D Efimov

A Aleksandrov

Analysis of robustness of homogeneous systems with time delays using Lyapunov-Krasovskii functionals

The paper is devoted to stability analysis of homogeneous time-delay systems applying the Lyapunov-Krasovskii theory, and a generic structure of the functional is given that suits for any homogeneous system of non-zero degree (and can also be used for any dynamics admitting a homogeneous approximation). The obtained stability conditions are utilized to evaluate the domain of attraction for the delayed twisting control algorithm.

I. INTRODUCTION

The influence of delays on the system stability and performance is important for many engineering systems

. The most of existing solutions deal with linear time-delay models, which is originated by complexity of stability analysis for the nonlinear case. Indeed, constructive variants of selection of a Lyapunov-Krasovskii functional or a Lyapunov-Razumikhin function are elaborated for linear systems (or close to them nonlinear dynamics) [5],

but for generic scenario it is rare to find an applicable recommendation.

Homogeneous dynamical systems take an intermediate place between linear and nonlinear ones [6]. The theory of homogeneous dynamical systems has been proposed for ordinary differential equations [7], [8], [9], [10], differential inclusions [11], [6], time-delay systems [12], [13] and partial differential equations [14]. The main feature of a homogeneous nonlinear system is that its local behavior is the same as the global one, the homogeneous stable/unstable systems admit homogeneous Lyapunov/Chetaev functions [10], [15], [16], [17], [18] and possess robustness properties with respect to external inputs and delays [19], [13], [20]. Since such a subclass of nonlinear systems (having global stability or convergence) is rather small, the concept of local homogeneity has been introduced [10], [21], [16], [22]. In [23], [24], [25], [26], [27], [13], [20] it has been shown that homogeneous systems have certain robustness of stability

, [20] using Lyapunov-Razumikhin theory.

In the present work, first, following the framework of [28], the same results are obtained by proposing a generic structure Lyapunov-Krasovskii functional for any homogeneous system with non-zero degree being asymptotically stable in the delayfree case. The presented approach allows the domain of stability/attraction to be evaluated. Second, the proposed Lyapunov-Krasovskii functional is used to estimate the domain of convergence for a delayed twisting algorithm. The outline of this paper is as follows. The preliminary definitions and the homogeneity for time-delay systems are given in Section II. The main result is presented in Section III. An example is considered in Section IV.

II. PRELIMINARIES

Consider an autonomous functional differential equation of retarded type [START_REF] Kolmanovsky | Stability of functional differential equations[END_REF]:

dx(t)/dt = f (x t ), t ≥ 0, (1)
where x(t) ∈ R n and x t ∈ C [-τ,0] is the state function, x t (s) = x(t + s), -τ ≤ s ≤ 0 (we denote by C [-τ,0] the Banach space of continuous functions φ : [-τ, 0] → R n with the uniform norm φ = sup -τ ≤ς≤0 |φ(ς)|, where | • | is the standard Euclidean norm); f : C [-τ,0] → R n ensures existence and uniqueness of solutions in forward time (such a requirement can be imposed even for discontinuous time-delay systems [START_REF] Fridman | Steady modes in relay control systems with time delay and periodic disturbances[END_REF], [START_REF] Shustin | Oscillations in a second-order discontinuous system with delay[END_REF]), f (0) = 0. For the system (1), denote the solution that corresponds to the initial functional condition x 0 ∈ C [-τ,0] by x(t, x 0 ) and the restriction of the solution on the interval [t -τ, t] by x x0 t (then x x0 t (s) = x(t + s, x 0 ) for -τ ≤ s ≤ 0), which is defined on time interval [-τ, T ) with T ∈ R + ∪ {+∞}. The representation (1) includes pointwise or distributed time-delay systems.

The upper right-hand Dini derivative of a locally Lipschitz continuous functional V : C [-τ,0] → R along the system [START_REF] Gu | Stability of Time-Delay Systems[END_REF] solutions is defined as follows for any φ ∈ C [-τ,0] :

D + V (φ) = lim sup h→0 + V (φ h ) -V (φ) h ,
where φ h ∈ C [-τ,0] for 0 < h < τ is given by

φ h =    φ(θ + h), θ ∈ [-τ, -h) φ(0) + f (φ)(θ + h), θ ∈ [-h, 0].
A continuous function σ : R + → R + belongs to class K if it is strictly increasing and σ(0) = 0; it belongs to class K ∞ if it is also unbounded. A continuous function β : R + × R + → R + belongs to class KL if β(•, r) ∈ K and β(r, •) is decreasing to zero for any fixed r > 0. The symbol 1, m is used to denote a sequence of integers 1, ..., m.

A. Stability definitions

Let A ⊂ C [-τ,0] be a bounded set, it is called forward invariant for (1) if x 0 ∈ A implies that x x0 t ∈ A for all t ≥ 0, denote distance to the set as x A = inf y∈A x -y . Let Ω ⊂ C [-τ,0] be an open neighborhood of a forward invariant set A.

Definition 1. The system (1) is said to be (a) stable at A in Ω if there is σ ∈ K such that for any x 0 ∈ Ω, the solution x x0 t is defined and x x0 t A ≤ σ( x 0 A ) for all t ≥ 0;

(b) asymptotically stable at A in Ω if it is stable in Ω and lim t→+∞ x x0 t A = 0 for any x 0 ∈ Ω. If Ω = C [-τ,0] , then the corresponding properties are called global stability/asymptotic stability of (1) at A.

For A = {0} the properties given in this definition are reduced to the standard ones [START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF] (see also the concept of ultimate boundedness in [START_REF] Hale | Theory of Functional Differential Equations[END_REF]).

B. Homogeneity

For any r i > 0, i = 1, n and λ > 0, define the dilation matrix Λ r (λ) = diag{λ ri } n i=1 and the vector of weights r = [r 1 , ..., r n ] , r max = max 1≤j≤n r j and r min = min 1≤j≤n r j .

For any r i > 0, i = 1, n and x ∈ R n the homogeneous norm can be defined, for example, as follows

|x| r = n i=1 |x i | ρ/ri 1/ρ , ρ ≥ r max .
For all x ∈ R n , its Euclidean norm |x| is related with the homogeneous one:

σ r (|x| r ) ≤ |x| ≤ σ r (|x| r )
for some σ r , σ r ∈ K ∞ [START_REF] Efimov | Homogeneous lyapunov functions: from converse design to numerical implementation[END_REF]. The homogeneous norm has an important property that is |Λ r (λ)x| r = λ|x| r for all x ∈ R n . Define S r = {x ∈ R n : |x| r = 1}.

For any r i > 0, i = 1, n and φ ∈ C [-τ,0] the homogeneous norm can be defined as follows

φ r = n i=1 φ i ρ/ri 1/ρ , ρ ≥ r max .
There exist two functions ρ r , ρ r ∈ K ∞ such that for all φ ∈ C [-τ,0] [START_REF] Efimov | Development of homogeneity concept for time-delay systems[END_REF]:

ρ r ( φ r ) ≤ φ ≤ ρ r ( φ r ).
The homogeneous norm in the Banach space has the same important property that Λ r (λ

)φ r = λ φ r for all φ ∈ C [-τ,0] . In C [-τ,0] the corresponding unit sphere is S r = {φ ∈ C [-τ,0] : φ r = 1}. Define B τ ρ = {φ ∈ C [-τ,0] : φ r ≤ ρ} as a closed ball of radius ρ > 0 in C [-τ,0] . Definition 2. [31] The function g : C [-τ,0] → R is called r-homogeneous if for any φ ∈ C [-τ,0] the relation g(Λ r (λ)φ) = λ d g(φ)
holds for some d ∈ R and all λ > 0.

The function

f : C [-τ,0] → R n is called r-homogeneous if for any φ ∈ C [-τ,0] the relation f (Λ r (λ)φ) = λ d Λ r (λ)f (φ)
holds for some d ≥ -r min and all λ > 0.

In both cases, the constant d is called the degree of homogeneity.

The introduced notion of weighted homogeneity in C

[-τ,0] is reduced to the standard one in R n if τ → 0.
An advantage of homogeneous systems described by ordinary differential equations is that any its solution can be obtained from another solution under the dilation and a suitable time re-parametrization. A similar property holds for functional homogeneous systems: Proposition 1. [START_REF] Efimov | Weighted homogeneity for time-delay systems: Finite-time and independent of delay stability[END_REF] Let x(t, x 0 ) be a solution of the r-homogeneous system (1) with the degree d for an initial condition

x 0 ∈ C [-τ,0] , τ ∈ (0, +∞). For any λ > 0 the functional differential equation dy(t)/dt = f (y t ), t ≥ 0 (2) with y t ∈ C [-λ -d τ,0] , has a solution y(t, y 0 ) = Λ r (λ)x(λ d t, x 0 ) 1 for all t ≥ 0 with the initial condition y 0 ∈ C [-λ -d τ,0] , y 0 (s) = Λ r (λ)x 0 (λ d s) for s ∈ [-λ -d τ, 0].
The following results have also been obtained in [START_REF] Efimov | Weighted homogeneity for time-delay systems: Finite-time and independent of delay stability[END_REF], [START_REF] Zimenko | A note on delay robustness for homogeneous systems with negative degree[END_REF]:

Lemma 1.
[13] Let the system (1) be r-homogeneous with degree d = 0 and globally asymptotically stable at the origin for some delay 0 < τ 0 < +∞, then it is globally asymptotically stable at the origin IOD.

For the next two results we assume that f in (1) is a uniformly continuous function.

Lemma 2. [13] Let f (x t ) = F (x(t), x(t -τ ))
in (1) and the system (1) be r-homogeneous with degree d > 0 and globally asymptotically stable at the origin for τ = 0, then for any ρ > 0 there is 0 < τ 0 < +∞ such that (1) is asymptotically stable at the origin in B τ ρ for any delay 0 ≤ τ ≤ τ 0 .

Lemma 3.

[20] Let f (x t ) = F (x(t), x(t -τ )) in (1) and the system (1) be r-homogeneous with degree d < 0 and globally asymptotically stable at the origin for τ = 0, then for any ρ > 0 there is 0 < τ 0 < +∞ such that (1) is globally asymptotically stable at B τ ρ for any delay 0 ≤ τ ≤ τ 0 .

Thus, (1) is locally robustly stable with respect to a sufficiently small delay if it is r-homogeneous and stable in the delay-free case. In [START_REF] Efimov | Weighted homogeneity for time-delay systems: Finite-time and independent of delay stability[END_REF], [START_REF] Zimenko | A note on delay robustness for homogeneous systems with negative degree[END_REF], the Lyapunov-Razumikhin approach was used to prove these results.

Finally, Young's inequality claims that for any a, b ∈ R + and γ > 0, δ > 0:

a γ b δ ≤ 1 p a γp + p -1 p b δp p-1
for any p > 1, while Hölder's inequality for any f, g : I → R with I ⊂ R ensures that for any p > 1. Another related result can be obtained using the properties of homogeneous functions and a specially defined homogeneous norm:

Lemma 4. Let a, b ∈ R + and > 0, α > 0, β > 0, γ > 0, δ > 0 be given, then

a α + b β -a γ b δ ≥ 0 provided that 1) max{a α , b β } ≥ 1 1- γ α -δ β and γ α + δ β < 1, 2) max{a α , b β } ≤ 1 1- γ α -δ β and γ α + δ β > 1.
Proof. Define weights r 1 = 1 α and r 2 = 1 β and a corresponding homogeneous norm

|(a, b)| r = max{a α , b β },
then the scalar functions a α + b β and a γ b δ are r-homogeneous (with r = [r 1 , r 2 ] ) of degrees 1 and γ α + δ β , respectively. Therefore, the following inequality is satisfied for a, b ∈ R + :

a γ b δ ≤ s|(a, b)| γ α + δ β r , where s = sup |(a,b)|r=1 a γ b δ = 1,
then the property

a α + b β -a γ b δ ≥ 0 for any a, b ∈ R + and c > 0, α > 0, β > 0, γ > 0, δ > 0 is verified provided that |(a, b)| r -|(a, b)| γ α + δ β r ≥ 0. Hence, if γ α + δ β < 1, then this inequality is satisfied for all |(a, b)| r ≥ 1 1- γ α -δ
β , and if

γ α + δ β > 1, then for |(a, b)| r ≤ 1 1- γ α -δ β .

III. MAIN RESULTS

Consider a variant of the system (1):

ẋ(t) = F (x(t), x(t -τ )), (3)
where F : R 2n → R n is differentiable with respect to the first argument, and introduce the following hypotheses:

Assumption 1. The system (3) is r-homogeneous of degree ν ≥ -r min , i.e., for all λ > 0, x ∈ R n and z ∈ R n :

F (Λ r (λ)x, Λ r (λ)z) = λ ν Λ r (λ)F (x, z),
and sup

|ξ|r≤1,|ζ|r≤1 |F (ξ, ζ)| < +∞, sup |ξ|r≤1,|ζ|r≤1 ∂F (ξ, ζ) ∂ξ < +∞.
Assumption 2. The system (3) is globally asymptotically stable at the origin if τ = 0.

Under these assumptions [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], there is a twice continuously differentiable Lyapunov function V : R n → R + for (3), which can be selected to be r-homogeneous of degree µ ≥ -ν, then for all x ∈ R n :

V (Λ r (λ)x) = λ µ V (x) ∀λ > 0, c 1 |x| µ r ≤ V (x) ≤ c 2 |x| µ r , (4) dV (x) dx F (x, x) ≤ -ς|x| ν+µ r
for some c 1 > 0, c 2 > 0 and ς > 0.

Remark 1. If the system (3) is r-homogeneous with degree ν, then a direct computation shows that it is also r-homogeneous for r = r -1 max (r 1 , . . . , r n ) T with degree ν = ν rmax . Hence, without loosing generality in the following we will assume that r max = 1.

To proceed we need the following auxiliary result: Lemma 5. Let Assumption 1 be satisfied, then for any twice continuously differentiable and r-homogeneous of degree µ ≥ 2 function V : R n → R + :

dV (x) dx F (x, z) ≤ w max{|x| µ-1 r |z| ν+1 r , |x| µ+ν r }, F (x, q) d 2 V (x) dx 2 F (x, z) ≤ v max{|x| µ+2ν r , |x| µ-2 r |z| 2ν+2 r , |x| µ-2 r |q| 2ν+2 r }, dV (x) dx ∂F (x, z) ∂x F (x, q) ≤ u max{|x| µ+2ν r , |x| µ-1 r |z| 2ν+1 r , |x| µ-1 r |q| 2ν+1 r },
where

w = sup y∈Sr dV (y) dy sup |ξ|r≤1,|ζ|r≤1 |F (ξ, ζ)| , v = sup y∈Sr d 2 V (y) dy 2 sup |ξ|r≤1,|ζ|r≤1 |F (ξ, ζ)| 2 , u = sup y∈Sr dV (y) dy sup |ξ|r≤1,|ζ|r≤1 ∂F (ξ, ζ) ∂ξ F (ξ, ζ) .
Proof. To substantiate the first property define λ = max{|z| r , |x| r }, then using homogeneity of V and F we obtain

dV (x) dx F (x, z) = |x| µ r λ ν dV (y) dy Λ -1 r (|x| r )Λ r (λ)F (ξ, ζ) for y = Λ -1 r (|x| r )x, ξ = Λ -1 r (λ)x and ζ = Λ -1 r (λ)z.
Using the diagonal structure of the matrix Λ -1 r (|x| r )Λ r (λ) we get:

dV (x) dx F (x, z) ≤ w|x| µ r λ ν    λ |x|r rmax |z| r > |x| r 1 |z| r ≤ |x| r ,
and since it has been imposed r max = 1, we substantiated that

dV (x) dx F (x, z) ≤ w max{|x| µ-1 r |z| ν+1 r , |x| µ+ν r }.
For the second inequality define

y = Λ -1 r (|x| r )x, ξ = Λ -1 r (λ)x, θ = Λ -1 r (λ)q and ζ = Λ -1 r (λ)z for λ = max{|z| r , |q| r , |x| r }, then by homogeneity: F (x, q) d 2 V (x) dx 2 F (x, z) = |x| µ r λ 2ν F (ξ, θ) Λ r (λ)Λ -1 r (|x| r ) d 2 V (y) dy 2 Λ -1 r (|x| r )Λ r (λ)F (ξ, ζ) ≤ v|x| µ r λ 2ν    λ |x|r 2 max{|z| r , |q| r } > |x| r 1 max{|z| r , |q| r } ≤ |x| r
, where r max = 1 was substituted on the last step, then the desired property is proven:

F (x, q) d 2 V (x) dx 2 F (x, z) ≤ v max{|x| µ+2ν r , |x| µ-2 r λ 2ν+2 } = v max |x| µ+2ν r , |x| µ-2 r max{|z| 2ν+2 r , |q| 2ν+2 r } = v max{|x| µ+2ν r , |x| µ-2 r |z| 2ν+2 r , |x| µ-2 r |q| 2ν+2 r }.
For the last inequality, again denote

y = Λ -1 r (|x| r )x, ξ = Λ -1 r (λ)x, θ = Λ -1 r (λ)q and ζ = Λ -1 r (λ)z for λ = max{|z| r , |q| r , |x| r }, then by homogeneity dV (x) dx ∂F (x, z) ∂x F (x, q) = dV (y) dy Λ -1 r (|x|r)Λr(λ) ∂F (ξ, ζ) ∂ξ F (ξ, θ) |x| µ r λ 2ν ,
and once more using the diagonal structure of the matrix Λ -1 r (|x| r )Λ r (λ) we get:

dV (x) dx ∂F (x, z) ∂x F (x, q) ≤ u|x| µ r λ 2ν    λ |x|r max{|z| r , |q| r } > |x| r 1 max{|z| r , |q| r } ≤ |x| r ≤ u max{|x| µ+2ν r , |x| µ-1 r |z| 2ν+1 r , |x| µ-1 r |q| 2ν+1 r } as required.
Note that ν ≥ -r min and r min ≤ r max = 1 as we assumed (due to Remark 1), then for µ > 2 all powers in the obtained estimates are positive (the latter estimate for the case ν < 0 we can rewrite as

max{|x| µ+2ν r , |x| µ 2 -1 r |z| 2ν+ µ 2 +1 r , |x| µ 2 -1 r |q| 2ν+ µ 2 +1 r } since |x| µ 2 r < max{|z| µ 2 r , |q| µ 2
r } in the latter two cases). Now we are in position to formulate the main result: Theorem 1. Let assumptions 1 and 2 hold for (3). Then for any value of the delay τ > 0 the properties

W (x t ) > 0, D + W (x t ) < 0 are satisfied for x t ∈ Ω if ν < 0 or for x t ∈ Ω if ν > 0, where W (x t ) = V (x(t)) + b t t-τ |x(s)| ρ r ds + g t t-τ (s -t + τ )|x(s)| ρ r ds + t t-τ dV (x(t)) dx F (x(t), x(s))ds, is a Lyapunov-Krasovskii functional for (3); b > τ ρ max{4v(ν + 1), u(µ + 4ν + 2)}, g > 0, max µ(ν + 1), 2(ν + 1) µ + ν ν + 2 , 2(ν + µ 2 ) 2 + ν + ( ν 2 + 1)µ ν + 1 + µ 2 , (ν + µ) µ 2 + 2ν + 1 ν + µ 2 + 1 < ρ < ν + µ, if ν < 0 or ν + µ < ρ < min µ(ν + 1), 2(ν + 1) µ + ν ν + 2 , (2ν + 1)(ν + µ) ν + 1
if ν > 0, and V is a twice continuously differentiable r-homogeneous Lyapunov function for the case τ = 0 with degree

µ >    2 ν < 0 ν + 2 ν > 0 ;
ς, c 1 and w, v, u are defined in (4) and Lemma 5, respectively;

Ω = Ω 1 ∩ Ω 2 , Ω = Ω 1 ∩ Ω 2 , Ω 1 = {φ ∈ C [-τ,0] : max{|φ(0)| µ r , 0 -τ |φ(s)| ρ r ds} ≥ 1 }, Ω 2 = {φ ∈ C [-τ,0] : max{|φ(0)| ν+µ r , 0 -τ |φ(s)| ρ r ds} ≥ 2 }, Ω 1 = {φ ∈ C [-τ,0] : max{|φ(0)| µ r , 0 -τ |φ(s)| ρ r ds} ≤ 1 }, Ω 2 = {φ ∈ C [-τ,0] : max{|φ(0)| ν+µ r , 0 -τ |φ(s)| ρ r ds} ≤ 2 }
where

1 = max        wτ min{c1, b} - µ ν ,   wτ ρ-ν-1 ρ min{c1, b}   1 1 µ -ν+1 ρ        , 2 = max    3 b + gτ min{ς, g} ν+µ ν+µ-ρ , 3vτ min{ς, g} - ν+µ ν , 3v ρ -2(ν + 1) ρ min{ς, g} τ (ρ-2(ν+1))(ν+µ) (ν+2)ρ-2(ν+1)(µ+ν) ,     3uτ ρ- µ 2 -2ν-1 ρ min{ς, g}     1 1- µ 2 -1 ν+µ - µ 2 +2ν+1 ρ , 3u ρ -µ 2 -2ν -1 ρ min{ς, g} τ (ρ- µ 2 -2ν-1)(ν+µ) (ν+µ)(ρ- µ 2 -2ν-1)-( µ 2 -1)ρ ,   3vτ ρ-2ν-2 ρ min{ς, g}   1 1- µ-2 ν+µ -2ν+2 ρ , 3uτ min{ς, g} - ν+µ ν    , 1 = min        wτ min{c1, b} - µ ν ,   wτ ρ-ν-1 ρ min{c1, b}   1 1 µ -ν+1 ρ        , 2 = min    3 b + gτ min{ς, g} ν+µ ν+µ-ρ , 3vτ min{ς, g} - ν+µ ν , 3v ρ -2(ν + 1) ρ min{ς, g} τ (ρ-2(ν+1))(ν+µ) (ν+2)ρ-2(ν+1)(µ+ν) ,   3uτ ρ-2ν-1 ρ min{ς, g}   1 ν+1 ν+µ -2ν+1 ρ , 3u ρ -2ν -1 ρ min{ς, g} τ (ρ-2ν-1)(ν+µ) (ν+1)ρ-(2ν+1)(ν+µ) ,   3vτ ρ-2ν-2 ρ min{ς, g}   1 1- µ-2 ν+µ -2ν+2 ρ , 3uτ min{ς, g} - ν+µ ν    .
Proof. To investigate stability of the nonlinear time-delay system (3) consider a candidate Lyapunov-Krasovskii functional W (x t ) as given in the formulation of the theorem, where b > 0, g > 0 and ρ > 0 are parameters selected below, and V is a homogeneous Lyapunov function for the case τ = 0, which exists due to Assumption 2 and satisfies (4) with µ > 2 if ν < 0 or µ > ν + 2 for ν > 0. Let us investigate positive definiteness of W . Obviously W (0) = 0 and

W (x t ) ≥ c 1 |x(t)| µ r + b t t-τ |x(s)| ρ r ds -w t t-τ max{|x(t)| µ-1 r |x(s)| ν+1 r , |x(t)| µ+ν r }ds,
where the first property from Lemma 5 was utilized. Therefore, the following inequalities have to hold for positivity of W :

c 1 |x(t)| µ r + b t t-τ |x(s)| ρ r ds -wτ |x(t)| µ+ν r ≥ 0, c 1 |x(t)| µ r + b t t-τ |x(s)| ρ r -w|x(t)| µ-1 r |x(s)| ν+1 r ds ≥ 0,
and the former is valid if

max{|x(t)| µ r , t t-τ |x(s)| ρ r ds} - wτ min{c 1 , b} |x(t)| µ+ν r ≥ 0 and, hence, for      max{|x(t)| µ r , t t-τ |x(s)| ρ r ds} ≥ wτ min{c1,b} -µ ν ν < 0 max{|x(t)| µ r , t t-τ |x(s)| ρ r ds} ≤ wτ min{c1,b} -µ ν ν > 0
the first inequality is true. For the latter, using Hölder's inequality, we obtain:

t t-τ |x(s)| ν+1 r |x(t)| µ-1 r ds ≤ t t-τ |x(s)| ρ r ds ν+1 ρ t t-τ |x(t)| ρ(µ-1) ρ-ν-1 r ds ρ-ν-1 ρ = τ ρ-ν-1 ρ |x(t)| µ-1 r t t-τ |x(s)| ρ r ds ν+1 ρ
.

Then for positivity of W the property has to satisfy:

c 1 |x(t)| µ r + b t t-τ |x(s)| ρ r ds -wτ ρ-ν-1 ρ |x(t)| µ-1 r t t-τ |x(s)| ρ r ds ν+1 ρ ≥ 0
that is true provided the conditions of Lemma 4 are verified for

a = |x(t)| r , b = t t-τ |x(s)| ρ r ds, α = µ, β = 1, γ = µ -1, δ = ν+1 ρ and = wτ ρ-ν-1 ρ
min{c1,b} , which can be formulated as:

max{|x(t)| µ r , t t-τ |x(s)| ρ r ds} ≥ wτ ρ-ν-1 ρ min{c1,b} 1 1- µ-1 µ -ν+1 ρ with ν+1 ρ < 1 µ for ν < 0, or max{|x(t)| µ r , t t-τ |x(s)| ρ r ds} ≤ wτ ρ-ν-1 ρ min{c1,b} 1 1- µ-1 µ -ν+1 ρ with ν+1 ρ > 1 µ for ν > 0.
Hence, taking ρ > µ(ν + 1) for ν < 0 or ρ < µ(ν + 1) for ν > 0 ensures positive definiteness of W outside of a vicinity of the origin Ω 1 or locally at zero in Ω 1 , respectively.

The derivative of the Lyapunov-Krasovskii functional W (t) = W (x t ) with respect to time for the system (3) takes the form:

Ẇ (t) = D + W (x t ) = dV (x(t)) dx(t) F (x(t), x(t -τ )) + b[|x(t)| ρ r -|x(t -τ )| ρ r ] +g[τ |x(t)| ρ r - t t-τ |x(s)| ρ r ds] + dV (x(t)) dx(t) F (x(t), x(t)) - dV (x(t)) dx(t) F (x(t), x(t -τ )) + H(x t ) ≤ -ς|x(t)| ν+µ r + (b + gτ )|x(t)| ρ r -b|x(t -τ )| ρ r -g t t-τ |x(s)| ρ r ds + H(x t ),
where the property (4) was used and

H(x t ) = t t-τ F (x(t), x(t -τ )) d 2 V (x(t)) dx(t) 2 F (x(t), x(s)) + dV (x(t)) dx(t) ∂F (x(t), x(s)) ∂x(t) F (x(t), x(t -τ ))ds.
Following Lemma 5,

|H(x t )| ≤ t t-τ v max{|x(t)| µ+2ν r , |x(t)| µ-2 r |x(s)| 2ν+2 r , |x(t)| µ-2 r |x(t -τ )| 2ν+2 r } +u max{|x(t)| µ+2ν r , |x(t)| µ 2 -1 r |x(t -τ )| 2ν+ µ 2 +1 r , |x(t)| µ 2 -1 r |x(s)| 2ν+ µ 2 +1 r }ds for ν < 0 or |H(x t )| ≤ t t-τ v max{|x(t)| µ+2ν r , |x(t)| µ-2 r |x(s)| 2ν+2 r , |x(t)| µ-2 r |x(t -τ )| 2ν+2 r } +u max{|x(t)| µ+2ν r , |x(t)| µ-1 r |x(t -τ )| 2ν+1 r , |x(t)| µ-1 r |x(s)| 2ν+1
ς 3 |x(t)| ν+µ r + b 2 |x(t -τ )| ρ r + t t-τ g 3 |x(s)| ρ r -v max{|x(t)| µ+2ν r , |x(t)| µ-2 r |x(s)| 2ν+2 r , |x(t)| µ-2 r |x(t -τ )| 2ν+2 r }ds ≥ 0 if the series of inequalities ς 3 |x(t)| ν+µ r + g 3 t t-τ |x(s)| ρ r ds -vτ |x(t)| µ+2ν r ≥ 0, ς 3 |x(t)| ν+µ r + b 2 |x(t -τ )| ρ r + g 3 t t-τ |x(s)| ρ r ds -vτ |x(t)| µ-2 r |x(t -τ )| 2ν+2 r ≥ 0, ς 3 |x(t)| ν+µ r + t t-τ g 3 |x(s)| ρ r -v|x(t)| µ-2 r |x(s)| 2ν+2
ς 3 |x(t)| ν+µ r -v ρ -2(ν + 1) ρ τ |x(t)| (µ-2)ρ ρ-2(ν+1) r + g 3 t t-τ |x(s)| ρ r ds + b 2 -v 2(ν + 1) ρ τ |x(t -τ )| ρ r ≥ 0 that for b 2 ≥ v 2(ν+1) ρ τ follows from max{|x(t)| ν+µ r , t t-τ |x(s)| ρ r ds} -3v ρ -2(ν + 1) ρ min{ς, g} τ |x(t)| (µ-2)ρ ρ-2(ν+1) r ≥ 0, which is true for max{|x(t)| ν+µ r , t t-τ |x(s)| ρ r ds} ≥ 3v ρ-2(ν+1) ρ min{ς,g} τ (ρ-2(ν+1))(ν+µ) (ν+2)ρ-2(ν+1)(µ+ν)
, ρ > 2(ν + 1) µ+ν ν+2 and ν < 0, or

max{|x(t)| ν+µ r , t t-τ |x(s)| ρ r ds} ≤ 3v ρ-2(ν+1) ρ min{ς,g} τ (ρ-2(ν+1))(ν+µ) (ν+2)ρ-2(ν+1)(µ+ν)
, 2(ν + 1) < ρ < 2(ν + 1) µ+ν ν+2 and ν > 0. To check the third property, by applying Hölder's inequality to the term

t t-τ |x(s)| 2ν+2 r |x(t)| µ-2 r
ds we derive:

ς 3 |x(t)| ν+µ r + g 3 t t-τ |x(s)| ρ r ds -vτ ρ-2ν-2 ρ |x(t)| µ-2 r t t-τ |x(s)| ρ r ds 2ν+2 ρ ≥ 0,
which is true provided the conditions of Lemma 4 are verified for a

= |x(t)| r , b = t t-τ |x(s)| ρ r ds, α = ν +µ, β = 1, γ = µ-2, δ = 2ν+2 ρ and = 3vτ ρ-2ν-2 ρ
min{ς,g} , and that can be formulated as:

max{|x(t)| ν+µ r , t t-τ |x(s)| ρ r ds} ≥ 3vτ ρ-2ν-2 ρ min{ς,g} 1 1- µ-2 ν+µ -2ν+2 ρ with (ν + µ) 2ν+2 ν+2 < ρ for ν < 0, or max{|x(t)| ν+µ r , t t-τ |x(s)| ρ r ds} ≤ 3vτ ρ-2ν-2 ρ min{ς,g} 1 1- µ-2 ν+µ -2ν+2 ρ with (ν + µ) 2ν+2 ν+2 > ρ for ν > 0. Finally, for ν < 0 the property ς 3 |x(t)| ν+µ r + b 2 |x(t -τ )| ρ r + t t-τ g 3 |x(s)| ρ r -u max{|x(t)| µ+2ν r , |x(t)| µ 2 -1 r |x(t -τ )| 2ν+ µ 2 +1 r , |x(t)| µ 2 -1 r |x(s)| 2ν+ µ 2 +1 r }ds ≥ 0,
follows the series of inequalities:

ς 3 |x(t)| ν+µ r + g 3 t t-τ |x(s)| ρ r ds -uτ |x(t)| 2ν+µ r ≥ 0, ς 3 |x(t)| ν+µ r + b 2 |x(t -τ )| ρ r + g 3 t t-τ |x(s)| ρ r ds -uτ |x(t)| µ 2 -1 r |x(t -τ )| 2ν+ µ 2 +1 r ≥ 0, ς 3 |x(t)| ν+µ r + t t-τ g 3 |x(s)| ρ r -u|x(t)| µ 2 -1 r |x(s)| 2ν+ µ 2 +1 r ds ≥ 0.
The first inequality can be treated directly, and it is satisfied for max{|x

(t)| ν+µ r , t t-τ |x(s)| ρ r ds} ≥ 3uτ min{ς,g} -ν+µ ν
. For the second case, applying Young's inequality we obtain

ς 3 |x(t)| ν+µ r -uτ ρ -µ 2 -2ν -1 ρ |x(t)| ( µ 2 -1)ρ ρ- µ 2 -2ν-1 r + g 3 t t-τ |x(s)| ρ r ds + b 2 -u µ 2 + 2ν + 1 ρ τ |x(t -τ )| ρ r ≥ 0 that for b 2 ≥ u µ 2 +2ν+1 ρ τ it is followed by max{|x(t)| ν+µ r , t t-τ |x(s)| ρ r ds} -3u ρ -µ 2 -2ν -1 ρ min{ς, g} τ |x(t)| ( µ 2 -1)ρ ρ- µ 2 -2ν-1 r ≥ 0,
which is true for

max{|x(t)| ν+µ r , t t-τ |x(s)| ρ r ds} ≥ 3u ρ -µ 2 -2ν -1 ρ min{ς, g} τ (ρ- µ 2 -2ν-1)(ν+µ) (ν+µ)(ρ- µ 2 -2ν-1)-( µ 2 -1)ρ and ρ > max{ µ 2 + 2ν + 1, 2(ν+ µ 2 ) 2 +ν+( ν 2 +1)µ ν+1+ µ 2 }.
Finally, for the third property using Hölder's inequality, we get

ς 3 |x(t)| ν+µ r + g 3 t t-τ |x(s)| ρ r ds -uτ ρ- µ 2 -2ν-1 ρ |x(t)|
γ = µ 2 -1, δ = µ 2 +2ν+1 ρ and = 3uτ ρ- µ 2 -2ν-1 ρ min{ς,g}
, and that can be formulated as:

max{|x(t)| ν+µ r , t t-τ |x(s)| ρ r ds} ≥   3uτ ρ- µ 2 -2ν-1 ρ min{ς, g}   1 1- µ 2 -1 ν+µ - µ 2 +2ν+1 ρ with (ν + µ) µ 2 +2ν+1 ν+ µ 2 +1 < ρ. For ν > 0 the property ς 3 |x(t)| ν+µ r + b 2 |x(t -τ )| ρ r + t t-τ g 3 |x(s)| ρ r -u max{|x(t)| µ+2ν r , |x(t)| µ-1 r |x(t -τ )| 2ν+1 r , |x(t)| µ-1 r |x(s)| 2ν+1 r }ds ≥ 0,
follows the series of inequalities:

ς 3 |x(t)| ν+µ r + g 3 t t-τ |x(s)| ρ r ds -uτ |x(t)| 2ν+µ r ≥ 0, ς 3 |x(t)| ν+µ r + b 2 |x(t -τ )| ρ r + g 3 t t-τ |x(s)| ρ r ds -uτ |x(t)| µ-1 r |x(t -τ )| 2ν+1 r ≥ 0, ς 3 |x(t)| ν+µ r + t t-τ g 3 |x(s)| ρ r -u|x(t)| µ-1 r |x(s)| 2ν+1 r ds ≥ 0.
The first inequality as before is satisfied for max{|x

(t)| ν+µ r , t t-τ |x(s)| ρ r ds} ≤ 3uτ min{ς,g} -ν+µ ν
. For the second case, applying Young's inequality we obtain

ς 3 |x(t)| ν+µ r -uτ ρ -2ν -1 ρ |x(t)| (µ-1)ρ ρ-2ν-1 r + g 3 t t-τ |x(s)| ρ r ds + b 2 -u 2ν + 1 ρ τ |x(t -τ )| ρ r ≥ 0 that for b 2 ≥ u 2ν+1 ρ τ is followed by max{|x(t)| ν+µ r , t t-τ |x(s)| ρ r ds} -3u ρ -2ν -1 ρ min{ς, g} τ |x(t)| (µ-1)ρ ρ-2ν-1 r ≥ 0, which is true for max{|x(t)| ν+µ r , t t-τ |x(s)| ρ r ds} ≤ 3u ρ-2ν-1 ρ min{ς,g} τ (ρ-2ν-1)(ν+µ) (ν+1)ρ-(2ν+1)(ν+µ) , 2ν + 1 < ρ < (2ν+1)(ν+µ) ν+1
. Finally, for the third case using Hölder's inequality, we get

ς 3 |x(t)| ν+µ r + g 3 t t-τ |x(s)| ρ r ds -uτ ρ-2ν-1 ρ |x(t)| µ-1 r t t-τ |x(s)| ρ r ds 2ν+1 ρ ≥ 0,
which is true provided the conditions of Lemma 4 are verified for

a = |x(t)| r , b = t t-τ |x(s)| ρ r ds, α = ν +µ, β = 1, γ = µ-1, δ = 2ν+1 ρ and = 3uτ ρ-2ν-1 ρ
min{ς,g} , and that can be formulated as

max{|x(t)| ν+µ r , t t-τ |x(s)| ρ r ds} ≤ 3uτ ρ-2ν-1 ρ min{ς,g} 1 ν+1 ν+µ -2ν+1 ρ with (2ν+1)(ν+µ) ν+1
> ρ. Therefore, we established the conditions that

Ẇ (t) < 0 for x t ∈ Ω 2 with ν < 0 or for x t ∈ Ω 2 with ν > 0.
Hence, the parameters of the Lyapunov-Krasovskii functional have to respect the constraints:

b > τ ρ max{4v(ν + 1), u(µ + 4ν + 2)}, g > 0.
Next, taking into account the admissible set of values for µ, the restrictions imposed on ρ can be summarized as follows for ν < 0:

max µ(ν + 1), 2(ν + 1) µ + ν ν + 2 , 2(ν + µ 2 ) 2 + ν + ( ν 2 + 1)µ ν + 1 + µ 2 , (ν + µ) µ 2 + 2ν + 1 ν + µ 2 + 1 < ρ < ν + µ,
or for ν > 0:

max{ν + µ, 2(ν + 1)} < ρ < min{µ(ν + 1), 2(ν + 1) µ + ν ν + 2 , (2ν + 1)(ν + µ) ν + 1 }.
It is straightforward to check that for both signs of degree ν there is a non-empty interval of admissible values for ρ under the given restrictions on µ. Consequently, the properties

W (x t ) > 0, D + W (x t ) < 0
are obtained for x t ∈ Ω with ν < 0 or for x t ∈ Ω with ν > 0, where the expressions for the sets Ω, Ω are given in the formulation of the theorem.

Note that V can be just twice differentiable having bounded d 2 V (y)

dy 2
for y ∈ S r (this property is used to calculate v in Lemma 5).

Remark 2. The local convergence to the origin in the case of ν > 0 or the global convergence to its neighborhood for ν < 0 follows from the result of Theorem 1 using the standard Lyapunov or Lyapunov-Krasovskii arguments [START_REF] Lin | A smooth converse Lyapunov theorem for robust stability[END_REF], [START_REF] Hale | Theory of Functional Differential Equations[END_REF], [START_REF] Fridman | On input-to-state stability of systems with time-delay: A matrix inequalities approach[END_REF].

Qualitatively, this theorem confirms the results of [START_REF] Aleksandrov | On the asymptotic stability of solutions of nonlinear systems with delay[END_REF], [START_REF] Aleksandrov | Delay-independent stability of homogeneous systems[END_REF], [START_REF] Efimov | Weighted homogeneity for time-delay systems: Finite-time and independent of delay stability[END_REF], [START_REF] Zimenko | A note on delay robustness for homogeneous systems with negative degree[END_REF], but it also has some quantitative estimates on the domains of stability (which are included in the sets Ω and Ω) and gives the explicit relations between all parameters. Contrarily [START_REF] Efimov | Weighted homogeneity for time-delay systems: Finite-time and independent of delay stability[END_REF], [START_REF] Zimenko | A note on delay robustness for homogeneous systems with negative degree[END_REF], there is no requirement on continuity of F with respect to both arguments in our results. In addition, a universal expression W (x t ) of a Lyapunov-Krasovskii functional is presented, which can be used for any homogeneous systems of non-zero degree (it also suits for ones admitting local homogeneous approximations [START_REF] Efimov | Homogeneity for time-delay systems[END_REF], [START_REF] Efimov | Development of homogeneity concept for time-delay systems[END_REF]), while the estimates derived in the proof of Theorem 1 can be strengthened as

α |x(t)| µ r + t t-τ |x(s)| ρ r ds ≤ W (x t ) ≤ α |x(t)| µ r + t t-τ |x(s)| ρ r ds , Ẇ (t) ≤ -α |x(t)| µ+ν r + |x(t -τ )| ρ r + t t-τ |x(s)| ρ r ds
for all x t ∈ Ω with ν < 0 or for all x t ∈ Ω with ν > 0, for suitably defined 0 < α ≤ α and α > 0, which is equivalent to

Ẇ ≤ -
x t ≤ σ r (|x(t)| r ) + t t-τ σ r (κ max{λ 1+r -1 min , λ 1+r -1 max })ds ≤ σ r (|x(t)| r ) + t t-2τ σ r (2κ (2|x(s)| r ) 1+r -1 min ) + σ r (2κ (2|x(s)| r ) 1+r -1 max )ds,
which implies that for an extended space C [-2τ,0] the set Ω also can be presented using the conventional norm.

Remark 3. It is important to highlight, that the parameter b of the Lyapunov-Krasovskii functional W can be selected proportionally to max{τ, τ ω } with ω < ρ-ν-1 ρ , then the domain of convergence estimated in Ω, Ω becomes also proportional to τ . In other words, the obtained estimates imply that when the value of delay τ is approaching zero, the domain of convergence to the origin for ν > 0 grows to infinity, while the zone globally attracting the trajectories for ν < 0 shrinks to the origin. Remark 4. Denote 1 = [1, . . . , 1] , then 1-homogeneous systems are homogeneous in the standard (Euler) sense. The result of Theorem 1 in the case of standard homogeneity and positive degree was obtained in [START_REF] Aleksandrov | Complete type functionals for some classes of homogeneous diffrential-difference systems[END_REF]. Nevertheless, the result proven in Theorem 1 for the case of weighted homogeneity has sense, since a transformation of r-homogeneous dynamics to the corresponding 1-homogeneous can be tricky. Indeed, there exist transformations of coordinates making an r-homogeneous system 1-homogeneous [START_REF] Korchak | Design of generalized homogeneous Lyapunov-Krasovskii functions[END_REF], e.g.,

y i = |x i | qr -1 i sign(x i ), i = 1, n.
However, such a transformation may not preserve stability properties of (3). For an example, consider a double integrator stabilized by a delayed feedback that satisfies the conditions of assumptions 1 and 2 for r = [1, 1 + ν] with the degree ν ∈ [-0.5, 0):

ẋ1 (t) = x 2 (t), ẋ2 (t) = -k 1 |x 1 (t -τ )| 1+2ν sign(x 1 (t -τ )) -k 2 |x 2 (t -τ )| 1+2ν 1+ν sign(x 2 (t -τ )).
The derivative of F with respect to the first (non-delayed) argument is well-defined. Consider a change of coordinates:

y 1 = x 1 , y 2 = |x 2 | 1 1+ν sign(x 2 ),
then in the new coordinates we obtain:

ẏ1 (t) = |y 2 (t)| 1+ν sign(y 2 (t)), ẏ2 (t) = -|y 2 (t)| -ν k 1 1 + ν |y 1 (t -τ )| 1+2ν sign(y 1 (t -τ )) -|y 2 (t)| -ν k 2 1 + ν |y 2 (t -τ )| 1+2ν sign(y 2 (t -τ )),
which is 1-homogeneous of degree 1 + ν, but it has a continuum of equilibria on the line y 2 = 0 (a change of the time t = t 0 |y 2 (s)| -ν ds is hard to apply here due to the presence of delay). The requirement on boundedness of the derivative of F with respect to the first argument (imposed in Assumption 1) can be relaxed by considering a special form of the system: Corollary 1. Let F (x, z) = F 1 (x) + F 2 (z) for some functions F 1 , F 2 : R n → R n , the system (3) be r-homogeneous of degree ν ≥ -r min , sup |ξ|r≤1,|ζ|r≤1 |F (ξ, ζ)| < +∞ and Assumption 2 hold. Then for any value of the delay τ > 0 the properties

W (x t ) > 0, D + W (x t ) < 0 are satisfied for x t ∈ Ω if ν < 0 or for x t ∈ Ω if ν > 0, where W (x t ) = V (x(t)) + b t t-τ |x(s)| ρ r ds + g t t-τ (s -t + τ )|x(s)| ρ r ds + t t-τ dV (x(t)) dx F 2 (x(s))ds,
is a Lyapunov-Krasovskii functional for (3); the meaning and constraints imposed on parameters b, g and ρ are the same as in Theorem 1 under substitution u = 0.

Proof. The proof follows exactly the steps of the proof of Theorem 1 taking into account that

H(x t ) = t t-τ F (x(t), x(t -τ )) d 2 V (x(t)) dx(t) 2 F 2 (x(s))ds
and there is no term proportional to u in the analysis.

Let us demonstrate applicability of the presented results on a real case.

IV. APPLICATION

In the paper [START_REF] S´anchez | A constructive Lyapunov function design method for a class of homogeneous systems[END_REF], for a super-twisting differentiation/observation algorithm, ẋ1 (t) = -k 1 |x 1 (t)|sign(x 1 (t)) + x 2 (t), ẋ2 (t) = -k 2 sign(x 1 (t)),

x(t) = [x 1 (t), x 2 (t)] ∈ R 2 , where 0 < k 2 < k 1 < +∞ are the observer gains, the following Lyapunov function has been proposed:

V (x) = η 1 x 2 1 + η 2 x 1 x 2 |x 2 | + η 3 x 4 2 ,
for η 1 , η 3 ∈ R + and η 2 ∈ R, which is continuously differentiable, and its second derivative is discontinuous (it contains sign(x 2 )) but bounded on any compact set, hence, the constant v can be estimated. It is straightforward to check that the An example of the system trajectory is shown in Fig. 1, where behavior of x 1 (t) and x 2 (t) are depicted using red solid and blue dash lines, respectively. The system is converging to a limit cycle. An example of the comportment of the norms y(t) = max{|x(t)| ν r , t t-τ |x(s)| ρ r ds} and ψ(t) = max{|x(t)| ν+µ r , t t-τ |x(s)| ρ r ds} are presented using red solid and blue dash lines, respectively, in Fig. 2. As we can conclude from these results of simulation, the obtained estimates are rather moderate, and for a concrete example the methodology has to be properly adjusted to reduce the conservativeness.

V. CONCLUSIONS

Considering the homogeneous systems with non-zero degree, which are asymptotically stable without delay, it is proposed a corresponding generic structure of a Lyapunov-Krasovskii functional, which allows also evaluation of the domain of convergence in the presence of any constant delay. In particular, we show that for the systems with a positive degree, the domain of convergence to the origin is inversely proportional to the value of the delay; while for the system with a negative degree, the size of the vicinity of the origin, which globally attracts all trajectories, is proportional to the value of delay. This Lyapunov-

  r }ds for ν > 0, and let us investigate the conditions of negative definiteness of Ẇ in the domains of positive definiteness of W for different signs of degree ν. First,

-

  provided the conditions of Lemma 4 are verified for a = |x(t)| r , b = t t-τ |x(s)| ρ r ds, α = ν + µ, β = 1,

  system and the Lyapunov function are r-homogeneous of degrees ν = -1 and µ = 4, respectively, with r =[START_REF] Hale | Theory of Functional Differential Equations[END_REF][START_REF] Gu | Stability of Time-Delay Systems[END_REF] . Assume that the actual output injection exhibits a delay τ > 0:ẋ1 (t) = -k 1 |x 1 (t -τ )|sign(x 1 (t -τ )) + x 2 (t), ẋ2 (t) = -k 2 sign(x 1 (t -τ )),then all conditions of Corollary 1 are satisfied, and such a F (x(t), x(t -τ )) is continuously differentiable with respect to the first argument.

Figure 1 .Figure 2 .η 1 = 4 , η 2 = - 2 , η 3 = 5 .

 1214225 Figure 1. Behavior of trajectories of the system

  = sup |ξ|r≤1,|ζ|r≤1 |F (ξ, ζ)| r and λ = max{|x(s)| r , |x(s -τ )| r } we obtain:

				αW 1+ ν µ
	for some α > 0 dependent on Ω or Ω. Note that
		t
	max{|x(t)| ν+µ r	,	|x(s)| ρ r ds} ≤ max{1, τ } max{ x t	ν+µ r	, x t	ρ r },
		t-τ
	then the set Ω can be re-formulated using the standard norm. Similarly, x t = |x(t -θ)| for some θ ∈ [-τ, 0], then
			t	t
	|x(t -θ)| = |x(t) -	ẋ(s)ds| ≤ |x(t)| +	| ẋ(s)|ds
			t-θ	t-τ
			t
	≤ σ r (|x(t)| r ) +	σ r (|F (x(s), x(s -τ ))| r )ds,
			t-τ
	next defining κ

If time is scaled t → λ d t then the argument of f : C [-τ,0] → R n in (1) is also scaled to f : C [-λ -d τ,0] → R n in (2).

This work was partially supported by the Government of Russian Federation (Grant 08-08), by the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898, and by the Russian Foundation for Basic Research (grant no. 19-01-00146-a).

Krasovskii functional can be useful for analysis of convergence of discretization of solutions of homogeneous delayed nonlinear systems by using Euler method, which is a direction of future research.


Henri Gouin
email: henri.gouin@univ-cezanne.fr

Thermocapillary fluid and adiabatic waves near the critical point THERMOCAPILLARY FLUID AND ADIABATIC WAVES NEAR THE CRITICAL POINT

Keywords: Exceptional waves, isentropic motions, fluid interfaces, extended van der Waals theory PACS, MSC Numbers : 64.60.Ht, 74Jxx, 47.35.Fg, 64.70.Fx, 47.35.Pq, 47.55.N-, 46.40.Cd

Isothermal interfacial zones are investigated starting from a local energy which can be considered as the sum of two terms: one corresponding to a medium with a uniform composition equal to the local one and a second one associated with the non-uniformity of the fluid. The additional term can be approximated by a gradient expansion, typically truncated to the second order. A representation of the energy near the critical point therefore allows the study of interfaces of nonmolecular size. Capillary layer and bulk phases are not considered independently. Obviously, this model is simpler than models associated with the renormalizationgroup theory. Nevertheless, it has the advantage of extending easily well-known results for equilibrium cases to the dynamics of interfaces. The equation of state of a one-component system may be expressed as a relation among the energy, entropy and matter density, α, s and ρ in the form α = α(ρ, s). Now, let α(ρ, s) be the analytic α as it might be given by a mean-field theory. In the simplest case, in an extended van der Waals theory 18 , the volume internal energy ε is proposed with a gradient expansion depending not only on grad ρ but also on grad s (the associated fluid is called thermocapillary fluid 3 ):

ε = f (ρ, s, grad ρ, grad s) (1)
With an energy in the form (1), we obtain the equations of motion of conservative movements for nonhomogeneous fluid near its critical point. For such a medium, it is not possible to obtain shock waves. The idea of studying interface motions as localized travelling waves in a multi-gradient theory is not new and can be traced throughout many problems of condensed matter and phase-transition physics [START_REF] Gouin | Symposium on waves in liquid/gas and liquid/vapor two-phase systems[END_REF] .

Here, adiabatic waves are considered and a new kind of waves appears. The waves are associated with the spatial second derivatives of entropy and matter density. In Cahn and Hilliard's model 1 , the direction of solitary waves was along the gradient of density. For this new kind of adiabatic waves, the direction of propagation is normal to the gradient of densities. In the case of a thick interface, the waves are tangential to the interface and the wave celerity is expressed depending on thermodynamic conditions at the critical point.

Equations of motion of thermocapillary fluids 1.Conservative motions

For conservative motions of perfect fluids -that is to say without heat fluxes or viscosity-the specific entropy of each particle remains constant along trajectories ( ṡ = 0, where ˙denotes the material derivative). A convenient method allowing to obtain the equations of motion comes from the Hamilton principle. The notations are in Serrin [START_REF] Serrin | Encyclopedia of physics VIII/I. Mathematical principles in classical fluid mechanics[END_REF] : in a fixed coordinate system, the components of a vector (covector) a are denoted by a i , (a i ), where i = 1, 2, 3. In order to describe the fluid motion analytically, we refer to the coordinates x ≡ (x 1 , x 2 , x 3 ) as the particle's position (Eulerian variables).

The corresponding reference position is denoted by X ≡ (X 1 , X 2 , X 3 ) (Lagrangian variables). The motion of a fluid is classically represented by the transformation x = ϕ(t, X) or x i = ϕ i (t, X), where t is the time. It is assumed that ϕ possesses an inverse X = φ(t, x) and continuous derivatives up to the second order except at certain surfaces.

We denote by z = t x the time-space variables; u ≡ ∂ϕ ∂t (t, X) and

V ≡ 1 u
are respectively the velocity and the time-space velocity. Let us consider a mobile surface Σ t defined in the physical space D t occupied by the fluid. Let us denote by g the celerity of Σ t , n its normal vector,

N = -g n and v = N * V ≡ n * u -g the fluid velocity with respect to Σ t ,
where * is the transposition on D t . The Lagrangian of a thermocapillary fluid is written

L = 1 2 ρu * u -ε -ρΩ,
where Ω is the extraneous force potential defined as a function of z and ε is the internal energy per unit volume. Between times t 1 and t 2 , Hamilton's action is

a = t2 t1 Dt

L dvdt

A variation of the particle motion comes from a family of virtual motions X = ψ(t, x, r) where r denotes a small parameter close to 0. The real motion is associated with r = 0 ψ(t, x, 0) = φ(t, x) and a virtual displacement is expressed in the form 7 δX = ∂ψ ∂r r=0

(2)

Expression (2) corresponds to the dual of the variation given by Serrin 14 in page 145. The density and the specific entropy verify respectively

ρ det F = ρ 0 (X), (3)
s = s 0 (X), (4)
where ρ 0 and (for conservative motions) s 0 are defined into a reference space D 0 and at time t fixed, F is the Jacobian of ϕ. If ε given by relation ( 1) is differentiable,

δε = ε ,ρ δρ + ε ,s δs + Φ i δρ ,i + Ψ i δs ,i ,
and consequently, the theory introduces two new vectors Φ and Ψ such that :

Φ i = ε ,ρ,i and Ψ i = ε ,s,i (In the case of compressible fluids, scalars ε ,ρ and (1/ρ) ε ,s are the specific enthalpy and the Kelvin temperature). Due to the fact the fluid is isotropic, grad ρ and grad s are taken into account by their scalar products only [START_REF]New perspectives in thermodynamics[END_REF] . Let us denote β = (grad ρ) 2 , χ = grad ρ . grad s, γ = (grad s) 2 and in variables ρ, s, β, χ, γ, ε = g(ρ, s, β, χ, γ)

Consequently,

Φ = C grad ρ + D grad s, Ψ = D grad ρ + E grad s, (5)
with

C = 2 ε ,β , D = ε ,χ , E = 2 ε ,γ
Variations of Hamilton's action are deduced from classical method of variational calculus 7 ,

δa = t2 t1 Dt 1 2 u * u δρ + ρu * δu -ε ,ρ δρ -ε ,s δs -Φ * δ grad ρ -Ψ * δ grad s -Ω δρ dvdt
Due to the definition of virtual displacement in (2), δ grad ρ = grad δρ, δ grad s = grad δs and δu = -F ⌢ δX

Due to the fact the virtual displacement δX and its derivatives are assumed null at the boundary of D t , integration by parts using Stokes' formula yields

δa = t2 t1 Dt 1 2 u * u -ε ,ρ + div Φ -Ω δρ + div Ψ -ε ,s δs + ρ ⌢ u * F δX dvdt
Taking into account relations 7 δρ = ρ div 0 δX + 1 detF

∂ρ 0 ∂X δX, (6)
where div 0 denotes the divergence operator in D 0 and

δs = ∂s 0 ∂X δX, (7)
we get finally

δa = t2 t1 D0 ρ 0 ⌢ u * F -grad 0 m -θ grad 0 s 0 δX dv 0 dt
where grad 0 is the gradient in D 0 , and

θ = 1 ρ (ε ,s -div Ψ) , h = ε ,ρ -div Φ, m = 1 2 u * u -h -Ω Then, Hamilton's principle yields ⌢ u * F = grad 0 m + θ grad 0 s 0 Let us note that Γ * + u * ∂u ∂x F = ⌢ u * F ,
where Γ is the acceleration vector; we obtain the equation of motion in the form

Γ = θ grad s -grad(h + Ω), (8)
which is the extension of relation (29.8) in Serrin [START_REF] Serrin | Encyclopedia of physics VIII/I. Mathematical principles in classical fluid mechanics[END_REF] . It is easy to prove that by algebraic calculus, this equation is equivalent to the balance of momentum 3,4 ∂ ∂t

(ρu * ) + div(ρ uu * -σ) + ρ grad Ω = 0 (9)
with σ j i = -(P -ρ div Φ) δ j i -Φ j ρ ,i -Ψ j s ,i where P = ρ ε ,ρ -ε (in the case of compressible fluids, P denotes the pressure). Relation ṡ = 0 is equivalent to the balance of energy 3,4 ∂e ∂t

+ div (e -σ)u -div U -ρ ∂Ω ∂t = 0 with U = ρ Φ + ṡ Ψ and e = 1 2 ρu * u + ε + ρΩ.

Properties of conservation for isentropic thermocapillary fluids

Conclusions obtained in ( 2,[START_REF] Serrin | Encyclopedia of physics VIII/I. Mathematical principles in classical fluid mechanics[END_REF] ) are easily extended to thermocapillary fluids.

They are deduced from Eq. ( 8). Let us recall the main results:

J = C u * dx
denotes the circulation of the velocity on a closed curve convected by the fluid. Then,

dJ dt = C θ ds (10)
We obtain: Kelvin's circulation theorem: the circulation of the velocity on a closed and isentropic curve is constant. For any motion of isentropic thermocapillary fluid, it is possible to introduce scalar potentials with the following evolutions 2 :

κ = m ≡ 1 2 u * u -h -Ω, τ = 0, ξ = 0, ς = 0, ṡ = 0, (11)
such that the velocity field is written

u = grad κ + ξ grad s + τ grad ς (12)
From eqs. ( 11) and ( 12), one deduces the classification given by Casal 2 for conservative flow of perfect fluids.

Oligotropic motions: they are motions for which iso-entropy surfaces are surfaces of vorticity. The circulation of the velocity on an isentropic closed curve convected by the fluid is null. Eq. ( 12) yields u = grad κ + ξ grad s Homentropic motions: s is constant in the fluid and Eq. ( 12) yields

u = grad κ + τ grad ς
The Cauchy theorem is extended without difficulty:

d dt rot u ρ = ∂u ∂x rot u ρ If H = 1 2 u * u + h
+ Ω, we deduce from Eq. ( 8), the extended Crocco-Vazsonyi equation for steady thermocapillary fluid motions :

rot u × u = θ grad s -grad H
Conservation laws expressed with Kelvin's theorems are associated to the group of permutation of particles of the same entropy. This group keeps the equations of motion invariant. As in ( 6) it is possible to associate an expression of the Noether theorem to this group. It is natural to conjecture such results for general fluids endowed with an internal energy which is a functional of matter and entropy densities.

Isothermal motions

Now, let us consider the case when only the total variation of the total entropy in D t is zero 3 (virtual displacements δX conserve the total entropy of D t ). Then,

δ Dt ρs dv = 0
There exists a constant Lagrange multiplier T 0 such that the new Lagrangian

L = 1 2 ρ u * u -ε -ρ Ω + T 0 ρ s yields δa = t2 t1 Dt δ( 1 2 ρ u * u -ε -ρ Ω + T 0 ρ s) dvdt = 0
and consequently, as in section 1.1,

δa = t2 t1 Dt 1 2 u * u -ε ,ρ + T 0 s + div Φ -Ω δρ + + div Ψ -ε ,s + T 0 ρ δs + ρ ⌢ u * F δX dvdt = 0
Here δs is any scalar field in D t and δρ is given by relation (6). Finally, we obtain

θ = T 0 , (13a)
Γ = -grad(µ + Ω), (13b)
where µ = ε ,ρ -T 0 s-div Φ is the chemical potential of the thermocapillary fluid. Eqs (13a), (13b) are the equations of motion of an isothermal capillary fluid. Let us remark that we obtain also Kelvin's circulation theorem: The circulation of the velocity on a closed and isotherm curve is constant.

Liquid-vapor interface near its critical point

The critical point associated with the equilibrium of two bulks of a fluid corresponds to the limit of their coexistence. The interface between the phases disappears when that point is reached. The thickness of the interface increases as the critical point is approached and it becomes infinite when the interface itself disappears. Much of what has been done on the theory of the near-critical interface has been within the framework of the van der Waals theory 18 , so much the present understanding of the properties of those interfaces comes from that theory or from suitable extended version of it [START_REF] Rowlinson | Molecular theory of capillarity[END_REF] . As its critical point is approached, the gradients of densities are then small. The present point of view (the interfacial region may be treated as matter in bulk, with local energy density that is that of a uniform fluid of composition equal to the local one, with an additional term arising from the non-uniformity, expressed by a gradient expansion truncated in second order) is then most likely to be successful and even qualitatively accurate.

In the following, we consider the case when

ε = ρ α(ρ, s) + 1 2 C (grad ρ) 2 + 2D grad ρ . grad s + E (grad s) 2 (14)
where α denotes the specific internal energy of the fluid in uniform composition, C, D, E are constants and CE -D 2 > 0.

If D = 0 and E = 0, we are back to the Cahn and Hilliard model of capillarity 1 . If not, we deduce

h = h 0 -(C ∆ ρ + D ∆ s), θ = T - 1 ρ (D ∆ ρ + E ∆ s),
where h 0 ≡ α + ρ α ′ ρ and T ≡ α ′ s are respectively the specific enthalpy and the Kelvin temperature of the homogeneous fluid of matter density ρ and specific entropy s. Let us note that vectors Φ and Ψ are always given by expression ( 5), but here coefficients C, D and E are constant. At phase equilibrium, Eq. ( 13a) is verified when T 0 is the temperature in the bulks. If we neglect the body forces and denote µ 0 ≡ α + ρ α ′ ρ -sT 0 , we obtain

C ∆ ρ + D ∆ s = µ 0 -µ 1 , (15a)
D ∆ ρ + E ∆ s = ρ (T -T 0 ), (15b)
where

µ 1 = µ(ρ l , s l ) = µ(ρ v , s v ) is constant (ρ l , s l , ρ v , s v are
the densities in the liquid and the vapor bulks). In one-dimensional problems, ρ = ρ(y) and equations of equilibrium are associated with the system

C ρ ′′ + D s ′′ = ε ′ ρ -s T 0 -µ 1 , (16a)
D ρ ′′ + E s ′′ = ε ′ s -ρ T 0 , (16b)
where ′′ denotes the second derivative with respect to the space variable y. Near the critical point of the fluid, we use the representation of ρ α(ρ, s) in relation (14) in the form

ρ α = B 2A 2 A(ρ -ρ c ) 2 + η 2 + η 2 + µ c ρ + T c η -p c (17)
given by Rowlinson and Widom in ( 12) when η = ρs is the entropy per unit volume in which s c = 0, A and B are two positive constants associated with the critical conditions and µ c , T c , p c are respectively the values of the chemical potential, the temperature and the pressure at the critical point.

It is easy to verify that this expression is equivalent to the chemical potential µ 0 of a compressible fluid (case when C = D = E = 0) in the form :

µ 0 (ρ, T 0 ) = µ c + B (ρ -ρ c ) 3 -A (T 0 -T c )(ρ -ρ c ),
where µ c is the value of µ 0 for the critical conditions.

2.1. Asymptotic analysis of system ( 16) near the critical point

Due to relation (17), system ( 16) yields,

C ρ ′′ + D s ′′ = 2 B (ρ -ρ c ) 3 + 2 B A ρ s (ρ -ρ c ) + B A (ρ -ρ c ) 2 s + 2 B A 2 ρ s 2 + (T c -T 0 ) s + µ c -µ 1 , D ρ ′′ + E s ′′ = B A ρ (ρ -ρ c ) 2 + 2 B A 2 ρ 2 s + (T c -T 0 ) ρ
To consider the physical scales associated with the interfacial sizes, we look at the change of variables

Y = ǫ y, ρ(y) -ρ c = ǫ n1 R(Y ), s(y) = ǫ n2 S(Y ),
where 0 < ǫ ≪ 1, and n 1 , n 2 are two positive constants. We suppose the coefficients C, D and E to have finite, non-vanishing limiting values at the critical point. Then the main part of system ( 16) leads to

ǫ 2 Cǫ n1 d 2 R dY 2 + Dǫ n2 d 2 S dY 2 = ǫ 3n1 2 B R 3 + ǫ n1+n2 2 B A ρ c R S ǫ 2n1+n2 B A R 2 S +ǫ 2n2 2 B A 2 ρ c S 2 +ǫ n2 (T c -T 0 ) S +µ c -µ 1 , ǫ 2 Dǫ n1 d 2 R dY 2 + Eǫ n2 d 2 S dY 2 = ǫ 2n1 B A ρ c R 2 +ǫ n2 2 B A 2 ρ 2 c S +(T c -T 0 )ρ c
It is easy to verify that the solution is associated with n 1 = 1, n 2 = 2 and µ c = µ 1 . Near the critical point, in densities ρ, s and variable y, system (16) leads to an approximation in the form :

C ρ ′′ = ε ′ ρ -s T 0 -µ 1 , (18a)
ε ′ s -ρ T 0 = 0 (18b)
By other arguments, this asymptotic analysis gives the same results as in ( 12), page 254, where Rowlinson and Widom compared the magnitudes of different terms in a system similar to system (16): the magnitude of D s ′′ is negligible compared with the typical magnitude of C ρ ′′ and the magnitude of D ρ ′′ + E s ′′ is negligible with respect to the magnitude of ε ′ s -ρ T 0 .

2.2. Integration of system (16) in the approximation of the critical point

The approximation of the system (16) yields Eq. (18b) which is equivalent to θ = T 0 and consequently,

2 ρs = A 2 B (T 0 -T c ) -A (ρ -ρ c ) 2 (19)
Taking Eq. (18a) into account, we obtain

C ρ ′′ = B (ρ -ρ c ) 3 -A (T c -T 0 )(ρ -ρ c ) + µ c -µ 1 (20)
and in the following µ 1 = µ c such that Eq. ( 20) is the classical equation of the density profile of a liquid-vapor interface [START_REF] Rowlinson | Molecular theory of capillarity[END_REF] . Integration of Eq. ( 20) yields

1 2 Cρ ′2 = B 4 (ρ -ρ c ) 4 - A 2 (T c -T 0 )(ρ -ρ c ) 2 + f 0 ,
where f 0 is a constant which verifies for a planar liquid-vapor interface

f 0 = A 2 4 B (T c -T 0 ) 2 ,
and we obtain

1 2 Cρ ′2 = √ B 2 (ρ -ρ c ) 2 - A 2 √ B (T c -T 0 ) 2 (21)
Eq. ( 21) yields the profile of matter density in the interfacial layer 12

ρ(y) = ρ c + 1 2 (ρ l -ρ v ) tanh( y 2ζ )
with

ρ l = ρ c + A B (T c -T 0 ) 1 2 , ρ v = ρ c - A B (T c -T 0 ) 1 2 , ζ = C 2A (T c -T 0 ) 1 2
, where ζ is the characteristic length of the interfacial layer. Moreover, the surface tension of the interfacial layer is :

σ = +∞ -∞ Cρ ′2 (y) dy ≡ √ C 3B 2A(T c -T 0 )
3. Weak discontinuity in conservative motions

Conditions of a weak discontinuity

To the equation of conservative motions given by Eq. ( 8), we have to add the equation of balance of mass

∂ρ ∂t + div(ρu) = 0, (22)
the equation of conservation of the specific entropy ṡ = 0 and relation (17).

Weak discontinuities of isentropic motions correspond to ρ, s, ∂ρ ∂x , ∂s ∂x , continuous through the wave surfaces. As in Hadamard 9 , we denote by [ ] the jump of a tensorial quantity through a surface of discontinuity Σ t . Consequently, with Hadamard's tensorial framework, there exits two Lagrange multipliers λ 1 , λ 2 such that :

∂ρ ∂z = 0 ⇒ ∂ ∂z ( ∂ρ ∂z ) * = λ 1 NN * ⇒ ∂ ∂x ( ∂ρ ∂x ) * = λ 1 nn * , (23a)
∂s ∂z = 0 ⇒ ∂ ∂z ( ∂s ∂z ) * = λ 2 NN * ⇒ ∂ ∂z ( ∂s ∂z ) * = λ 2 nn * , (23b)
λ 2 NN * V + NΞ * ∂s ∂z * = 0 ⇔ N λ 2 v + ∂s ∂x H = 0
or,

λ 2 v + ∂s ∂x H = 0 (25)
From Rankine-Hugoniot condition associated to Eq. ( 9), we obtain the compatibility condition: [-g ρ u * + ρ n * uu * -n * σ] = 0, and the continuity of ρ, ρ ,i , s, s ,i , yields [ div Φ ] = 0, or

[ C∆ρ + D∆s ] = 0 ⇔ C λ 1 + D λ 2 = 0 (26)
Consequently, there exists a scalar Lagrange multiplier λ 3 such that ∂ ∂x

(C∆ρ + D∆s) = λ 3 n * (27)
Equation of motion ( 8) yields

[ u ] = [ θ ] grad s -λ 3 n, or ρ v H = -(Dλ 1 + Eλ 2 ) grad s -ρ λ 3 n
By projection on the normal n to Σ t and taking relation (24) into account, we obtain

ρ λ 3 + n * grad s (D λ 1 + E λ 2 ) = 0
By projection on the tangent plane to Σ t and taking relation (24) into account, we obtain

ρ v H = -(D λ 1 + E λ 2 ) grad tg s, (28)
where grad tg s denotes the tangential part of grads in Σ t . Elimination of H in the relation (28) comes from relation (25), and we get

D (grad tg s) 2 λ 1 + E (grad tg s) 2 -ρ v 2 λ 2 = 0
Consequently, we obtain a system of three linear equations with respect to the variables λ 1 , λ 2 , λ 3 ,

           Cλ 1 + Dλ 2 = 0, D (n * grads) λ 1 + E (n * grads) λ 2 + ρ λ 3 = 0, D (grad tg s) 2 λ 1 + E (grad tg s) 2 -ρ v 2 λ 2 = 0
The compatibility of these three equations yields

ρ v 2 = (CE -D 2 ) (grad tg s) 2 C ( 29
)
Scalar v is the celerity of the surfaces for isentropic weak discontinuity of acceleration in an isothermal fluid interface near its critical point. ). An isentropic wave surface of acceleration is orthogonal to isodensity layers and its celerity of value v is tangential to the dividing surface.

Celerity of isentropic waves of acceleration

The temperature in liquid and vapor bulks is T 0 . Then, relation (19)

yields grad A(ρ -ρ c ) 2 + 2 ρ s = 0 ⇔ A(ρ -ρ c ) grad ρ + grad ρs = 0
The value ρ = ρ c of the matter density in the interface corresponds to the maximum value of grad ρ (see Eq. ( 20), when µ c = µ 1 ). The matter density ρ c is characteristic of the interfacial matter. For such a value, grad ρs = 0 and s = (A 2 /2Bρ c ) (T 0 -T c ). Consequently,

grad s = A 2 2Bρ 2 c (T c -T 0 ) grad ρ, and
v 2 = (CE -D 2 ) A 4 (grad tg ρ) 2 (T c -T 0 ) 2 4 C B 2
v 2 = (CE -D 2 ) A 6 (T c -T 0 ) 4 8 C 2 B 3 ρ 5 c (30)
In the interfacial layer, grad ρ is normal to iso-density surfaces. The isentropic waves of acceleration associated with a weak discontinuity shear the interfacial layer (see Fig. 1). The wave celerity, which is proportional to (T c -T 0 ) 2 , vanishes at the critical point and can be calculated numerically by means of a state equation.

Results and discussion

Dynamics of liquid-vapor interfaces is easily studied in mechanics by means of second gradient theory which is an extension of the Landau-Ginzburg models in physics. The theory is associated with continuous variations of the matter density through the interfacial layer and was initiated by van der Waals, improved by Rocard in gas theory [START_REF] Rocard | Thermodynamique, V[END_REF] and Cahn and Hilliard in physical chemistry 1 . Rowlinson and Widom [START_REF] Rowlinson | Molecular theory of capillarity[END_REF][START_REF] Widom | Phase transitions and critical phenomena[END_REF] pointed out that the model can be extended by taking into account not only the strong variations of matter density through the interfacial layer but also the variations of entropy. Due to the fact the variation of matter density leads the variations of entropy (see Eq. ( 19)), this extension seems at first sight purely formal and yields the same results that the classical van der Waals model 18 does. Solitary waves, motions in the normal direction to fluid interfaces are not involved in an additive dependance of the entropy gradient. In this paper, we see the dependance of entropy gradient is necessary for isentropic waves of acceleration along the interfaces: the fact the internal energy depends not only on the gradient of matter density but also on the gradient of entropy, yields a new kind of waves which does not appear in the simpler models by van der Waals, Cahn and Hilliard, Rocard and many others 10,17,16... : this kind of waves does not appear when the gradient of entropy is not taken into account. It is easy to see they are exceptional waves in the sense of Lax 13 and they appear only in, at least, two-dimension spaces. Recent experiments in space laboratories, for carbonic dioxide near its critical point have showed the possibility of such waves.

λ 1 =

 1 [ ∆ρ ] and λ 2 = [ ∆s ], (23c) From [V] = 0, we deduce ∂V ∂z = Ξ N * with N * = (-g, n *) and Ξ = 0 H is a vector Lagrange multiplier, where H is a 3-vector of D t . Then,[ u] = ∂V ∂z V = v H, with v = N * V ≡ n * ugHere, v denotes the velocity of the fluid with respect to the wave surface of acceleration Σ t . Equation of mass conservation (22) is equivalent to

Figure 1 .

 1 Figure 1. The interfacial layer has a real thickness of the order of the characteristic length ζ. The average matter density ρc corresponds to a dividing surface, (Rowlinson & Widom, chapter 3). An isentropic wave surface of acceleration is orthogonal to isodensity layers and its celerity of value v is tangential to the dividing surface.

ρ 5 c

 5 Due to relation (21), we obtain when ρ = ρ c , C (grad tg ρ) 2 = A 2 2 B (T c -T 0 ) 2 , and consequently,

Acknowledgments

This work was performed under the auspices of GDR 2258 CNRS/CNES by French Spatial Agency. I am grateful to Professor Ruggeri for his comments and invitation to the "Wascom 2003" and to Professor Gavrilyuk for his criticisms.


Martin Dres

Boris Jung

Nicolas Molinari

Federico Manna

Bruno-Pierre Dubé

Gerald Chanques

Thomas Similowski

Samir Jaber

Alexandre Demoule

Respective contribution of intensive care unit-acquired limb muscle and severe diaphragm weakness on weaning outcome and mortality: a post hoc analysis of two cohorts

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Gaëtan Texier

Rodrigue S Allodji

Loty Diop

Jean-Baptiste Meynard

Liliane Pellegrin

Herve Chaudet

Hervé Chaudet

Rodrigue S Alldoji

Correction to: Using decision fusion methods to improve outbreak detection in disease surveillance


We designed, fabricated and characterized a microresonator in the terahertz range to increase light matter interaction using near-field optics. The goal of our device is to confine a THz electric field in a volume smaller than the diffraction limit to enhance light matter interaction with subwavelength samples. We aim at field confinement on a length scale of a few tens of nanometers.

I. INTRODUCTION

nhancing the interaction between matter and THz radiation is a major requirement for the study of macromolecules, as many of them have responses in this range but are only available in small volume, far below the free space diffraction limit of a THz beam. Improving the quality of THz spectroscopy experiments has application in biology since it has been proven to be a viable approach for the characterization of biological molecular structures such as large proteins, RNA or DNA [START_REF] Globus | THz-Spectroscopy of Biological Molecules[END_REF][START_REF] Castro-Camus | Conformational changes of photoactive yellow protein monitored by terahertz spectroscopy[END_REF]. Light can be confined with a dielectric resonant structure such as the so called photon cages [START_REF] Sieutat | Strong confinement of light in low index materials: the Photon Cage[END_REF], or other photonic crystal cavities [START_REF] Descharmes | Observation of Backaction and Self-induces Trapping in a Planar Hollow Photonic Crystal Cavity[END_REF]. But reaching even smaller confinement volume compared to the wavelength requires using metallic structures. Confining THz radiations in a subwavelength volume to enhance light matter interaction has been done with planar metallic waveguide coupled with antennas, allowing the analysis of samples with volume of a few nL [START_REF] Peretti | Broadband terahertz lightmatter interaction enhancement for precise spectroscopy of thin films and micro-samples[END_REF][START_REF] Mitryukovskiy | Shining the Light to Terahertz Spectroscopy of nL-Volume Biological Samples[END_REF][START_REF] Peretti | Device for Light-Matter Interaction Enhancement in the Full THz Range for Precise Spectroscopy of Small Volume Samples[END_REF].

Another approach to surpass the diffraction limit uses near field optic to confine a non-propagating field around a nanoscale metallic object, for example the scanning tip of a SNOM (scanning near-field optical microscopy) [START_REF] Mastel | Terahertz nanofocusing with cantilevered terahertzresonant antenna tips[END_REF]. Here we propose the use of split-ring resonators as a way to achieve the same level of confinement as a SNOM scanning tip while being compact enough to be integrated in a micro-fluidic chip.

II. DESIGN

A split ring resonator (SRR) is a sub wavelength resonant metallic structure analogous to a LC circuit [START_REF] Froncisz | The loop-gap resonator: a new microwave lumped circuit ESR sample structure[END_REF]. The most basic example of SRR is a metallic loop with a small gap, where the loop is acting as an inductance, and the gap is acting as a capacitance. This kind of structure has a resonant mode with an electric field localized in the capacitive area. Since the equivalent capacity of a parallel plate capacitance is at the same time proportional to the surface area of the two plates and inversely proportional to the distance between the said plates, this capacitive area can be designed arbitrarily small while keeping the same capacity, by decreasing concordantly both variables. Same capacity implies same resonance frequency, which is convenient for tuning the design, and most importantly same energy stored in this arbitrarily small volume. This makes split ring resonators a promising way to achieve very high spatial confinement of an electric field in the Terahertz and millimeter wave range.

The only theoretical limitation for the confinement volume arises when the size of the gap is small enough that electron cross it through quantum tunneling, making it effectively conductive and shunting the capacitance. This effect is significant only for gap sizes below 5nm [START_REF] Scholl | Observationof Quantum Tunneling between Two Plasmonic Nanoparticles[END_REF]. Hence this will not be an actual limitation for enhancing light matter interaction with biological molecular structures having potential response in the THz range, most of which are larger than 5nm.

The SRR design tested here has a single gap and two inductive loops, such that the magnetic fields from both loops are cancelling each other in far field. This reduces the radiative losses of the resonator, and therefore increase its quality factor. Our SRR are 200nm thick, made out of gold on a quartz substrate. The substrate is only 150µm thick to allow microscopic observation from below.

III. FABRICATION AND CHARACTERIZATION

The resonators are made starting from a bare 150µm thick quartz substrate, applying a positive resist coating, and writing the SRR pattern on the coating with e-beam lithography. After resist development, 20nm of Titanium and 200nm of gold are then deposited on the sample by evaporation. Finally, the resist is chemically removed and the excess metal is lifted-off, leaving only the SRR on the quartz. This technology showed very good results, with excellent agreement between the wanted shape and the scanning electron microscope (SEM) observations (see figure 1). Moreover, gap size down to about 25nm have been reached, which is promising for extreme field confinement experiments. The fabricated SRR are characterized with a THz time domain spectroscopy (TDS) setup (see figure 4). It allows to measure the temporal and spectral characteristics of the resonance mode, such as the resonance frequency and the quality factor. Those parameters are extracted directly from the temporal response with the fit@TDS software developed by our team [START_REF] Peretti | THz-TDS time trace analysis for physical parameters extraction[END_REF].

In order to better characterize the spatial properties of the resonance, the electric field at different point around the resonator will be measured using a THz SNOM. This will allow us to get a map of the electric field and to measure the actual confinement volume of the field at the resonance.

IV. CONCLUSION

We have gathered all the elements needed to study the limit of electric field confinement in a split ring resonator, potentially reaching a few tens of nanometer scale at 2.5 THz, which represent a relative length of λ/4000. Such a confinement needs to be further characterized, but is very promising to reach high enhancement of light matter interaction at very small scale such as predicted in numerical simulation (FDTD software from Lumerical, see figure 3). This will allow the derivation of the Purcell factor and subsequent conclusion of possible strong coupling or other quantum effects. This work will constitute a powerful tool for the study of the susceptibility of nanometer sized elements with a terahertz response, such as many biological samples like viruses and proteins. In addition, it could be extended to more fundamental experiments aiming at exploring quantum effects in the terahertz range.

EFig. 1 .

 1 Fig. 1. SEM image of the fabricated SRR.

Fig. 2 .

 2 Fig. 2. SEM close view of a ~25-30nm gap.

Fig. 3 .

 3 Fig. 3. Simulated electric field map at the resonance in SRR with a 500nm gap. A very strong electric field is predicted in the gap.

Fig. 4 .

 4 Fig. 4. Measured transmission spectrum of an array of resonator with TDS, relative to the transmission of a bare quartz substrate.

IEMN, Univ. Lille, Villeneuve d'Ascq, France Terahertz radiation confinement using metallic resonators.