recognition using wearable sensors[END_REF]. Among its various applications, people counting in indoor environments is of great interest to many people-centric, Internet of Things (IoT) applications including crowd control, traffic management and smart home energy management. Classical solutions to this problem are devised into two broad categories: image-based and non-image based solutions. Image-based counting solutions use specific hardware (e.g., camera) for people counting. In addition to being an expensive approach, this method has many drawbacks such as Line-of-Sight (LOS) requirement, lighting conditions requirements, and privacy problems.

Non-image based solutions perform counting based on wireless signals (infrared, ultrasound, Wi-Fi, etc.). These solutions alleviate the previous drawbacks as they can perform in a Non-Line-of-Sight (NLOS) environment. They do not require lighting conditions, they significantly reduce privacy concerns and they are economical and practical solutions since they do not require additional equipment to be installed. Among the different radio technologies, Wi-Fi emerges as a promising and effective solution in wireless sensing. This can be best achieved by utilizing the Channel State Information (CSI) values captured from Wi-Fi signals highlighting the different multipaths distortions based on human activity [START_REF] Yang | From rssi to csi: Indoor localization via channel response[END_REF].

Feature extraction combined with Machine Learning (ML) models have received great attention in activity recognition and people counting during the last decade [START_REF] Khalili | Wi-fi sensing: Applications and challenges[END_REF]. In particular, Deep Learning (DL) has paved the way as an appealing option in extracting useful features for different classification tasks while adapting to real-world imperfections. One of the earliest work on counting people using DL is found in [START_REF] Liu | Wicount: A deep learning approach for crowd counting using wifi signals[END_REF]. The authors explored the correlation between people count (up to five people) and Wi-Fi CSI variations using a fully connected feed-forward neural network with two hidden layers. Their model achieved an accuracy of 78% when neither action nor position restrictions are imposed on the volunteers. The work was extended in [START_REF] Liu | Deepcount: Crowd counting with wifi via deep learning[END_REF] where a Convolutional Neural Network (CNN) combined with Long Short Term Memory (LSTM) DL architecture was adopted to resolve the dependencies of number of people and CSI. With this architecture, the accuracy was improved to 84.6%. In another work on counting humans using DL, the authors in [START_REF] Talaat | Crosscount: A deep learning system for device-free human counting using wifi[END_REF] leveraged Recurrent Neural Network (RNN) to count people walking in an indoor environment by placing the transmitter and receiver behind the walls. The authors tested their system for a total number of 10 simultaneous people reaching an accuracy of 59%. In a similar work, the authors in [START_REF] Sobron | A deep learning approach to device-free people counting from wifi signals[END_REF] proposed a CNN architecture to extract features from the received CSI information to link it with the number of moving people. The model achieved an average accuracy of 71% for counting up to five people in different indoor locations.

The above mentioned work considered the relationship between CSI waveforms and people counting. Doppler radar systems have been also proposed to capture the human body motion. Opportunistic monitoring using passive Doppler radar was considered in [START_REF] Li | Passive radar for opportunistic monitoring in e-health applications[END_REF]. With a wireless energy transmitter and a radar sensor, the authors generated Doppler spectrograms to extract the person's activity from a set of actions. In a similar work, the authors in [START_REF] Al-Rahbi | Human activity recognition using channel state information[END_REF] studied Doppler scalograms for different human activities including walking, clapping, falling and sitting.

In this paper, we propose a novel 3D-CNN model able to extract features from range-Doppler images to count the number of people present in an indoor environment. The experiment is carried out using a commercial Wi-Fi pulse Doppler radar that operates in the 5 GHz frequency band. This radar can be integrated in any classical Wi-Fi access point that uses the same frequency band. To the best of our knowledge, this work is the first to count people based on Doppler and micro-Doppler frequencies measured from movements such as walking and breathing. In contrast to [START_REF] Liu | Wicount: A deep learning approach for crowd counting using wifi signals[END_REF]- [START_REF] Sobron | A deep learning approach to device-free people counting from wifi signals[END_REF] that measure the CSI between two locations, in this paper, and for the best of our knowledge, we propose the first solution to use a single device able to generate range-Doppler images for feature extraction to classify people count. We achieve an accuracy better than the ones attained in the previous presented literature. Different from [START_REF] Li | Passive radar for opportunistic monitoring in e-health applications[END_REF], [START_REF] Al-Rahbi | Human activity recognition using channel state information[END_REF] that limit the work to one person only, our model extends the presence detection up to four people.

The rest of the paper is organized as follows. Section II describes the pulse Doppler radar used in this work and details the signal processing steps. In Section III, we detail the architecture of the 3D-CNN used in this work. The evaluation of our proposed model is reported in Section IV. Finally, Section V concludes the paper.

II. SYSTEM DESIGN AND IMPLEMENTATION

In this section, we first present the hardware description and the performance of the pulse Doppler radar used in this work to sense the environment. Then, we detail the signal processing steps performed to generate the range-Doppler images capturing the presence of people. Finally, we describe the experimental setup carried out in this work. The overview of the system design block diagram is summarized in Fig. 1.

MIMO Radar Transceiver

A. Wi-Fi radar overview and performance

The measurements in this paper have been carried out by an active pulse Doppler radar based on Celeno 802.11ac Wi-Fi chipset in the 5 GHz frequency band [START_REF]Celeno Wi-Fi Doppler imaging technology[END_REF]. This radar contains one transmitter and 4 receiving antennas. Its antenna elements geometry corresponds to a uniform linear array. For calibration purposes, a reference channel that usually contains the signal from source is acquired by one of the receiving antennas' port. The radar uses 80 MHz of bandwidth to generate Wi-Fi radio pulses with a period of 1 ms (can be modified), then it collects the back scattered echos in the form of CSI data of the shifted Doppler frequencies. We summarize in Table I the radar radio characteristics. Celeno radar has the potential to detect objects as far as 10 meters in LOS and 5-6 meters in NLOS depending on obstructing materials. It has a range resolution of ∆r = c/2B = 1.875 m. The Doppler frequency resolution ∆F d is a function of the observation time frame T f rame , ∆F d = 1/T f rame . In order to be able to detect people with different movement activities (e.g. walking and breathing), a fine Doppler resolution is required. In the next section, we discuss how to adjust the Doppler resolution to detect the sensitivity from different actions.

B. CSI processing

The presence of people in an indoor environment can be detected using pulse Doppler radar by either sensing movement or detecting breathing. These features can be obtained by measuring the Doppler frequencies received from the reflected signals of moving objects (in this case people). We divide the signal processing into two parts:

1) Reduce spectral leakage: We plot in Fig. 2, the average Power Delay Profile (PDP) over the different receive antennas during a total time duration T . The radar samples its total 80 MHz bandwidth into 48 sample points. We then add zero padding of size 3 × 48 to resolve the waveform signal. This results in 48 × 4 = 192 samples. These samples can also be

r i (τ ) = h i (τ ) × s(τ ) = h i (τ ) × r 4 (τ ) (1a) R i (F ) = FFT(r i (τ )) = H i (F ) × R 4 (F ) (1b) PDP(τ ) = h i (τ ) = iFFT(H i (F )) (1c)
where τ is the propagation delay, r i is the received signal at the i-th receiver, s is the transmitted pulse and h i is the channel gain between the object and receiver i. Since the transmitter is directly connected to the port of r 4 , s(τ ) can be substituted by r 4 (τ ). FFT and iFFT are the Fast Fourier Transform and inverse Fast Fourier Transform, respectively. In order to reduce spectral leakage and side lobes that might be masking actual echos, we apply a window in the frequency domain for the range axes. The choice of the window has a major impact on the side lobes attenuation, and thus on reducing the spectral leakage. Many windows exist in the literature with different characteristics (i.e., highest side lobe level, side lobe fall-off, etc.) [START_REF] Harris | On the use of windows for harmonic analysis with the discrete fourier transform[END_REF]. In our work, we applied a Hanning window on the range axes. This window has a wide peak and low side lobes. In Fig. 3, we plot the average PDP with a Hanning window applied on the range axes. Compared to Fig. 2, we can observe the different delayed echos that were masked by the side lobes due to spectral leakage.

For each transmitted pulse, the radar captures the CSI from its returned signal in the form of amplitudes and phases over the 192 sample points. For a total of N transmitted pulses, we denote by C i the captured CSI by the i-th receiver. It can be expressed as follows:

C i =     (a 1,1 , φ 1,1 ) . . . (a 1,192 , φ 1,192 ) . . . . . . . . . (a N,1 , φ N,1 ) . . . (a N,192 , φ N,192 )     (2)
where a n,d and φ n,d are the amplitude and phase received from the n-th pulse at the d-th distance point. A range-Doppler image I i can be generated from the above captured C i by performing a Fast Fourier Transform (FFT) over T f rame consisting of F pulses. A Chebyshev window is applied in the time domain with a 75 dB side attenuation to reduce spectral leakage and capture low Doppler frequencies that are around the DC component. We can write a range-Doppler image as follows:

I i =     |b 1,1 | . . . |b 1,192 | . . . . . . . . . |b F,1 | . . . |b F,192 |     (3)
where |b f,d | is the amplitude of the Doppler frequency f d at distance point d.

From the total N transmitted pulses, we generate S Doppler images each containing F frequency bins. The number of images can be calculated as follows:

S = N -F F (4)
These images will be used later as the input of our deep learning architecture.

2) Time-Frequency analysis: The Doppler images generated in the previous section are sampled in time. Thus, a sequence of these images show the variations of the Doppler frequencies in time. To address the time dependency variations, we use Short Time Fourier Transform (STFT). The idea is to pass a sliding Chebyshev window over the received signal to capture the frequency components. The window of width T f rame slides on the time axis with a step duration t step .

In order to detect human presence, a high Doppler resolution is required to capture slow movements. Breathing for instance results in a Doppler frequency of less then 1 Hz. We set the radar sampling frequency f s = 200 Hz. We then set the window size to F = 1024 pulses and a time step t step = 130 ms. This is equivalent to a time frame duration of T f rame = 1024/200 = 5.12 seconds. This gives us a Doppler frequency resolution of ∆f d = 1/T f rame = 0.195 Hz with a maximal Doppler frequency of f max d = ±f s /2 = ±100 Hz. Such resolution will allow us to capture low Doppler frequencies from respiration as well as higher frequencies from movements such as walking.

In order to reduce the process complexity of the deep learning process, we limit the size of the Doppler images to 64 frequency bins and 64 range points. This results in an image of size 64×64. We do this by grouping different frequency bins to obtain a logarithmic scale on the frequency axes that spans low frequency components for micro movement detection (e.g., breathing detection) and high frequency components for macro movement detection (e.g., walking). We illustrate in Fig. 4 some examples of generated images from different scenarios.

C. Experimental setup and datasets collection

We conducted our experiment in an apartment living room representing a domestic indoor environment. The detail of the room plan is described in Fig. 5. This room has furniture: a dining table, a small table, a bed, a closet, a desk and chairs.

The furniture is all made from wood. The radar is placed on the floor as shown in the Fig. 5 and facing the area of interest.

The blue dotted area represents the coordinates where people are allowed to move. The radar is placed in this location to have the area in interest in LOS.

We collected data samples from different scenarios. We conducted 12 cases of measurements: 1) empty room, 2) one sitting person, 3) one walking person, 4) two sitting persons, 5) one sitting person and one walking person, 6) three sitting persons, 7) one sitting person and two walking persons, 8) two sitting persons and one walking person, 9) four sitting persons, 10) four walking persons, 11) one sitting person and three walking persons and 12) two sitting persons and two walking persons.

We do not limit or force the volunteers to take fixed positions or fixed activities. The volunteers are free to move inside the dotted area with no restrictions on the distance between them.

III. 3D-CNN ARCHITECTURE

This section highlights the deep learning architecture which learns from the constructed images the number of people present inside the room. This architecture is based on CNN and in particular 3D-CNN [START_REF] Goodfellow | Deep learning[END_REF].

Radar Closet

Table

Desk

Dining table

The units are in cms Fig. 5: Room plan and measurement locations.

CNN is a class of deep learning neural networks that is used in image processing and classification [START_REF] Wu | Introduction to convolutional neural networks[END_REF]. CNN networks have shown very good performances in objects recognition by automatically extracting features from raw inputs and ability to learn localized patterns through weights sharing and pooling. One important characteristic of CNN is that the extraction of features is shift-invariant. This gives CNN the ability to recognize patterns at different locations of the input [START_REF] Yang | Advances in Knowledge Discovery and Data Mining: 23rd Pacific-Asia Conference[END_REF]. This characteristic is important in our classification problem as the presence signal can happen at any period of time.

The main component of a CNN is the convolution layer. It is composed of several filters or kernels that are small in size. The goal of these filters is to extract features from the image that is the input of the network. In a 3D-CNN, The filters have the shape of (size 1, size 2, size 3) which is an matrix of weights, and each point in the filter is a neuron. These filters are applied to the input. In our problem, the input is a sequence of images. It has the shape of (Distance, Doppler frequency, time). The output of the convolution layer is called feature maps having a shape of (Distance, Doppler frequency, time, filters number). The pooling layer acts as a down-sampling layer to reduce the system computational complexity.

We consider a CNN composed of 3 convolution layers. Each convolution layer is followed by a max pooling layer. The output of the last max pooling layer serves as input for a classical Multi-Layer Perceptron (MLP) network. The role of the MLP is to classify the extracted information (features map) from the previous convolution layers. We illustrate in Fig. 6 the architecture of our convolution network. The input consists of a sequence of 38 images, each having a size of 64x64. This size represents the points in Doppler frequency and distance. In other words, we have 64 Doppler frequencies and 64 distances points. The number of images or frames can be regarded as a time dimension. The convolution layers (C 1 , C 2 and C 3 ) consist of 64, 25 and 16 filters, respectively. Their kernel sizes are 3 × 3 × 10, 3 × 3 × 9 and 3 × 3 × 8. Each convolution layer is followed by a max pooling layer (P 1 , P 2 , and P 3 ) of size (2 × 2 × 2). The output of the last pooling layer (P 3 ) enters an MLP network of size 1000 neurons and then another MLP network of size 200. The output of the last MLP network (M 3 ) is the output of our network identifying to which class the input images belongs. In order to reduce overfitting, we add a dropout layer between the MLP layers with a value of 0.15. The choice of the hyperparameters are based on extensive simulations. These values resulted in having the optimal accuracy.

IV. EVALUATION AND RESULTS

In this section, we describe the datasets used for training and validation in our deep learning architecture. For training the 3D-CNN, we use the Keras library with TensorFlow 1.5 back-end in Python 3.6. The training and testing phase have been supported by two NVIDIA Quadro P5000 with CUDA Toolkit 9.1 and cuDNN 7.1 with a batch size set to 64.

A. Evaluation setup

We consider 944 samples for training and validation. Each sample consists of 38 range-Doppler images. Given that the images are generated from sliding windows of size 5.12 seconds with a time step t step = 130 ms, each sample of 38 images is 10.06 seconds long. We detail in Table II the measurements taken for the different scenarios. We refer to 0P the case of empty room (zero person), as for 1P, 2P, 3P and 4P the cases where there are one to four persons. Then, we distinguish between a person sitting on a chair (s) and a walking person (w). For each scenario, we conduct different measurements. The total amount of measurement time is 158 minutes.

Because we have a limited number of training samples, we artificially increase our dataset by making translations on the range axes in the range-Doppler images. This is a possible solution since in this work we are not interested at what range (position) the frequencies were captured. Thus, we perform four translations for each image resulting in augmenting our dataset by a factor of four. Then, we adopt K-Fold cross validation method to train and validate the data collected. This method splits the collected data into K sets. One of the k sets is used for testing (validation), while the remaining k -1 sets are used for training. The cross validation is repeated K times with each of the k sets is used exactly once as the validation data. The overall performance is then evaluated by averaging over the K sets. In this work, we set K = 5.

B. Benchmark algorithms

We compare our 3D-CNN architecture with two baseline methods: Fully Connected Back Propagation (FCBP) neural network and Gaussian Naive Bayes (GaussianNB). FCBP is a fully connected MLP neural network which contains a lot of tunable parameters that is suited for people counting. We use a FCBP with two hidden layers. The first hidden layer contains 300 neurons whereas the second hidden layer contains 100 neurons. The input layer is the sequence of 38 images vectorized and reshaped to one dimension vector of size 64 × 64 × 38.

Naive Bayes model is a classical linear classification algorithm that belongs to the family of probabilistic classifiers [START_REF] Leung | Naive bayesian classifier[END_REF]. It is a supervised learning method based on applying Bayes theorem with strong independence (naive) assumption between features. GaussianNB assumes that the collected data are distributed according to a Gaussian (or normal) distribution. Naive Bayes classifiers do not require a lot of data to train and they are simple to implement. The input of this model is similar to the FCBP model.

C. Results

In Table III, we compare the average classification accuracy obtained by our proposed 3D-CNN architecture with FCBP and GaussianNB. This average is obtained over the 5 experiments (Ex.#) from the 5-fold cross validation. From the obtained results, we observe that 3D-CNN outperforms the other two baseline models with an average accuracy of 89.62%, while FCBP and GaussianNB achieve 73.51% and 45.35%, respectively. The lower performance of the GaussianNB model is due to the assumption that the features are independent. Thus, it processes the range-Doppler images in the sample as one vector with independent data features. The importance of the correlation between the range-Doppler images in one sample can be well distinguished with the other two deep learning architectures as the average classification accuracy almost doubled. The accuracy achieved by FCBP shows that it is powerful enough to count people. However, 3D-CNN handles images sequence better by extracting features from the applied filters, and uses these features for classification. We present in Fig. 7 the confusion matrix obtained from our 3D-CNN model describing the performance of our classifier. The horizontal axis corresponds to the actual class whereas the vertical axes corresponds to the predicted class.

V. CONCLUSION

In this paper, we present a novel DL model based on 3D convolutional networks to solve the problem of people counting with a pulse Doppler Wi-Fi radar. Taking into account the complexity of the environment and various cases such as sitting and walking people, we achieved a good counting accuracy of 89% for up to four people in LOS with no constraints on the actions taken by the volunteers. For future work, we focus on extending the model to sense human presence and count people in a NLOS scenario and to be able to track the positions of the people.

Fig. 1 :

 1 Fig. 1: Block diagram system overview.

Fig. 2 :

 2 Fig. 2: Average power delay profile over the different receive antennas.

Fig. 3 :

 3 Fig. 3: Average power delay profile over the different receive antennas with Hanning window on the range axes.

Fig. 4 :

 4 Fig. 4: Examples of different range-Doppler images with different scenarios.

Fig. 6 :

 6 Fig. 6: 3D-CNN architecture illustrating the input images, the output classes and the different intermediate layers.

Fig. 7 :

 7 Fig. 7: The confusion matrix for 3D-CNN network architecture.

TABLE I :

 I Celeno radar radio characteristics.

	Carrier frequency	5.54 GHz
	Bandwidth	80 MHz
	Pulse duration	40 µ s
	Pulse period	1 ms
	EIRP	11 dBm
	Directivity	30

• lobe at 3 dB and 60 • lobe at 6 dB

TABLE II :

 II Summary of the measurements dataset from different scenarios.

	Scenario	0P	1P (s)	1P (w)	2P (s)	2P (1s,1w)	3P (s)	3P (1s,2w)	3P (2s,1w)	4P (s)	4P (w)	4P (1s,3w)	4P (2s,2w)	Total
	Number of samples	94	100	100	100	100	50	100	100	50	50	50	50	944
	Measurement time (mins)	15.7 16.7 16.7 16.7	16.7	8.4	16.7	16.7	8.4	8.4	8.4	8.4	158

TABLE III :

 III Classification accuracy obtained from our proposed 3D-CNN architecture and other baseline methods.

	Architecture/ Method	Ex. 1	Ex. 2	Ex. 3	Ex. 4	Ex. 5	Acc.
	3D-CNN	89.15%	87.95%	92.19%	87.28%	91.52%	89.62%
	FCBP	73.54%	69.84%	73%	78.83%	72.34%	73.51%
	GaussianNB	42.72%	42.35%	47.6%	48%	46.1%	45.35%