Denis Efimov

Alexander Y Aleksandrov

D Emov

Lyapunov-Krasovskii functional for discretized homogeneous systems

come

Introduction

Presence of delays in the modern interconnected plants is hard to neglect, especially for networked and distributed large-scale systems. Delays and lags may inuence the system stability, as well as quality of transients, see [START_REF] Gu | Stability of Time-Delay Systems[END_REF][START_REF] Hale | Theory of Functional Dierential Equations[END_REF][START_REF] Kolmanovsky | Stability of functional dierential equations[END_REF]. Synthesis of control and estimation algorithms, which take into account uncertain or time-varying delays, is a popular subject of research [START_REF]Recent Results on Nonlinear Delay Control Systems[END_REF][START_REF] Karafyllis | Predictor Feedback forDelay Systems: Implementations and Approximations[END_REF][START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]. For stability analysis of time-delay systems, a Lyapunov-Krasovskii functional or a Lyapunov-Razumikhin function can be used [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF], and there are plenty of results in the domain, e.g., [START_REF] Monaco | Nonlinear average passivity and stabilizingcontrollers in discrete-time[END_REF][START_REF] Gielen | Lyapunov methods for time-invariant delay dierence inclusions[END_REF], mainly related with linear dynamics [START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF]. The popularity of the latter is based on a wide choice of available Lyapunov-Krasovskii functionals and Lyapunov-Razumikhin functions, which is also the case for some classes of nonlinear plants, as port-Hamiltonian systems [START_REF] Schier | Stability of a class of delayed port-Hamiltonian systems with application to microgrids with distributed rotational and electronic generation[END_REF], for instance.

Another promising class of canonical nonlinear dynamical models, which disposes a variety of tools, is composed by homogeneous systems that take an intermediate place between linear and nonlinear ones [START_REF] Bernuau | On homogeneity and its application in sliding mode[END_REF].

The theory of homogeneous dynamical systems has been established, rst, for ordinary dierential equations [START_REF] Bacciotti | Liapunov Functions and Stability in Control Theory[END_REF][START_REF] Bhat | Geometric homogeneity with applications to nite-time stability[END_REF][START_REF] Kawski | Progress in systems and control theory: New trends in systems theory[END_REF][START_REF] Zubov | On systems of ordinary dierential equations with generalized homogenous right-hand sides[END_REF], and next extended to dierential inclusions [START_REF] Filippov | Stability of dierential equations with discontinuous and multi-valued right-hand parts[END_REF][START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF][START_REF] Bernuau | On homogeneity and its application in sliding mode[END_REF], time-delay systems [START_REF] Emov | Development of homogeneity concept for time-delay systems[END_REF][START_REF] Emov | Weighted homogeneity for time-delay systems: Finite-time and independent of delay stability[END_REF], discrete-time systems [START_REF] Sanchez | On homogeneity of discrete-time systems: stability and convergence rates[END_REF] and partial dierential equations [START_REF] Polyakov | On homogeneous distributed parameter systems[END_REF]. The main feature of a homogeneous nonlinear model is that its local behavior is the same as the global one, the homogeneous stable/unstable systems admit homogeneous Lyapunov/Chetaev functions [START_REF] Zubov | On systems of ordinary dierential equations with generalized homogenous right-hand sides[END_REF][START_REF] Kanevskiy | Design of homogeneous Lyapunov-Krasovskii functions[END_REF][START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector eld[END_REF][START_REF] Emov | Oscillations conditions in homogenous systems[END_REF][START_REF] Emov | On necessary conditions of instability and design of destabilizing controls[END_REF][START_REF] Emov | Homogeneous lyapunov functions: from converse design to numerical implementation[END_REF] and possess robustness properties with respect to external inputs and delays [START_REF] Bernuau | Verication of iss, iiss and ioss properties applying weighted homogeneity[END_REF][START_REF] Aleksandrov | Delay-independent stability of homogeneous systems[END_REF][START_REF] Emov | Weighted homogeneity for time-delay systems: Finite-time and independent of delay stability[END_REF][START_REF] Zimenko | A note on delay robustness for homogeneous systems with negative degree[END_REF]. Using the concept of local homogeneity [START_REF] Zubov | On systems of ordinary dierential equations with generalized homogenous right-hand sides[END_REF][START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF][START_REF] Emov | Oscillations conditions in homogenous systems[END_REF][START_REF] Emov | Conditions of oscillations and multi-homogeneity[END_REF]] a generic dynamical system may be locally approximated by a homogeneous one, which allows the set of methods developed for this class of dynamics to be applied at large. It has been shown that homogeneous systems have certain robustness of stability with respect to delays [START_REF] Diblik | Asymptotic equilibrium for homogeneous delay linear dierential equations with l-perturbation term[END_REF][START_REF] Asl | Analytical solution of a system of homogeneous delay dierential equations via the Lambert function[END_REF][START_REF] Mazenc | Global asymptotic stabilization for chains of integrators with a delay in the input[END_REF][START_REF] Bokharaie | D-stability and delay-independent stability of homogeneous cooperative systems[END_REF][START_REF] Aleksandrov | On the asymptotic stability of solutions of nonlinear systems with delay[END_REF][START_REF] Emov | Weighted homogeneity for time-delay systems: Finite-time and independent of delay stability[END_REF][START_REF] Zimenko | A note on delay robustness for homogeneous systems with negative degree[END_REF]. In particular, for the case of positive degree it has been proven that homogeneity and global asymptotic stability in the delay-free case imply local asymptotic stability for any value of delay. For the case of negative degree, under similar restrictions, for any delay, global asymptotic stability with respect to a compact set containing the origin is obtained. The latter results have been obtained in [START_REF] Emov | Weighted homogeneity for time-delay systems: Finite-time and independent of delay stability[END_REF][START_REF] Zimenko | A note on delay robustness for homogeneous systems with negative degree[END_REF] using Lyapunov-Razumikhin theory, and in [START_REF] Aleksandrov | Complete type functionals for some classes of homogeneous dirential-dierence systems[END_REF][START_REF] Emov | Analysis of robustness of homogeneous systems with time delays using Lyapunov-Krasovskii functionals[END_REF] by applying Lyapunov-Krasovskii approach.

In the present work, following the results of [START_REF] Aleksandrov | On stability of solutions to one class of nonlinear dierence systems[END_REF][START_REF] Aleksandrov | Complete type functionals for some classes of homogeneous dirential-dierence systems[END_REF][START_REF] Emov | Analysis of robustness of homogeneous systems with time delays using Lyapunov-Krasovskii functionals[END_REF], we consider a discrete-time system obtained after discretization (using the explicit Euler method) of a (locally) homogeneous system with non-zero degree, which is asymptotically stable in the delay-free case. The stability conditions are derived by proposing a generic structure of Lyapunov-Krasovskii functional for this class of dynamics. The presented approach allows the convergence rates and the domain of stability/attraction to be evaluated. It is demonstrated that for a homogeneous timedelay system (asymptotically stable for zero delay) the Euler discretization preserves the rates of decay, which is not the case for non-delayed homogeneous dynamics [START_REF] Emov | Realization and discretization of asymptotically stable homogeneous systems[END_REF].

The outline of this paper is as follows. The preliminary denitions and the homogeneity for delay-free systems are given in Section 2. The problem statement and the main results are presented in Section 3, their * D.

Preliminaries

In the sequel, R denotes the eld of real numbers, and R n is the n-dimensional Euclidean space; R + = {s ∈ R : s ≥ 0}. The Euclidean norm |x| will be used for vectors x ∈ R n .

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

Convergence conditions

We will need the following results estimating the rate of convergence for discrete-time scalar nonlinear systems:

Lemma 1. [40] If a sequence {v k } +∞ k=0 satises the condition 0 ≤ v k+1 ≤ v k -αv 1+ k with α > 0, > 0, v 0 ∈ [0, α --1 (1+ ) -1 ], then v k ≤ v 0 (1 + α v 0 k) -1 for k = 0, 1, . . .
In a similar way, it can be proven that the following lemma is valid:

Lemma 2. If a sequence {v k } N k=0 with N > 0 satises the condition 0 ≤ v k+1 ≤ v k -αv 1- k , for k = 0, 1, . . . , N , where α > 0, ∈ (0, 1), v 0 ≥ α -1 , then v k ≤ v 0 (1 -α v - 0 k) -1 for k = 0, 1, . . . , N with N ≤ N 0 = v 0 -α α .
Proof. Let us use the same arguments as in the proof of Lemma 1 [START_REF] Aleksandrov | On stability of solutions to one class of nonlinear dierence systems[END_REF].

Note that the restriction

v k+1 ≥ 0 follows v k ≥ α -1
for all k = 0, 1, . . . , N , then the upper estimate veries this condition for v 0 -α α ≥ k. By induction, the estimate obviously holds for k = 0, and assuming that it is satised for all k ≤ l for some integer l ∈ [1, N 0 -1) let us demonstrate that it is valid in such a case for k = l + 1. Hence,

v l+1 ≤ v l (1 -αv - l ) ≤ v 0 (1 -α v - 0 l) -1 1 - αv - 0 1 -α v - 0 l = v 0 (1 -α v - 0 (l + 1)) -1 g(z, l), where z = α v - 0 and g(z, l) = 1 -zl 1 -z(l + 1) -1 1 - 1 z 1 -zl .
It is necessary to establish the conditions providing g(z, l) ≤ 1 for any z > 0 and l ∈ [1, N 0 -1). Note that the restriction on l and N 0 implies that (α (l + 1)) -1 < v 0 , and we have 1 -z(l + 1) > 0 for any l ∈ [1, N 0 -1)

(thus, 1 -zl > 0 also). For g(z, 0) = 1 1-z -1
1 -z we get that g(z, 0) ≤ 1 (indeed, g(0, 0) = 1 and ∂g(z,0) ∂z

= z 2 -1 (1-z) 1+ < 0 for z ∈ (0, 1), which is the case for v 0 > α -1 > ( α) -1
), then

∂g(z, l) ∂l = - z 3 (1 -) 2 (1 -zl) 2 (1 -z(l + 1)) 1 -zl 1 -z(l + 1) -1
and it is always negative for l ∈ [1, N 0 -1), which implies the desired result.

In the conditions of this lemma, we have that from any initial value v 0 satisfying the introduced restrictions, the corresponding sequence converges to the zone

v k ≤ α -1
for k ≤ N 0 , and it stays there if additionally 0 ≤ v k+1 ≤ v k for all k = 0, 1, . . .

Homogeneity

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

For any r and x ∈ R n the homogeneous norm can be dened, for example, as follows

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

σ r (|x| r ) ≤ |x| ≤ σ r (|x| r )
for some σ r , σ r ∈ K ∞ [START_REF] Emov | Homogeneous lyapunov functions: from converse design to numerical implementation[END_REF]. The homogeneous norm has an important property that is |Λ r (λ)x| r = λ|x| r for all x ∈ R n . Dene S r = {x ∈ R n : |x| r = 1} and B r = {x ∈ R n : |x| r ≤ 1} as the unit sphere and the unit ball in the homogeneous norm, respectively.

Denition 1. [START_REF] Zubov | On systems of ordinary dierential equations with generalized homogenous right-hand sides[END_REF][START_REF] Bacciotti | Liapunov Functions and Stability in Control Theory[END_REF] The function g : R n → R is called rhomogeneous if for any x ∈ R n the relation

g(Λ r (λ)x) = λ d g(x)
holds for some d ∈ R and all λ > 0.

The vector eld f : R n → R n is called rhomogeneous if for any x ∈ R n the relation

f (Λ r (λ)x) = λ d Λ r (λ)f (x)
holds for some d ≥ -r min and all λ > 0.

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

A system ẋ = f (x) is called homogeneous if f is rhomogeneous. An advantage of homogeneous systems is that any its solution can be obtained from another solution under the dilation and a suitable time reparametrization, and that homogeneous asymptotically stable system admits a homogeneous Lyapunov function [START_REF] Zubov | On systems of ordinary dierential equations with generalized homogenous right-hand sides[END_REF][START_REF] Bacciotti | Liapunov Functions and Stability in Control Theory[END_REF]. If f (x) = Ax for some A ∈ R n×n , then it is rhomogeneous of zero degree for r = [1, ..., 1] , and the properties of scalability of solutions and existence of a homogeneous (quadratic) Lyapunov function are well-known for linear systems, then homogeneity allows these useful features to be extended on a wider class of dynamics.

Lemma 3. Let h : R n → R be χ-Hölder continuous on B r with constant L > 0 and rhomogeneous function of

degree ν ≥ χr min , then |h(x) -h(z)| ≤ L max{1, |x| ν-χrmin r , |z| ν-χrmin r }|x -z| χ for all x, z ∈ R n .
Proof. By denition of homogeneity h(Λ(λ)x) = λ ν h(x) for any x ∈ R n and λ > 0. Take any x, z ∈ R n and introduce λ = max{|x| r , |z| r }.

If λ ≤ 1, then |h(x) -h(z)| ≤ L|x -z| χ
by the properties of L, and the required estimate holds. Let λ > 1, then there exist ξ, ζ

∈ B r such that x = Λ(λ)ξ and z = Λ(λ)ζ, hence |h(x) -h(z)| = |h(Λ(λ)ξ) -h(Λ(λ)ζ)| = λ ν |h(ξ) -h(ζ)| ≤ L|ξ -ζ| χ λ ν = L|Λ -1 (λ)x -Λ -1 (λ)z| χ λ ν ≤ L|Λ -1 (λ)| χ |x -z| χ λ ν ,
and recalling that

|Λ -1 (λ)| ≤ λ -rmin λ > 1 λ -rmax λ ≤ 1
we obtain the desired result.

This lemma shows that if a homogeneous function is Lipschitz or Hölder continuous in B r with a constant L, then it possesses this property on any compact set X ⊂ R n with the constant L max{1, sup x∈X |x| ν-χrmin r }.

Useful inequalities

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

a γ b δ ≤ 1 p a γp + p -1 p b δp p-1
for any p > 1. Jensen's inequality for any convex κ ∪ : R → R or concave κ ∩ : R → R function reads:

κ ∪ n i=1 x i n ≤ n i=1 κ ∪ (x i ) n , κ ∩ n i=1 x i n ≥ n i=1 κ ∩ (x i ) n
for any x 1 , . . . , x n ∈ R. In addition,

n i=1 x i γ ≤ n i=1
x γ i for any x 1 , . . . , x n ∈ R + and γ ∈ [0, 1).

Another related result can be obtained using the properties of homogeneous functions and a specially dened homogeneous norm: Lemma 4. [START_REF] Emov | Analysis of robustness of homogeneous systems with time delays using Lyapunov-Krasovskii functionals[END_REF] Let a, b ∈ R + and > 0, α > 0, β > 0, γ > 0, δ > 0 be given, then

a α + b β -a γ b δ ≥ 0 provided that 1) max{a α , b β } ≥ 1 1- γ α -δ β and γ α + δ β < 1, 2) max{a α , b β } ≤ 1 1- γ α -δ β and γ α + δ β > 1.

Main Results

In this section, rst, the system of interest will be introduced with the basic hypotheses. Next, two cases will be analyzed with positive and negative homogeneous degrees.

Problem statement

Consider the dierence system

x(k + 1) = x(k) + hF (x(k), x(k -m)), (1)
where

x(k) ∈ R n and x (k) = x T (k), x T (k -1), . . . , x T (k -m) T ∈ R n(m+1) is the augmented state vector, m
is a nonnegative integer representing delay; h > 0 is a discretization step, k = 0, 1, . . . is the discrete time; the vector function F : R 2n → R n is locally bounded in R 2n . Denote by x k, x (0) a solution of (1) with initial conditions x (0) ∈ R n(m+1) , and x (k) (x (0) ) as the corresponding state trajectory.

The system (1) is a discrete-time counterpart of the continuous-time system

ż(t) = F (z(t), z(t -mh)) (2)
obtained by applying the explicit Euler discretization method with the step h.

Assumption 1. The vector eld F is rhomogeneous of degree ν ∈ (-r min , 0) ∪ (0, +∞):

F (Λ(λ)ξ, Λ(λ)ζ) = λ ν Λ(λ)F (ξ, ζ) ∀ξ, ζ ∈ R n , ∀λ > 0,
and uniformly continuous with respect to the rst argument in B r : for any L > 0 there exists χ > 0 such that

|F (φ + ϕ, ψ) -F (φ, ψ)| ≤ L for any φ, ψ ∈ B r with |ϕ| ≤ χ.
This continuity hypothesis is satised if, for example, F is continuous in R 2n . We will need also stability of the corresponding delay-free system:

Assumption 2. The zero solution of

ż(t) = F (z(t), z(t))
is asymptotically stable.

Following [START_REF] Zubov | On systems of ordinary dierential equations with generalized homogenous right-hand sides[END_REF][START_REF] Bacciotti | Liapunov Functions and Stability in Control Theory[END_REF], let V (z) be a twice continuously dierentiable rhomogeneous Lyapunov function of the degree µ > r max constructed for the delay-free (homogeneous) system from Assumption 2 and satisfying the conditions of the Lyapunov asymptotic stability theorem [START_REF] Bacciotti | Liapunov Functions and Stability in Control Theory[END_REF].

Hence, it is assumed that a dynamical system (2) is given, which is asymptotically stable and homogeneous with non-zero degree for the delay-free case in continuous time (assumptions 1 and 2). Under these restrictions and for any mh ≥ 0, the origin is asymptotically stable for (2) with ν > 0 and there is a vicinity of the origin that globally attracts all trajectories of (2) with ν < 0 [START_REF] Emov | Development of homogeneity concept for time-delay systems[END_REF][START_REF] Zimenko | A note on delay robustness for homogeneous systems with negative degree[END_REF]. Let us consider its discretization (1) obtained by the explicit Euler method, and the objective is to determine the conditions and the kind of the discrete-time system (1) stability under introduced hypotheses. To this end, the Lyapunov-Krasovskii theory will be used.

Note that the case of ν = 0 is excluded by Assumption 1, hence, in the sequel the case of linear dynamics in [START_REF] Gu | Stability of Time-Delay Systems[END_REF], which is well studied already [START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF], is not addressed. Recall that in the delay-free continuous-time setting, an asymptotically stable homogeneous system has a faster converge rate than any linear dynamics close to the origin (nite-time stability) for a negative degree and at innity (nearly xed-time stability) for a positive degree [START_REF] Bernuau | On homogeneity and its application in sliding mode[END_REF].

Lyapunov-Krasovskii functional

Choose a Lyapunov-Krasovskii functional candidate for the system (1) as follows

W x (k) = V (x(k)) + h ∂V (x(k)) ∂z T m i=1 F (x(k), x(k -i)) +h m i=1 β i |x(k -i)| ρ r , (3)
where ρ, β 1 , . . . , β m are positive parameters, whose values will be dened later. It is a functional of the complete type [START_REF] Kharitonov | Lyapunov-Krasovskii approach to the robust stability analysis of time-delay systems[END_REF], which is based on the Lyapunov function V of the delay-free continuous-time counterpart of (1), and its feature consists in the introduction of a sign-indenite middle term, whose advantages can be eectively utilized due to the homogeneity [START_REF] Emov | Analysis of robustness of homogeneous systems with time delays using Lyapunov-Krasovskii functionals[END_REF]. We will use the following norm in the sequel:

x (k) = max{|x(k)| r , |x(k -1)| r , . . . , |x(k -m)| r }.
Using the homogeneity of V and F we obtain:

h m i=1 β i |x(k -i)| ρ r -a 3 h n j=1 |x(k)| µ-rj r m i=1 (|x(k)| ν+rj r + |x(k -i)| ν+rj r ) +a 1 |x(k)| µ r ≤ W x (k) ≤ a 2 |x(k)| µ r + h m i=1 β i |x(k -i)| ρ r (4) +a 3 h n j=1 |x(k)| µ-rj r m i=1 (|x(k)| ν+rj r + |x(k -i)| ν+rj r ),
where a 1 , a 2 , a 3 are the positive constants, which may be evaluated as follows:

a 1 = inf y∈Sr V (y), a 2 = sup y∈Sr V (y), a 3 = sup φ,ψ∈Br ∂V (φ) ∂φ T F (φ, ψ) .
Consider the dierence ∆W = W x (k+1) -W x (k) of W along the solutions of (1):

∆W = h ∂V (x(k) + hθ 1k F (x(k), x(k -m))) ∂z T F (x(k), x(k -m)) +h m i=1 β i |x(k -i + 1)| ρ r -h m i=1 β i |x(k -i)| ρ r +h ∂V (x(k) + hF (x(k), x(k -m))) ∂z T m i=1 F (x(k + 1), x(k -i + 1)) -h ∂V (x(k)) ∂z T m i=1 F (x(k), x(k -i)),
where θ 1k ∈ (0, 1) comes from application of the Mean Value Theorem to the dierence V (x(k + 1)) -V (x(k)).

Further,

∆W = h ∂V (x(k)) ∂z T F (x(k), x(k -m)) +h ∂V (x(k) + hθ 1k F (x(k), x(k -m))) ∂z - ∂V (x(k)) ∂z T F (x(k), x(k -m)) +hβ 1 |x(k)| ρ r -hβ m |x(k -m)| ρ r + h m-1 j=1 (β j+1 -β j )|x(k -j)| ρ r +h ∂V (x(k) + hF (x(k), x(k -m))) ∂z - ∂V (x(k)) ∂z T m i=1 F (x(k + 1), x(k -i + 1)) +h ∂V (x(k)) ∂z T m i=1 F (x(k + 1), x(k -i + 1)) - m i=1 F (x(k), x(k -i + 1)) +h ∂V (x(k)) ∂z T m i=1 F (x(k), x(k -i + 1)) - m i=1 F (x(k), x(k -i))
and making a direct sum of the rst and the last terms we obtain:

∆W = h ∂V (x(k)) ∂z T F (x(k), x(k)) +hβ 1 |x(k)| ρ r -hβ m |x(k -m)| ρ r + h m-1 j=1 (β j+1 -β j )|x(k -j)| ρ r +h ∂V (x(k) + hθ 1k F (x(k), x(k -m))) ∂z - ∂V (x(k)) ∂z T F (x(k), x(k -m)) +h ∂V (x(k) + hF (x(k), x(k -m))) ∂z - ∂V (x(k)) ∂z T m i=1 F (x(k + 1), x(k -i + 1)) +h ∂V (x(k)) ∂z T m i=1 F (x(k + 1), x(k -i + 1)) - m i=1 F (x(k), x(k -i + 1)) .
Hence, due to the homogeneity of V and F the following estimate can be substantiated:

∆W ≤ -a 4 h|x(k)| µ+ν r + hβ 1 |x(k)| ρ r -hβ m |x(k -m)| ρ r + h m-1 j=1 (β j+1 -β j )|x(k -j)| ρ r +ha 5 n l=1 ∂V (x(k) + hθ 1k F (x(k), x(k -m))) ∂z l - ∂V (x(k)) ∂z l (|x(k)| ν+r l r + |x(k -m)| ν+r l r ) +ha 5 n l=1 ∂V (x(k) + hF (x(k), x(k -m))) ∂z l - ∂V (x(k)) ∂z l m i=1 (|x(k + 1)| ν+r l r + |x(k -i + 1)| ν+r l r ) +ha 6 n l=1 |x(k)| µ-r l r m i=1 |F l (x(k + 1), x(k -i + 1)) -F l (x(k), x(k -i + 1))| ,
where a 4 , a 5 , a 6 are positive constants, which may be derived as Note that in such a case a 3 ≤ a 5 a 6 .

a 4 = -sup y∈Sr ∂V (
For brevity of notation denote λ = x (k) and recall that Λ(λ) = diag{λ r1 , . . . , λ rn } (we assume that λ = 0).

Then, applying again the Mean Value Theorem we obtain:

∂V (x(k) + hθ 1k F (x(k), x(k -m))) ∂z l - ∂V (x(k)) ∂z l (|x(k)| ν+r l r + |x(k -m)| ν+r l r ) ≤ a 5 h n s=1 ∂ 2 V (x(k) + hθ 2kl F (x(k), x(k -m))) ∂z l ∂z s ×(|x(k)| ν+rs r + |x(k -m)| ν+rs r )(|x(k)| ν+r l r + |x(k -m)| ν+r l r ) = a 5 h n s=1 λ µ-r l -rs ∂ 2 V (Λ -1 (λ)x(k) + hθ 2kl λ ν F (Λ -1 (λ)x(k), Λ -1 (λ)x(k -m))) ∂z l ∂z s ×(|x(k)| ν+rs r + |x(k -m)| ν+rs r )(|x(k)| ν+r l r + |x(k -m)| ν+r l r ) ≤ 4a 5 hλ 2ν+µ n s=1 ∂ 2 V (Λ -1 (λ)x(k) + hθ 2kl λ ν F (Λ -1 (λ)x(k), Λ -1 (λ)x(k -m))) ∂z l ∂z s ≤ 4a 5 h m p=0 |x(k -p)| 2ν+µ r n s=1 ∂ 2 V (Λ -1 (λ)x(k) + hθ 2kl λ ν F (Λ -1 (λ)x(k), Λ -1 (λ)x(k -m))) ∂z l ∂z s ,
where θ 2kl ∈ (0, 1).

In a similar way, we obtain for θ 3kl ∈ (0, 1):

∂V (x(k) + hF (x(k), x(k -m))) ∂z l - ∂V (x(k)) ∂z l m i=1 (|x(k + 1)| ν+r l r + |x(k -i + 1)| ν+r l r ) ≤ 2a 5 hλ ν+µ-r l n s=1 ∂ 2 V (Λ -1 (λ)x(k) + hθ 3kl λ ν F (Λ -1 (λ)x(k), Λ -1 (λ)x(k -m))) ∂z l ∂z s × mλ ν+r l Λ -1 (λ)x(k) + hλ ν F (Λ -1 (λ)x(k), Λ -1 (λ)x(k -m)) ν+r l + m i=1 |x(k -i + 1)| ν+r l r ≤ 2ma 5 h m p=0 |x(k -p)| 2ν+µ r n s=1 ∂ 2 V (Λ -1 (λ)x(k) + hθ 3kl λ ν F (Λ -1 (λ)x(k), Λ -1 (λ)x(k -m))) ∂z l ∂z s × Λ -1 (λ)x(k) + hλ ν F (Λ -1 (λ)x(k), Λ -1 (λ)x(k -m)) ν+r l + 1 .
Dene ω = max{|x(k + 1)| r , λ}, recall that λ = 0 and that due to imposed restrictions on F we get

n l=1 |x(k)| µ-r l r m i=1 |F l (x(k + 1), x(k -i + 1)) -F l (x(k), x(k -i + 1))| = n l=1 |x(k)| µ-r l r ω ν+r l m i=1 |F l (Λ -1 (ω)x(k + 1), Λ -1 (ω)x(k -i + 1)) -F l (Λ -1 (ω)x(k), Λ -1 (ω)x(k -i + 1))| = n l=1 |x(k)| µ-r l r ω ν+r l m i=1 |F l (Λ -1 (ω)x(k) + Λ -1 (ω)hF (x(k), x(k -m)), Λ -1 (ω)x(k -i + 1)) -F l (Λ -1 (ω)x(k), Λ -1 (ω)x(k -i + 1))| = n l=1 |x(k)| µ-r l r ω ν+r l m i=1 |F l (Λ -1 (ω)x(k) +hλ ν Λ -1 (ω)Λ(λ)F (Λ -1 (λ)x(k), Λ -1 (λ)x(k -m)), Λ -1 (ω)x(k -i + 1)) -F l (Λ -1 (ω)x(k), Λ -1 (ω)x(k -i + 1))| ≤ Lm n l=1 |x(k)| µ-r l r ω ν+r l ≤ Lm n l=1 |x(k)| µ-r l r max{|x(k) + hF (x(k), x(k -m))| ν+r l r , λ ν+r l } ≤ Lm n l=1 |x(k)| µ-r l r λ ν+r l |Λ -1 (λ)x(k) + hλ ν F (Λ -1 (λ)x(k), Λ -1 (λ)x(k -m))| ν+r l r + 1 ≤ Lm m p=0 |x(k -p)| ν+µ r n l=1 |Λ -1 (λ)x(k) + hλ ν F (Λ -1 (λ)x(k), Λ -1 (λ)x(k -m))| ν+r l r + 1
where the rst inequality is satised since Λ -1 (ω)x(k), Λ -1 (ω)x(k -i + 1) ∈ B r for any L > 0 and provided that

hλ ν |Λ -1 (ω)Λ(λ)F (Λ -1 (λ)x(k), Λ -1 (λ)x(k -m))| ≤ χ
for some χ > 0 using the uniform continuity of F from Assumption 1 (we select a gain L > 0 suciently small as we need, then we can nd χ > 0 corresponding to this L). The latter property is valid if

ha 5 λ ν ≤ χ. (5)
Therefore, combining all obtained estimates we conclude that

∆W ≤ -a 4 h|x(k)| µ+ν r + hβ 1 |x(k)| ρ r -hβ m |x(k -m)| ρ r + h m-1 j=1 (β j+1 -β j )|x(k -j)| ρ r +4a 2 5 h 2 m p=0 |x(k -p)| 2ν+µ r n l=1 n s=1 ∂ 2 V (Λ -1 (λ)x(k) + hθ 2kl λ ν F (Λ -1 (λ)x(k), Λ -1 (λ)x(k -m))) ∂z l ∂z s +2ma 2 5 h 2 m p=0 |x(k -p)| 2ν+µ r n l=1 n s=1 ∂ 2 V (Λ -1 (λ)x(k) + hθ 3kl λ ν F (Λ -1 (λ)x(k), Λ -1 (λ)x(k -m))) ∂z l ∂z s × Λ -1 (λ)x(k) + hλ ν F (Λ -1 (λ)x(k), Λ -1 (λ)x(k -m)) ν+r l + 1 +Lma 6 h m p=0 |x(k -p)| ν+µ r n l=1 |Λ -1 (λ)x(k) + hλ ν F (Λ -1 (λ)x(k), Λ -1 (λ)x(k -m))| ν+r l r + 1 .
Recall that Λ -1 (λ)x(k), Λ -1 (λ)x(k -m) ∈ B r , then the following upper estimate on the increment of W is derived: Call back that L can be chosen arbitrary small by decreasing the value of λ ν due to (5).

∆W ≤ -a 4 h|x(k)| µ+ν r + hβ 1 |x(k)| ρ r -hβ m |x(k -m)| ρ r + h m-1 j=1 (β j+1 -β j )|x(k -j)| ρ r +4n 2 a 2 5 h 2 g 1 (λ) m p=0 |x(k -p)| 2ν+µ r +2mna 2 5 h 2 g 1 (λ) m p=0 |x(k -p)| 2ν+µ

Stability results

Now we are in position to formulate the main results of this section.

Positive degree

Theorem 1. Let assumptions 1, 2 be valid with ν > 0, β j+1 < β j for all j = 1, . . . , m -1, β 1 ≤ a4

4 and ρ = µ + ν. Then for any h > 0 and integer m ≥ 0 there exists D > 0 such that for the Lyapunov-Krasovskii functional (3) and the system (1) the relations Proof. Note that µ ν rmax + 1 > ν + µ since we selected µ > r max , and ρ < µ ν rmax + 1 due to the selection of ρ. Thus, using estimates (4) we obtain that

a 1 2 |x(k)| µ r + h 2 m j=1 β j |x(k -j)| ν+µ r ≤ W ≤ (a 2 + 2a 3 hnm)|x(k)| µ
a 1 2 |x(k)| µ r + h 2 m j=1 β j |x(k -j)| ρ r ≤ W ≤ a 2 |x(k)| µ r + h m j=1 β j |x(k -j)| ρ r +2a 3 hnm m p=0 |x(k -p)| ν+µ r for x (k) ≤ D 1 with D 1 ≤ min 1≤j≤n        a 1 4a 3 hnm ν -1 ,   2na 3 h ρ-ν-r j ρ min{ a1 2m , β j }   µ -1 1- µ-r j µ - ν+r j ρ , h -ρ -1   2na 3 h ρ-ν-r j ρ min{ a1 2m , β j }   ρ -1 1- µ-r j µ - ν+r j ρ       
, where the computations for the upper estimate are straightforward (it holds without a restriction on x (k) ) and Lemma 4 was used for the lower one. Note that for ν > 0, for any D 2 > 0 the functions g 1 (λ) and g 2 (λ) admit an upper bound for all λ = x (k) ≤ D 2 and for all x (k) ≤ D 2 (and additionally L > 0 can be made arbitrary small by decreasing the value of D 2 using ( 5)), where g 3 (D 2 ) and g 4 (D 2 ) are the respective upper bounds for g 3 and g 4 . For ρ = ν + µ and any D 2 > 0,

g 1 (λ) ≥ a 7 = max 1≤l,s≤n sup φ∈Br ∂ 2 V (φ) ∂z l ∂z s , g 2 (λ) ≥ a 8 = sup φ∈Br |φ| , then a 9 ≤ g 3 (λ) ≤ g 3 (D 2 ), a 10 ≤ g 4 (λ) ≤ g 4 (D 2 ),
the estimate ( 6) can be rewritten as follows:

∆W ≤ - h 2   a 4 |x(k)| µ+ν r + β m |x(k -m)| ρ r + m-1 j=1 (β j -β j+1 )|x(k -j)| ρ r   for all x (k) ≤ D 2 , which is a solution of the inequality D 2 ≤ min{ a4 2 , β m , β 1 -β 2 , . . . , β m-1 -β m } 4hg 3 (D 2 ) ν -1
, and for

β 1 ≤ a 4 4 , g 4 (D 2 ) ≤ min{ a4 2 , β m , β 1 -β 2 , . . . , β m-1 -β m } 4 .
These inequalities always have a solution with respect to D 2 since for D 2 → 0 the right-hand side of the former stays separated from zero converging to

min{ a4 2 , β m , β 1 -β 2 , . . . , β m-1 -β m } 4ha 9 ν -1
, while g 4 (D 2 ) can be made as small as needed with D 2 → 0 recalling (5). Thus, with D = min{D 1 , D 2 } and = min{a 4 , β m , β 1 -β 2 , . . . , β m-1 -β m } the statements of the theorem are veried.

An approach to evaluate the values of D (the bound on domain of positive deniteness of W and negative deniteness of ∆W ) and (the convergence rate) is suggested in the proof of Theorem 1. Remark 1. As we can conclude from the proof of Theorem

1, if h → +∞ or m → +∞, then D → 0 (since 1 - µ-rj µ - ν+rj ρ
< 0), which corresponds to the results obtained in [START_REF] Emov | Weighted homogeneity for time-delay systems: Finite-time and independent of delay stability[END_REF][START_REF] Emov | Analysis of robustness of homogeneous systems with time delays using Lyapunov-Krasovskii functionals[END_REF] that with the growth of the delay, the domain of convergence is shrinking.

Remark 2. If we assume that F is χ -Hölder continuous on B r with constant L > 0 with respect to the rst argument, χ ∈ (0, 1], and for µ > (1 + χ )r max (see also Lemma 3), then the conclusion of Theorem 1 can be obtained for µ + ν < ρ < (1 + χ )ν + µ and without a restriction on β 1 .

Using the result of Lemma 1 the following time estimate on decay of solutions in (1) can be obtained:

Corollary 1. Let all conditions of Theorem 1 be valid, then there exist δ > 0 and α > 0 such that

W x (k) (x (0) ) ≤ W x (0) 1 + α ν µ W ν µ x (0) k -µ ν
for all k ≥ 0 and all x (0) ≤ δ.

Proof. Consider the Lyapunov-Krasovskii functional (3) constructed for the system (1) in Theorem 1. Since all conditions of the theorem are valid, there exists D > 0 such that for all x (k) ≤ D:

W 1+ ν µ ≤ c   |x(k)| µ+ν r + m j=1 |x(k -j)| (ν+µ) 2 µ r   ≤ c m p=0 |x(k -p)| ν+µ r , where c = (2m + 2) ν µ max{a 2 , h max 1≤j≤m β j } + 2a 3 hnm 1+ ν µ , c = c max{1, D ν+ ν 2 µ },
and Jensen's inequality was used on the rst step, while simple relation s η+ ≤ D s η , which is true for any s ∈ [0, D] and η, > 0, was utilized on the last one. Therefore,

W (x (k+1) ) ≤ W (x (k) ) -αW 1+ ν µ (x (k) )
,

where α = h 2c . Choose δ > 0 such that if x (0) ≤ δ, then x (k) (x (0) ) ≤ D for k ≥ 0 and α(1+ ν µ )W ν µ (x (0) ) ≤ 1,
and with the aid of Lemma 1 we obtain the required estimate.

Note that a1

2 |x(k)| µ r ≤ W (x (k) ), then the time rate of convergence of x(k) is of order k -ν -1
, which is the same as for the continuous-time delay-free system.

Negative degree

For this part we introduce an additional restriction on V and assume that µ > max{r max , 1 + r min }.

Theorem 2. Let assumptions 1, 2 be valid with -r min < ν < 0, β j+1 < β j for all j = 1, . . . , m -1, β 1 < a4 4 and ρ = µ + ν. Then for any h > 0 and integer m ≥ 0 there exists D > 0 such that for the Lyapunov-Krasovskii functional (3) and the system (1) the relations

a 1 2 |x(k)| µ r + h 2 m j=1 β j |x(k -j)| ν+µ r ≤ W ≤ (a 2 + 2a 3 hnm)|x(k)| µ r +h m j=1 (β j + 2a 3 nm)|x(k -j)| ν+µ r , ∆W ≤ - h 2 m p=0 |x(k -p)| ν+µ r , > 0 are satised for all x (k) ≥ D.
Proof. Note that µ ν rmax + 1 < ν + µ since we selected µ > r max and ν < 0, and ρ > µ ν rmax + 1 due to the selection of ρ. Thus, using (4) the upper estimate can be derived in the same way as in Theorem 1, and the lower estimate holds provided that

min{a 1 , β min } max{|x(k)| µ r , h m i=1 |x(k -i)| ρ r } ≥ 2a 3 h   nm|x(k)| ν+µ r + n j=1 |x(k)| µ-rj r m i=1 |x(k -i)| ν+rj r   ,
where β min = min{β 1 , . . . , β m }. Dening d 1 = 4a3h min{a1,βmin} this inequality can be decomposed on

max{|x(k)| µ r , h m i=1 |x(k -i)| ρ r } ≥ d 1 nm|x(k)| ν+µ r , max{|x(k)| µ r , h m i=1 |x(k -i)| ρ r } ≥ d 1 n j=1 |x(k)| µ-rj r m i=1 |x(k -i)| ν+rj r .
This former can be resolved directly, and Lemma 4 can be used for the latter since

m i=1 |x(k -i)| ν+rj r ≤ m 1-ν+r j ρ m i=1 |x(k -i)| ρ r ν+r j ρ
by applying Jensen's inequality for µ > r max . Hence, these inequalities are valid for

max{|x(k)| µ r , h m i=1 |x(k -i)| ρ r } ≥ max 1≤j≤n (d 1 nm) -ν -1 , 2d 1 nm(hm) -ν+r j ρ 1 1- µ-r j µ - ν+r j ρ , (d 1 nm) -µ ν ,
and there exists D 1 > 0 such that x (k) ≥ D 1 is included in the above domain. Note that for ν < 0, for any D 2 > 0 the functions g 1 (λ) and g 2 (λ) admit an upper bound for all λ = x (k) ≥ D 2 and

g 1 (λ) ≥ a 7 , g 2 (λ) ≥ a 8 , then a 9 ≤ g 3 (λ) ≤ g 3 (D 2 ), a 10 ≤ g 4 (λ) ≤ g 4 (D 2 ),
for all x (k) ≥ D 2 , where g 3 (D 2 ) and g 4 (D 2 ) are the respective upper bounds for g 3 and g 4 , and the constants a 7 , a 8 , a 9 and a 10 are dened in the proof of Theorem 1 (similarly, g 4 (D 2 ) is proportional to L that can be made arbitrary small by increasing the value of D 2 using ( 5)). For any D 2 > 0, the estimate ( 6) can be rewritten as follows:

∆W ≤ - h 2   a 4 |x(k)| µ+ν r + β m |x(k -m)| ρ r + m-1 j=1 (β j -β j+1 )|x(k -j)| ρ r   provided that 1 2   a 4 |x(k)| µ+ν r + β m |x(k -m)| ρ r + m-1 j=1 (β j -β j+1 )|x(k -j)| ρ r   ≥ β 1 |x(k)| ρ r +hg 3 (D 2 ) m p=0 |x(k -p)| 2ν+µ r + g 4 (D 2 ) m p=0 |x(k -p)| ν+µ r .
Since ρ = ν + µ, verifying the terms with the same powers leads to:

β 1 + g 4 (D 2 ) ≤ a 4 4 , g 4 (D 2 ) ≤ min{β m , β 1 -β 2 , . . . , β m-1 -β m } 4 ,
which can be ensured by increasing the value of D 2 as it was explained above. Then the last property to

establish is m p=0 |x(k -j)| ν+µ r ≥ d 2 m p=0 |x(k -p)| 2ν+µ r for d 2 = 4hg 3 (D 2 ) min{a 4 , β m , β 1 -β 2 , . . . , β m-1 -β m } .
Applying Jensen's inequality we get

m p=0 |x(k -p)| 2ν+µ r ≤ (m + 1) -ν ν+µ m p=0 |x(k -j)| ν+µ r 2ν+µ ν+µ due to ν + µ ≥ 1 for µ > 1 + r min , then m p=0 |x(k -j)| ν+µ r ≥ (m + 1)d -ν+µ ν 2
is the condition to check, and using again Jensen's inequality we can show that there exists D 2 > 0 such that the above relation is satised for x (k) ≥ D 2 (as in the proof of Theorem 1, increasing the value of D 2 will push g 3 (D 2 ) to its lower bound a 9 ). Thus, with D = max{D 1 , D 2 } and = min{a 4 , β m , β 1 -β 2 , . . . , β m-1 -β m } the statements of the theorem are veried.

Since as in theorems 1 or 2, the analysis will be performed either for x (k) ≤ D (ν > 0) or for x (k) ≥ D (ν < 0), then the above hypothesis we will not need globally, and this formulation is given for all R n(m+1) for brevity only.

Remark 5. Under Assumption 3, the system (7) admits homogeneous approximations at the origin or at innity in the form of (1) for σ > ν > 0 or σ < ν < 0, respectively. These scenarios will be analyzed in the sequel of the section.

To study the stability of ( 7), the Lyapunov-Krasovskii functional (3) can again be applied. The lower and upper estimates (4) save their meaning. Hence, consider the increment of the functional W along the solutions of ( 7):

∆W = h ∂V (x(k) + θ 1k ∆x(k)) ∂z T (F (x(k), x(k -m)) + Q(k, x (k) )) +h m j=1 β j |x(k -j + 1)| ρ r -h m j=1 β j |x(k -j)| ρ r +h ∂V (x(k) + ∆x(k)) ∂z T m i=1 F (x(k + 1), x(k -i + 1)) -h ∂V (x(k)) ∂z T m i=1 F (x(k), x(k -i)) = h ∂V (x(k)) ∂z T F (x(k), x(k)) + h ∂V (x(k)) ∂z T Q(k, x (k) ) +h ∂V (x(k) + θ 1k ∆x(k)) ∂z - ∂V (x(k)) ∂z T (F (x(k), x(k -m)) + Q(k, x (k) )) +hβ 1 |x(k)| ρ r -hβ m |x(k -m)| ρ r + h m-1 j=1 (β j+1 -β j )|x(k -j)| ρ r +h ∂V (x(k) + ∆x(k)) ∂z - ∂V (x(k)) ∂z T m i=1 F (x(k + 1), x(k -i + 1)) +h ∂V (x(k)) ∂z T m i=1 F (x(k + 1), x(k -i + 1)) - m i=1 F (x(k), x(k -i + 1)) ,
where ∆x(k) = h F (x(k), x(k -m)) + Q(k, x (k) ) , θ 1k ∈ (0, 1) and we used the same tricks as before. Applying Assumption 3 and since (as previously, denote λ = x (k) for brevity) Repeating the same operations as in the previous section and using the equality

∂V (x(k)) ∂z T Q(k, x (k) ) ≤ a 6 c n i=1 |x(k)| µ-ri
Q(k, x (k) ) = λ σ Λ(λ)ς k with ς k ∈ Γ = {z ∈ R n : |z| ≤ (m + 1)c} we derive: ∂V (x(k) + θ 1k ∆x(k)) ∂z l - ∂V (x(k)) ∂z l (|x(k)| ν+r l r + |x(k -m)| ν+r l r + m p=0 |x(k -p)| σ+r l r ) ≤ hw 1 m p=0 |x(k -p)| 2ν+µ r + m p=0 |x(k -p)| σ+ν+µ r + m p=0 |x(k -p)| 2σ+µ r × n s=1 ∂ 2 V (Λ -1 (λ)x(k) + hθ 2kl [λ ν F (Λ -1 (λ)x(k), Λ -1 (λ)x(k -m)) + λ σ ς k ]) ∂z l ∂z s , ∂V (x(k) + ∆x(k)) ∂z l - ∂V (x(k)) ∂z l m i=1 (|x(k + 1)| ν+r l r + |x(k -i + 1)| ν+r l r ) ≤ m max{a 5 , c}h 2 m p=0 |x(k -p)| 2ν+µ r + (m + 1) m p=0 |x(k -p)| σ+ν+µ r × n s=1 ∂ 2 V (Λ -1 (λ)x(k) + hθ 3kl [λ ν F (Λ -1 (λ)x(k), Λ -1 (λ)x(k -m)) + λ σ ς k ]) ∂z l ∂z s × Λ -1 (λ)x(k) + h[λ ν F (Λ -1 (λ)x(k), Λ -1 (λ)x(k -m)) + λ σ ς k ] ν+r l + 1 , n l=1 |x(k)| µ-r l r m i=1 |F l (x(k + 1), x(k -i + 1)) -F l (x(k), x(k -i + 1))| ≤ Lm m p=0 |x(k -p)| ν+µ r n l=1 |Λ -1 (λ)x(k) + h[λ ν F (Λ -1 (λ)x(k), Λ -1 (λ)x(k -m)) + λ σ ς]| ν+r l r + 1 ,
where θ 2kl , θ 3kl ∈ (0, 1), w 1 = max{a 5 , c} max{4, 4(m + 1), (m + 1) 2 }, and for any L > 0 the latter inequality is satised under the constraint

h[a 5 λ ν + λ σ (m + 1)c] ≤ χ (8)
with some χ > 0 due to uniform continuity of F from Assumption 1. Combining with the estimates calculated in the previous section we conclude: (λ) + 1 g 5,l (λ),

∆W ≤ -a 4 h|x(k)| µ+ν r + hβ 1 |x(k)| ρ r -hβ m |x(k -m)| ρ r + h m-1 j=1 (β j+1 -β j )|x(k -j)| ρ r (9) +ha 6 c(m + 1)n m p=0 |x(k -p)| σ+µ r + hg 9 (λ) m p=0 |x(k -p)| ν+µ r + h 2 (g 7 (λ) + 2g 8 (λ)) m p=0 |x(k -p)| 2ν+µ r +h 2 (g 7 (λ) + (m + 1)g 8 (λ)) m p=0 |x(k -p)| σ+ν+µ r + h 2 g 7 (λ) m p=0 |x(k -p)| 2σ+µ r , where g 5,l (λ) = sup φ,ψ∈Br sup ς∈Γ n s=1 ∂ 2 V (φ + h[λ ν F (φ, ψ) + λ σ ς]) ∂z l ∂z s , g 6 (λ) = sup φ,ψ∈Br sup ς∈Γ |φ + h[λ ν F (φ, ψ) + λ σ ς k ]| , g 7
g 9 (λ) = La 6 m n l=1 σ -1 r (g 6 (λ)) ν+r l + 1 .
Thus, in [START_REF] Gielen | Lyapunov methods for time-invariant delay dierence inclusions[END_REF] with respect to (6), there are three additional terms containing the powers proportional to σ (their appearance is related with the disturbing term Q).

Theorem 3. Let assumptions 1, 2 and 3 be valid for σ > ν > 0. Then for any h > 0 and integer m ≥ 0 there exists D > 0 such that the zero solution of ( 7) is asymptotically stable for all x (0) ≤ D.

Proof. The proof of this result, rst, follows the steps of the proof of Theorem 1 to establish the same properties of W with β j+1 < β j for all j = 1, . . . , m -1, β 1 ≤ a4 4 and ρ = µ + ν (the terms containing the powers with σ, since σ > ν, can be treated as ones with µ + 2ν previously). Second, the result of Corollary 1 can be used to prove asymptotic convergence to the origin. Theorem 4. Let assumptions 1, 2 and 3 be valid for -r min ≤ σ < ν < 0. Then for any h > 0 and integer m ≥ 0 there exists D > 0 such that all trajectories of (7) with x (0) ≥ D converge to the set {ξ ∈ R n(m+1) : ξ ≤ D}.

Proof. The proof is based on application of the arguments used in the proof of Theorem 2 (for W with β j+1 < β j for all j = 1, . . . , m -1, β 1 < a4 4 and ρ = µ + ν) and on Corollary 2.

Simulations

Consider a double integrator nominal system:

ẋ1 (t) = x 2 (t) + Q 1 (t, x(t), x(t -h), . . . , x(t -mh)), ẋ2 (t) = u(t) + Q 2 (t, x(t), x(t -h), . . . , x(t -mh)), where x 1 (t), x 2 (t) ∈ R, x(t) = [x 1 (t), x 2 (t)] , u(t) ∈ R is the control, t ≥ 0, Q 1 , Q 2 : R m+2 → R are the disturbances.
This system can be stabilized by a feedback [START_REF] Bhat | Geometric homogeneity with applications to nite-time stability[END_REF]:

u(t) = -k 1 |x 1 (t -mh)| 1+2ν sign(x 1 (t -mh)) -k 2 |x 2 (t -mh)| 1+2ν 1+ν sign(x 2 (t -mh)),
where k 1 > 0 and k 2 > 0 are the control gains, ν ∈ (-0.5, 0) ∪ (0, +∞) is a tuning parameter. In the delay-free case with m = 0, and without perturbations Q 1 and Q 2 , this control makes the closed-loop system rhomogeneous of degree ν with the weights r = [1, 1+ν] ,

the assumptions 1 and 2 are satised (the control is continuous for ν > -0.5), and discretization of this controlled dynamics leads to the system of interest [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF].

For simulations let To apply our results it is enough to check assumptions 1 and 2, which are essential. Indeed, if we drop Assumption 1 by considering in the above system u(t) = -k 1 |x 1 (t -mh)| 1+2ν sign(x 1 (t -mh)) -k 2 |x 2 (t -mh)| 1+2ν 1+3ν sign(x 2 (t -mh)),

where the power for the variable x 2 was slightly modied, ν > -1

3 , then the closed loop dynamics is still globally asymptotically stable with m = 0 and Q 1 (•) = Q 2 (•) = 0 (to verify this claim, a Lyapunov function 2 can be utilized together with the LaSalle invariance principle). However, for ν = -0.2 and the rest values of parameters being unchanged, the system (7) admits a locally asymptotically stable oscillating trajectory with unbounded solutions for suciently large initial conditions, see Fig. 3. Similar unstable behavior can be obtained by considering a homogeneous system with ν = 0 (the scenario not studied in this work) and a big enough discretization step h, since it is well-known that for linear systems the explicit Euler discretization method can be applied for a suciently small h only.

Conclusion

The problem of stability analysis for a discrete-time time-delay system, which is derived from Euler discretization of homogeneous continuous-time model, was studied by applying the Lyapunov-Krasovskii theory. A generic structure of the Lyapunov-Krasovskii functional was proposed that can be utilized for any homogeneous system of non-zero degree. An extension of this results to the class of dynamics admitting a homogeneous approximation was developed. The obtained stability conditions were illustrated by simulation of a discretized delayed locally homogeneous planar system of negative and positive degrees. The presented results demonstrate that the explicit Euler method preserves stability properties for the selected class of homogeneous systems with non-zero degree for any value of the discretization step. Extension of this approach to the case of innite dimensional systems with m → +∞ can be a direction of future research.

  y) ∂y T F (y, y), a 5 = sup φ,ψ∈Br |F (φ, ψ)| , a 6 = sup y∈Sr ∂V (y) ∂y .

∂ 2 V

 2 (φ + hλ ν F (φ, ψ)) ∂z l ∂z s , g 2 (λ) = sup φ,ψ∈Br |φ + hλ ν F (φ, ψ)| ; and nally ∆W ≤ -a 4 h|x(k)| µ+ν r + hβ 1 |x(k)| ρ r -hβ m |x(k -m)| ρ r -β j )|x(k -j)| ρ r +h 2 g 3 (λ)

,

  + 2a 3 nm)|x(k -j)| ν+µ r all x (k) ≤ D.

λ+ h m- 1 j=1(∂V

 1 σ+ri ≤ a 6 c(m + 1)n m p=0 |x(k -p)| σ+µ r we obtain:∆W ≤ -a 4 h|x(k)| µ+ν r + hβ 1 |x(k)| ρ r -hβ m |x(k -m)| ρ r β j+1 -β j )|x(k -j)| (x(k) + ∆x(k)) ∂z l -∂V (x(k)) ∂z l m i=1 (|x(k + 1)| ν+r l r + |x(k -i + 1)| ν+r l r ) +ha 6 n l=1 |x(k)| µ-r l r m i=1|F l (x(k + 1), x(k -i + 1)) -F l (x(k), x(k -i + 1))| .

  (λ) = max{a 5 , c}w 1 n l=1 g 5,l (λ), g 8 (λ) = a 5 m max{a 5 , c} n l=1 g ν+r l 6

  (then we select |x| r = |x 1 |+|x 2 | 1 1+ν), and the system is globally nite-time stable for ν ∈ (-0.5, 0) or globally nearly xed-time stable for ν > 0. Appearance of the delay m > 0 models the information measurement and processing lags in the control feedback. Obviously,F (φ, ψ) = φ 2 -k 1 |ψ 1 | 1+2ν sign(ψ 1 ) -k 2 |ψ 2 | 1+2ν 1+ν sign(ψ 2 )

k 1 = 1 , 1

 111 k 2 = 2, h = 0.01, m = 10, (k, x (k) ) = c sin(kh)m p=0 |x(k -p)| σ+r1 r , Q 2 (k, x (k) ) = c cos(2kh) a choice of Q 1 and Q 2 Assumption 3 is valid.Thus, all conditions of theorems 3, 4 are veried. For ν = -0.2 and c = 0.1 the behavior of the norm |x(k)| r for three dierent initial conditions (they are assumed to be constant for negative values of the time) is presented in Fig.1in logarithmic scale. For ν = 0.2 and c = 10 -2 the respective behavior of |x(k)| r is given in Fig.2. As we can see from these plots, the convergence of the trajectories is faster than exponential (nite-time) in Fig.1and slower in Fig.2, which conrms the results of corollaries 1, 2.

Figure 1 :

 1 Figure 1: Trajectories of |x(k)| r for ν < 0

Figure 2 :

 2 Figure 2: Trajectories of |x(k)| r for ν > 0

Figure 3 :

 3 Figure 3: Trajectories of |x(k)| r for non-homogeneous case

  Emov is with Univ. Lille, Inria, CNRS, UMR 9189 -CRIStAL, F-59000 Lille, France and ITMO University, 49 av. Kronverkskiy, 197101 Saint Petersburg, Russia. A. Aleksandrov is with Saint Petersburg State University, 7-9 Universitetskaya nab., 199034 Saint Petersburg, Russia. This work was partially supported by the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898, and by the Russian Foundation for Basic Research (grant no. 19-01-00146-a). extension to locally homogeneous systems is developed in Section 4. An illustrative example is considered in

	Section 5.

Remark 3. For h → +∞ or m → +∞ the constants d 1 and d 2 , whose values are given in the proof of Theorem 2, will also grow to innity implying the same property of D (it is a function of d 1 and d 2 ). This result for the continuous-time systems was established in [START_REF] Zimenko | A note on delay robustness for homogeneous systems with negative degree[END_REF][START_REF] Emov | Analysis of robustness of homogeneous systems with time delays using Lyapunov-Krasovskii functionals[END_REF]. Similarly to the continuous-time case, in theorems 1 or 2, h → 0 implies the growth of D to innity (ν > 0) or its decay to zero (ν < 0). Corollary 2. Let all conditions of Theorem 2 be valid, then there exist positive numbers δ 1 , δ 2 and α such that

for all k ≥ 0 and all x (0) ≥ δ 1 .

Proof. Consider the Lyapunov-Krasovskii functional (3) constructed for the system (1) in Theorem 2. Since all conditions of the theorem are valid, there exists D > 0 such that for all x (k) ≥ D:

and inequality s η-≤ D -s η was used in the second case, which is true for any s ≥ D and η, > 0. Therefore,

,

then with the aid of Lemma 2 we obtain the estimate

. From another side, W (x (k+1) ) ≤ W (x (k) ) while W (x (k) ) > c(m + 1) max{D µ , D ν+µ } = δ 2 , which leads to the proposed estimate. Remark 4. The obtained results in this section can be extended to a homogeneous system with multiple delays:

Due to properties of the Lyapunov-Krasovskii functional (3), the results of corollaries 1 and 2 imply asymptotic stability of the zero solution of (1) with the domain of attraction x (0) ≤ δ and convergence of all trajectories of (1) with initial conditions in {ξ ∈ R n(m+1

Perturbed System

Consider a perturbed version of the system (1):

where Q : R + × R n(m+1) → R n is a locally bounded function, which satises the condition: Assumption 3. For all x (k) ∈ R n(m+1) , k ≥ 0 and i = 1, . . . , n,


Joonas Rokka
email: rokka@em-lyon.com

Consumer Culture Theory's Future in Marketing

Keywords: Consumer Culture Theory, CCT, marketing, branding

This commentary offers a view into the contributions of Consumer Culture Theory (CCT) in marketing and charts promising future avenues for research and marketing practices building a culturally sensitive and reflexive approach. After highlighting pioneering CCT perspectives, an outline for future directions in marketing is offered emphasizing the assembling of experiences, shaping of brands' symbolic universes, institutional and creative market processes, and networked and algorithmic mediation of consumption ideologies and desires. Overall, CCT's future looks promising in its commitment and ability to foster critical, contextually sensitive, and reflexive cultural insights into marketing -an important foundation for marketing strategy and practices.

INTRODUCTION

"Every so often there comes along a new symbol, one that makes a leap from the past into the present and that has power because it captures the spirit of the present and makes other on-going symbols old-fashioned. -Sidney J. [START_REF] Levy | Symbols for Sale[END_REF] Since the early days of marketing thought, scholars have recognized the power of brands as symbols that carry socially resonant cultural ingredients, including building blocks for consumers' identity construction, social affiliation, meaning-making, and emotional life more broadly. As observed and underlined early on by Sidney J. [START_REF] Levy | Symbols for Sale[END_REF] "Symbols for Sale" essay in Harvard Business Review, the idea of culture as a constantly evolving system of meanings is crucial, for example, for understanding how brands are used by consumers to express taste, lifestyle ideals, class, and gender within their specific contexts. This insight was soon echoed and distilled by several pioneering scholars seeking to explore and develop alternative understandings of marketing and consumer behavior, notably, by problematizing prior economic, individualist, behaviorist, and information-processing foundations of the discipline (e.g. [START_REF] Arnould | River magic: Extraordinary experience and the extended service encounter[END_REF][START_REF] Arnould | Market-oriented Ethnography: Interpretation Building and Marketing Strategy Formulation[END_REF][START_REF] Belk | Possessions and the Extended Self[END_REF][START_REF] Brown | Postmodern Marketing[END_REF][START_REF] Cova | Community and consumption: Towards a definition of the "linking value" of products or services[END_REF][START_REF] Firat | Liberatory Postmodernism and the Reenchantment of Consumption[END_REF][START_REF] Fournier | Consumers and Their Brands: Developing Relationship Theory in Consumer Research[END_REF][START_REF] Holbrook | The Experiential Aspects of Consumption: Consumer Fantasies, Feelings, and Fun[END_REF][START_REF] Mick | Consumer Research and Semiotics: Exploring the Morphology of Signs, Symbols, and Significance[END_REF][START_REF] Peñaloza | Atravesando Fronteras/Border Crossings: A Critical Ethnographic Exploration of the Consumer Acculturation of Mexican Immigrants[END_REF][START_REF] Schouten | Subcultures of Consumption: An Ethnography of the New Bikers[END_REF][START_REF] Scott | The Bridge from Text to Mind: Adapting Reader-Response Theory to Consumer Research[END_REF][START_REF] Sherry | Servicescapes: the concept of place in contemporary markets[END_REF][START_REF] Thompson | Putting Consumer Experience Back into Consumer Research: The Philosophy and Method of Existential-Phenomenology[END_REF].

Following this creative disruption, a movement of culturally-oriented marketing and consumer research was set forth drawing inspiration from a variety of interdisciplinary perspectives and methodologies from the humanities and social sciences -including sociology, anthropology, semiotics, cultural studies, literature, and historical studies -in other words, fields less familiar to marketing scholars and business practitioners. Fifteen years ago the movement and community of scholars become institutionalized under its own symbol CCT -Consumer Culture Theory -as suggested by Eric Arnould and Craig Thompson (2005;2018), and has since been qualified and classified as a thriving scholarly label, also in the discipline's leading journals, such as Journal of Marketing and Journal of Consumer Research.

The objective of this commentary is, first, to explain how CCT's research orientation, perspectives, and programs have gradually evolved and focused on understanding current marketing and consumer behavior phenomena in the recent years (for insightful reviews, Arnould [START_REF] Arnould | Consumer Culture Theory[END_REF][START_REF] Askegaard | Towards an Epistemology of Consumer Culture Theory: Phenomenology and the Context of Context[END_REF][START_REF] Askegaard | Consumer Culture Theory: The Ironies of History[END_REF][START_REF] Moisander | From CCT to CCC: Building Consumer Culture Community[END_REF][START_REF] Thompson | Discursivity, Difference, and Disruption: Genealogical Reflections on the CCT Heteroglossia[END_REF]. Second, while only few articles to date have cast a light explicitly on CCT's influence in marketing theory and practice, to advance the field, I will sketch an outline of relevant and promising future directions.

CCT's unique conceptual and methodological tools beneficial for marketing scholars, practitioners -and for marketing students -have been discussed and disseminated in a range of influential and pedagogical marketing textbooks and popularized chapters (e.g. [START_REF] Belk | Qualitative Consumer & Marketing Research[END_REF][START_REF] Bode | Marketing and consumer research: An uneasy relationship[END_REF][START_REF] Holt | How Brands Become Icons: The Principles of Cultural Branding[END_REF][START_REF] Kozinets | Netnography: Doing ethnographic research online[END_REF][START_REF] Moisander | Qualitative Marketing Research: A Cultural Approach[END_REF]Schroeder & Salzer-Mörling, 2006;Visconti et al., 2020;Zwick & Cayla, 2011). Yet, more work is still needed to systematize these efforts. As many have argued, much of the contemporary marketing work and practices remain "largely a mystery" (Zwick & Cayla, 2011, p 3.), as textbooks have tended to treat marketing rather on an abstract level and providing formal descriptions of marketing processes. One area also missing is detailed accounts of CCT inspired marketing practices addressing focal marketing problems -not to mention the grass-roots level collaborations and exchanges with companies that CCT scholars have done over the past 25 years. Towards this end, I wish to conclude with a brief articulation of what kind of managerial knowledge and reflexivity CCT can offer for the future marketing practitioner.

CCT'S ORIGINS AND INFLUENCES AS A SUBDISCIPLINE

While broader history of the Consumer Culture Theory (CCT) tradition is beyond the scope of this short commentary, it is important to situate and understand the paradigmatic landscape in which CCT was first articulated. CCT emerged as a subdiscipline into the broader consumer research tradition which, on the other hand, was born as a subdiscipline of marketing at the end of 1960s [START_REF] Bode | Marketing and consumer research: An uneasy relationship[END_REF]). These shifts were facilitated in particular by the urgent need to widen the unnecessarily narrow perspective on consumer behavior that had prevailed until the 1970s and beyond -one of company's and marketers' perspective, in turn informed by microeconomic and information processing theories [START_REF] Bode | Marketing and consumer research: An uneasy relationship[END_REF]Zwick & Cayla, 2011). Equally shaping the marketing and consumer research landscape since its inception were the ideological pressures in mainstream social sciences toward a logical empiricist epistemology, with an emphasis on quantification, and specification of causal relationships [START_REF] Tadajewski | Remembering Motivation Research: Toward an Alternative Genealogy of Interpretive Consumer Research[END_REF].

These had an important impact on how consumer research was being conducted and theorizedas witnessed not least by Sidney J. Levy who already early on sought to introduce novel perspectives in this regard, but often in vain.

The "cultural turn" in social sciences in the early 1980s influenced several key scholars in marketing and consumer research and finally opened the field to adopt and experiment with interpretive, sociological, qualitative, and naturalistic approaches and methods. These CCT pioneers demonstrated the need for understanding consumer behavior as embedded in cultural and social relations, and as influenced by complex symbolic, material, experiential, and emotional worlds (e.g. [START_REF] Belk | The Sacred and Profane in Consumer Behavior: Theodicy on the Odyssey[END_REF][START_REF] Belk | The Fire of Desire: A Multi-Sited Inquiry into Consumer Passion[END_REF][START_REF] Holbrook | The Experiential Aspects of Consumption: Consumer Fantasies, Feelings, and Fun[END_REF][START_REF] Levy | Interpreting Consumer Mythology: A Structural Approach to Consumer Behavior[END_REF][START_REF] Mick | Consumer Research and Semiotics: Exploring the Morphology of Signs, Symbols, and Significance[END_REF][START_REF] Thompson | Putting Consumer Experience Back into Consumer Research: The Philosophy and Method of Existential-Phenomenology[END_REF]. These early contributions carved the intellectual space and theoretical-methodological ramifications that continue to inspire CCT research.

Yet it was not until 2005 that Arnould and Thompson outlined CCT initiative as a scholarly brand and as a framework to comprehend the characteristic aims, topical focus, and ways of investigation of the heterogeneous collective of researchers that it touches. Four heuristic research programs useful in outlining and classifying CCT-oriented work were suggested, which I will briefly describe in the following. Common threads in these programs were both their orientation towards addressing similar kinds of questions and dynamics in consumer culture phenomena -so call "family resemblances" between related studies -but also how these programs resonate with broader interdisciplinary interests and conversations [START_REF] Arnould | Consumer Culture Theory[END_REF].

First program is the exploration of Consumer Identity Projects. This includes investigations into how consumers -as active rather than passive market actors -integrate and enact various cultural resources, including brands' symbols, myths, and ideologies into their own identity work in negotiating a sense of self and narratives of meaningful identity goals.

Characteristic to this stream of work has been to illuminate the inherent complexity, tensions, and constraints in consumers' identity negotiation -for instance, in relation to gender, ethnicity, age, class, status, environmental, or financial pressures [START_REF] Arsel | Demythologizing Consumption Practices: How Consumers Protect their Field-Dependent Capital from Devaluing Marketplace Myths[END_REF][START_REF] Holt | Man-of-Action Heroes: The Pursuit of Heroic Masculinity in Everyday Consumption[END_REF][START_REF] Luedicke | Consumer Identity Work as Moral Protagonism: How Myth and Ideology Animate a Brand-Mediated Moral Conflict[END_REF][START_REF] Peñaloza | Atravesando Fronteras/Border Crossings: A Critical Ethnographic Exploration of the Consumer Acculturation of Mexican Immigrants[END_REF][START_REF] Peñaloza | Living US Capitalism: The Normalization of Credit/Debt[END_REF][START_REF] Thompson | Speaking of Fashion: Consumers' Uses of Fashion Discourses and the Appropriation of Countervailing Cultural Meanings[END_REF]. One concrete implication of this work for marketing is the shift in thinking about consumers as producers of cultural meanings (relevant to their unfolding identity projects), and consequently that neither brands nor consumers are entirely in control of the cultural meanings being produced [START_REF] Fournier | Consumers and Their Brands: Developing Relationship Theory in Consumer Research[END_REF]Schroeder & Salzer-Mörling, 2006). Therefore, as highlighted convincingly by [START_REF] Holt | How Brands Become Icons: The Principles of Cultural Branding[END_REF], in order to be culturally relevant brands need to carefully analyze how they can help consumers' identity work by offering them resonant identity myths -that is, sets of compelling stories and narratives -useful in addressing and negotiating their desires and anxieties, for instance about femininity or masculinity [START_REF] Holt | Man-of-Action Heroes: The Pursuit of Heroic Masculinity in Everyday Consumption[END_REF].

A second focus has been examinations of Marketplace Cultures, that is, the influences of immediate social-cultural contexts and material environments in which consumption takes place. This work highlights above all how grassroot-level cultural formations of often small-scale collectives, networks, or communities partake in the co-creation, negotiation, and diffusion of distinct market and consumption practices, performances, and meanings. Here, CCT's role has been important in conceptualizing and theorizing the notions of brand community [START_REF] Muñiz | Brand Communities[END_REF][START_REF] Schau | How Brand Community Practices Create Value[END_REF], sub-cultures [START_REF] Schouten | Subcultures of Consumption: An Ethnography of the New Bikers[END_REF], consumer tribes [START_REF] Cova | Community and consumption: Towards a definition of the "linking value" of products or services[END_REF], online communities [START_REF] Kozinets | Netnography: Doing ethnographic research online[END_REF], and co-creative market practices that shape new market formation and evolution [START_REF] Giesler | Conflict and Compromise: Drama in Marketplace Evolution[END_REF][START_REF] Hietanen | Market practices in countercultural market emergence[END_REF]. On the other hand, this stream has highlighted how specific marketplace resources and myths become central features of "collectively shared identities" [START_REF] Peñaloza | Consuming the American West: Animating Cultural Meaning at a Stock Show and Rodeo[END_REF][START_REF] Thompson | Marketplace Mythology and Discourses of Power[END_REF], and how they are ritualized, materialized, and expressed in different consumption spaces and environments [START_REF] Canniford | Purifying Practices: How Consumers Assemble Romantic Experiences of Nature[END_REF][START_REF] Sherry | Servicescapes: the concept of place in contemporary markets[END_REF][START_REF] Stevens | An embodied approach to consumer experiences: the Hollister brandscape[END_REF]. This includes the study of how various market actors including so called "influencers" mediate taste and aesthetic appreciation of consumers [START_REF] Arsel | Taste Regimes and Market-Mediated Practice[END_REF][START_REF] Pomiès | Taste. Consumption Markets & Culture[END_REF]. Crucial marketing implications of this stream overall underline how culture is not an external influence factor or variable that may or may not impact consumption and marketing practices, but is always inseparable from consumption and marketing practices. As explained well by Visconti et al. (2020), it thus becomes misleading to speak of consumption and marketing as an "effect" of culture, even if it has been considered so in more conventional cross-cultural marketing approach. Instead marketing strategies actively -albeit often implicitly -draw from and co-create cultural marketplace resources (e.g. stories, myths, identities, lifestyle ideals) as they seek to engage and interact with consumers.

Third CCT research program studies the Sociohistoric Patterning of Consumption, highlighting especially how different (macro-level) structural and historical influences pattern consumption and identities available for consumers. These accounts are specifically valuable in that they problematize the "agentic" consumer subject. Influential in this work has been Bourdieu's theories that propose social class and cultural capital can largely explain the distribution of tastes and consumption practices (e.g. [START_REF] Holt | Does cultural capital structure American consumption[END_REF]. This CCT stream has inspired scholars to examine notably the relations of power, dominance, marginalization, and social distinction -for example, of class, gender, ethnicity [START_REF] Crockett | Paths to Respectability: Consumption and Stigma Management in the Contemporary Black Middle Class[END_REF][START_REF] Crockett | The Role of Normative Political Ideology in Consumer Behavior[END_REF][START_REF] Peñaloza | We're Here, We're Queer, and We're Going Shopping! A Critical Perspective on the Accommodation of Gays and Lesbians in the U.S. Marketplace[END_REF][START_REF] Schroeder | Marketing images of gender: A visual analysis[END_REF][START_REF] Üstüner | Toward a Theory of Status Consumption in Less Developed Countries[END_REF] -reproduced in consumer culture, not least by marketers. Other related works have advanced our understandings of how consumers' engagement in consumption practices is socially (and historically) rather than individually shaped and established [START_REF] Holt | Poststructuralist Lifestyle Analysis: Conceptualizing the Social Patterning of Consumption[END_REF][START_REF] Karababa | Early Modern Ottoman Coffeehouse Culture and the Formation of the Consumer Subject[END_REF]. Key marketing insights drawn from this are important, for instance, toward explaining how innovative products proliferate [START_REF] Sandicki | Veiling in Style: How Does a Stigmatized Practice Become Fashionable[END_REF][START_REF] Shove | Consumers, Producers and Practices: Understanding the invention and reinvention of Nordic walking[END_REF] and how value is created through socially shared practices [START_REF] Schau | How Brand Community Practices Create Value[END_REF], and how these can also be influenced [START_REF] Maciel | Taste Engineering: An Extended Consumer Model of Cultural Competence Constitution[END_REF].

Finally, the fourth CCT program focuses on Mass-Mediated Marketplace Ideologies and

Consumers' Interpretive Strategies. Considering the high-speed circulation of news, advertising, and entertainment not least spurred by digital platforms, CCT has sought to unpack the ways in which consumers read, make sense, "decode", respond, and sometimes resist marketing messages [START_REF] Scott | The Bridge from Text to Mind: Adapting Reader-Response Theory to Consumer Research[END_REF]. Building on critical media theories [START_REF] Murray | The Critical Imagination: Emancipatory Interests in Consumer Research[END_REF], these scholars have also examined how consumer ideologies proliferate through advertising and other media. On the one hand, the studies highlight consumers as active "interpretive agents" who may formulate criticism, perform resistance, boycott, and even at times attack brands [START_REF] Holt | Why Brands Cause Trouble? A Dialectical Theory of Consumer Culture and Branding[END_REF]. On the other hand, as highlighted by Askegaard and Linnet (2012), a key strength of CCT theorizing has been the recognition of how both the macro-social (structural) and micro-social (individual) contexts are simultaneously at play and feed into each other. This means that while consumers may appear as individual and autonomous agents as they go about constructing personalized narratives about their consumption, they at the same time tacitly embrace and repeat consumption ideologies, cultural narratives and scripts circulating in media representations of identity and lifestyle ideals -for instance, about "what is a good life". CCT work has notably paid attention to the negotiation of ideological influences of globalization processes [START_REF] Thompson | The Starbucks Brandscape and Consumers' (Anticorporate) Experiences of Glocalization[END_REF][START_REF] Kjeldgaard | The Glocalization of Youth Culture: The Global Youth Segment as Structures of Common Difference[END_REF] as well as localized systems of meaning, not least within branded servicescapes. In these ways, CCT has contributed to the long-standing marketing debates, including the standardization or localization of marketing and communication strategies.

CCT -FUTURE DIRECTIONS IN MARKETING

Given that several long-term technological, sociocultural, and geopolitical developments are necessarily shaping the future of marketing (for a recent review; Rust, 2020) -especially about what marketing will be in the future -CCT is particularly well placed in addressing these ongoing challenges. The inherent interdisciplinary foundations of CCT work will undoubtedly be an asset in this regard. To explore relevant and promising pathways for CCT in marketing, the following section highlights key research areas that call for future attention: i) assembling of experiences, ii) shaping of brands' symbolic universes, iii) institutional and creative market processes, and iv) networked and algorithmic mediation of consumption ideologies and desires.

The listing, which is also summarized in Table 1, is by no means exhaustive but an attempt towards articulating and structuring key tendencies and agenda readily visible across numerous ongoing CCT writings, exchanges, forums, and seminars.

Assembling of Experiences

While customer experience and journey have been highlighted among the most important marketing objectives (e.g. [START_REF] Lemon | Understanding Customer Experience Throughout the Customer Journey[END_REF], it is striking that that relatively little is still known about how consumers and marketers assemble these experiences as informed by social and cultural dynamics. Thus, there seems to be a tendency to continue to frame the customer experience (and also journey) as happening in some evident isolation and detachment from their sociocultural context. As explained insightfully by [START_REF] Askegaard | Towards an Epistemology of Consumer Culture Theory: Phenomenology and the Context of Context[END_REF], the same tendency can be found in many studies examining consumer identity projects, due to too much empirical and methodological focus on the individual's phenomenological experience (mental context). Such a focus is unnecessarily constraining for fostering a more nuanced understanding of how consumers' experience -as well as unfolding personalized narratives of it -is formed always in relation to broader society-level and historical context(s), including nationality, gender, age, class, lifestyle, ethnicity, geographic location, political leanings, etc.

For example, understanding a "simple" consumer experience in a holiday resort as being somehow neutral to the contextual influences becomes immediately limiting and ill-defined. As consumers and marketers/service providers enter into an active encounter and interaction with each other they always carry with them particular sociocultural understandings -as well as "affective/emotional charges" -which can significantly influence the ways in which unfolding experiences are structured both for the consumers and service providers. Even the issue of how well the participants in this encounter master a particular language, or are able to follow social norms and rules, may prove an important pivot in impacting the entire experience itself. Such an exchange is likely to include a multitude of intersubjective struggles that may impact the participants in different ways (Cayla & Bhatnagar, 2019). Thus, research into consumer experiences that account for the influences of the "context of the context" [START_REF] Askegaard | Towards an Epistemology of Consumer Culture Theory: Phenomenology and the Context of Context[END_REF]) are especially needed. This includes continuing attempts towards addressing the difficult issue in extant research that consumers or marketers themselves are not particularly wellequipped or able to consciously reflect on their contextual influences (which are internalized and pre-reflexive). This requires considerable analytical insight and critical assessment by the researcher, as encouraged by CCT work.

The second CCT future direction is to focus much more on the work of marketers and service providers themselves (Zwick & Cayla, 2011) in assembling experiences for consumers.

While CCT has been preoccupied with investigating the "lived experiences" of consumers, far less empirical and theoretical attention has been guided to the production side of experiences (e.g. [START_REF] Haytko | Firm-to-Firm and Interpersonal Relationships: Perspectives form Advertising Agency Account Managers[END_REF]. For example, in CCT and marketing generally, ethnographies of service workers or marketers are particularly rare, not to mention across different sociocultural and national (especially, non-Western) contexts -although rare exceptions can be found (Cayla & Bhatnagar, 2019;[START_REF] Üstüner | How marketplace performances produce interdependent status games and contested forms of symbolic capital[END_REF]. Examining this diversity of contexts in which experiences are assembled and marketing/service work is conducted, require further understanding about how providing "good service" or "good experience" is culturally and socially structured. In addition, more work is needed in relation to advancing our knowledge of the role of "working consumers" in assembling experiences [START_REF] Cova | Critical perspectives on consumers' role as 'producers': Broadening the debate on value co-creation in marketing processes[END_REF]. This major shift is increasingly propelled by the sharing/platform economy tendencies.

A third future direction is to address specifically the creation of experiences from novel perspectives. Here, CCT work has already begun to chart how consumers and marketers "assemble" experiences through the analytical frameworks assemblage theory [START_REF] Canniford | Assembling Consumption: Research Actors, Networks and Markets[END_REF][START_REF] Canniford | Purifying Practices: How Consumers Assemble Romantic Experiences of Nature[END_REF][START_REF] Epp | Outsourcing Parenthood? How Families Manage Care Assemblages Using Paid Commercial Services[END_REF][START_REF] Rokka | Heterotopian selfies: how social media destabilizes brand assemblages[END_REF], actor-network theory [START_REF] Giesler | How Doppelgänger Brand Images Influence the Market Creation Process: Longitudinal Insights from the Rise of Botox Cosmetic[END_REF]Martin & Schouten, 2013), and practice theory provide [START_REF] Woermann | Timeflow: How Consumption Practices Shape Consumers Temporal Experiences[END_REF]. What is specific to these novel theorizations is how they cast new light on the heterogeneous material and semiotic capacities of things, bodies, narratives, and spaces (of different kind and scale) that affect the forging and shape of consumption assemblages [START_REF] Canniford | Assembling Consumption: Research Actors, Networks and Markets[END_REF]. What these views share in common is their commitment to investigate the "messy" boundaries commonly established between consumers and marketers, consumption and production, objects and subjects, and local and global interconnections.

Although consumer experiences [START_REF] Holbrook | The Experiential Aspects of Consumption: Consumer Fantasies, Feelings, and Fun[END_REF][START_REF] Joy | Speaking of Art as Embodied Imagination: A Multisensory Approach to Understanding Aesthetic Experience[END_REF] and experiential consumption have been at the heart of marketing oriented CCT work all along, it is still important to bear in mind that we continue to work with a notion that is not very sharply conceptualized or understood [START_REF] Carù | Revisiting Consumption Experience: A More Humble but Complete View of the Concept[END_REF]. Novel and promising areas of theorization can be found, for example, in insights into embodiment of experiences [START_REF] Scott | Selling Pain to the Saturated Self[END_REF]Stevens et al., 2018), temporality/flow [START_REF] Husemann | Consumer Deceleration[END_REF][START_REF] Woermann | Timeflow: How Consumption Practices Shape Consumers Temporal Experiences[END_REF], atmospheres/emplacement [START_REF] Higgins | Therapeutic Servicescapes and Market-Mediated Performances of Emotional Suffering[END_REF], and ideological/political influences on the assembling of experiences. A further limitation characterizing existing research is its over-reliance on "fixed" textually-based treatments and representations of experiences, despite their increasingly visual, affective, multisensory, and flowing nature [START_REF] Rokka | Screening marketing: videography and the expanding horizons of filmic research[END_REF].

With an eye on most recent social and cultural disruptions and developments (especially the post-covid era), it is also evident that issues of gender, ethnicity, class, immigration, poverty, wellbeing, and vulnerability are likely to shape the agenda and challenges in assembling of experiences.

Shaping of Brands' Symbolic Universes

Early branding models were based on mass-media brand-building anchored in the understanding of brands as transferring meanings and representations distributed by the broadcast media. These approaches tended to conceive brands as "abstract associations" of key benefits, values, and product attributes. Soon psychologically oriented models were challenged and complemented by investigations into the interactions and relationships that consumers have with "their brands", setting the stage for emotional, interactional, and social branding models [START_REF] Fournier | Consumers and Their Brands: Developing Relationship Theory in Consumer Research[END_REF]. For decades, CCT work has shed new light particularly on how consumers adapt, re-mix, resist, and "author" brands' meanings in various localized contexts, sometimes even producing competing doppelgänger brand images [START_REF] Giesler | How Doppelgänger Brand Images Influence the Market Creation Process: Longitudinal Insights from the Rise of Botox Cosmetic[END_REF][START_REF] Thompson | Emotional branding and the strategic value of the doppelgänger brand image[END_REF]. Above all, these works illuminate brands' "linking value;" how consumers build affiliation with social groupings and shared identities [START_REF] Cova | Community and consumption: Towards a definition of the "linking value" of products or services[END_REF]). Yet, perhaps the most systematic effort in building a strategic cultural model of branding so far has been suggested by [START_REF] Holt | How Brands Become Icons: The Principles of Cultural Branding[END_REF]Schroeder & Saltzer-Mörling, 2006). The cultural branding principles underline especially how brands can leverage their symbolic and identity value to consumers, that is, how brands can perform powerful cultural myths that help consumers address their desires, tensions, and anxieties stemming from cultural contradictions in the society.

In essence, what seems to be still under-developed is a dynamic understanding of how brands are shaped and built as specific kinds of complex "symbolic universes", akin to what [START_REF] Askegaard | Brand as a global ideoscape[END_REF] called brand ideoscapes. Brand as an ideoscape provides the ideological basis for the establishment of new meaning systems, new practices, and new identity forms for consumers. A look into the symbolic universes of luxury brands, for instance, quickly reveals that many globally leading brands operate less according to a meaning transfer model that seeks to fix brand meanings, symbols, and image, but rather a model that produces unique experiential and highly aestheticized imaginary or "dream-like" worlds [START_REF] Dion | Retail luxury strategy: assembling charisma through art and magic[END_REF]. This suggests that brands should be analyzed and theorized even more in terms of the powerful aesthetic, affective, and embodied encounters they create -that are not least driven by the growth of visually driven digital platforms such as Instagram. Such encounters do not produce "fixed brand images" but complex aesthetic, visionary, and ideological universes and emotional atmospheres in which consumers can indulge in and participate in their coconstruction. Less is known also about how these universes are designed, performed, and managed by the marketers, and how they have (or not) the capacity to trigger various effects on consumers. Few theorizations explain how the endlessly unfolding streams of images, videos, and technologically mediated encounters (AR, VR, AI, for example) are curated in ways that produce compelling brand universes, and how these may compete, create loyalty, or "stickiness" [START_REF] Siebert | Customer Experience Journeys: Loyalty Loops Versus Involvement Spirals[END_REF]. CCT work that would address such branding and marketing processes -as well as what specific cultural competences they entail -in a more systematized way would be a much-needed future direction.

Future CCT theorizing is also well-positioned in investigating less conventional brand forms that proliferate in radically new but related ways, as evidenced in the case of "Gilets Jaunes" Yellow vest movement in France, or "Black Lives Matter" protests in the USA. These exemplify instances of ideologically-driven brands and idea branding that are founded on organically unfolding movements, mediatized and polarized images, and increasingly decentralized (or non-existing), liquid organizational forms. What is likely to be gained from such investigations is a better understanding of how brands' symbolic universes operate also in ways that escape our current understanding of the branding process facilitated by the marketer.

Institutional and Creative Market Processes

Linking with the earlier CCT focus on examining the various market forms and systems such as brand and online communities, there is an ongoing demand for novel and alternative insights into the institutional and creative market processes (for a review, [START_REF] Giesler | Market system dynamics[END_REF].

First, there is an important proliferation of alternative market forms including access/platform-based, hybrid, and circular economies of collaborative networks [START_REF] Bardhi | Access-Based Consumption: The Case of Car Sharing[END_REF][START_REF] Figueiredo | The Systemic Creation of Value Through Circulation in Collaborative Consumer Networks[END_REF]Scaraboto, 2015) that stem from complex social, technological, and environmental changes. These changes are not yet well understood nor are their implications for different market actors. Notable is the nascent CCT research stream that examines creative consumer-driven movements as facilitators of new market forms and social change (Golnhofer et al., 2019;Kjelgaard et al. 2017;Martin & Schouten, 2013;[START_REF] Weijo | Consumer Movements and Collective Creativity: The Case of Restaurant Day[END_REF]. It is of crucial importance to pay further attention to how consumers and collective movements play a role in triggering social innovations that problematize existing social and market structures and creatively re-envision them. These have until now focused especially on issues linking with the breadth or limits of current market alternatives, but in the future could also include a broader set of social and environmental causes and collective movements that are still relatively poorly understood.

Second, CCT work has only recently begun to address the institutional perspectives and analytical frameworks to theorize market systems and their change. This work is visible in analyses of market legitimization dynamics [START_REF] Humphreys | Megamarketing: The Creation of Markets as a Social Process[END_REF], consumers' institutional work [START_REF] Dolbec | Refashioning a Field? Connected Consumers and Institutional Dynamics in Markets[END_REF][START_REF] Scaraboto | Frustrated Fatshionistas: An Institutional Theory Perspective on Consumer Quests for Greater Choice in Mainstream Markets[END_REF], market-driving entrepreneurs/companies [START_REF] Ertimur | Navigating the Institutional Logics of Markets: Implications for Strategic Brand Management[END_REF]Humphreys & Carpenter, 2019;[START_REF] Maciel | Collaborative Market Driving: How Peer Firms Can Develop Markets Through Collective Action[END_REF]. This is a fruitful ground for novel theorizations that illuminate the interplay of (historical) institutional forces that hold market systems together [START_REF] Press | Ideological Challenges to Changing Strategic Orientation in Commodity Agriculture[END_REF] as well as emergent disruptions, often stemming from adjacent institutional fields [START_REF] Kjellberg | Joint markets: How adjacent markets influence the formation of regulated markets[END_REF]. What is specifically exciting about this work is how it unpacks long-standing trajectories of institutionalized takenfor-granted social, political, historical, and economic logics (of worth/valuation) and how they become destabilized and recalibrated as a result of institutional entrepreneurship, recruitment, and creative collaborative engagement.

Networked and Algorithmic Mediation of Ideologies and Desires

As highlighted, CCT scholars have invested in explaining broader social and cultural structuring of consumption ideologies and how they are "lived" through and translated in consumers everyday experiences and identity struggles. What is new in this picture, however, is the changing technological and media landscape that foregrounds shifts toward platform economy principles and big data-driven digitalized consumer culture. As observed by [START_REF] Thompson | The 'big data' myth and the pitfalls of 'thick data' opportunism: on the need for a different ontology of markets and consumption[END_REF], CCT has a particularly important role in addressing this (often unquestioned) shift by companies and society at large towards managerial frameworks that privilege and seek most -if not allanswers from big data analytics. These tendencies promote novel marketing perspectives but also myths [START_REF] Darmody | Manipulate to empower: Hyper-relevance and the contradictions in the age of surveillance capitalism[END_REF][START_REF] Thompson | The 'big data' myth and the pitfalls of 'thick data' opportunism: on the need for a different ontology of markets and consumption[END_REF] regarding how digital traces of consumer data including social media, mobile applications, self-monitoring, and online searchers/purchases are aggregated and employed towards predictive and automated models for increased marketing efficiency, impact, and returns. What is less known, however, is how these new "intelligent" technologies influence cultural production and with what implications.

On the one hand, CCT research has already highlighted how new digital technologies promote "liquefying tendencies" in consumer culture [START_REF] Bardhi | Liquid Consumption[END_REF]. This means that the more consumption practices are digitalized (access-based, ephemeral, and de-materialized), the less rigid and more fluid the social structures that produce them may seem, or at least be experienced by consumers. Similarly, [START_REF] Kozinets | Networks of Desire: How Technology Increases Our Passion to Consume[END_REF] highlight the role of networked and machinic recommendation systems -such as Instagram, Facebook -in facilitating consumers' desires even to the direction of increasing consumer empowerment. On the other hand, as convincingly argued by [START_REF] Darmody | Manipulate to empower: Hyper-relevance and the contradictions in the age of surveillance capitalism[END_REF], it is also likely that despite platform economy's promise for increased consumer agency, the new logic also invites evermore control and surveillance over the datafied consumers. The authors suggest that the digital marketing era is in reality resting on the contradiction that increasing marketer control produces an autonomous and agentic consumer subject.

It would thus be crucial to advance CCT work by theorizing and empirically investigating how the networked and algorithmic systems embedded in the everyday lives of consumers transform identities, behaviors, and the broader cultural and social dynamics. For example, little is known about how social and cultural processes are mediated by marketer-controlled algorithms that recursively filter and feed inputs based on consumers' behaviors and thus, for example, influence tastes [START_REF] Airoldi | Digital traces of taste: methodological pathways for consumer research[END_REF]. This offers a new and important CCT field to study, inviting a multitude of (re-)inquiries into the production of consumer ideologies and desires, as well as their (dis/empowering) effects.

CCT scholars are in a solid position to comment and advance theories that foster alternative understandings and approaches to the big-data driven marketing practices, which despite evident advantages run the danger of devaluing contextualized cultural and social insights, and alternative sorts of (less voluminous) data as the basis of marketing actions. CCT work is thus called to shed light on how ethnographic, embodied, and critical social-cultural knowledge are needed in navigating and addressing new market environments -perhaps more than ever before?

DISCUSSION: FOSTERING REFLEXIVE MARKETING THINKING

As evoked by the outlined future directions above, CCT can help foster different kinds of reflexive skills, tools, and competences highly relevant for future marketers and marketing scholars.

First of all, there is a demand for reflexive understandings of the consumption and market systems as "entities" whose connections stretch far beyond the immediate and seemingly isolated consumer-marketer encounters and interactions. This invites a greater awareness of the level and scale of the assemblage processes of "social-material life" as a whole, that is shaped by a multitude of forces. These include "external" institutional forces and constraints that often implicitly guide the assembling of experiences, brands, and market systems. For example, important financial and regulatory constraints -imposed not least by the stock markets, company owners, and governments -influence what kinds of market offerings, experiences, and brands are developed, introduced, and also how they are managed. There are the constraints of human labor and uneven wellbeing, as well as those created by surveillance and platform economy. Crucially, there are constraints of the environment and sustainable future, but also those linking with social and cultural dynamics that reproduce discrimination, marginalization, and unequal treatment of people of different color, gender, age, nationality, and class. In short, we need new reflexive approaches that problematize our preconceptions of "what we think we know" and recognize these forces to better understand where people (and marketers/service workers) really come from, and also what kinds struggles they go through. Such critical contextual knowledge should inform the marketing practices but also how we produce knowledge about consumers and market systems in general.

Second, there is a specific need for reflexivity in addressing the current shift towards the big-data driven marketing practices and obsession. On the one hand, future CCT scholars and marketers should actively question the limits of these approaches that are often referred to as having nearly "magical" powers -at least in the level of ongoing managerial discourse. Notably, it should be brought to attention how the employment of large volumes of "decontextualized" data can also be weak in explaining and answering context and situation-dependent questions about "why" and "how" consumers do what they do, but also how they "feel" about it. Big data is also likely to be unhelpful in addressing affective/emotional, embodied, and multisensory nature of experiences or knowledge -a key emphasis of CCT work. On the other hand, CCT's future in marketing also requires examining, using, and critically engaging with the big data, algorithms, AI, smart-objects, and understanding their heterogenous outcomes and implications from the point of view of different stakeholders.

In these ways, future CCT research is likely to help marketers to continuously reflect on the "right" questions and perspectives that need to examined, and hopefully pave ways towards more sustainable social and market innovations.

CONCLUSION

While CCT emerged originally as a corrective in the marketing discipline, to highlight the consumers' perspective and illuminate how the marketer's views of the consumer had been too limited and narrow, I propose that it is also time now to re-direct some CCT attention back to examinations of the marketer and marketing practice. This would be beneficial, as I have highlighted, for better understanding how marketers assemble and produce experiences, how they craft branded symbolic universes, how they engage -not least with the consumer -in processes of market creation and innovation, and how they create networked and algorithm-mediated communication systems, thus propagating consumption ideologies and desires. In addition, it is important to direct further attention to and explore how CCT can help in fostering a reflexive mind-set, skills, tools, and knowledge for future marketers marketing scholars.

ACKNOWLEDGEMENTS

This commentary article is anchored on insights stemming from frequent and ongoing exchanges within the CCT community. The paper has also benefitted from the kind support and highlyvalued feedback from Craig Thompson, Søren Askegaard, Lisa Peñaloza, Eric Arnould, John Schouten, and Jeff Murray, and the members of Lifestyle Research Center at EMLYON Business School. Writing this commentary would not have been possible without this help.