A V Kazeykina

A large time asymptotics for the solution of the Cauchy problem for the Novikov-Veselov equation at negative energy with non-singular scattering data

In the present paper we are concerned with the Novikov-Veselov equation at negative energy, i.e. with the (2 + 1)-dimensional analog of the KdV equation integrable by the method of inverse scattering for the two-dimensional Schrödinger equation at negative energy. We show that the solution of the Cauchy problem for this equation with non-singular scattering data behaves asymptotically as const t 3/4 in the uniform norm at large times t. We also prove that this asymptotics is optimal.

Introduction

In the present paper we consider the Novikov-Veselov equation

∂ t v = 4Re(4∂ 3 z v + ∂ z (vw) -E∂ z w), ∂ z w = -3∂ z v, v = v, E ∈ R, E < 0, (1.1) v = v(x, t), w = w(x, t), x = (x 1 , x 2 ) ∈ R 2 , t ∈ R,
where

∂ t = ∂ ∂t , ∂ z = 1 2 ∂ ∂x 1 -i ∂ ∂x 2 , ∂ z = 1 2 ∂ ∂x 1 + i ∂ ∂x 2 .
We will say that (v, w) is a rapidly decaying solution of (1.1) if

• v, w ∈ C(R 2 × R), v(•, t) ∈ C 3 (R 3 ),
(1.2a)

• |∂ j x v(x, t)| q(t) (1 + |x|) 2+ε , |j| 3, for some ε > 0, w(x, t) → 0, |x| → ∞,

(1.2b)

• (v, w) satisfies (1.1).

(1.2c)

Note that if v(x, t) = v(x 1 , t), w(x, t) = w(x 1 , t), then (1.1) is reduced to the classic KdV equation. In addition, (1.1) is integrable via the inverse scattering method for the twodimensional Schrödinger equation

Lψ = Eψ, L = -∆ + v(x), x = (x 1 , x 2 ), E = E f ixed .
(1.3)

In this connection, it was shown (see [M], [START_REF] Novikov | Finite-zone, two-dimensional, potential Schrödinger operators. Explicit formula and evolutions equations[END_REF], [START_REF] Novikov | Finite-zone, two-dimensional Schrödinger operators. Potential operators[END_REF]) that for the Schrödinger operator L from (1.3) there exist appropriate operators A, B (Manakov L-A-B triple) such that (1.1) is equivalent to

∂(L -E) ∂t = [L -E, A] + B(L -E),
where [•, •] is the commutator. Note that both Kadomtsev-Petviashvili equations can be obtained from (1.1) by considering an appropriate limit E → ±∞ (see [ZS], [G2]).

We will consider the Cauchy problem for equation (1.1) with the initial data v(x, 0) = v 0 (x), w(x, 0) = w 0 (x).

(1.4)

We will assume that the function v 0 (x) satisfies the following conditions

• v 0 = v0 , (1.5a)

• E = ∅, where E is the set of zeros of the Fredholm determinant ∆ of (2.7) for equation (2.6) with v(x) = v 0 (x), (1.5b)

• v 0 ∈ S(R 2 ), where S denotes the Schwartz class.

(1.5c)

As for the function w 0 (x), which plays an auxiliary role, we will assume that it is a continuous function decaying at infinity and determined using ∂ z w 0 (x) = -3∂ z v 0 (x) from (1.1). Condition (1.5b) is equivalent to non-singularity of scattering data for v 0 (x). Conditions (1.5) define the class of initial values for which the direct and inverse scattering equations (2.4), (2.9)-(2.11), with time dynamics given by (2.15), are everywhere solvable and the corresponding solution v of (1.1) belongs to C ∞ (R 2 , R). We will call such solution (v(x, t), w(x, t)), constructed from (v 0 (x), w 0 (x)) via the inverse scattering method, an "inverse scattering solution" of (1.1).

The main result of this paper consists in the following: we show that for the "inverse scattering solution" v(x, t) of (1.1), (1.4), where E < 0 and v(x, 0) = v 0 (x) satisfies (1.5), the following estimate holds

|v(x, t)| const(v 0 ) ln(3 + |t|) (1 + |t|) 3/4
, t ∈ R, uniformly on x ∈ R 2 .

(1.6)

We show that this estimate is optimal in the sense that for some initial values v(x, 0) and for some lines x = ωt, ω ∈ S 1 , the exact asymptotics of v(x, t) along these lines is const (1+|t|) 3/4 as |t| → ∞ (where the constant is nonzero). Note that de facto the "inverse scattering solution" is the rapidly decaying solution in the sense of (1.2).

This work is a continuation of the studies on the large time asymptotic behavior of the solution of the Cauchy problem for the Novikov-Veselov equation started in [START_REF] Kazeykina | A large time asymptotics for transparent potentials for the Novikov-Veselov equation at positive energy[END_REF] for the case of positive energy E. It was shown in [START_REF] Kazeykina | A large time asymptotics for transparent potentials for the Novikov-Veselov equation at positive energy[END_REF] that if the initial data (v 0 (x), w 0 (x)) satisfy the following conditions:

• (v 0 (x), w 0 (x)) are sufficiently regular and decaying at |x| → ∞,

• v 0 (x) is transparent for (1.3) at E = E f ixed > 0, i.e. its scattering amplitude f is identically zero at fixed energy,

• the additional "scattering data" b for v 0 (x) is non-singular, then the corresponding solution of (1.1),(1.4) can be estimated as

|v(x, t)| const • ln(3 + |t|) 1 + |t| , t ∈ R uniformly on x ∈ R 2 .
This estimate implies, in particular, that there are no localized soliton-type traveling waves in the asymptotics of (1.1) with the"transparent" at E = E f ixed > 0 Cauchy data from the aforementioned class, in contrast with the large time asymptotics for solutions of the KdV equation with reflectionless initial data. It was shown in [N2] that all soliton-type (traveling wave) solutions of (1.1) with E > 0 must have a zero scattering amplitude at fixed energy; in addition it was proved in [N2] that for the equation (1.1) with E > 0 no exponentially localized soliton-type solutions exist (even if the scattering data are allowed to have singularities). However, in [G1], [G2] a family of algebraically localized solitons (traveling waves) was constructed de facto (see also [START_REF] Kazeykina | Large time asymptotics for the Grinevich-Zakharov potentials[END_REF]). We note that for the case E < 0, though the absence of exponentially-localized solitons has been proved (see [START_REF] Kazeykina | Absence of exponentially localized solitons for the Novikov-Veselov equation at negative energy[END_REF]), the existence of bounded algebraically localized solitons is still an open question.

Note that studies on the large time asymptotics for solutions of the Cauchy problem for the Kadomtsev-Petviashvili equations were fulfilled in [MST], [HNS], [K].

The proofs provided in the present paper are based on the scheme developed in [START_REF] Kazeykina | A large time asymptotics for transparent potentials for the Novikov-Veselov equation at positive energy[END_REF], the stationary phase method (see [Fe]). These proofs include, in particular, an analysis of some cubic algebraic equation depending on a complex parameter.

This work was fulfilled in the framework of research carried out under the supervision of R.G. Novikov.

2 Inverse "scattering" transform for the two-dimensional Schrödinger equation at a fixed negative energy

In this section we give a brief description of the inverse "scattering" transform for the twodimensional Schrödinger equation ( 1.3) at a fixed negative energy E (see [GN], [N1], [G2]).

First of all, we note that by scaling transform we can reduce the scattering problem with an arbitrary fixed negative energy to the case when E = -1. Therefore, in our further reasoning we will assume that E = -1.

Let us consider potentials v(x) for the problem (1.3) satisfying the following conditions

v = v, |v(x)| q (1 + |x|) 2+ε , x ∈ R 2 , (2.1)
for some fixed q and ε > 0. Then it is known that for λ ∈ C\(0 ∪ E), where E is the set of zeros of the modified Fredholm determinant ∆ for equation (2.6), (2.2) there exists a unique continuous solution ψ(z, λ) of (1.3) with the following asymptotics

ψ(z, λ) = e -1 2 (λz+z/λ) µ(z, λ), µ(z, λ) = 1 + o(1), |z| → ∞. (2.3)
Here the notation z = x 1 + ix 2 is used. These solutions are known as the Faddeev solutions for the Schrodinger equation (1.3), E = -1, see for example [Fa], [N1].

The function µ(z, λ) satisfies the following integral equation

µ(z, λ) = 1 + ζ∈C g(z -ζ, λ)v(ζ)µ(ζ, λ)dReζdImζ (2.4) g(z, λ) = - 1 2π 2 ζ∈C exp(i/2(ζ z + ζz)) ζ ζ + i(λ ζ + ζ/λ) dReζdImζ, (2.5) where z ∈ C, λ ∈ C\0. In terms of m(z, λ) = (1 + |z|) -(2+ε)/2 µ(z, λ) equation (2.4) takes the form m(z, λ) = (1 + |z|) -(2+ε)/2 + ζ∈C (1 + |z|) -(2+ε)/2 g(z -ζ, λ) v(ζ) (1 + |ζ|) -(2+ε)/2 m(ζ, λ)dReζdImζ, (2.6) where z ∈ C, λ ∈ C\0. In addition, A(•, •, λ) ∈ L 2 (C × C), |TrA 2 (λ)| < ∞, where A(z, ζ, λ)
is the Schwartz kernel of the integral operator A(λ) of the integral equation (2.6). Thus, the modified Fredholm determinant for (2.6) can be defined by means of the formula: ln ∆(λ) = Tr(ln(I -A(λ)) + A(λ))

(2.7)

(see [GK] for more precise sense of such definition).

Taking the subsequent members in the asymptotic expansion (2.3) for ψ(z, λ), we obtain (see [N1]):

ψ(z, λ) = exp - 1 2 λz + z λ 1 -2πsgn(1 -λ λ)× × iλa(λ) z -λ 2 z + exp - 1 2 1 λ -λ z + 1 λ -λ z λb(λ) i( λ2 z -z) + o 1 |z| , (2.8) |z| → ∞, λ ∈ C\(E ∪ 0).
The functions a(λ), b(λ) from (2.8) are called the "scattering" data for the problem (1.3), (2.1) with E = -1.

The function µ(z, λ), defined by (2.4), satisfies the following properties:

µ(z, λ) is a continuous function of λ on C\(0 ∪ E); (2.9) ∂µ(z, λ) ∂ λ = r(z, λ)µ(z, λ), (2.10a) r(z, λ) = r(λ) exp 1 2 λ - 1 λ z -λ - 1 λ z , (2.10b) r(λ) = πsgn(1 -λ λ) λ b(λ) (2.10c) for λ ∈ C\(0 ∪ E); µ → 1, as λ → ∞, λ → 0.
(2.11)

The function b(λ) possesses the following symmetries:

b - 1 λ = b(λ), b 1 λ = b(λ), λ ∈ C\0.
(2.12)

In addition, the following theorem is valid:

Theorem 2.1 (see [GN], [N1], [G2]). (ii) Let b be a function on C, such that b ∈ S( D-) and the symmetry properties (2.12) hold.

Then the equations of inverse scattering (2.16)-(2.18) are uniquely solvable and the corresponding potential v(z) satisfies the following properties:

v ∈ C ∞ (C), v = v, |v(z)| → 0 when |z| → ∞.
Let us denote by T the unit circle on the complex plane:

T = {λ ∈ C : |λ| = 1}. (2.13)
Then, in addition, it is known that under the assumptions of item (i) of Theorem 2.1 the function b is continuous on C and its derivative ∂ λ b(λ) is bounded on C, though discontinuous, in general, on T . Finally, if (v(z, t), w(z, t)) is a solution of equation (1.1) with E = -1 in the sense of (1.2), then the dynamics of the "scattering" data is described by the following equations (see [GN])

a(λ, t) = a(λ, 0), (2.14) b(λ, t) = exp λ 3 + 1 λ 3 -λ3 - 1 λ3 t b(λ, 0). (2.15)
The reconstruction of the potential v(z, t) from these "scattering" data is based on the following scheme.

1. Function µ(z, λ, t) is constructed as the solution of the following integral equation

µ(z, λ, t) = 1 - 1 π C r(z, ζ, t)µ(z, ζ, t) dReζdImζ ζ -λ . (2.16) 2. Expanding µ(z, λ, t) as λ → ∞, µ(z, λ, t) = 1 + µ -1 (z, t) λ + o 1 |λ| , (2.17)
we define v(z, t) as v(z, t) = -2∂ z µ -1 (z, t).

(2.18)

3. It can be shown that Lψ = Eψ where

L = -4∂ z ∂ z + v(z, t), v(z, t) = v(z, t), E = -1 ψ(z, λ, t) = e -1 2 (λz+z/λ) µ(z, λ, t), λ ∈ C, z ∈ C, t ∈ R.
3 Estimate for the linearized case

Consider I(t, z) = C f (ζ) exp(S(ζ, z, t))dReζdImζ, J(t, z) = 3 C ζ ζ f (ζ) exp(S(ζ, z, t))dReζdImζ, (3.1) where z ∈ C, t ∈ R, f (ζ) ∈ L 1 (C), S is defined by S(λ, z, t) = 1 2 λ - 1 λ z -λ - 1 λ z + t λ 3 + 1 λ 3 -λ3 - 1 λ 3 . (3.2)
We will also assume that f (ζ) satisfies the following conditions

f ∈ C ∞ ( D+ ), f ∈ C ∞ ( D-), (3.3a) ∂ m λ ∂ n λ f (λ) = O (|λ| -∞ ) as |λ| → ∞, O (|λ| ∞ )
as |λ| → 0, for all m, n 0, (3.3b)

where

D + = {λ ∈ C : 0 < |λ| 1}, D -= {λ ∈ C : |λ| 1}, (3.4) and D+ = D + ∪ T , D-= D -∪ T with T defined by (2.13). Note that if v(z, t) = I(t, z), w(z, t) = J(t, z), where (|ζ| 3 + |ζ| -3 )f (ζ) ∈ L 1 (C) as a function of ζ,
and, in addition,

f (ζ) = f (-ζ) and/or f (ζ) = -|ζ| -4 f 1 ζ , then v, w satisfy the linearized Novikov-Veselov equation (1.1) with E = -1. Besides, if f (ζ) = π|1 -ζ ζ| 2|ζ| 2 b(ζ), (3.5)
where b(ζ) is the scattering data for the initial functions (v 0 (z), w 0 (z)) of the Cauchy problem (1.1), (1.4), then the integrals I(t, z), J(t, z) of (3.1) represent the approximation of the solution (v(z, t), w(z, t)) under the assumption that v ≪ 1. The goal of this section is to give, in particular, a uniform estimate of the large-time behavior of the integral I(t, z) of (3.1).

For this purpose we introduce parameter u = z t and write the integral I in the following form

I(t, u) = C f (ζ) exp(tS(u, ζ))dReζdImζ, (3.6)
where

S(u, ζ) = 1 2 ζ - 1 ζ ū -ζ - 1 ζ u + ζ 3 -ζ3 + 1 ζ 3 - 1 ζ3 . (3.7)
We will start by studying the properties of the stationary points of the function S(u, ζ) with respect to ζ. These points satisfy the equation

S ′ ζ = ū 2 - u 2ζ 2 + 3ζ 2 - 3 ζ 4 = 0. (3.8)
The degenerate stationary points obey additionally the equation

S ′′ ζζ = u ζ 3 + 6ζ + 12 ζ 5 = 0. (3.9) We denote ξ = ζ 2 and Q(u, ξ) = u 2 - ū 2ξ + 3ξ - 3 ξ 2 .
For each ξ, a root of the function Q(u, ξ), there are two corresponding stationary points of

S(u, ζ), ζ = ± √ ξ.
The function S ′ ζ (u, ζ) can be represented in the following form

S ′ ζ (u, ζ) = 3 ζ 4 (ζ 2 -ζ 2 0 (u))(ζ 2 -ζ 2 1 (u))(ζ 2 -ζ 2 2 (u)).
(3.10)

We will also use hereafter the following notations:

U = {u = -6(2e -iϕ + e 2iϕ ) : ϕ ∈ [0, 2π)} and U = {u = re iψ : ψ = Arg(-6(2e -iϕ + e 2iϕ )), 0 r |6(2e iϕ + e -2iϕ )|, ϕ ∈ [0, 2π)},
the domain limited by the curve U .

Lemma 3.1 (see [START_REF] Kazeykina | A large time asymptotics for transparent potentials for the Novikov-Veselov equation at positive energy[END_REF]).

1. If u = -18e 2πik 3 , k = 0, 1, 2, then ζ 0 (u) = ζ 1 (u) = ζ 2 (u) = e πik 3 (3.11)
and S(u, ζ) has two degenerate stationary points, corresponding to a third-order root of the function

Q(u, ξ), ξ 1 = e 2πik 3 . 2. If u ∈ U ( i.e. u = -6(2e -iϕ + e 2iϕ ) ) and u = -18e 2πik 3 , k = 0, 1, 2, then ζ 0 (u) = ζ 1 (u) = e -iϕ/2 , ζ 2 (u) = e iϕ .
(3.12)

Thus S(u, ζ) has two degenerate stationary points, corresponding to a second-order root of the function Q(u, ξ), ξ 1 = e -iϕ , and two non-degenerate stationary points corresponding to a first-order root, ξ 2 = e 2iϕ .

3. If u ∈ intU = U\∂U, then ζ i (u) = e -iϕ i for some real ϕ i , and

ζ i (u) = ζ j (u) for i = j. (3.13)
In this case the stationary points of S(u, ζ) are non-degenerate and correspond to the roots of the function Q(u, ξ) with absolute values equal to 1.

4. If u ∈ C\U, then ζ 0 (u) = (1 + ω)e -iϕ/2 , ζ 1 (u) = e iϕ , ζ 2 (u) = (1 + ω) -1 e -iϕ/2 (3.14)
for certain real values ϕ and ω > 0.

In this case the stationary points of the function S(u, ζ) are non-degenerate, and correspond to the roots of the function Q(u, ξ) that can be expressed as

ξ 0 = (1 + τ )e -iϕ , ξ 1 = e 2iϕ , ξ 2 = (1 + τ ) -1 e -iϕ , (1 + τ ) = (1 + ω) 2 .
Formula (3.10) and Lemma 3.1 give a complete description of the stationary points of the function S(u, ζ).

In order to estimate the large-time behavior of the integral having the form

I(t, u, λ) = C f (ζ, λ) exp(tS(u, ζ))dReζdImζ (3.15)
(where S(u, ζ) is an imaginary-valued function) uniformly on u, λ ∈ C, in the present and the following sections we will use the following general scheme.

1. Consider D ε , the union of disks with a radius of ε and centers in singular points of function f (ζ, λ) and stationary points of S(u, ζ) with respect to ζ.

2. Represent I(t, u, λ) as the sum of integrals over D ε and C\D ε :

I(t, u, λ) = I int + I ext , where
I int = Dε f (ζ, λ) exp(tS(u, ζ))dReζdImζ, I ext = C\Dε f (ζ, λ) exp(tS(u, ζ))dReζdImζ.
(3.16)

3. Find an estimate of the form

|I int | = O (ε α ) , as ε → 0 (α 1)
uniformly on u, λ, t.

4. Integrate I ext by parts using Stokes formula

I ext = - 1 2it ∂Dε f (ζ, λ) exp(tS(u, ζ)) S ′ ζ (u, ζ) d ζ- - 1 2it T \Dε (f + (ζ, λ) -f -(ζ, λ)) exp(tS(u, ζ)) S ′ ζ (u, ζ) d ζ - 1 t C\Dε f ′ ζ (ζ, λ) exp(tS(u, ζ)) S ′ ζ (u, ζ) dReζdImζ+ + 1 t C\Dε f (ζ, λ) exp(tS(u, ζ))S ′′ ζζ (u, ζ) (S ′ ζ (u, ζ)) 2 dReζdImζ = - 1 t (I 1 + I 2 + I 3 -I 4 ), (3.17) where f ± (ζ, λ) = lim δ→0 f (ζ(1 ∓ δ), λ), ζ ∈ T and
T is defined by (2.13).

5.

For each I i find an estimate of the form

|I i | = O 1 ε β , as ε → 0. 6. Set ε = 1 (1 + |t|) k , where k(α + β) = 1, which yields the overall estimate |I(t, u, λ)| = O 1 (1 + |t|) α α+β , as |t| → ∞.
Using this scheme we obtain, in particular, the following result Lemma 3.2. Let a function f satisfy assumptions (3.3) and, additionally,

f | T ≡ 0, T = {λ ∈ C : |λ| = 1}.
(3.18)

Then the integral I(t, u) of (3.6) can be estimated

|I(t, u)| const(f ) ln(3 + |t|) (1 + |t|) 3/4 for t ∈ R (3.19)
uniformly on u ∈ C.

Note that condition (3.18) is satisfied if f has the special form (3.5). A detailed proof of Lemma 3.2 is given in Section 6.

Estimate for the non-linearized case

In this section we prove estimate (1.6) for the solution v(x, t) of the Cauchy problem for the Novikov-Veselov equation at negative energy with the initial data v(x, 0) satisfying properties (1.5).

We proceed from the formulas (2.17), (2.18) for the potential v(z, t) and the integral equation (2.16) for µ(z, λ, t).

We write (2.16) as µ(z, λ, t) = 1 + (A z,t µ)(z, λ, t), (4.1)

where

(A z,t f )(λ) = ∂ -1 λ (r(λ) exp(itS(u, λ))f (λ)) = - 1 π C r(ζ) exp(tS(u, ζ))f (ζ) ζ -λ dReζdImζ and S(u, ζ) is defined by (3.7), u = z t .
Equation (4.1) can be also written in the form

µ(z, λ, t) = 1 + A z,t • 1 + (A 2 z,t µ)(z, λ, t). (4.2)
According to the theory of the generalized analytic functions (see [V]), equations (4.1), (4.2) have a unique bounded solution for all z, t. This solution can be written as

µ(z, λ, t) = (I -A 2 z,t ) -1 (1 + A z,t • 1). (4.3)
Equation ( 4.3) implies the following formal asymptotic expansion

µ(z, λ, t) = (I + A 2 z,t + A 4 z,t + . . .)(1 + A z,t • 1). (4.4)
We also introduce functions ν(z, λ, t) = ∂ z µ(z, λ, t) and η(z, λ, t) = ∂ z µ(z, λ, t). In terms of these functions the potential v(z, t) is obtained by the formula v(z, t) = -2ν -1 (z, t), (4.5)

where ν -1 (z, t) is defined by expanding ν(z, λ, t) as |λ| → ∞:

ν(z, λ, t) = ν -1 (z, t) λ + o 1 |λ| , |λ| → ∞.
The pair of function ν(z, λ, t), η(z, λ, t) satisfy the following system of differential equations:

     ∂ν(z, λ, t) ∂ λ = ∂ z r(z, λ, t)µ(z, λ, t) + r(z, λ, t)η(z, λ, t), ∂η(z, λ, t) ∂ λ = ∂ z r(z, λ, t)µ(z, λ, t) + r(z, λ, t)ν(z, λ, t).
(4.6)

Equations (4.6) can also be written in the integral form

ν(z, λ, t) = (B z,t µ)(z, λ, t) + (A z,t η)(z, λ, t), η(z, λ, t) = (B z,t µ)(z, λ, t) + (A z,t ν)(z, λ, t), (4.7)
where operator B z,t is defined

(B z,t f )(λ) = ∂ -1 λ (∂ z r(z, λ, t)f (λ)) = - 1 π C ∂ z r(z, ζ, t)f (ζ) ζ -λ dReζdImζ. (4.8)
Thus for the function ν(z, λ, t) we obtain equation

ν = (B z,t + A z,t B z,t )µ + A 2 z,t ν,
or the following formal asymptotic expansion

ν = (I + A 2 z,t + A 4 z,t + . . .)((B z,t + A z,t B z,t )(I + A 2 z,t + A 4 z,t + . . .)(1 + A z,t • 1)).
(4.9)

We will write this formula in the form

ν = B z,t • 1 + A z,t B z,t • 1 + R z,t (λ).
Lemma 4.1. Let f (λ, z, t) be an arbitrary testing function such that

|f | c f (1 + |t|) δ , |∂ λ f | c f (1 + |t|) δ ∀λ ∈ C, z ∈ C, t ∈ R
with some positive constant c f independent of λ, z, t and some δ 0. Then:

1. The following estimates hold for B z,t • f :

(B z,t • f )(λ) = β 1 (z, t) λ + o 1 |λ| for λ → ∞,
where

β 1 (z, t) = 1 π C ∂ z r(z, ζ, t)f (ζ, z, t)dReζdImζ,
and

|β 1 (z, t)| β1 c f ln(3 + |t|) (1 + |t|) 3/4+δ ; (4.10) in addition, |(B z,t • f )(λ)| β2 c f ln(3 + |t|) (1 + |t|) 1/2+δ for λ ∈ T, (4.11)
where T is defined by (2.13), and

|(B z,t • f )(λ)| β3 c f (1 + |t|) 1/4+δ ∀λ ∈ C.
(4.12)

The following estimates hold for

A z,t • B z,t • f : (A z,t • B z,t • f )(λ) = α 1 (z, t) λ + o 1 |λ| for λ → ∞,
where

α 1 (z, t) = - 1 π 2 C dReζdImζ r(z, ζ, t) C ∂ z r(z, η, t) η -ζ f (η, z, t)dReη dImη, (4.13) and |α 1 (z, t)| α1 c f (1 + |t|) 3/4+δ ; (4.14)
in addition,

|(A z,t • B z,t • f )(λ)| α2 c f (1 + |t|) 1/2+δ ∀λ ∈ C.
(4.15)

3. The following estimates for A n z,t • f hold:

(A n z,t • f )(λ) = γ n (z, t) λ + o 1 |λ| for λ → ∞, where γ n (z, t) = 1 π C r(z, ζ, t)(A n-1 z,t • f )(ζ)dReζdImζ and |γ n (z, t)| (γ 1 ) n c f (1 + |t|) δ+ 1 5 ⌈ n-1 2 ⌉+ 2 5 , (4.16)
where ⌈s⌉ denotes the smallest integer following s. In addition,

|(A n z,t • f )(λ)| (γ 2 ) n c f (1 + |t|) δ+ 1 5 ⌈ n 2 ⌉ ∀λ ∈ C. (4.17)
4. The following estimate holds for R z,t :

R z,t (λ) = q(z, t) λ + o 1 |λ| for λ → ∞, (4.18) and |q(z, t)| q(c f ) (1 + |t|) 9/10 . (4.19)
A detailed proof of Lemma 4.1 is given in Section 6.

From formulas (4.5), (4.9) and Lemma 4.1 follows immediately the following theorem.

Theorem 4.1. Let v(x, t) be the "inverse scattering solution" of the Cauchy problem for the Novikov-Veselov equation (1.1) with E = -1 and the initial data v(x, 0) = v 0 (x) satisfying (1.5). Then

| v(x, t)| const(v 0 ) ln(3 + |t|) (1 + |t|) 3/4 , x ∈ R 2 , t ∈ R.
5 Optimality of estimates (1.6) and (3.19)

In this section we show that estimates (1.6) and (3.19) are optimal in the following sense: there exists such a line z = ût that along this line I(z, t) from (3.19) behaves asymptotically as const (1+|t|) 3/4 with come nonzero constant; there exist such initial data satisfying (1.5) that the corresponding solution v(z, t) behaves asymptotically as const (1+|t|) 3/4 as |t| → ∞, where the constant is nonzero.

Optimality of the estimate for the linearized case

Let us consider the integral To calculate the exact asymptotic behavior of I(t, û) we will use the classic stationary method as described in [Fe]. First of all, we note that f (ζ) is continuous, but not continuously differentiable on C. Thus we will consider separately the integrals

I(t, u) = C f (ζ) exp(tS(u, ζ))dReζdImζ, ( 5
I + (t) = D + f (ζ) exp(tS(ζ))dReζdImζ, I -(t) = D - f (ζ) exp(tS(ζ))dReζdImζ,
where D + and D -are defined in (3.4).

Let us introduce the partition of unity

ψ 1 (ζ) + ψ 0 (ζ) + ψ -1 (ζ) ≡ 1, such that 0 ψ i 1, ψ i ∈ C ∞ (C), ψ ±1 (ζ) ≡ 1 in some neighborhood of ζ = ±1
, respectively, and ψ ±1 (ζ) ≡ 0 everywhere outside some neighborhood of ζ = ±1 respectively. Then it is known (see [Fe]) that

I + (t) = D + f (ζ)ψ 1 (ζ) exp(tS(ζ))dReζdImζ+ D + f (ζ)ψ -1 (ζ) exp(tS(ζ))dReζdImζ+O 1 |t| = = I + + (t) + I - + (t) + O 1 |t| as |t| → ∞.
First we will estimate I + + (t). We note that for the phase S

I + + (t) = ∆ + f (x + iy) ψ1 (x + iy) exp(3itxy(x 2 -y 2 ))|∂ η ϕ(x + iy)| 2 dxdy,
where f = f • ϕ, ψ1 = ψ 1 • ϕ and ∆ + = {(x, y) ∈ R 2 : x < 0}, i.e. ∆ + is the half-plane containing the image of D + ∩ B ε (1) under the transformation x + iy = ϕ -1 (ζ).

The integral I + + (t) can be written in the form

I + + (t) = +∞ -∞ dc exp(3itc) γc∩∆ + f (x + iy) ψ1 (x + iy)|∂ η ϕ(x + iy)| 2 dω S ,
where dω S is the Gelfand-Leray differential form, defined as

dS ∧ dω S = dx ∧ dy (5.4)
and in the particular case under study equal to

dω S = -(x 3 -3xy 2 )dx + (3x 2 y -y 3 )dy (x 2 + y 2 ) 3 ;
γ c is an oriented contour consisting of points of the set {S(x, y) = c} with the orientation chosen so that (5.4) holds.

As ψ1 (x + iy) is equal to zero outside some B R (0), a disk of radius R centered in the origin, then there exists such c * > 0 that the set {S(x, y) = c} lies outside B R (0) for any c < -c * , c > c * . Thus the integral I + + (t) can be written

I + + (t) = c * -c * dc exp(3itc) γc∩∆ + f (x + iy) ψ1 (x + iy)|∂ η ϕ(x + iy)| 2 dω S .
Performing the change of variables

x → c 1/4 x, y → c 1/4 y for c > 0, x → (-c) 1/4 x, y → (-c) 1/4 y for c < 0 yields

I + + (t) = c * 0 dc exp(3itc) c 1/2 F + (c) + 0 -c * dc exp(3itc) (-c) 1/2 F -(c),
where

F + (c) = γ + ∩{c 1/4 (x,y)∈∆ + } F(c, x, y)dω S , F -(c) = γ -∩{(-c) 1/4 (x,y)∈∆ + } F(-c, x, y)dω S ,
(5.5) γ + , γ -are oriented sets consisting of points of the sets {S(x, y) = 1}, {S(x, y) = -1} correspondingly with orientation chosen so that (5.4) holds and

F(c, x, y) = f (c 1/4 (x + iy)) ψ1 (c 1/4 (x + iy))|∂ η ϕ(c 1/4 (x + iy))| 2 .
For any fixed positive c the integrals in (5.5) converge because the set {S(x, y) = 1} is separated from zero and, consequently, the denominator does not vanish, and because ψ1 is a function with a bounded support and thus the domains of integration in (5.5) are, in fact, bounded. Besides, since ∆ + is a conic set, the functions F + (c) and F -(c) can be expressed as follows (5.6) In some neighborhood U 0 of c 1/4 (x + iy) = 0 containing the support of ψ1 the function f (c 1/4 (x + iy)) can be represented as

F + (c) = γ + ∩∆ + F(c, x, y)dω S , F -(c) = γ -∩∆ + F(-c, x, y)dω S ,
f (c 1/4 (x + iy)) = f (1) + 6 -1/4 [∂ ζ f D + (1)(x + iy) + ∂ζf D + (1)(x -iy)]c 1/4 + g(c 1/4 (x + iy)),
where

∂ ζ f D + (1) = lim ζ∈D + ζ→1 ∂ ζ f , ∂ζf D + (1) = lim ζ∈D + ζ→1
∂ζf and, in addition, g is a function that can be estimated |g(c 1/4 (x + iy))| K|c 1/4 (x + iy)| 1+α , c 1/4 (x + iy) ∈ U 0 (5.7)

with some constants α > 0, K > 0. Using (5.2), we note that f (1) = 0,

∂ ζ f D + (1) = ∂ζf D + (1) def = f ′ D + (1) and thus f (c 1/4 (x + iy)) = 6 -1/4 f ′ D + (1)xc 1/4 + g(c 1/4 (x + iy)), c 1/4 (x + iy) ∈ U 0 .
Taking into account (5.3) we obtain (5.8) where γ = 6 -3/4 and g is a function satisfying an estimate similar to (5.7).

F(c, x, y) = γf ′ D + (1)xc 1/4 + g(c 1/4 (x + iy)), c 1/4 (x + iy) ∈ U 0 ,
It follows then that the functions F ± (c) behave asymptotically as

F ± (c) = γf ′ D + (1)J ± ∆ + (±c) 1/4 + R(c), when c → 0, where J + ∆ + = γ + ∩∆ + xdω S , J - ∆ + = γ -∩∆ + xdω S
and R(c) denotes the remainder. The integrals J ± ∆ + converge because the set {S(x, y) = ±1} represents a combination of curves which do not pass through zero and converge either to the lines |y| = |x| or to the coordinate axes with velocities |y| = 1 |x| 3 and |x| = 1 |y| 3 correspondingly. The remainder R(c) behaves asymptotically as o c 1/4 because we can estimate

|R(c)| Kc 1/4(1+α) (γ + ∪γ -)∩∆ + |x + iy| 1+α |dω S |.
and the integral converges due to the properties of the set {S(x, y) = ±1} explained above. Thus I + + (t) behaves asymptotically as (see [START_REF] Fedoryuk | Asymptotics: integrals and series[END_REF]Chapter III,§1])

I + + (t) = γ 3 3/4 f ′ D + (1)Γ 3 4 J + ∆ + exp iπ3 8 + J - ∆ + exp - iπ3 8 1 t 3/4 + o 1 t 3/4 ,
where Γ is the Gamma function.

Let us perform the same procedure for I - + (t). In this case we will define P (ζ) as

P (ζ) = -9ζ -9 ζ + ζ 3 + 1 ζ 3 -16 and obtain P (ζ) = ρ(ζ)(ζ + 1) 4 , ρ(ζ) = ζ 2 -4ζ + 1 ζ 3 .
Since ρ(-1) = -6, we will define the following transformation ζ → η in the neighborhood of ζ = -1: η = (-ρ(ζ))

I - + (t) = ∆ - f (x + iy) ψ-1 (x + iy) exp(-3itxy(x 2 -y 2 ))|∂ η ϕ(x + iy)| 2 dxdy,
where ∆ -= {(x, y) ∈ R 2 : x > 0} and the functions f , ψ-1 , ϕ are defined similarly to the case of I + + (t). The integral I - + (t) can also be written

I - + (t) = +∞ -∞ dc exp(-3itc) γc∩∆ - f (x + iy) ψ1 (x + iy)|∂ η ϕ(x + iy)| 2 dω S ,
where γ c and dω S are the same as for the case of I + + (t). Performing further the same procedure as for the case of I + + (t) and taking into account that

J + ∆ + = -J + ∆ -, J - ∆ + = -J - ∆ -,
we obtain the following asymptotic expansion for I - + (t):

I - + (t) = - γ 3 3/4 f ′ D + (-1)Γ 3 4 J + ∆ + exp - iπ3 8 + J - ∆ + exp iπ3 8 1 t 3/4 + o 1 t 3/4 .
Considering the case of I -(t) we note that in order to get an asymptotic representation for

I + -(t) and I - -(t) we need to replace D + → D -, ∆ + → ∆ -, ∆ -→ ∆ + in the formulas for I + + (t) and I - + (t) correspondingly. Taking into account that f ′ D + (1) = -f ′ D -(1), f ′ D + (-1) = -f ′ D -(-1
), we obtain

I(t) = C t 3/4 + o 1 t 3/4 , where C = 2γ 3 3/4 Γ 3 4 f ′ D + (1) J + ∆ + exp iπ3 8 + J - ∆ + exp - iπ3 8 - f ′ D + (-1) J + ∆ + exp - iπ3 8 + J - ∆ + exp iπ3 8 .
(5.9)

Thus we have shown that the linear approximation of the solution v(z, t) of (1.1), when z = -18t, behaves asymptotically as C t 3/4 when t → ∞. Note that on the set γ + ∪γ -the differential form dω S is positive. Thus J ± ∆ + are some negative constants, and expressions

J + ∆ + exp iπ3 8 + J - ∆ + exp -iπ3 8 , J + ∆ + exp -iπ3 8 + J - ∆ + exp iπ3 8
do not vanish. On the other hand, from (2.12) and (5.2) it follows that 1). Thus, in the general case the constant C from (5.9) is nonzero. We have proved the optimality of the estimate (1.6) in the linear approximation.

f ′ D + (1)/f ′ D + (-1) = b(1)/b(

Optimality of the estimate for the non-linear case

Now we show that for certain initial values v(z, 0) the corresponding solution v(z, t) of (1.1) behaves asymptotically as c t 3/4 along the line z = -18t for some c = 0. Let us show that the integral (4.13) with f ≡ 1 and z = -18t behaves as const t 3/4 .

We can represent α 1 (-18t, t) in the form (5.1) with f

(ζ, t) = r(ζ, t)ρ(ζ, t), where ρ(ζ, t) = - 1 π 2 C ∂ z r(η, z, t)| z=-18t η -ζ dReηdReζ.
In other terms,

f (ζ) = f (ζ, t) = sgn(1 -ζ ζ) ζ b(ζ)∂ -1 ζ π|1 -ζ ζ| 2|ζ| 2 b(ζ) exp(tS(ζ)) .
We proceed following the scheme of estimate for I(t) until formula (5.7). Then we represent

ψ1 (c 1/4 (x + iy))|∂ η ϕ(c 1/4 (x + iy))| 2 = 1 √ 6 + kc 1/4 (x + iy) + h(c 1/4 (x + iy)),
where k is some coefficient and h(c 1/4 (x + iy)) satisfies an estimate of type (5.7). Consequently, for F(c, x, y) we can write

F(c, x, y) = f (1)(6 -1/2 +kc 1/4 (x+iy))+6 -1/4 [∂ ζ f D + (1)(x+iy)+∂ζ f D + (1)(x-iy)]c 1/4 +g(c 1/4 (x+iy)),
where g(c 1/4 (x + iy)) satisfies an estimate of type (5.7). This allows us to obtain in the end

α 1 (-18t, t) = l 1 (f (±1)) t 1/2 1 + const t 1/4 + l 2 (f ζ (±1), fζ (±1)) t 3/4 + o 1 t 3/4 , t → ∞, (5.10)
where l 1 (f (±1)) is a linear combination of the limit values of f as ζ tends to 1 and -1 from inside and outside of the unit circle, and l 2 (f ζ (±1), fζ(±1)) is a linear combination of the limit values of f ζ and fζ as ζ tends to 1 and -1 from inside and outside of the unit circle. Now let us consider the potential v θ corresponding to the scattering data θb, where θ ∈ R is some small parameter. In a way similar to which (5.10) was obtained it can be shown that

|f (±1, t)| c 1 θ 2 t 1/2 , |f ζ (±1, t)| c 2 θ 2 , |fζ(±1, t)| c 2 θ 2
for sufficiently large values of t, where c 1 , c 2 , c 3 are some constants independent of t and θ. When θ → 0 and t → ∞, the linear approximation of v θ behaves as O θ t 3/4 , while the expression α 1 (-18t, t) behaves as O θ 2 t 3/4 (it can be shown that the member o 1 t 3/4 in (5.10) depends quadratically on θ).

Finally, from (4.9) and Lemma 4.1 it follows that for θ small enough and for z = -18t,

v θ (z, t) = C θ t 3/4 + o 1 t 3/4 , t → ∞,
where C θ is some nonzero constant. Thus we have shown that the estimate (1.6) is optimal.

6 Proofs of Lemma 3.2 and Lemma 4.1

Proof of Lemma 3.2. The proof follows the scheme described in Section 3 and is carried out separately for four cases depending on the values of the parameter u. In all the reasonings that follow we denote by D ε the union of disks with the radius ε centered in the stationary points of S(u, ζ) and we denote by T the unit circle on the complex plane:

T = {λ ∈ C : |λ| = 1}; (6.1)
in addition, const will denote an independent constant and const(f ) will denote a constant depending only on function f . Case 1. u ∈ U In this case all the stationary points belong to T and due to assumptions (3.3) and (3.18) of Lemma 3.2 we can estimate

|f (ζ)| const(f )ε for ζ ∈ D ε . (6.2)
Now we estimate the integral I int (as in (3.16)) as follows

|I int | = Dε f (ζ) exp(tS(u, ζ))dReζdImζ const(f ) • ε Dε dReζdImζ const(f )ε 3 .
The estimate for I ext (as in (3.16)) is proved as follows.

We note that the function S ′ ζ (u, ζ) can be estimated as

|S ′ ζ (u, ζ)| const ε 3 0 |ζ| 4 for ζ ∈ C\D ε 0 , and
|S ′ ζ (u, ζ)| const ρ 3 |ζ| 4 for ζ ∈ ∂D ρ , ε ρ ε 0 . (6.3)
Similarly, we can estimate

S ′′ ζζ (u, ζ) (S ′ ζ (u, ζ)) 2 const |ζ| 4 ε 4 0 for ζ ∈ C\D ε 0 , and
S ′′ ζζ (u, ζ) (S ′ ζ (u, ζ)) 2 const |ζ| 4 ρ 4 0 for ζ ∈ ∂D ρ , ε ρ ε 0 .
(6.4)

Thus we obtain the following estimate for I 1 from (3.17)

|I 1 | 1 2 ∂Dε |f (ζ)| |S ′ ζ (u, ζ)| |d ζ| const ε ε 3 ∂Dε |ζ| 4 |d ζ| const(f ) ε ε 2 (1 + ε) 4 const(f ) ε .
Due to assumption (3.18) of Lemma 3.2 the integral I 2 from (3.17) is equivalent to zero. When estimating I 3 and I 4 from (3.17) we fix some independent ε 0 > 0 and integrate separately over D ε 0 \D ε and C\D ε 0 :

|I 3 | Dε 0 \Dε f ′ ζ (ζ) exp(tS(u, ζ)) S ′ ζ (u, ζ) dReζdImζ + C\Dε 0 f ′ ζ (ζ) exp(tS(u, ζ)) S ′ ζ (u, ζ) dReζdImζ const(f ) ε 0 ε ρ ρ 3 dρ + const C\Dε 0 |f ′ ζ (ζ)||ζ 4 |dReζdImζ const(f ) ε , |I 4 | Dε 0 \Dε f (ζ) exp(tS(u, ζ))S ′′ ζζ (u, ζ) (S ′ ζ (u, ζ)) 2 dReζdImζ+ + C\Dε 0 f (ζ) exp(tS(u, ζ))S ′′ ζζ (u, ζ) (S ′ ζ (u, ζ)) 2 dReζdImζ const(f ) ε 0 ε ρ 2 ρ 4 dρ + const C\Dε 0 |f (ζ)||ζ 3 |dReζdImζ const(f ) ε .
Setting finally ε =

1 (1+|t|) 1/4 yields I(t, u) const(f ) (1 + |t|) 3/4
uniformly on u ∈ U.

Case 2. u ∈ C\U and ω from (3.14) satisfies ω 0 < ω 1+ω < 1ω 1 for some fixed independent positive constants ω 0 and ω 1 (i.e. the roots ζ 0 , ζ 2 from (3.14) are separated from T , defined by (6.1), and the root ζ 2 is separated from the origin)

In this case the we can estimate

|S ′ ζ (u, ζ)| constρ |ζ| 4 for ζ ∈ ∂D ρ , S ′′ ζζ (u, ζ) (S ′ ζ (u, ζ)) 2 const|ζ| 4 ρ 2 for ζ ∈ ∂D ρ .
Using these estimates and proceeding as in case 1, we obtain

|I int | const(f )ε 2 , |I 1 | const(f ), I 2 ≡ 0, |I 3 | const(f ), |I 4 | const(f ) ln 1 ε .
Setting ε = 1 1+|t| , we obtain that

I(t, u) const(f ) ln(3 + |t|) 1 + |t|
uniformly for the considered values of the parameter u. Case 3. u ∈ C\U and ω 1+ω < ω 0 (i.e. the roots ζ 0 and ζ 2 from (3.14) lie in some neighborhood of T from (6.1)) Lemma 6.1. For any t t 0 with some fixed t 0 > 0 and any ω > 0 one of the following conditions holds

(a) 0 < ω < 2 (1+|t|) 1/4 ; (b) ω > 1 (1+|t|) 1/8 ; (c) ∃n : 1 (1+|t|) γ n+1 /(2+2γ n+1 ) < ω < 2 (1+|t|) γn /(2+2γ n+1 ) , where γ n+1 = 2 3 γ n + 1 3 , γ 1 = 1 3 .
Proof. We note that

γ n+1 2 + 2γ n+1 → 1 4 , n → ∞; γ n 2 + 2γ n+1 < γ n 2 + 2γ n ; γ 1 2 + 2γ 2 < 1 8 .
Thus the intervals from the cases (a), (b), (c) ∀n ∈ N cover the whole range 0 < ω < +∞.

We will prove the result separately for three different cases depending on the value of parameter ω

(a) 0 < ω < 2ε = 2 (1+|t|) 1/4
In this case estimates (6.2), (6.3), (6.4) hold and so the reasoning of the case 1 can be carried out to obtain that

I(t, u) const(f ) (1 + |t|) 3/4
uniformly for the considered values of the parameter u satisfying

0 < ω < 2 (1 + |t|) 1/4 . (6.5) (b) ω > ε 1/3 = 1 (1+|t|) 1/8
In this case we estimate |I int | const(f )ε 2 . Further, we note that the derivative of the phase is estimated as

|S ′ ζ (u, ζ)| const εω 2 |ζ| 4 for ζ ∈ ∂D ε . (6.6)
Thus for I 1 we obtain |I 1 | const(f ) 1 ε 2/3 . In order to estimate the integral I 3 we use the following estimate of the derivative S ′ ζ for ζ ∈ ∂D ρ when ε ρ ε 0 :

|S ′ ζ (u, ζ)| constρω 2 |ζ| 4 , if ρ < ω, |S ′ ζ (u, ζ)| constρ 3 |ζ| 4 , if ρ > ω. (6.7) It allows to derive |I 3 | const(f ) 1 ε 2/3
. Finally, we proceed to the study of the integral I 4 . We use the following estimates

S ′′ ζζ (u, ζ) (S ′ ζ (u, ζ)) 2 const|ζ| 4 ρ 2 ω 2 and |f (ζ)| const(f ) ω, if ρ < ω, |f (ζ)| const(f ) ρ, if ρ > ω. (6.8)
After integration we obtain the estimate

|I 4 | const(f ) 1 ε 2/3 .
Setting finally ε = 1 (1+|t|) 3/8 , we obtain

I(t, u) const(f ) (1 + |t|) 3/4
uniformly for the considered values of the parameter u satisfying

ω > 1 (1 + |t|) 1/8 . (6.9) (c) ε γ n+1 < ω < 2ε γn , where ε = 1 (1+|t|) 1/(2+2γ n+1 ) and γ n+1 = 2 3 γ n + 1 3 , γ 1 = 1 3 (note that γ n → 1)
We proceed similarly to the case (b). Evidently, I int can be estimated |I int | const(f )ε 2+γn . Employing the estimate (6.6) we obtain |I 1 | const(f ) ε γn ε 2γ n+1 . Using (6.7) in order to estimate I 3 we obtain |I 3 | const(f ) 1 ε γ n+1 . Finally, to estimate I 4 we use (6.8) and

S ′′ ζζ (S ′ ζ (u, ζ)) 2 const|ζ| 4 ρ 2 ω 2 , ρ < ω, const|ζ| 4 ρ 3 ω , ρ > ω to obtain |I 4 | const(f ) ε γn ln(1/ε) ε 2γ n+1 . Setting ε = 1 (1+|t|) 1/(2+2γ n+1 ) yields |I(t, u)| const(f ) ln(3 + |t|)) (1 + |t|) 3/4
uniformly for the considered values of the parameter u satisfying

1 (1 + |t|) γ n+1 /(2+2γ n+1 ) < ω < 2 (1 + |t|) γn/(2+2γ n+1 ) .
(6.10)

Finally, from Lemma 6.1 it follows that we have proved the required estimate uniformly on the values of parameter u ∈ C\U such that ω 1+ω < ω 0 . Case 4. u ∈ C\U and ω 1+ω > 1ω 1 (i.e. the roots ζ 2 , -ζ 2 lie in the ω 1 -neighborhood of the origin)

This case is treated similarly to the previous one. We denote ω = Proof of Lemma 4.1.

1. The proof of inequality (4.10) repeats the proof of Lemma 3.2. The proof of inequality (4.11) also follows the scheme of the proof of Lemma 3.2. In this case we take D ε to be the union of disks of the radius ε with centers in the stationary points of S(u, ζ) and in the point λ.

For the case when λ ∈ T , where T is defined by (6.1), an estimate weaker than (4.11) can be obtained via a simplified reasoning. Indeed, I int , as in (3.16) 2. In order to obtain estimates (4.14), (4.15) we proceed according to the scheme outlined in Section 3. In this case the integral I 2 does not annul. On the other hand, when the variable of integration belongs to T , the estimate (4.11) on the integrand of I 2 is stronger than the estimate (4.12) for the general case. Thus we obtain for α 1 (z, t)

|I int | O ε 2 (1 + |t|) δ+ 1 4 , |I 1 | O 1 (1 + |t|) δ+ 1 4 ε 2 , |I 2 | O 1 (1 + |t|) δ+ 1 4 ε 3 , |I 3 | O 1 (1 + |t|) δ ε , |I 4 | O 1 (1 + |t|) δ+ 1 4 ε 2 .
Setting ε = 1 (1+|t|) 1/4 yields the required estimate. The estimate (4.15) is obtained similarly.

3. We will give the scheme of the proof for estimate (4.16). The estimate (4.17) is obtained similarly.

We will prove (4.16) by induction. Suppose that (4.16) holds for all n = 1, 2, . . . , N .

Then following the scheme of Section 3 and taking into account that ∂ λ (A n z,t • f )(λ) = (A n-1 z,t • f )(λ), we obtain for n = N + 1:

|I int | O ε (1 + |t|) δ+ 1 5 ⌈ n-1 2 ⌉ , |I 1 | O 1 (1 + |t|) δ+ 1 5 ⌈ n-1 2 ⌉ ε 3 , |I 2 | O 1 (1 + |t|) δ+ 1 5 ⌈ n-1 2 ⌉ ε 4 , |I 3 | O 1 (1 + |t|) δ+ 1 5 ⌈ n-2 2 ⌉ ε 3 , |I 4 | O 1 (1 + |t|) δ+ 1 5 ⌈ n-1 2 ⌉ ε 3
.

Setting ε = 1 (1+|t|) 1/5 we obtain the required estimate. 4. We represent R z,t (λ) as the sum of the following members R z,t (λ) = B(A + A 2 + A 3 + . . .) • 1 + AB(A + A 2 + A 3 + . . .) • 1+ + (A + A 2 + A 3 + . . .)AB(I + A + A 2 + . . .) • 1 = R 1 z,t (λ) + R 2 z,t (λ) + R 3 z,t (λ). The convergence of the series at sufficiently large times follows from the estimate (4.17). Now let

R i z,t (λ) = q i (z, t) λ + o 1 |λ| , as λ → ∞.
From (4.10) and (4.17) it follows that |q 1 (z, t)| q1 (c f ) ln(3+|t|) (1+|t|) 3/4+1/5 . From (4.14) and (4.17) we obtain that |q 2 (z, t)| q2 (c f ) (1+|t|) 3/4+1/5 . Finally, from (4.15), (4.16) and (4.17) it follows that |q 3 (z, t)| q3 (c f ) (1+|t|) 1/2+2/5 . This yields the required estimate.

( i )

 i Let v satisfy (1.5). Then the scattering data b(λ) for the potential v(z) satisfy properties (2.12) and b ∈ S( D-), where D -= {λ ∈ C : |λ| > 1}, D-= D -∪ ∂D -and S denotes the Schwartz class.

  .1) where f (ζ) = π|1ζ ζ| 2|ζ| 2 b(ζ), (5.2) with b(ζ) satisfying (3.3), and S(u, ζ) is defined by (3.7), for u = û = -18. As shown in Lemma 3.1 for this value of parameter u the phase S(û, ζ) = S(ζ) has two degenerate stationary points ζ = ±1 of the third order.

  (ζ) the following representation is valid S(ζ) = P (ζ) -P ( ζ), where P (ζ) is a holomorphic function defined by P (ζ) = -9ζ -P ζ (ζ) = S ζ (ζ) = 3 ζ 4 (ζ -1) 3 (ζ + 1) 3 . Note that P (ζ) can be written in the form P (ζ) = ρ(ζ)(ζ -1) 4 , where ρ(ζ) = ζ 2 +4ζ+1 ζ 3 and lim ζ→1 ρ(ζ) = 0. For the function ρ(ζ) the expression (ρ(ζ)) 1/4 can be uniquely defined in some neighborhood of ζ = 1. Further, we define the transformation ζ → η: η = (ρ(ζ)) 1/4 (ζζ = ϕ(η) is defined in some small neighborhood of η = 0. In terms of the new variable η the phase can be represented S(ζ) = η 4 -η4 . Now if we denote x = Reη, y = Imη, the integral I + + (t) becomes

  ζ ∈ D ρ , to obtain the necessary estimates.

  , can be estimated |I int | O ε (1+|t|) δ . Using estimates (6.3), (6.4) and |ζ -λ| ρ for ζ ∈ ∂D ρ (6.11) we obtain that |I ext | O

	1 (1+|t|) 1+δ ε 3 . Setting ε =	1 (1+|t|) 1/4 we get the estimate (4.12).

CMAP, Ecole Polytechnique, Palaiseau, 91128, France; email: kazeykina@cmap.polytechnique.fr


Oscillating system design applying universal formula for control

The problem of oscillating system design applying the homogeneity approach is studied. The Anti-control Lyapunov Function (ALF) is introduced as a counterpart of Control Lyapunov Function (CLF) for the control design that destabilizes a nonlinear system. A universal anti-control formula is proposed. Next, the universal control formulas based on ALF and CLF are used to design an oscillating system. Efficiency of the proposed approach is demonstrated on example.

I. INTRODUCTION

The problem of a stabilizing control design for nonlinear dynamical systems was intensively studied during the last few decades [START_REF] Kokotović | Constructive nonlinear control: a historical perspective[END_REF]. A number of approaches have been developed: method of CLF [START_REF] Artstein | Stabilization with relaxed control[END_REF], [START_REF] Sontag | A "universal" construction of arstein's theorem on nonlinear stabilization[END_REF], feedback linearization approach [START_REF] Isidori | Nonlinear control systems: An Introduction[END_REF], passification design method, flatness-based control [START_REF] Fliess | On differentially flat non linear systems[END_REF], [START_REF] Fliess | Flatness and defect of non-linear systems: introductory theory and examples[END_REF], backstepping, nested saturation design or forwarding [START_REF] Krstić | Nonlinear and Adaptive Control Design[END_REF], [START_REF] Sepulchre | Constructive nonlinear control[END_REF]. Almost all these techniques have been devoted to nonlinear systems stabilization with respect to an equilibrium or a reference trajectory. Another promising topic deals with the problem of nonlinear systems stabilization with respect to a set [START_REF] Fradkov | Nonlinear and adaptive control of complex systems[END_REF], [START_REF] Rumyantsev | Stability and stabilization of motion with respect to part of variables[END_REF], [START_REF] Shiriaev | Virtual constraints: a tool for orbital stabilization of nonlinear systems theory[END_REF], [START_REF] Shiriaev | Stabilization of invariant sets for nonlinear non-affine systems[END_REF], [START_REF] Skjetne | Nonlinear maneuvering with gradient optimization[END_REF], [START_REF] Vorotnikov | Partial Stability and Control[END_REF] (part of variables or output). Such problem arises in oscillations or synchronization control, energy level stabilization in mechanical systems, maneuvering problem or in robotic applications.

In spite of these achievements the design problem of a system with a predefined number of limit cycles, their locations and frequencies of oscillation remains rather challenging and unsolved. This problem can be considered as an inverse of the Hilbert's 16th problem that addresses the issue of evaluation of an upper bound for the number of limit cycles in polynomial vector fields of degree n and investigation of their relative positions (for a planar system).

Partly such a lack of solutions can be linked with the absence of an apparatus for complex nonlinear oscillating system analysis and, hence, design. Almost all conventional design techniques for nonlinear systems are based on the Lyapunov second method, that is hard to apply for oscillating or chaotic system analysis. Recently revisited the Yakubovich's oscillation framework gives some promising hints and methods for complex oscillatory system analysis The first author is with University of Bordeaux, IMS-lab, Automatic control group 351 cours de la libération, 33405 Talence, France, denis.efimov@ims-bordeaux.fr. The second author is with LAGIS (UMR-CNRS 8146), Ecole Centrale de Lille, BP 48, Cité Scientifique, 59651 Villeneuve-d'Ascq, France and ALIEN project at INRIA -LNE, Parc Scientifique de la Haute Borne 40, avenue Halley Bât.A, Park Plaza 59650 Villeneuve d'Ascq, France, wilfrid.perruquetti@inria.fr.

and design [START_REF] Yakubovich | Frequency oscillations conditions in nonlinear systems with stationary single nonlinearity[END_REF], [START_REF] Efimov | Oscillatority of nonlinear systems with static feedback[END_REF]. This concept of oscillations covers all types of oscillating behavior: from periodical to chaotic. Combination of this approach with the homogeneous system theory simplifies its applicability conditions [START_REF] Efimov | Oscillations conditions in homogenous systems[END_REF]. In this work we are going to develop these frameworks and apply the homogeneity approach for synthesis of oscillating and chaotic systems.

All notations are summarized in Section 2. The preliminary definitions are given in Section 3. The control design is investigated in Section 4. Example of application of the proposed technique follows the theoretical development in Section 5.

II. NOTATIONS

• R + = {x ∈ R : x > 0}, where R is the set of real number.

• the sequence i = 1, . . . , n is denoted as i = 1, n.

• a continuous function α : R + → R + belongs to the class K if α(0) = 0 and the function is strictly increasing. The function α : R + → R + belongs to the class K ∞ if α ∈ K and it is increasing to infinity. • |s| is stated for the absolute value of s ∈ R, ||•|| denotes a norm on R n (the Euclidean norm by default if another is not stated explicitly). • the notation DV (x)f (x) stands for the directional derivative of a continuously differentiable function V with respect to a vector field f evaluated at the point x, and for Dini derivative in the direction of f for a locally Lipschitz function V :

DV (x)f (x) = lim inf t→0+ V (x + tf (x)) -V (x) t .

III. PRELIMINARIES

Consider the following nonlinear system:

ẋ = f (x), (1)
where x ∈ R n is the state vector, f : R n → R n is a locally Lipschitz continuous function. For an initial condition x 0 ∈ R n the corresponding solution x(t, x 0 ) of the system (1) is defined at least locally in time t ≥ 0.

A. Homogeneity

In the following, for any positive real number λ and any strictly positive numbers r i , i ∈ 1, n called weights one can define:

• the vector of weights r = (r 1 , . . . , r n ) T , • the dilation matrix Λ r = diag{λ ri } n i=1 , note that for x ∈ R n we have Λ r x = (λ r1 x 1 , . . . , λ ri x i , . . . , λ rn x n ) T .

• let r = n i=1 r i , then the r-homogeneous norm of x ∈ R n is defined by:

||x|| r = n i=1 |x i | r r i 1 r
. Definition 1. A function h : R n → R is r-homogeneous with degree d r,h ∈ R if for all x ∈ R n we have (see [START_REF] Hermes | Nilpotent and high-order approximations of vector field systems[END_REF]):

λ -d r,h h(Λ r x) = h(x).
When such a property holds, we write deg r (h) = d r,h .

Let us note that for any positive real number λ:

||Λ r x|| r = λ||x|| r , thus deg r (||•|| r ) = 1. Let us introduce the following compact set S r = {x ∈ R n : ||x|| r = 1}.
Lemma 2. For all x ∈ R n , the following holds

min x =1 ||x|| r × x ≤ ||x|| r ≤ max x =1 ||x|| r × x m 1 ||x|| r ≤ x ≤ m 2 ||x|| r ,
where

m 1 = 1 max x =1 ||x||r , m 2 = 1 min x =1 ||x||r , x ∈ S r =⇒ m r ≤ x ≤ M r .
where m r = min x∈Sr ||x||r

max x =1 ||x||r , M r = max x∈Sr ||x||r min x =1 ||x||r . Proof: Since ||x|| r is r-homogeneous of degree 1 we have ||x|| r = x x x r
, using continuity of ||x|| r and compactness of the unit ball one get the first relation. The two other ones comes directly from the first relation.

The homogeneity notion can be also defined for vector fields or ordinary differential systems as follows. Definition 3. A vector field f : R n → R n is r-homogeneous with degree d r,f ∈ R, with d r,f > -min i∈{1,...,n} r i if for all x ∈ R n we have (see [START_REF] Hermes | Nilpotent and high-order approximations of vector field systems[END_REF]):

λ -d r,f Λ -1 r f (Λ r x) = f (x),
which is equivalent for i-th component of f i being rhomogeneous function of degree r i + d r,f . When such a property holds, we write deg r (f ) = d r,f . The system (1) is r-homogeneous of degree d r,f if the vector field f is homogeneous of degree d r,f . Theorem 4. [START_REF] Rosier | Homogeneous lyapunov function for homogeneous continuous vector field[END_REF] For the system (1) with r-homogeneous and continuous function f the following properties are equivalent:

• the system (1) is asymptotically stable;

• there exists a continuously differentiable rhomogeneous Lyapunov function V : R n → R + such that for all x ∈ R n ,

α 1 (||x||) ≤ V (x) ≤ α 2 (||x||), DV (x)f (x) ≤ -α(||x||), λ -d V (Λ r x) = V (x), d ≥ 0,
for some α 1 , α 2 ∈ K ∞ and α ∈ K.

B. Local homogeneity

The r-homogeneity concept presented in definitions 1,3 and Theorem 4 is introduced for some r and all λ > 0.

Restricting the set of admissible values for λ we can introduce local homogeneity [START_REF] Efimov | Oscillations conditions in homogenous systems[END_REF] (this property is in fact has been already defined in the homogeneity in the bi-limit [START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF], which consist in local homogeneity at the origin and at infinity).

Definition 5. A function h : R n → R with h(0) = 0 is (r 0 , λ 0 , h 0 )-homogeneous with degree d r0,h0 ∈ R with h 0 (0) = 0 if for all x ∈ S r0 we have: lim λ→λ0 λ -d r 0 ,h 0 h(Λ r0 x) -h 0 (x) = 0. A vector field f : R n → R n with f (0) = 0 is (r 0 , λ 0 , f 0 )- homogeneous with degree d r0,f0 > -min i∈{1,...,n} (r 0i ) and f 0 (0) = 0 if for all x ∈ S r0 we have: lim λ→λ0 λ -d r 0 ,f 0 Λ -1 r0 f (Λ r0 x) -f 0 (x) = 0,
The system (1) is (r 0 , λ 0 , f 0 )-homogeneous with degree d r0,f0 ∈ R if the vector field f is (r 0 , λ 0 , f 0 )-homogeneous with degree d r0,f0 ∈ R.

The coefficients r 0i > 0, i ∈ 1, n are called the weights, d r0,h0 (respectively d r0,f0 ) is the degree of homogeneity (it may depend on λ 0 ) and h 0 (respectively f 0 ) is the approximating function of h (respectively f ) at λ 0 .

If the pairs of functions (h, h 0 ), and (f, f 0 ) are continuous, then for any ε > 0 there exist

λ ε ≤ λ 0 ≤ λ ε such that for all λ ∈ [λ ε , λ ε ]: sup x∈Sr 0 λ -d r 0 ,h 0 h(Λ r0 x) -h 0 (x) ≤ ε, sup x∈Sr 0 λ -d r 0 ,f 0 Λ -1 r0 f (Λ r0 x) -f 0 (x) ≤ ε.
In the paper [START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF] this definition has been introduced for λ 0 = 0 and λ 0 = ∞ (the function h is called homogeneous in the bi-limit if it is simultaneously (r 0 , 0, h 0 )-homogeneous and (r ∞ , +∞, h ∞ )-homogeneous). The system (1) can be also homogeneous in more than two limits Definition 6. The function g (respectively the vector field f , the system (1)) is homogeneous in the multi-limit if there exist a finite number of triplets (r i , λ i , g i (respectively f i )) for which the function (respectively the vector field f , the system (1)) is (r i , λ i , g i (respectively f i ))-locally homogeneous for each index i.

The following formulas give an example for choice of locally approximating functions for any 0 < λ 0 < +∞ and x ∈ S r [START_REF] Efimov | Oscillations conditions in homogenous systems[END_REF]:

h 0 (x) = λ -d0 0 h(Λ r,0 x), f 0 (x) = λ -d0 0 Λ -1 r,0 f (Λ r,0 x),
where Λ r,0 = diag{λ ri 0 } n i=1 . Moreover, the approximating functions can be chosen homogeneous:

h 0 (x) = ||x|| d r λ -d0 0 h(Λ r,0 Λ -1 ||x|| x), f 0 (x) = ||x|| d r λ -d0 0 Λ ||x|| Λ -1 r,0 f (Λ r,0 Λ -1 ||x|| x), (2)
where

Λ ||x|| = diag{||x|| ri r } n i=1
, provided that h 0 (0) and f 0 (0) are well defined. The proposed formulas do not cover two limit cases with λ 0 = 0 and λ 0 = +∞. For the case λ 0 = 0 at least one variant of the approximating dynamics can be pointed out for differentiable functions h and f with the property h(0

) = 0, f (0) = 0: h 0 (x) = g (0)x, f 0 (x) = f (0)x for r i = 1, i = 1, n.

C. Conditions of oscillation in the sense of Yakubovich

The function g :

R n → R is called monotone if the condition x 1 ≤ x 1 , . . . , x n ≤ x n implies that everywhere either g(x 1 , . . . , x n ) ≤ g(x 1 , . . . , x n ) or g(x 1 , . . . , x n ) ≥ g(x 1 , . . . , x n ).
Definition 7. [START_REF] Yakubovich | Frequency oscillations conditions in nonlinear systems with stationary single nonlinearity[END_REF], [START_REF] Efimov | Oscillatority of nonlinear systems with static feedback[END_REF] The solution x(t, x 0 ) with x 0 ∈ R n of the system (1) is called [π -, π + ]-oscillation with respect to the output ψ = η(x) (where η : R n → R is a continuous monotone function) if the solution is defined for all t ≥ 0 and lim t→+∞ ψ(t) = π -; lim t→+∞ ψ(t) = π + ;

-∞ < π -< π + < +∞.
The solution x(t, x 0 ) with x 0 ∈ R n of the system (1) is called oscillating, if there exist some output ψ and constants π -, π + such that x(t, x 0 ) is [π -, π + ]-oscillation with respect to the output ψ. The forward complete system (1) is called oscillatory, if for almost all x 0 ∈ R n the solutions x(t, x 0 ) of the system are oscillating. The oscillatory system (1) is called uniformly oscillatory, if for almost all x 0 ∈ R n for corresponding solutions x(t, x 0 ) there exist output ψ and constants π -, π + not depending on initial conditions.

In other words the solution x(t, x 0 ) is oscillating if the output ψ(t) = η(x(t, x 0 )) is asymptotically bounded and there is no single limit value of ψ(t) for t → +∞. The term "almost all solutions" is used to emphasize that generally the system (1) has a nonempty set of equilibrium points, thus there exists a set of initial conditions with zero measure such that the corresponding solutions are not oscillating. The notion of oscillations in the sense of Yakubovich is rather generic including periodical oscillations (limit cycles), quasi-periodical, recurrent and chaotic trajectories. The oscillating trajectories could be repelling being oscillating. The trajectories also could be unbounded, it is required to find a function of the state vector, that is bounded and admits certain requirements introduced in Definition 7.

Theorem 8. [START_REF] Efimov | Oscillatority of nonlinear systems with static feedback[END_REF] Let the system (1) have two locally Lipschitz continuous Lyapunov functions V 1 and V 2 fulfilling the following inequalities for all x ∈ R n :

υ 1 (||x||) ≤ V 1 (x) ≤ υ 2 (||x||), υ 3 (||x||) ≤ V 2 (x) ≤ υ 4 (||x||), υ 1 , υ 2 , υ 3 , υ 4 ∈ K ∞ and for some 0 < X 1 < υ -1 1 • υ 2 • υ -1 3 • υ 4 (X 2 ) < +∞: DV 1 (x)f (x) > 0 for all 0 < ||x|| < X 1 and x / ∈ Ξ; DV 2 (x)f (x) < 0 for all ||x|| > X 2 and x /
∈ Ξ, where Ξ ⊂ R n is a set with zero Lebesgue measure containing all equilibriums of the system, and

Ω ∩ Ξ = ∅, Ω = {x : υ -1 2 • υ 1 (X 1 ) ≤ ||x|| ≤ υ -1 3 • υ 4 (X 2 )}.
Then the system (1) is oscillatory.

Theorem 9.

[8] Let the system (1) be (r j ,λ j ,f j )homogeneous for j = 1, 2, the functions f : R n → R n and f j : R n → R n , j = 1, 2 be continuous and the locally approximating dynamical systems ẋ = f j (x), j = 1, 2 have r j -homogeneous and continuously differentiable Lyapunov functions

V j : R n → R + , α 1,j (||x||) ≤ V j (x) ≤ α 2,j (||x||), α 1,j , α 2,j ∈ K ∞ for all x ∈ R n and j = 1, 2. Let Ξ ⊂ R n
be the set containing all equilibriums of the system (1). Let one of the following conditions hold:

• (i) a 1 = inf y∈Sr 1 DV 1 (y)f 1 (y) > 0, a 2 =
-sup y∈Sr 2 DV 2 (y)f 2 (y) > 0 and the sets

X j = {x ∈ R n : α -1 1,j • α 2,j (m 2 λ j ) < ||x|| < α -1
1,j • α 2,j (m 1 λ j )}, j = 1, 2 are connected and non empty where

sup y∈Sr j |DV j (y)[λ -d Λ -1 rj f (Λ rj y)-f j (y)]| < a j , j = 1, 2
for all λ ∈ (λ j , λ j ), λ j ≤ λ j ≤ λ j (such constants λ j , λ j exist due to homogeneity assumption), the constants m 1 , m 2 are defined in Lemma 2, and

Ω ∩ Ξ = ∅, Ω = Ω 1 ∩ Ω 2 , Ω 1 = R n \{x ∈ R n : ||x|| ≤ α -1 1,1 • α 2,1 (m 1 λ 1 )}, Ω 2 = {x ∈ R n : ||x|| ≤ α -1 1,2 • α 2,2 (m 2 λ 2 )}; • (ii) a 1 = -sup y∈Sr 1 DV 1 (y)f 1 (y) > 0, a 2 =
inf y∈Sr 2 DV 2 (y)f 2 (y) > 0 and the sets X j , j = 1, 2 are connected and non empty where λ j ≤ λ j ≤ λ j are defined as for the case (i), and

Ω ∩ Ξ = ∅, Ω = {x ∈ R n : α -1 1,1 • α 2,1 (m 1 λ 1 ) ≤ ||x|| ≤ α -1 1,2 • α 2,2 (m 2 λ 2 )
}. Then the system (1) has oscillating trajectories into the set Ω.

After introduction of these preliminary notions let us proceed with the main result presentation.

IV. CONTROL DESIGN

Consider an affine in control nonlinear system

ẋ = f (x) + G(x)u, x ∈ R n , u ∈ R m , (3)
where f and the columns of G are locally Lipschitz continuous vector fields, f (0) = 0. There exist a lot of approaches devoting to the stabilizing control u construction [START_REF] Krstić | Nonlinear and Adaptive Control Design[END_REF], [START_REF] Sepulchre | Constructive nonlinear control[END_REF] for (3), and some methods of anti-control design [START_REF] Chen | Controlling Chaos and Bifurcations in Engineering Systems[END_REF], [START_REF] Chen | Introduction to anti-control of discrete chaos: theory and applications[END_REF], [START_REF] Vanecek | Control Systems: From Linear Analysis to Synthesis of Chaos[END_REF] (the controls granting the closed loop system with instability property). Among approaches for stabilizing control design it is worth to mention CLF method [START_REF] Artstein | Stabilization with relaxed control[END_REF], [START_REF] Sontag | A "universal" construction of arstein's theorem on nonlinear stabilization[END_REF], [START_REF] Efimov | Universal formula for output asymptotic stabilization[END_REF], [START_REF] Efimov | A condition of clf existence for affine systems[END_REF] that gives a universal formula for the control laws. For the homogeneous system (3) this approach has been developed in [START_REF] Faubourg | Control lyapunov functions for homogeneous Žjurdjevic-quinn¡ systems[END_REF], [START_REF] Moulay | Stabilization via homogeneous feedback controls[END_REF], [START_REF] Moulay | Finite time stability and stabilization of a class of continuous systems[END_REF], [START_REF] Nakamura | Homogeneous stabilization for input affine homogeneous systems[END_REF].

Definition 10. [26] A continuously differentiable and positive definite function

V : R n → R + is called a CLF for the system (3) if for all x ∈ R n \ {0}, inf u∈R m {a(x) + B(x) T u} < 0, where a(x) = DV (x)f (x), B(x) = [DV (x)G(x)] T . Such a CLF satisfies the Small Control Property (SCP) if for each > 0 there is a δ > 0 such that, if x = 0 satisfies ||x|| < δ, then there is some ||u|| < such that a(x) + B(x) T u < 0.
It is possible to show [START_REF] Sontag | A "universal" construction of arstein's theorem on nonlinear stabilization[END_REF] that a continuously differentiable and positive definite function V : R n → R + is a CLF for the system (3) if for all x ∈ R n \ {0} the property Such an ALF satisfies the Small Control Property (SCP) if for each > 0 there is a δ > 0 such that, if x = 0 satisfies ||x|| < δ, then there is some ||u|| < such that

a(x) + B(x) T u > 0.
Similarly, it is possible to conclude that a continuously differentiable and positive definite function

V : R n → R + is an ALF, if for all x ∈ R n \ {0} the property a(x) > 0 if ||B(x)|| = 0
holds. The SCP property for ALF is equivalent to the limit one:

lim ||x||→0 a(x) ||B(x)|| ≥ 0.
Lemma 12.

[17] If for the system (3) there exists a CLF V : R n → R + , then the control

u(x) = -φ 1 [a(x), ||B(x)||]B(x), φ 1 (a, b) = a+ p √ |a| p +b 2q b 2 if b = 0; 0 if b = 0 (4)
for any 2q ≥ p > 1, q > 1 is continuous for all x ∈ R n \ {0} and ensures the system stabilization. If the function a and all elements of the vector function B are r-homogeneous with degrees d a and d B respectively:

a(Λ r x) = λ da a(x); B i (Λ r x) = λ d B B i (x), i = 1, m,
then the control ( 4) is elementwise r-homogeneous with degree d a -d B provided that it is possible to choose 2d B q = d a p. If the CLF V is r-homogeneous with degree d V and the vector field f and all columns of the matrix function G are r-homogeneous with degrees d f and d G respectively:

f (Λ r x) = λ d f Λ r f (x); G i (Λ r x) = λ d G Λ r G i (x), i = 1, m,
then the control ( 4) is elementwise r-homogeneous with degree d f -d G and the closed loop system (3), ( 4) is rhomogeneous with degree

d f provided that 2(d G + d V )q = (d f + d V )p.
If furthermore V satisfies the SCP, then the feedback control (4) is also continuous at the origin.

The last part of this lemma has been proven in [START_REF] Moulay | Stabilization via homogeneous feedback controls[END_REF] for the case m = 1 only. Lemma 13. If for the system (3) there exists an ALF V : R n → R + , then the control

u(x) = -φ 2 [a(x), ||B(x)||]B(x), φ 2 (a, b) = a-p √ |a| p +b 2q b 2 if b = 0; 0 if b = 0 (5)
for any 2q ≥ p > 1, q > 1 is continuous for all x ∈ R n \ {0} and ensures the system instability. If the function a and all elements of the vector function B are r-homogeneous with degrees d a and d B respectively:

a(Λ r x) = λ da a(x); B i (Λ r x) = λ d B B i (x), i = 1, m,
then the control ( 5) is elementwise r-homogeneous with degree d a -d B provided that it is possible to choose 2d B q = d a p. If the ALF V is r-homogeneous with degree d V and the vector field f and all columns of the matrix function G are r-homogeneous with degrees d f and d G respectively:

f (Λ r x) = λ d f Λ r f (x); G i (Λ r x) = λ d G Λ r G i (x), i = 1, m,
then the control ( 5) is elementwise r-homogeneous with degree d f -d G and the closed loop system (3), ( 5) is rhomogeneous with degree

d f provided that 2(d G + d V )q = (d f + d V )p.
If furthermore V satisfies the SCP, then the feedback control ( 5) is also continuous at the origin.

Proof: The proof is excluded due to space limitations.

Together with the local homogeneity concept, these two lemmas may be used to propose a universal control formula for the system (3) stabilization/destabilization at a specified sphere around the origin. Theorem 14. Assume that for the system (3):

(i) there exists a r-homogeneous CLF (ALF)

V 0 : R n → R + , d V = deg r (V 0 );
(ii) the function f is (r, λ 0 , f 0 )-homogeneous with degree d f ;

(iii) the columns G i , i = 1, m of the matrix function G are (r, λ 0 , G i 0 )-homogeneous with degree d G ; (iv) there exist 2q ≥ p > 1, q > 1 such that 2(d G +d V )q = (d f + d V )p.

Then the system (3) with the control (4) (the control ( 5)) for a(x) = DV 0 (x)f 0 (x), B(x) = [DV 0 (x)G 0 (x)] T is (r, λ 0 , F 0 )-homogeneous with degree d f , where the vector field F 0 (x) = f 0 (x) + G 0 (x)u(x) is stable (unstable).

Proof: For the system ẋ = f 0 (x)+G 0 (x)u all conditions of the lemma 12 (the lemma 13) are satisfied, the system ẋ = F 0 (x) is stable (unstable) and r-homogeneous with degree d f . Let us show, that the system (3) with the control (4) (the control [START_REF] Efimov | A condition of clf existence for affine systems[END_REF]) is (r, λ 0 , F 0 )-homogeneous with degree d f . For F (x) = f (x) + G(x)u(x) by definition we have

λ -d f Λ -1 r F (Λ r x) = λ -d f Λ -1 r [f (Λ r x) + G(Λ r x)u(Λ r x)] = = λ -d f Λ -1 r f (Λ r x) + λ -d G Λ -1 r G(Λ r x)λ d G -d f u(Λ r x)]. Since u(Λ r x) = λ d f -d G u(x), lim λ→λ0 λ -d f Λ -1 r f (Λ r x) -f 0 (x) = 0, lim λ→λ0 λ -d G Λ -1 r G(Λ r x) -G 0 (x) = 0
we obtain lim λ→λ0 λ -d f Λ -1 r F (Λ r x) -F 0 (x) = 0. For 0 < λ 0 < +∞, a variant of homogeneous approximating functions f 0 and G i 0 , i = 1, m is given in [START_REF] Artstein | Stabilization with relaxed control[END_REF]. Any other approach (backstepping, forwarding or feedback linearization) generating a r-homogeneous control may substitute the CLF/ALF controls (4), [START_REF] Efimov | A condition of clf existence for affine systems[END_REF] in this theorem under conditions (i)-(iii). Owing the framework of homogeneity in the multilimit, Theorem 14 provides an approach to design oscillating systems.

Corollary 15. Let for some r k , k = 0, N , 0 < N < +∞ and 0 ≤ λ 0 < . . . < λ N ≤ +∞ for the system (3) the following properties be true:

(i) there exist r 2j -homogeneous CLFs (ALFs) V 2j : R n → R + and r 2j+1 -homogeneous ALFs (CLFs)

V 2j+1 : R n → R + for j = 0, N/2, define h 2j = 1, h 2j+1 = 2 (h 2j = 2, h 2j+1 = 1) and d V,k = deg r k (V k ) for k = 0, N ; (ii) the function f is (r k , λ k , f k )--homogeneous with de- gree d f,k , k = 0, N ; (iii) the columns G i , i = 1, m of the matrix function G are (r k , λ k , G i k )-homogeneous with degree d G,k , k = 0, N ; (iv) for all k = 0, N there exist 2q k ≥ p k > 1, q k > 1 such that 2(d G,k + d V,k )q k = (d f,k + d V,k )p k ;
(v) the vector fields f and G i , i = 1, m are linearly independent for all x ∈ R n \ {0}.

Then the system (3) with the control

u(x) = - N k=0 [ k (||x|| k , λ k )φ h k [a k (x), ||B k (x)||]B k (x)] , (6)
a k (x) = DV k (x)f k (x), B k (x) = [DV k (x)G k (x)] T , has different oscillating trajectories into the sets X k = {x ∈ R n : λ k < ||x|| < λ k+1 }, k = 0, N -1, where the continuous weighting functions k (λ k , λ k ) = 1 and N k=0 k (||x|| k , λ k ) = 1 for any x ∈ R n .
Proof: If the vector fields f and G i , i = 1, m are linearly independent for all x ∈ R n \ {0}, then for any control the system has the single equilibrium at the origin. For each k = 0, N the conditions of Theorem 14 hold and the system (3) with the control u k 6) is (r k , λ k , F k )-homogeneous for F k (x) = f k (x) + G k (x)u k (x). The oscillating properties follow by application of Theorem 9 (all conditions of this theorem hold for each paire 2j, 2j + 1, j = 0, N/2).

(x) = φ h k [a k (x), ||B k (x)||]B k (x)
Note that there exist constants 0 < χ k < +∞, k = 0, N such that the choice

k (||x|| k , λ k ) = e -χ k ν k (||x||r k ,λ k ) 2 , k = 0, N (7)
is admissible, where

ν k (s, λ) = s -λ, k = 0, N and if λ N = +∞ then ν N (s, +∞) = 0 if s -λ N ≥ 0; s -λ N if s -λ N < 0 for some λ N -1 < λ N < +∞.
The term e -χ k ν k (||x||r k ,λ k ) 2 can be replaced with any other type of weighting functions (polynomial, for instance). For this choice of the functions k , k = 0, N , the normalization condition holds with some error (dependent on χ k ), however the arguments of Theorem 9 remain valid.

Example 16. Consider a bilinear planar system:

ẋ = f (x) + g(x)u, x ∈ R 2 , u ∈ R, f (x) = [x 2 -2x 1 ] T , g(x) = [x 1 x 2 ] T .
The vector fields f and g are linearly independent for all x ∈ R 2 \ {0}. Choose λ 0 = 0 and r 0 = [1 1], then due to linearity of f and g we obtain f 0 (x) = f (x), g 0 (x) = g(x). Take V 0 (x) = (x 1 +x 2 ) 2 +x 2 2 , then a 0 (x) = 2(x 1 +x 2 )(x 2 -4x 1 ) -4x 1 x 2 , b 0 (x) = 2(x 1 + x 2 ) 2 + 2x 1 x 2 and V 0 is a CLF. Since d f,0 = 0, d g,0 = 0 and d V,0 = 2, pick q 0 = 1 and p 0 = 2. Next, choose λ 1 = 2 and r 1 = [1 2]. According to (2) we obtain f 1 (x) = [x 2 -0.5||x|| 2 r1 x 1 ] T , g 1 (x) = g(x)

for d f,1 = 1 and d g,1 = 0.

For d V,1 = 4 take V 1 (x) = x 2 2 + (x 1 |x 1 | + x 2 ) 2 , then a 1 (x) = 2(x 1 |x 1 | + x 2 )(2|x 1 |x 2 -0.5||x|| 2 r1 x 1 ) -||x|| 2 r1 x 1 x 2 , b 1 (x) = 2(x 1 |x 1 | + x 2 )(2|x 1 |x 1 + x 2 ) + 2x 2 2
and V 1 is an ALF. Therefore, q 1 = 1.25 and p 1 = 2 is an admissible choice. Finally, let λ 2 = +∞ and r 2 = [1 1], then again f 2 (x) = f (x), g 2 (x) = g(x), V 2 (x) = (x 1 + x 2 ) 2 + x 2 2 is a CLF, a 2 (x) = a 0 (x), b 2 (x) = b 0 (x), q 2 = 1 and p 2 = 2. In the control (6) we put [START_REF] Efimov | Oscillatority of nonlinear systems with static feedback[END_REF] with χ k = 5, k = 0, 1, 2. The results of the system simulation are shown in Fig. 1, the system has two limit cycles (the inner is unstable and the outer one is stable).

V. CONCLUSION

The paper extends the CLF approach for anti-control analysis and design, for that the ALF concept is introduced. The universal formula for destabilizing control algorithms is proposed. Applying the local homogeneity concept, the control law for stabilization/destabilization of nonlinear systems around a sphere is presented. Finally, the universal formulas for stabilizing/destabilizing controls are used for oscillating system design with multiple limit cycles. The results are illustrated by example of computer simulation.

  a(x) < 0 if ||B(x)|| = 0 holds. The SCP property is equivalent to the following one: lim ||x||→0 a(x) ||B(x)|| ≤ 0. Now we are in position to introduce the new anti-control (destabilizing) Lyapunov function. Definition 11. A continuously differentiable and positive definite function V : R n → R + is called an Anti-control Lyapunov Function (ALF) for the system (3) if for all x ∈ R n \ {0}, sup u∈R m {a(x) + B(x) T u} > 0.

  is stable or unstable and locally homogeneous. The function k (||x|| k , λ k ) equals 1 for ||x|| r k = λ k and it uniformly decays to zero on leaving this set (due to the normalization condition N k=0 k (||x|| k , λ k ) = 1). Therefore, the closed-loop system (3), (

Fig. 1 .

 1 Fig. 1. Trajectories of the closed-loop system.


HOMOGENEITY FOR TIME-DELAY SYSTEMS

The notion of homogeneity is extended to the time-delay nonlinear systems. It is shown that under some conditions the stability of homogeneous functional systems on a sphere implies the global stability of the system. The notion of local homogeneity is introduced, the relations between stability of the locally approximating dynamics and the original time-delay system are established. An example of the proposed theory application is given.

INTRODUCTION

For nonlinear systems, behavior of a homogeneous system trajectories can be extended to the whole state space based on their behavior on a suitably defined sphere around the origin [START_REF] Rothschild | Hypoelliptic differential operators and nilpotent groups[END_REF]. Thus the state vector rescaling does not change the system behavior. As it has been shown during the last two decades this property can be used for stability analysis [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF][START_REF] Bacciotti | Liapunov Functions and Stability in Control Theory[END_REF][START_REF] Hermes | Homogeneous coordinates and continuous asymptotically stabilizing feedback controls[END_REF][START_REF] Hong | Finite-time stabilization and stabilizability of a class of controllable systems[END_REF][START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF], systems approximation (Hermes, 1991a), stabilization (Bhat and Bernstein, 2001;[START_REF] Grüne | Homogeneous state feedback stabilization of homogeneous systems[END_REF][START_REF] Kawski | Homogeneous feedback stabilization[END_REF][START_REF] Moulay | Finite time stability and stabilization of a class of continuous systems[END_REF][START_REF] Sepulchre | Stabilizability does not imply homogeneous stabilizability for controllable systems[END_REF] and observation [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF]. Analysis and synthesis of homogeneous systems is a little bit simpler, for example, the Lyapunov function has to be constructed on the unit sphere only (on the whole state space it can extended using rescaling). Moreover, it has been shown that for stability/instability analysis the Lyapunov function of a homogeneous system is also homogeneous [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF][START_REF] Efimov | Oscillations Conditions in Homogenous Systems[END_REF].

In the work [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF] the homogeneity in the bilimit has been introduced, that is homogeneity with different weights and approximating functions at a vicinity of the origin and far outside. Recently the bi-limit homogeneity has been extended to the local homogeneity [START_REF] Efimov | Oscillations Conditions in Homogenous Systems[END_REF], when the system is similar to a homogeneous one on a sphere only. In this case the homogeneous systems theory allows us to analyze locally the original nonlinear system behavior.

All this theory has been developed for continuous timeinvariant nonlinear dynamical systems. Another important class of systems includes ones described by differential equations with time-delayed states. The presence of delays is usual in many applications (Chiasson and Loiseau, 2007;[START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF]. Analysis of the delay influence on the system stability is critical for many natural and human-developed systems [START_REF] Gu | Stability of Time-Delay Systems (Control Engineering)[END_REF][START_REF] Kolmanovsky | Stability of functional differential equations[END_REF]. The goal of this work is to extend homogeneity approach to the nonlinear time-delay systems. The problem is that the former ones operate in the infinite dimensional state space. Extension of the standard results [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF] to this class of systems needs complete revision of the homogeneous systems apparatus. In [START_REF] Asl | Analytical Solution of a System of Homogeneous Delay Differential Equations via the Lambert Function[END_REF][START_REF] Dublik | Asymptotic equilibrium for homogeneous delay linear differential equations with l-perturbation term[END_REF] the homogeneous (linear) delay differential equation have been studied, in [START_REF] Bokharaie | D-Stability and Delay-Independent Stability of Homogeneous Cooperative Systems[END_REF] the cooperative and homogeneous systems have been analyzed. In all these works the homogeneity theory has not been extended to the functional spaces, the papers are based on homogeneity of nondelayed parts of differential equations.

The outline of this work is as follows. The preliminary definitions and the system equations are given in Section 2. The homogeneous norm, an extended definition of homogeneity for time-delay systems and sufficient stability conditions are presented in Section 3. The local homogeneity theory is studied in Section 4. An example is presented in Section 5.

PRELIMINARIES

Consider the autonomous functional differential equation:

( ) / [ ( )] d t dt t τ = x fx , 0 t ≥ , (1)
where

n R ∈ x is the state vector, ( ) ( ), t t s τ = + x x 0 s -τ ≤ ≤ ∈ [ 0] n C -τ,
is the extended (lifted) state vector

(we denote by [ ] n C a b , , 0 a b ≤ < ≤ +∞ the Banach space of continuous functions :[ , ] n a b R ϕ → with the norm || || sup | ( ) | a b ≤ς≤ ϕ = ϕ ς , where | | ⋅ is the standard Euclidean norm); : [ 0] n n C R -τ, → f is a locally Lipschitz continuous function, (0) 0 = f
. We assume that all solutions of the sys-tem (1) satisfy the initial condition

0 (0) [ 0] n C τ = ∈ -τ, x x
. It is known from the theory of functional differential equations (Gu, et al., 2003), that under above assumptions the system (1) has a unique solution 0 ( , ) t x x satisfying initial condition 0 x , which is defined on some finite interval [0, )

T . The upper right hand Dini derivative of the Lipschitz continuous functional : [ 0] n V C R -τ, → along the system (1) solutions is defined as follows for any

[ 0] n C ϕ ∈ -τ, : 0 1 ( ) limsup [ ( ) ( )] h h D V V V h + + → ϕ = ϕ -ϕ , where [ 0] n h C ϕ ∈ -τ, for 0 h < < τ is given by { ( ), [ , ) (0) ( )( ), [ , 0]. h h h h h ϕ θ + θ ∈ -τ - ϕ = ϕ + ϕ θ+ θ∈ - f The notation [ (0)] ( ) DV τ τ x f x
stands for directional derivative of the continuously differentiable function : n V R R → with respect to the vector field f . Qualify a continuous function

+ + → σ R R :
belongs to class K if it is strictly increasing and ( ) 0 0 = σ ; belongs to class ∞ K if it is also radially unbounded.

HOMOGENEITY

For any 0 For any 0

i r > , 1, i n = and n R ∈ x
the homogeneous norm can be defined as follows ( )

1/ / 1 | | | | i n r i i x ρ ρ = = ∑ r x , 1 n i i r = ρ = ∏ . For all n R ∈ x
, its Euclidean norm | | x is related with the homogeneous one via some constants , R + σ σ ∈ Since the dilation matrix r Λ transformation is linear, it can be applied to functional arguments. Indeed, for any 0

i r > , 1, i n = and [ ] n C a b ϕ ∈
, , 0 a b ≤ < ≤ +∞ the homogeneous norm can be defined as follows ( )

1/ / 1 || || || || i n r i i ρ ρ = ϕ = ϕ ∑ r , 1 n i i r = ρ = ∏ . L e m m a 1 . For all [ ] n C a b ϕ ∈
, , there exist two functions

, ∞ ρ ρ ∈ r r K such that (|| || ) || || (|| || ) ρ ϕ ≤ ϕ ≤ρ ϕ r r r r . P r o o f . By definition 2 1 || || sup | ( ) | sup | ( ) | n a b a b i i ≤ς≤ ≤ς≤ = ϕ = ϕ ς = ϕ ς ∑ . Thus 2 1 || || || || n i i= ϕ ≤ ϕ ∑ and 1 0.5 max ( ) i n i r s c s ≤ ≤ ρ ρ ρ = for some 0 c ρ > dependent on ρ , 2 1 || || || || n i i n = ϕ ≤ ϕ ∑ and 1 0.5 min 1 ( ) i n i r s c n s ≤ ≤ ρ - ρ ρ = , 0 c ρ > . ■ Therefore,
ϕ = λ ϕ r r r Λ for all [ ] n C a b ϕ ∈ , . Define the corre- sponding unit sphere { [ 0]: || || 1} n C = ϕ ∈ -τ, ϕ = r r S . D e f i n i t i o n 1 . The function : [ 0] n g C R -τ, → is called r -homogeneous ( 0 i r > , 1, i n = ), if for any [ 0] n C ϕ ∈ -τ, the relation ( ) ( ) d g g ϕ = λ ϕ r Λ holds for some 0 d ≥ and all 0 λ ≥ . The function : [ 0] n n C R -τ, → f is called r -homogeneous ( 0 i r > , 1, i n = ), if for any [ 0] n C ϕ ∈ -τ, the relation ( ) ( ) d ϕ = λ ϕ r r
f Λ Λf holds for some

1 min i n i d r ≤ ≤
≥and all 0 λ ≥ .

The constant d is called the degree of homogeneity. The introduced notion of homogeneity in [ 0] n C -τ, is reduced to the standard one in n R [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF] under vector argument substitution. The advantage of homogeneous systems described by non linear ordinary differential equations is that analysis of their stability can be performed on the unit sphere S r only [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF]) (the homogeneous system trajectories have similar behavior on any sphere defined by the norm | | ⋅ r ). It has been also shown that asymptotically stable/unstable homogeneous systems always have homogeneous Lyapunov functions [START_REF] Efimov | Oscillations Conditions in Homogenous Systems[END_REF][START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF]. In this work we would like to prove a similar (sufficient only) result for time-delay homogeneous systems.

For time-delay systems there exist two main techniques for stability analysis based on the Lyapunov approach. The first one is based on the Lyapunov-Krasovskii functional, another one is the Razumikhin stability theorem [START_REF] Gu | Stability of Time-Delay Systems (Control Engineering)[END_REF][START_REF] Kolmanovsky | Stability of functional differential equations[END_REF]. Unfortunately, due to peculiarities of the directional derivatives for functionals (see the definition above) the Lyapunov-Krasovskii approach is hard to develop using homogeneity. To explain the problem consider the following example. Let the functions 0 :

n n R R → f , 1 : n n R R → f , 0 : n g R R + → , 1 : n g R R + →
be rhomogeneous (with the same 0

i r > , 1, i n =
and similar de-gree

1 min i n i d r ≤ ≤ ≥ - for 0 f , 1 f and 0 v ≥ for 0 g , 1 g ). For 0 1 [ ( )] [ ( )] [ ( )] t t t τ = + -τ f x f x f x in (1) consider the Lyapunov- Krasovskii functional 0 1 [ ( )] [ ( )] [ ( )] t t V t g t g s ds τ -τ = + ∫ x x x ,
that satisfies the required homogeneity conditions. Assume that the structural condition

0 1 1 2 2 ( ) ( ) [ ( )] [ ( )] [ ( )] t g t g t g t = ∂ ∂ -τ ≤ -τ + x x x x f x x x
holds for some function 2 : n g R R → , then verification of the derivative V negative definiteness can be performed for ( )

t S ∈ r x
(that is relatively simple):

0 0 1 ( ) 1 1 0 0 1 2 ( ) ( ) { [ ( )] [ ( )]} [ ( )] [ ( )] ( ) [ ( )] [ ( )] [ ( )]. t t V g t t g t g t g t g t g t = = = ∂ ∂ + -τ + - -τ ≤ ≤ ∂ ∂ + x x x x x x f x f x x x x x f x x x
Let for ( ) t S ∈ r x the above expression verifies 0 V < . Consider the dilation transformation = r z Λ x influence on V :

0 0 ( ) 1 1 1 ( ) ( ) ( ) { [ ( )] [ ( )]} { [ ( )] [ ( )]}. d t V V g t t g t g t +ν = ν = =λ ∂ ∂ + -τ + λ - -τ r x x z Λ x x x f x f x x x
Thus this basic for homogeneous systems transformation scales differently the parts proportional to 0 g and 1 g , therefore in general case for some 0 λ ≥ it could be 0 V > .

Since the Razumikhin approach, based on Lyapunov-Razumikhin functions [START_REF] Gu | Stability of Time-Delay Systems (Control Engineering)[END_REF] defined on n R , gives a point-wise sufficient criteria for stability (not a functional one), it allows us to develop this approach using homogeneous arguments. Note, that if a function

: n V R R + →
is positive definite and radially unbounded, then there are functions 1 2 ,

∞ α α ∈K such that 1 2 (| |) ( ) (| |) V α ≤ ≤α x x x for all n R ∈ x .
T h e o r e m 1 . Let the function f in (1) be r -homogeneous with degree d and there exist r -homogeneous continuously differentiable Lyapunov-Razumikhin function

: n V R R + → with degree ν , 0 d ν + > such that (i) there exist functions , α γ ∈K ( ( ) s s γ > for all s R + ∈ ) such that for all φ ∈ r S [ ,0] max [ ( )] { [ (0)]} V V θ∈ -τ φ θ < γ φ ⇒ [ (0)] ( ) (| (0) |) DV φ φ ≤ -α φ f ; (ii) there exists function ′ γ ∈K such that ( ) ( ) s s ′ λγ ≤ γ λ for all , s R + λ ∈ .
Then the system (1) is globally asymptotically stable. :

[ ,0] max [ ( )] { [ (0)]} V V ν ν θ∈ -τ λ φ θ <λ γ φ ⇒ [ (0)] ( ) (| (0) |) d d DV ν+ ν+ λ φ φ ≤-λ α φ f ,
the inequalities and relations should not be changed for 0 λ ≥ . Owing the function γ properties, the inequality

{ [ (0)]} { [ (0)]} V V ν ν ′ λ γ φ ≤ γ λ φ holds. Formally inf { (| (0) |)} 0 φ∈ α φ = r S
, however having in mind that φ is a continuous function and a solution of (1), there exists

[ ,0] ,max [ ( )] { [ (0)]} inf { (| (0) |)} V V a θ∈ -τ φ∈ φ θ < γ φ = - α φ r S
and 0 a > . Therefore, due to homogeneity of the functions f and V we have:

[ ,0] max [ ( )] { [ (0)]} V V θ∈ -τ ′ φ θ < γ φ r r Λ Λ ⇒ [ (0)] ( ) d DV a ν+ φ φ ≤-λ r r Λ f Λ , or equivalently ( 1 || || | (0) | | (0) | - λ = ϕ ≥ ϕ ≥ σ ϕ r r r
),

[ ,0] max [ ( )] { [ (0)]} V V θ∈ -τ ′ ϕ θ < γ ϕ ⇒ [ (0)] ( ) (| (0) |) DV ′ ϕ ϕ ≤ -α ϕ f ,
where the function The condition (i) imposed in Theorem 1 on the system (1) behavior is the conventional Razumikhin condition (except in the homogeneous case it can be verified on the sphere r S only). The constraint (ii) on existence of the function ′ γ is new. Roughly speaking this requirement says that the function γ has to be "homogeneous" with degree 1. Another explanation is that the function γ is globally Lipschitz. For instance, this is the case if there exists

1 2 1 k k < < < +∞ such that 1 2 ( ) k s s k s ≤ γ ≤ .
R e m a r k 1 . Note that the full derivative

[ (0)] ( ) DV φ φ f is a function of φ ∈ r S , however under the condition [ ,0] max [ ( )] { [ (0)]} V V θ∈ -τ ′ φ θ < γ φ r r Λ Λ the inequality [ (0)] ( ) ( (0)) DV m φ φ < φ f holds for some : n m R R → .
If the function m is r -homogeneous (the functions V and f possess this property), then the property ( ( 0

)) (| (0) |) m φ ≤ -α φ in (ii) has to be verified for (0) S φ ∈ r only. □
The result of Theorem 1 means that the system stability can be checked on the sphere r S only for homogeneous systems with homogeneous Lyapunov-Razumikhin function. These facts may simplify the function V search and the system analysis applying numerical routines. The drawback is that in the space [ 0] n C -τ, this sphere r S is a rather complex object, especially the subset of φ ∈ r S where

[ ,0] max [ ( )] { [ (0)]} V V θ∈ -τ ′ ϕ θ < γ ϕ .

LOCAL HOMOGENEITY

A disadvantage of the global homogeneity introduced so far is that such systems possess the same behavior globally for all

[ 0] n C ϕ ∈ -τ, .
Thus the homogeneous systems are not really "nonlinear", they have similar diversity of operating modes as linear systems. In fact, the homogeneous systems are a generalization of linear ones from analysis and design points of view. Comparing with other nonlinear systems it is easier to find Lyapunov function for homogeneous ones. Additionally, since the homogeneous systems have global stability property, the local stability analysis at the origin using linearization verifies the global stability in this case. This is why finding a possibility to apply this approach for a broader class of nonlinear systems is very important. An approach to resolve this issue consists in local version of homogeneity introduction as in [START_REF] Efimov | Oscillations Conditions in Homogenous Systems[END_REF].

D e f i n i t i o n 2 . The function : [ 0] n g C R -τ, → , (0) 0 g = is called ( r , 0 λ , 0 g )-homogeneous ( 0 i r > , 1, i n = ; 0 : [ 0] n g C R -τ, → , 0 (0) 0 g = ) if for any ϕ ∈ r S the relation 0 0 0 li m ( ) ( ) 0 d g g - λ → λ λ ϕ - ϕ = r Λ
holds for some 0 0 d ≥ .

The system ( 1) is called ( r , 0 λ , 0 f )-homogeneous (

0 i r > , 1, i n = ; 0 : [ 0] n n C R -τ, → f , 0 ( 0 ) 0 = f ) if for any ϕ ∈ r S the relation 0 0 1 0 lim ( ) ( ) 0 d - - λ→λ λ ϕ -ϕ = r r Λ f Λ f holds for some 0 1 min i n i d r ≤ ≤ ≥ - . □
This property is called local homogeneity [START_REF] Efimov | Oscillations Conditions in Homogenous Systems[END_REF], it allows us to analyze local stability of the system (1) on the basis of the simplified system

0 ( ) / [ ( )] d t dt t τ = y f y , 0 t ≥ , (2)
called the local approximating dynamics for (1). For any 0 0 < λ < + ∞ the following formulas give a variant of homogeneous approximations 0 g and 0 f choice:

0 1 0 , 0 | | | | 0 ( ) || || ( ) d d g g - - ϕ ϕ = λ ϕ r r Λ Λ , 0 1 1 0 || || ,0 ,0 || || 0 ( ) || || ( ) d d - - - ϕ ϕ ϕ = ϕ λ ϕ r r r f Λ Λ f Λ Λ , where ,0 1 0 { } i r n i diag = = λ r Λ , || || 1 0 {|| || } i r n i i diag ϕ = = ϕ Λ and 0 d ≥ .
T h e o r e m 2 . Let the system (1) be ( r , 0 λ , 0 f )homogeneous for some

0 i r > , 1, i n =
, the function 0 f be continuous and r -homogeneous with the degree 0 d and there exist a r -homogeneous Lyapunov-Razumikhin function

0 : n V R R + →
with the degree 0 ν , 0 0 0 d ν + > such that:

(i) there exist functions , α γ ∈K ( ( ) s s γ > for all s R + ∈ ) such that for all φ ∈ r S [ ,0] 0 0 max [ ( )] { [ (0)]} V V θ∈ -τ φ θ < γ φ ⇒ 0 0 [ (0)] ( ) (| (0) |) DV φ φ ≤ -α φ f ; (ii) there exists function ′ γ ∈K such that ( ) ( ) s s ′ λγ ≤ γ λ for all , s R + λ ∈ .
Then 1) if 0 0 λ = , then there exists 0 ε < λ such that the system (1) is locally asymptotically stable with the domain of asymptotic stability containing the set

1 0 1 2 { [ 0]:|| || ( )} n X C - ε = ϕ∈ -τ, ϕ ≤ α α ρ λ r ;
2) if 0 λ = +∞ , then there exists 0 ε < λ < + ∞ such that the system ( 1) is globally asymptotically stable with respect to forward invariant set

1 1 2 { [ 0]:|| || ( )} n X C - ∞ ε = ϕ∈ -τ, ϕ ≤ α α ρ λ r ; 3) if 0 0 < λ < + ∞ , then there exist 0 0 ε ε < λ ≤ λ ≤ λ < +∞
such that the system ( 1) is asymptotically stable with respect to the forward invariant set X ∞ with region of attraction

1 1 2 1 1 2 { [ 0]: ( ) || || ( )} n X C - ε - ε = ϕ ∈ -τ, α α ρ λ < ϕ < α α ρ λ r r
provided that the set X is connected and X ≠ ∅ .

P r o o f . For the system (1) the coordinates transformation

ϕ = φ r Λ connects any [ 0] n C ϕ ∈
-τ, with some φ ∈ r S for suitably chosen 0 λ ≥ . According to definition of the function 0

V we have:

0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 [ (0)] ( ) [ (0)] ( ) [ (0)]{ ( ) [ ( ) ( )]} [ (0)] ( ) [ (0)] { ( ) ( )} [ (0)]{ ( ) [ ( ) ( ) ] } . d d d d DV DV DV DV DV DV +ν ν - +ν - - ϕ ϕ = φ φ = = φ φ + φ - φ = = λ φ φ + λ φ φ- λ φ =λ φ φ + λ φ-φ r r r r r r r r r r r f Λ f Λ Λ f Λ f Λ f Λ f Λ f Λ Λ f f Λ f Λ f
Due to continuity of the functions f , 0 f and the local homogeneity property definition for any 0 ε > there exist

0 ε ε λ ≤ λ ≤ λ such that 0 1 0 0 sup | [ (0)]{ ( ) ( )}| d DV - - φ∈ φ λ φ -φ ≤ ε r r r Λ f Λ f S for all ( , ) ε ε λ ∈ λ λ . From (i) we know that under the condi- tion [ ,0] 0 0 max [ ( )] { [ (0)]} V V θ∈ -τ φ θ < γ φ inequality 0 0 [ (0)] ( ) (| (0) |) DV φ φ ≤ -α φ f is satisfied. Since inf { (| (0) |)} 0 φ∈ α φ = r S
it seems that the additive disturbance caused by the dilation transformation can not be attenuated by α . However, the condition 0 [, 0 ]0 0

[ (0)] max [ ( )] { [ (0)]} V V V θ∈ -τ φ ≤ φ θ < γ φ
excludes from consideration some functions, in particular those with (0) 0 φ = . Moreover, taking in mind that φ is a continuous function and a solution of (1), there is

[ ,0] 0 0 ,max [ ( )] { [ (0)]} inf { (| (0) |)} V V a θ∈ -τ φ∈ φ θ < γ φ = - α φ r S
and 0 a > . Let the constants ε λ , ε λ be chosen to ensure that a > ε . Multiplying the inequality 0 0

[ ( )] { [ (0)]} V V φ θ < γ φ on 0 ν λ we get that it is equivalent to 0 0 [ ( )] { [ (0)]} V V ′ ϕ θ < γ ϕ for all
[ ,0] θ ∈ -τ . Then from (i), (ii) the following property holds for

0 0 1 ( ) ( ) ( ) d s s a ν + - ′ α = σ -ε r : [ ,0] 0 0 max [ ( )] { [ (0)]} V V θ∈ -τ ′ ϕ θ < γ ϕ ⇒ 0 0 [ (0)] ( ) (| (0) |) DV ′ ϕ ϕ ≤ -α ϕ f ,
where ϕ = φ r Λ , φ ∈ r S and ( , )

ε ε λ ∈ λ λ , i.e. for all { [ 0]: || || } n X C τ ε τ ε ∈ = ∈ -τ, λ < <λ r r x x x
in the system (1). If 0 0 λ = , then clearly 0 ε λ = and the system is locally asymptotically stable with the domain of asymptotic stability containing the set 0 X [START_REF] Gu | Stability of Time-Delay Systems (Control Engineering)[END_REF][START_REF] Kolmanovsky | Stability of functional differential equations[END_REF]. If 0 λ = +∞ , then ε λ = +∞ and the function 0 V has strictly negative time derivative for the system (1) into the set \ n R X ∞ . Thus the set X ∞ is forward invariant for (1) and borrowing arguments from [START_REF] Gu | Stability of Time-Delay Systems (Control Engineering)[END_REF][START_REF] Lin | A Smooth Converse Lyapunov Theorem for Robust Stability[END_REF], these facts imply the global asymptotic stability of the system (1) with respect to the set X ∞ . Finally, let 0 0 < λ < + ∞ and the set X be not empty, then the function 0 V is strictly decreasing into the set X and all trajectories 0 ( , ) t x x with initial conditions 0 X ∈ x reach for the set X ∞ , that implies the desired conclusion. ■

This result establishes the links between different variants of local homogeneity with stable approximating dynamics and the stability properties of the original system similarly (Efimov and Perruquetti, 2010). In [START_REF] Efimov | Oscillations Conditions in Homogenous Systems[END_REF] the case of unstable approximating dynamics has been also studied. Note that Remark 1 is valid for the condition (ii) verification.

C o r o l l a r y 1 . Let the system (1) be ( r , 0 λ , 0 f )homogeneous for some 0

i r > , 1, i n = , the function 0 : n n R R → f
be continuous and r -homogeneous with the degree 0 d and there exist a r -homogeneous Lyapunov-

Razumikhin function 0 : n V R R + →
with the degree 0 ν ,

0 0 0 d ν + > such that 0 0 ( ) ( ) (| |) DV ≤ -α x f x x for all S ∈ r x
. Then all conclusions of Theorem 2 hold.

P r o o f . If 0 : n n R R → f
, then the conditions (i) and (ii) of Theorem 2 can be reduced to 0

0 ( ) ( ) (| |) DV ≤ -α x f x x for all S ∈ r x
(the functions 0 f and 0 V are homogeneous). ■

EXAMPLE

Consider the system:

3 1 1 2 1 ( ) ( ) ( ) ( ) x t x t x t x t = - + - ; 3 2 2 1 2 2 ( ) ( ) ( ) ( ) ( ) x t kx t x t ax t x t = - - + -τ - ,
where k , a are some positive parameters, 0 τ > is a fixed time delay. This system is locally homogeneous in the bilimit:

1 0 λ = , 1 [0.5 0.5] = r , [ ] 1 1 2 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) T x t x t kx t x t ax t τ = - + - - + -τ f x , 1 0 d = , 2 2 1 1 2 ( ) ( ) ( ) V x t x t = + x , 1 1 v = ; 2 λ = +∞ , 2 [1 1] = r , 3 3 2 1 2 ( ) ( ) ( ) T x t x t τ ⎡ ⎤ = - - ⎣ ⎦ f x , 2 2 d = , 2 2 2 1 2 ( ) ( ) ( ) V x t x t = + x , 2 1 v = .
Straightforward calculations show that --> the system (2) at 1 0 λ = is asymptotically stable, that according to Theorem 2 implies local asymptotic stability of the system around the origin. Obviously for 2 λ = +∞ the system (2) is also asymptotically stable that ensures global convergence of the trajectories to a vicinity of the origin. Note that approximate behavior of the system in this case is defined by the time-delay free dynamics (Corollary 1). The results of simulation for 1 k = , 0.5 a = , 0.1 τ = presented in Fig. 1 demonstrate that actually the system is globally asymptotically stable.

CONCLUSION

The homogeneity notion is extended to time-delay systems (nonlinear differential equations with functional arguments). The sufficient conditions for stability of homogenous systems is presented. These conditions are based on the Razumikhin stability arguments. It is shown on a counterexample that development of the Lyapunov-Krasovskii approach is tricky for the homogeneous case. The definition of local homogeneity is proposed, relations between stability of the locally approximating dynamics and the original system are established. Efficiency of the proposed approach is demonstrated on academic example.

  P r o o f . The dilation transformation ϕ = φ with some φ ∈ r S for properly chosen 0 λ ≥ . Let us multiply the left hand part of the condition (i) by ν λ and the right hand part by d ν+ λ