Efimov Denis

Polyakov Andrey

On converse Lyapunov theorem for fixed-time input-to-state stability

Input-to-state stability is one of the most utilizable robust stability properties for nonlinear dynamical systems, while (nearly) fixed-time convergence is a kind of decay for trajectories of disturbance-free systems that is independent in initial conditions. The presence of both these features for a system can be checked by existence of a proper Lyapunov function. The objective of this work is to provide the conditions for a converse result that (nearly) fixed-time input-to-state stable systems admit a respective Lyapunov function. Similar auxiliary results for uniform finite-time stability and uniform (nearly) fixed-time stability are obtained.

I. INTRODUCTION

Analysis of stability properties of dynamical systems is an important theoretical problem having applications in many research and development domains [START_REF] Khalil | Nonlinear Systems[END_REF]. The method of Lyapunov functions constitutes the most popular approach for stability investigation [START_REF] Krasovskii | Stability of motion. Applications of Lyapunov's second method to differential systems and equations with delay[END_REF], [START_REF] Zubov | Methods of A.M. Lyapunov and Their Applications[END_REF], [START_REF] Khalil | Nonlinear Systems[END_REF], since existence of a proper Lyapunov function is an equivalent (necessary and sufficient) condition of respective stability properties. Establishment of such an equivalence is important: it formulates the canonical representations of Lyapunov functions and conditions to check in the applications (avoiding multiple similar framings). However, the issue of finding a Lyapunov function for a given nonlinear system still has no generic solution.

The problem becomes even more sophisticated if the system of interest is subject to external disturbances or/and the rates of convergence have to be established and quantified. A popular and powerful framework for robust stability analysis is created around the input-to-state stability (ISS) concept [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF], [START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF], which also disposes equivalent characterizations in terms of existence of different Lyapunov functions. For the rates of decay, recently, finite-time and (nearly) fixed-time ones got some popularity [START_REF] Roxin | On finite stability in control systems[END_REF], [START_REF] Bhat | Finite time stability of continuous autonomous systems[END_REF], [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF]: the former corresponds to an exact convergence to the equilibrium with a finite settling time dependent on initial conditions, while the latter is related with existence of the settling-time function independent on initial deviations (globally bounded). The equivalent Lyapunov characterizations of the finite-time stability are well known [START_REF] Bhat | Finite time stability of continuous autonomous systems[END_REF], while for the fixed-time stability they are quite new [START_REF] Lopez-Ramirez | Conditions for fixed-time stability and stabilization of continuous autonomous systems[END_REF]. The ISS systems having finite-time convergence rates in the disturbance-free setting are introduced in [START_REF] Hong | Finite-time input-to-state stability and applications to finite-time control design[END_REF] with sufficient conditions in terms of existence of Lyapunov functions, and its necessary counterpart is proposed in [START_REF] Lopez-Ramirez | Finite-time and fixed-time input-to-state stability: Explicit and implicit approaches[END_REF]. The sufficient characterization of fixed-time ISS is also given in [START_REF] Lopez-Ramirez | Finite-time and fixed-time input-to-state stability: Explicit and implicit approaches[END_REF] (see also a survey [START_REF] Efimov | Finite-time stability tools for control and estimation[END_REF]). This work aims on filling the gap by proposing the necessary Lyapunov conditions for (nearly) fixed-time ISS property.

To this end the strategy proposed in [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF] will be used: i) the existence of Lyapunov functions will be shown for uniformly (nearly) fixed-time stable systems (when the inputs take values in a compact set) in Section III; ii) it will be demonstrated that (nearly) fixed-time ISS property (introduced in Section IV) can be related with uniform (nearly) fixed-time stability of an auxiliary system (in Section V); iii) under additional mild restrictions these results will be united in Section VI. The difficulty of such a development is twofold: first, systems possessing finite/fixed-time convergence rates are not Lipschitz continuous at the origin; second, a Lyapunov characterization has to guarantee the respective accelerated convergence of disturbance-free trajectories.

Notation

Denote by R the set of real numbers and R + = {s ∈ R : s ≥ 0}. For x ∈ R n , x corresponds to its (Euclidean) norm.

The authors are with Inria, Univ. Lille, CNRS, UMR 9189 -CRIStAL, F-59000 Lille, France.

For a Lebesgue measurable function of time d : R + → R m , define the norms d [t0,t1) = ess sup s∈[t0,t1) d(s) for [t 0 , t 1 ) ⊂ R + and d ∞ = ess sup s≥0 d(s) , then the space of such functions with • ∞ < +∞ we further denote as L m ∞ . The class of continuous functions X → Y is denoted by C(X, Y) for two metric spaces X and Y.

Problem statement

So, our goal is to provide the converse Lyapunov function results for the ISS property in (nearly) fixed-time sense for the system:

ẋ(t) = f (x(t), d(t)), t > 0, x(0) = x 0 , (1)
where

x(t) ∈ R n and d(t) ∈ R m , d ∈ L m ∞ ; f ∈ C(R n × R m , R n
) ensures forward existence and uniqueness of solutions of the system at least locally in time, f (0, 0) = 0. For any x 0 ∈ R n and d ∈ L m ∞ the respective solution is denoted by x(t, x 0 , d) with x(0, x 0 , d) = x 0 .

The converse Lyapunov theorem for finite-time ISS has been obtained in [START_REF] Lopez-Ramirez | Finite-time and fixed-time input-to-state stability: Explicit and implicit approaches[END_REF] under additional restrictions:

Assumption 1. Let (i) f ∈ C(R n × R m , R n ) be locally Lipschitz continuous in (R n \ {0}) × R m ; (ii) there exist L ∈ C(R + , R + ) such that the inequality f (x, d) -f (x, 0) ≤ L( x ) d is satisfied for all (x, d) ∈ R n × R m .
In the sequel we assume that Assumption 1 is verified for (1).

II. PRELIMINARIES

A. Comparison functions

A function σ ∈ C(R + , R + ) belongs to class K if it is strictly increasing and σ(0) = 0; it belongs to class K ∞ if it is also unbounded. A function β ∈ C(R + × R + , R + ) belongs to class KL if β(•, r) ∈ K and β(r, •) is decreasing to zero for any fixed r > 0. A function χ ∈ C(R + × R + , R + ) belongs to class FKL if χ(•, 0) ∈ K, χ(r,
•) is decreasing to zero for any fixed r > 0 and there is T ∈ K such that χ(r, t) = 0 for all t ≥ T(r). A function χ ∈ FKL with sup r≥0 T(r) < +∞ belongs to the class FxKL.

Finally, χ ∈ C(R + × R + , R + ) belongs to class nFxKL if χ(•, 0) ∈ K, χ(r, •) is decreasing to zero for any fixed r > 0,
and there is T ∈ K such that χ(r, t) ≤ ρ for any ρ > 0, for all t ≥ T (ρ -1 ) and all r ∈ R + .

B. Derivatives and gradient

For g ∈ C(R, R), the upper right Dini derivative [START_REF] Driver | Analysis Tools with Applications[END_REF] is given by

D + g(t) = lim sup h→0 + g(t + h) -g(t) h for any t ∈ R. For V ∈ C(R n , R), the upper right Dini derivative in the direction of v ∈ R n at point x ∈ R n is defined as DV (x)v = lim sup h→0 + V (x + hv) -V (x) h .
For V ∈ C(R n , R), the derivative along the solutions of ( 1) is defined as

V (t) = D + V (x(t, x 0 , d))
for any t ≥ 0,

x 0 ∈ R n and d ∈ L m ∞ ; and if V is locally Lipschitz continuous in R n \ {0}, then V (x, d) := DV (x)f (x, d) for any x ∈ R n and d ∈ R m (in such a case V (t) = V (x(t, x 0 , d), d(t))
for almost all t ≥ 0). If V is locally Lipschitz continuous, then its generalized gradient is defined as follows [START_REF] Clarke | Generalized gradients and applications[END_REF]:

∂V (x) = co lim i→+∞ ∇V (x i ) : x i → x, ∇V (x i ) exists ,
where co{•} takes the convex hull. Definition 1. 1) The system (1) is called uniformly globally asymptotically stable (uGAS) at the origin [START_REF] Lin | A smooth converse Lyapunov theorem for robust stability[END_REF], if there exists

III. UNIFORM COMPLETE

β ∈ KL such that x(t, x 0 , d) ≤ β( x 0 , t) ∀t ≥ 0 for all x 0 ∈ R n and d ∈ D.
2) A uGAS system (1) is called uniformly finite-time stable (uFTS) at the origin if for all x 0 ∈ R n and d ∈ D there is

T x0,d ∈ [0, T (x 0 )] such that x(t, x 0 , d) = 0 ∀t ≥ T x0,d , x(t, x 0 , d) = 0 ∀t ∈ [0, T x0,d ),
where T : R n → R + is a locally bounded settling-time function T (x 0 ) = sup d∈D T x0,d .

3) A uGAS system (1) is called uniformly nearly fixed-time stable (unFxTS) at the origin if for any ρ > 0 there is

T ρ ∈ R + such that x(t, x 0 , d) ≤ ρ ∀t ≥ T ρ
for all x 0 ∈ R n and d ∈ D.

4) A unFxTS system (1) is called uniformly fixed-time stable (uFxTS) at the origin if additionally sup ρ>0 T ρ < +∞.

Equivalently, uFTS, unFxTS or uFxTS of (1) can be formulated as the uGAS property with β ∈ F KL, β ∈ nFxKL or β ∈ FxKL, respectively. Another definition of uFTS (for time-varying systems) can be found in [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF].

To formulate the converse Lyapunov function theorem for u(n)FxTS we will consider a special subclass of these systems: Definition 2. A u(n)FxTS system is called complete if for any x 0 ∈ R n \ {0} and any d ∈ D there exists a unique solution of the system ẋ * (t) = -f (x * (t), d(t)) with x * (0) = x 0 , denoted as x * (t, x 0 , d) and defined for t ∈ [0, τ x0,d ) such that lim sup Necessary and sufficient Lyapunov characterizations of uGAS property (for the case of locally Lipschitz continuous f ) can be found in [START_REF] Lin | A smooth converse Lyapunov theorem for robust stability[END_REF].

t→τ x 0 ,d x * (t, x 0 , d) = +∞,
Definition 3. V ∈ C(R n , R + ) is called unFxTS-Lyapunov function (LF) for the system (1) if there exist α 1 , α 2 ∈ K ∞ , α > 1 and c > 0 such that α 1 ( x ) ≤ V (x) ≤ α 2 ( x ), ∀x ∈ R n , V (t) ≤ -cV α (x(t, ξ, d))
for almost all t ≥ 0, all ξ ∈ R n and all d ∈ D. If α ∈ [0, 1), then such a V is called uFTS-LF.

For α = 1, V has at least asymptotic (exponential) convergence, which for α 1 , α 2 ∈ K ∞ allows only uGAS property of [START_REF] Bhat | Finite time stability of continuous autonomous systems[END_REF] to be concluded. If V is locally Lipschitz continuous in R n \ {0}, then the last condition of Definition 3 can be replaced with

V (x, d) ≤ -cV α (x)
for almost all x ∈ R n and all d ∈ D.

It is straightforward that existence of a uFTS/unFxTS-LF implies the respective stability property, this claim can be proven using standard arguments following the properties of LFs given in Definition 3. Note that uGAS property in (1) is realizable provided that f (0, d) = 0 for any d ∈ D, then Assumption 1 implies a uniform in the second argument local Lipschitz continuity of f on (R n \ {0}) × D.

Theorem 1. If the system (1) is complete unFxTS with a continuous escape-time function τ , then there exists a unFxTS-LF.

Proof. The argumentation is split on the following steps. 1) The result is recalled that for a locally Lipschitz f there is a Lyapunov function. 2) Since f is not Lipschitz continuous at the origin, it is shown how to construct an auxiliary system possessing this property globally. 3) It is demonstrated that a global Lyapunov function W for the auxiliary Lipschitz system is also one for the original dynamics (1). 4) Due to assumed completeness of unFxTS property, the system (1) in backward time is studied, and based on the escape-time function, a Lyapunov function V 1 is designed for big deviations of the state with desired upper estimate on the derivative. 5) The upper estimate on the derivative of W is refined, and a local Lyapunov

function V 2 is introduced based on W . 6) The Lyapunov functions V 1 and V 2 are united. I If f is uniformly in the second argument locally Lipschitz continuous on R n × D (i.e., there exists L ∈ C(R + , R + ) such that the inequality f (x 1 , d) -f (x 2 , d) ≤ L( x 1 + x 2 ) x 1 -x 2
is satisfied for all x 1 , x 2 ∈ R n and d ∈ D), since any unFxTS system is uGAS, then there is a smooth LF W ∈ C(R n , R + )

satisfying for all x ∈ R n and d ∈ D:

α 1 ( x ) ≤ W (x) ≤ α 2 ( x ), Ẇ (x, d) ≤ -η(W (x)) (2)
for some α 1 , α 2 ∈ K ∞ and η ∈ K (see, for example, Theorem 2.9 in [START_REF] Lin | A smooth converse Lyapunov theorem for robust stability[END_REF]). Define two positive constants, r and R, such that 0 < 2r < R, and denote

Ω r = {x ∈ R n : W (x) ≥ r}.
II If f is not Lipschitz at the origin (under Assumption 1), define functions (see also Section 8 in [START_REF] Daniel Liberzon | Universal construction of feedback laws achieving ISS and integral-ISS disturbance attenuation[END_REF])

g δ (v) = 1 v 2v v sup s≤ x ≤δ max i=1,...,n sup d∈D sup i∈∂fi(x,d) i ds, µ δ (v) := min 1, 1 + 1 v 2v v sup x ≤s,d∈D f (x, d) ds 1 + 1 δ 2δ δ sup x ≤s,d∈D f (x, d) ds b(v)v δ , b(v) :=    1 g δ (0) < +∞ g δ (δ) g δ (v)
otherwise for any v ∈ R + and some δ > R. Note that by design, D + µ δ (0) < +∞, µ δ differentiable on (0, δ) and globally Lipschitz continuous, increasing, bounded by 1, equals to 1 when v ≥ δ, strictly positive outside of the origin and µ δ (0) = 0. Define a vector field

f δ (x, d) :=    f (x, d), if x ≥ δ µ δ ( x )f (x, d), if x < δ
, which we claim is uniformly locally Lipschitz continuous by constitution on R n × D. Indeed, the function f possesses this property outside of the origin by the imposed hypothesis, and for almost all x < δ we have that

∂f δ (x, d) ∂x = ∂µ δ ( x )f (x, d) ∂x ≤ ∂µ δ ( x ) ∂x f (x, d) + µ δ ( x ) ∂f (x, d) ∂x ,
where

∂f (x, d) ∂x ∈     1 . . . n     , i ∈ ∂f i (x, d), i = 1, . . . , n and ∂f (x,d) ∂x ≤ g δ ( x ) by construction, then the direct computations show that ∂f δ (x, d) ∂x ≤ C < +∞ for almost all x < δ and all d ∈ D. Since f δ is Lipschitz continuous on R n \{0} then for any x 1 , x 2 satisfying x i < δ, i = 1, 2 and λ ∈ [0, 1] : λx 1 + (1 -λ)x 2 = 0 we have f δ (x 1 , d) -f δ (x 2 , d) = 1 0 (x 1 -x 2 ) ∂f δ (x 1 + tx 2 , d) ∂x dt ≤ C x 1 -x 2 .
On the other hand, for x < δ we get:

f δ (x, d) ≤ µ δ ( x )f (x, d) = f (x, d) 1 + 1 δ 2δ δ sup z ≤s,d∈D f (z, d) ds • 1 + 1 x 2 x x sup z ≤s,d∈D f (z, d) ds δ b( x ) x ≤ 1 + 1 x 2 x x sup z ≤s,d∈D f (z, d) ds δ x ≤ 1 + sup z ≤2 x ,d∈D f (z, d) δ x ≤ C x , C = 1 + sup z ≤2δ,d∈D f (z, d) δ . If ∃λ ∈ [0, 1] : λx 1 + (1 -λ)x 2 = 0, then λ x 1 -x 2 = x 2 and (1 -λ) x 1 -x 2 = x 1 , hence, f δ (x 1 , d) -f δ (x 2 , d) ≤ C x 1 + C x 2 = C (1 -λ) x 1 -x 2 + C λ x 1 -x 2 = C x 1 -x 2 .
So f δ is locally Lipschitz for all x ∈ R n and d ∈ D.

III Now let us consider the system

ẋ = f δ (x, d),
where f δ is, as above, a locally Lipschitz continuous function uniformly in the second argument, and it is unFxTS since (1) has this property (multiplication by a continuous bounded strictly positive scalar function µ δ does not influence the stability, it acts as a time re-scaling). Following the converse results on existence of uGAS LFs, for this auxiliary system there exists a

smooth W ∈ C(R n , R + ) such that (2) is verified for all x ∈ R n and d ∈ D, for some α 1 , α 2 ∈ K ∞ and η ∈ K. Note that in such a case 0 ≥ -η(W (x)) ≥ Ẇ (x, d) = DW (x)f δ (x, d) = µ δ ( x )DW (x)f (x, d) ≥ DW (x)f (x, d)
for all x < δ and d ∈ D. Therefore, an unFxTS system (1) admits an LF W satisfying (2) for all x ∈ R n and d ∈ D under Assumption 1.

IV Consider the system (1) in the "backward" time:

ẋ * (t) = -f (x * (t), d * (t)), t ≥ 0,
then due to the complete unFxTS property (see Definition 2), for any x 0 ∈ Ω r and d * ∈ D there is a unique solution x * (t, x 0 , d * ) that is defined on some bounded interval of time

t ∈ [0, τ x0,d * ) ⊆ [0, τ (x 0 )) (i.e., τ (x 0 ) ∈ [0, T r ] for all x 0 ∈ Ω r ,
where T r > 0 is a finite constant), and lim sup t→τ x 0 ,d * x * (t, x 0 , d * ) = +∞. For a continuous function τ , by its definition,

τ (x * (t, x 0 , d * )) ≤ max{0, τ (x 0 ) -t}, then D + τ (x * (t, x 0 , d * )) ≤ -1
for almost all t ∈ [0, τ x0,d * ), all x 0 ∈ Ω r and d * ∈ D. For any

x 0 ∈ Ω r , d * ∈ D and T ∈ [0, τ x0,d * ), there exists d ∈ D such that x * (t, x 0 , d * ) = x(T -t, x * (T, x 0 , d * ), d)
for all t ∈ [0, T ], or in other words, solution x of (1) in negative time argument coincides with x * on any finite interval of time for a properly selected inputs and x 0 ∈ Ω r . Hence,

∂τ (x(t, x 0 , d)) ∂t = D + τ (x(t, x 0 , d)) ≥ 1
for almost all t ≥ 0, any x 0 ∈ Ω r and d ∈ D. For x ∈ Ω r and α > 1, introduce a candidate LF function

V 1 (x) = ρτ (x) 1 1-α
with some ρ > 0 specified later. The function V 1 is radially unbounded due to the properties of τ and since α > 1, its time derivative computed along trajectories of (1) can be rewritten as follows:

V1 (t) = - ρ α -1 τ (x(t, x 0 , d)) α 1-α ∂τ (x(t, x 0 , d)) ∂t ≤ - ρ α -1 τ (x(t, x 0 , d)) α 1-α = - ρ 1-α α -1 V α 1 (x(t, x 0 , d))
for almost all t ≥ 0 such that x(t, x 0 , d) ∈ Ω r and d ∈ D.

V Moreover, consider the related to the dynamics of W a scalar comparison system in the backward time:

ẏ(t) = η(y(t)), t ∈ [0, τ η (y(0))]
with initial conditions y(0) ∈ (0, R]; the escape-time function τ η : (0, R] → R + to the level y = R is defined as follows

τ η (y(0)) = t * ≥ 0 : y(t * ) = R.
Then, by construction,

τ η (y) = R y ds η(s) , dτ η (y) dy = -1 η(y) < 0
for all y ∈ (0, R]. The latter meant τ η is a monotone function and

lim y→0 1 τ η (y) = lim y→0 1 R y ds η(s) = b ∈ [0, +∞).
Applying argumentation similar as for the function τ we derive

∂τ η (y(t)) ∂t = dτ η (y) dy y=y(t) ẏ(t) = -1 for all t ∈ [0, τ η (y 0 )]. Returning to the dynamics of W , if W (x(t, x 0 , d)) = W (t) = y(t), while Ẇ (t) ≤ -ẏ(t)
V 2 (x) = 1 τ η (W (x)) -b 1 α-1 for W (x) ≤ R, then V2 (x, d) ≤ - 1 α -1 1 τ η (W (x)) -b α α-1 = - 1 α -1 V α 2 (x) for 0 < W (x) ≤ R and d ∈ D.
VI Finally, let us unite the LF candidates V 1 and V 2 to obtain a strict LF V ∈ C(R n , R + ) as

V (x) =          V 2 (x) if W (x) ≤ r, V 1 (x) if W (x) ≥ 2r, λ(x)V 1 (x) + (1 -λ(x))V 2 (x) otherwise,
where λ(x) = r -1 (W (x) -r) with λ ≤ -r -1 η(W (x)) < 0, and ρ is chosen in a way to guarantee that V 1 (x) > V 2 (x) for all x ∈ Ω 2r \ Ω r . Such a design of V ensures all the requirements imposed on a unFxTS-LF due to the proven properties of V 1 , V 2 , and since

V ≤ - 1 α -1          V α 2 (x) if W (x) ≤ r, ρ 1-α V α 1 (x) if W (x) ≥ 2r, ρ 1-α λ(x)V α 1 (x) + (1 -λ(x))V α 2 (x) otherwise.
Hence, there exists c > 0 such that

V (t) ≤ -cV α (x(t, ξ, d))
for almost all t ≥ 0, all ξ ∈ R n \ {0} and d ∈ D as required.

For the sufficient part, if we assume that there is a unFxTS-LF, then the system (1) is unFxTS. Consider the system behavior in the backward time, then we obtain that

- ∂V (x) ∂x f (x, d) ≥ cV α (x),
from which we conclude that the escape-time function should be upper bounded by a continuous function, and hence it is also continuous at infinity (and it is difficult to substantiate continuity of τ used in Theorem 1).

Theorem 2. If the system (1) is uFTS with a continuous settling-time function T , then there exists a uFTS-LF.

Proof. For a continuous settling-time function T , by its definition we have:

T (x(t, x 0 , d)) ≤ max{0, T (x 0 ) -t} for all t ≥ 0, then ∂T (x(t, x 0 , d)) ∂t = D + T (x(t, x 0 , d)) ≤ -1 for all t ∈ [0, T x0,d ) and all x 0 ∈ R n and d ∈ D. Let V (x) = T 1 1-α (x) for some α ∈ [0, 1), then V (t) ≤ - 1 1 -α T α 1-α (x(t, x 0 , d)) = -cV α (x(t, x 0 , d)), c = 1 1 -α ,
for almost all t ≥ 0, all x 0 ∈ R n and d ∈ D. If such a LF V is radially unbounded, then it is uFTS-LF as needed; otherwise, as in the proof of Theorem 1 it can be combined with a LF that exists since a uFTS system is also UGAS.

Obviously, if there is a uFTS-LF V as in Definition 3, then the system is uFTS, and from the estimate for V we get that

T (x) ≤ V 1-α (x)
c(1-α) for all x ∈ R n , which implies continuity of T at the origin.

Remark 1. If in the conditions of theorems 1 and 2 we additionally assume that τ and T are locally Lipschitz continuous in R n \ {0}, respectively, then the obtained LFs V also inherit the same property.

The proofs of theorems 1 and 2 demonstrate the main technical difficulties of the studies problem: the lack of Lipschitz continuity at the origin in (1) and the special forms of upper estimates on V that should be established.

IV. ROBUST STABILITY

The stability concepts of interest and their Lyapunov characterizations are formulated in the following definitions completing [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF], [START_REF] Hong | Finite-time input-to-state stability and applications to finite-time control design[END_REF], [START_REF] Lopez-Ramirez | Finite-time and fixed-time input-to-state stability: Explicit and implicit approaches[END_REF]:

Definition 4. The system (1) is called ISS, if there exist β ∈ KL and γ ∈ K such that

x(t, x 0 , d) ≤ β( x 0 , t) + γ( d [0,t) ) for all x 0 ∈ R n , d ∈ L m ∞ and t ≥ 0. It is called FTS-ISS or (n)FxT-ISS if β ∈ F KL or β ∈ FxKL (β ∈ nFxKL), respectively.
If d = 0, then we recover the conventional FTS/(n)FxTS properties [START_REF] Efimov | Finite-time stability tools for control and estimation[END_REF] (since D = {0} is singleton, the uniformity required in Definition 1 can be omitted).

Definition 5.

[16] The system (1) is said to possess the asymptotic gain (AG) property, if there exists γ ∈ K such that

lim t→+∞ x(t, x 0 , d) ≤ γ( d ∞ ) for all x 0 ∈ R n and d ∈ L m ∞ .
Definition 6.

[16] The system (1) is said to have the global stability (GS) property, if there exist σ 1 , σ 2 ∈ K such that

x(t, x 0 , d) ≤ max{σ 1 ( x 0 ), σ 2 ( d ∞ )} for all x 0 ∈ R n , d ∈ L m ∞ and t ≥ 0.
A system (1), with a locally Lipschitz continuous f , has AG and GS properties if and only if it is ISS [START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF].

Definition 7. A V ∈ C(R n , R + )
, locally Lipschitz continuous on R n \ {0}, is called ISS-LF for the system (1) if there exist α 1 , α 2 ∈ K ∞ and η, ∈ K such that for almost all x ∈ R n and all d ∈ R m :

α 1 ( x ) ≤ V (x) ≤ α 2 ( x ), V (x) ≥ ( d ) =⇒ V (x, d) ≤ -η(V (x)). It is called FTS-ISS-LF, nFxT-ISS-LF or FxT-ISS-LF if η(s) = cs a , η(s) = cs b or η(s) = c 1 s a + c 2 s b , respectively, for c, c 1 , c 2 > 0, a ∈ [0, 1) and b > 1.
The relations between FTS-ISS-LF and FTS-ISS property have been investigated in [START_REF] Hong | Finite-time input-to-state stability and applications to finite-time control design[END_REF] and [START_REF] Lopez-Ramirez | Finite-time and fixed-time input-to-state stability: Explicit and implicit approaches[END_REF], sufficient and necessary parts, respectively.

Theorem 3. If there exists a (n)FxT-ISS-LF for the system (1), then it is (n)FxT-ISS with the AG γ = α -1 1 • . If additionally (1) is complete (n)FxTS for d ≡ 0, then the escape-time function is continuous at infinity; while in FxTS case, the settling-time function is continuous at the origin.

Proof. The proof of this theorem is rather standard and can be partly found in the existing literature [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF], [START_REF] Hong | Finite-time input-to-state stability and applications to finite-time control design[END_REF], [START_REF] Lopez-Ramirez | Finite-time and fixed-time input-to-state stability: Explicit and implicit approaches[END_REF]. For completeness of the paper, consider the FxT-ISS case. If there exists a FxT-ISS-LF for (1), then

α 1 ( x ) ≤ V (x) ≤ α 2 ( x ) ∀x ∈ R n for some α 1 , α 2 ∈ K ∞ , and x ≥ ( d ) implies that DV (x)f (x, d) ≤ -c 1 V a (x) -c 2 V b (x), (3)
where c 1 , c 2 > 0, a ∈ [0, 1) and b > 1, hence, the system is ISS. Further, the following steps are performed below: 1) an estimate is derived for the values of V sufficiently higher than the amplitude of d; 2) an estimate is obtained for small values of V ; 3) these estimates are combined; 4) the continuity of the settling-time and the escape-time functions is discussed.

I Let us define the set V d = {x : V (x) ≥ α 2 • ( d ∞ )}. We have α 2 ( x ) ≥ V (x) ≥ α 2 • ( d ∞ ) for any x ∈ V d ,
which implies that x ≥ ( d ) and by (3), R n \V d is an invariant and attractive set. Using the comparison lemma and direct integration, it is straightforward to obtain a class-FxKL function β(r, t) such that

x(t, x 0 , d) ≤ β( x 0 , t) while x(t, x 0 , d) ∈ V d , (4)
where β(r, t) = 0 for all t ≥ ∼ T β and all r ∈ R + , for some

∼ T β > 0. II If x / ∈ V d then V (x) < α 2 • ( d ∞ ) and therefore x ≤ ϑ( d ∞ ), where ϑ = α -1 1 • α 2 • . In addition, R n \V d is invariant so that x(t, x 0 , d) ≤ ϑ( d ∞ ) while x(t, x 0 , d) / ∈ V d , (5)
III Combining ( 4) and ( 5) gives

x(t, x 0 , d) ≤ β( x 0 , t) + ϑ( d ∞ ) ∀t ≥ 0,
for any x 0 ∈ R n and d ∈ L m ∞ , and FxT-ISS for ( 1) is obtained. IV The continuity at the origin of the settling-time function follows the continuity of its upper estimate derived for LF: i.e.,

DV (x)f (x, 0) ≤ -c 1 V a (x) for almost all x ∈ R n implies T (x 0 ) ≤ V 1-a (x0)
(1-a)c1 for any initial condition x 0 ∈ R n , then T is continuous at the origin (recall T (0) = 0) since V is continuous, V (0) = 0 and a ∈ [0, 1). In the complete (n)FxT-ISS case, for any x 0 ∈ R n \ {0} there is a solution of the system ẋ = -f (x, 0), and continuity of the escape-time function at infinity follows the estimates for LF: Remark 2. The stability properties with accelerated convergence were introduced iteratively over decades and, in some cases, with perplexing names. To clarify their roles and relations, note that if for d ≡ 0 the system (1) is globally asymptotically stable at the origin having unique solutions in forward time, then for any point x 0 ∈ R n there exists the trajectory passing by it, and we can define the time of convergence from this initial condition x 0 to the origin by T 0 (x 0 ), and the time of arrival of the respective trajectory to x 0 from infinity by T ∞ (x 0 ). The following table describes the relations between different properties used in this work:

-DV (x)f (x, 0) ≥ c 2 V b (x) for almost all x ∈ R n implies τ (x 0 ) ≤ V 1-b (x0) (b-1)c2 for any x 0 ∈ R n \ {0}, then τ (x 0 ) → 0 for |x 0 | → +∞
Stability for d = 0 T 0 (x 0 ) T ∞ (x 0 ) Stability for d ∈ L m ∞ Class of KL functions Examples for x, d ∈ R uGAS infinite infinite ISS KL f (x, d) = -x + d uFTS finite infinite FTS-ISS FKL f (x, d) = -3 √ x + d unFxTS infinite finite nFxTS-ISS nFxKL f (x, d) = -x 3 + d uFxTS finite finite FxTS-ISS FxKL f (x, d) = -3 √ x -x 3 + d V. A UNIFORMLY GAS SYSTEM
The result given in this section will be used to make a bridge between u(n)FxTS and (n)FxT-ISS properties.

Further consider an auxiliary system:

ż(t) = f (z(t), γ -1 • φ( z(t) )δ(t)), t ≥ 0, (6)
where z(t) ∈ R n is the state as before, z 0 = z(0) ∈ R n , γ ∈ K ∞ is the AG function given for (1) in Definition 5

(which exists assuming that (1) is ISS, and it can be always chosen from the class K ∞ , then the inverse is well-defined), φ(s) = 0.5 min(s, β -1 (0.25s, 0)) is a function from the class K ∞ where β also comes from the ISS property definition, and δ ∈ D is a uniformly bounded input. Obviously, the origin is an equilibrium of (6) since f (0, •) = 0. We can always modify the AG γ of ( 1) such that γ -1 • φ will be locally Lipschitz continuous away from zero. In the sequel this property is assumed to hold. In addition, for any z 0 ∈ R n and δ ∈ D the corresponding solution z(t, z 0 , δ) of ( 6) is defined on some interval of time [0, T max z0 ) and it coincides with the solution x(t, z 0 , d) of (1) for d(t) = γ -1 • φ( z(t, z 0 , δ) )δ(t) on this interval of time. Let us demonstrate a stronger relation between solutions of the systems (1) and ( 6) (as in [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF] for the pure ISS case): Proposition 1. The following relations are true: i) If the system (1) is (n)FxT-ISS with the decay β ∈ FxKL (β ∈ nFxKL) and AG γ, then the system (6) is u(n)FxTS.

ii) If the system (6) is u(n)FxTS with the decay β ∈ FxKL (β ∈ nFxKL), then the system (1) is (n)FxT-ISS with the same decay β and the AG β(φ -1 • γ(•), 0). Similar relations hold true for uFTS of ( 6) and FTS-ISS of (1) (the proof is the same as below).

Proof. i) If the system ( 1) is (n)FxT-ISS, then according to Definition 4 (by time invariance of the system we can replace in that definition 0 by t 0 and t by t -t 0 )

x(t) = x(t -t 0 , x(t 0 ), d) ≤ β( x(t 0 ) , t -t 0 ) + γ( d [t0,t) ), for all t ≥ t 0 , for some β ∈ FxKL (β ∈ nFxKL), γ ∈ K ∞ and all x(t 0 ) ∈ R n , d ∈ L m ∞ , t 0 ≥ 0. Since substituting d(t) = γ -1 • φ( x(t)
)δ(t) in the system (1) we get [START_REF] Hong | Finite-time input-to-state stability and applications to finite-time control design[END_REF], for the latter with z(t) = z(t -t 0 , z(t 0 ), δ) and any z(t 0 ) ∈ R n , δ ∈ D we obtain:

z(t) ≤ β( z(t 0 ) , t -t 0 ) + γ( sup s∈[t0,t) γ -1 • φ( z(s) )δ(s)||) ≤ β( z(t 0 ) , t -t 0 ) + γ( sup s∈[t0,t) γ -1 • φ( z(s) )) = β( z(t 0 ) , t -t 0 ) + φ( sup s∈[t0,t) z(s) ) for all t ∈ [t 0 , T max z(t0)
). Take t 0 = 0, then from the last inequality and the definition of φ,

sup s∈[0,t) z(s) ≤ sup s∈[0,t) {β( z 0 , s) + 0.5 sup σ∈[0,s) z(σ) } ≤ β( z 0 , 0) + 0.5 sup s∈[0,t) z(s) , leading to 0.5 sup s∈[0,t) z(s, z 0 , δ) ≤ β( z 0 , 0)
for any t ≥ 0, all z 0 ∈ R n and all δ ∈ D, which implies global uniform boundedness of the solutions of the system [START_REF] Hong | Finite-time input-to-state stability and applications to finite-time control design[END_REF].

Therefore, T max z0 = +∞ for all z 0 ∈ R n and all δ ∈ D. Choosing t 0 = t 2 and repeating the same manipulations we get:

sup s∈[ t 2 ,t) z(s) ≤ sup s∈[ t 2 ,t)
{β( z(0.5s) , 0.5s) + 0.5 sup

σ∈[ s 2 ,s) z(σ) } ≤ sup s∈[ t 2 ,t)
β(2β( z 0 , 0), 0.5s) + 0.5 sup

s∈[ t 4 ,t) z(s) ,
which is true for any t ≥ 0, all z 0 ∈ R n and all δ ∈ D. Then, for t → +∞ (i.e., considering the asymptotic behavior of ( 6))

it results in lim

t→+∞ sup s∈[ t 2 ,t) z(s) ≤ 0.5 lim t→+∞ sup s∈[ t 4 ,t) z(s)
that can be satisfied for lim t→+∞ z(t) = 0 only, and ( 6) is uGAS.

For β ∈ FxKL, denote by T ∈ K the settling time function satisfying β(s, T(s)) = 0 for any s ∈ R + . Note that locally, for sufficiently small s ∈ [0, 1], the function β(s, t) can be upper bounded by β(T -1 • max{0, 2T(s) -t}, 0) (with a bigger settling time), then we can always assume, with a slight ambiguity in the notation, that β is (locally) presented in a canonical form β(s, t) = χ • max{0, T(s) -t} for some χ ∈ K ∞ (then χ(s) ≥ β(T -1 (s), 0) and φ(s) = 0.5 min(s, T -1 • χ -1 (0.25s))).

Denote by T ∈ K another time decay function satisfying 2β(s,

T(s)) ≥ β(s, 0) (i.e., T(s) ≤ T(s) -χ -1 • 1 2 χ • T(s)), then
locally, for sufficiently small s ∈ [0, 1], there exists κ ∈ (0, 1) such that T(s) ≥ κT(s). Due to proven uGAS property, select arbitrary t 0 ∈ R + and δ ∈ D, and a sufficiently small z(t 0 ) ∈ R n such that β, T and T verify the above restrictions, then

z(t) ≤ sup s∈[t0,t) z(s) ≤ 2β( z(t 0 ) , 0) ≤ 4β( z(t 0 ) , T( z(t 0 ) ))
for all t ∈ [t 0 , t 0 + T( z(t 0 ) )), and for t = t 0 + T( z(t 0 ) ), by continuity:

z(t 0 + T( z(t 0 ) )) ≤ φ sup s∈[t0,t0+T( z(t0) )) z(s) ≤ T -1 • max{0, T( z(t 0 ) ) -T( z(t 0 ) )}.
Define iteratively the series of time instants t i = t i-1 + T( z(t i-1 ) ) for all i ≥ 1, then we get the estimate:

z(t 0 + i≥0 T( z(t i ) ) ≤ T -1 • max{0, T( z(t 0 ) ) - i≥0 T( z(t i ) )} ≤ T -1 • max{0, T( z(t 0 ) ) -κ i≥0 T( z(t i ) )},
and either κ i≥0 T( z(t i ) ) is equal or bigger than T( z(t 0 ) ) implying that z(t 0 +T( z(t 0 ) )) = 0 or i≥0 T( z(t i ) ) < κ -1 T( z(t 0 ) ) that due to the properties of T results in z(t 0 + i≥0 T( z(t i ) ) = 0, which was necessary to show.

In order to substantiate the nearly fixed-time convergence, recall existence of the convergence time T ∈ K such that β(s, T (ρ -1 )) ≤ ρ for arbitrary ρ > 0 for any s ∈ R + . With this notation, any function β(s, t) from class nFxKL can be upper bounded by 2β(s,0) 1+T -1 (t)β(s,0) for s > 0 sufficiently big (with a bigger convergence time), then we can always assume, with a slight ambiguity in the notation, that β is written in a canonical form β(s, t) = χ(s)

1+T -1 (t)χ(s) for some χ ∈ K ∞ and the argument of s sufficiently big (then χ(s) ≥ β(s, 0) ≥ s and φ(s) = 0.5 min(s, χ -1 (0.25s))). Denote by T : R + → R + an auxiliary time decay function satisfying 2β(s, T(s)) ≥ β(s, 0) (i.e., T -1 • T(s) ≤ 1 χ(s) ). In such a case, a time of convergence T : R + → R + can be defined from the interval 1 2 χ -1

1 1 χ(s) +T -1 •T(s) -1 -1 χ(s) ≤ T -1 • T (s) ≤ 2 χ -1 (0.5χ(s)) -1 χ(s)
and providing the property

χ(s) 1+T -1 •T (s)χ(s) ≤ 1 2 χ -1 χ(s)
1+T -1 •T(s)χ(s) (T is a decreasing function). Select arbitrary t 0 ∈ R + and δ ∈ D, and z(t 0 ) ∈ R n with z(t 0 ) ≥ ρ for a sufficiently big ρ > 0 such that β, T and T verify the above restrictions,

then z(t) ≤ sup s∈[t0,t) z(s) ≤ 2β( z(t 0 ) , 0) ≤ 4β( z(t 0 ) , T( z(t 0 ) ))
for all t ∈ [t 0 , t 0 + T ( z(t 0 ) )), and for t = t 0 + T ( z(t 0 ) ), by continuity:

z(t 0 + T ( z(t 0 ) )) ≤ β( z(t 0 ) , T ( z(t 0 ) )) + φ sup s∈[t0,t0+T ( z(t0) )) z(s) ≤ 1 2 χ -1 χ( z(t 0 ) ) 1 + T -1 • T( z(t 0 ) )χ( z(t 0 ) ) + 1 2 χ -1 χ( z(t 0 ) ) 1 + T -1 • T( z(t 0 ) )χ( z(t 0 ) ) = χ -1 1 1 χ( z(t0) ) + T -1 • T( z(t 0 ) )
.

Define iteratively the series of time instants t i = t i-1 + T ( z(t i-1 ) ) for all i ≥ 1, then we get the estimate:

z(t 0 + i≥0 T ( z(t i ) )) ≤ χ -1 1 1 χ( z(t0) ) + i≥0 T -1 • T( z(t i ) )
, which implies uniform convergence since i≥0 T ( z(t i ) ) and i≥0 T -1 • T( z(t i ) ) stay bounded while z(t i ) ≥ ρ, which due to proven uGAS property implies unFxTS of [START_REF] Hong | Finite-time input-to-state stability and applications to finite-time control design[END_REF].

ii) Conversely, let (6) be u(n)FxTS, then according to Definition 1:

z(t -t 0 , z(t 0 ), δ) ≤ β( z(t 0 ) , t -t 0 ), ∀t ≥ t 0 ,
for some β ∈ FxKL (β ∈ nFxKL) and all z(t 0 ) ∈ R n , δ ∈ D, t 0 ≥ 0. Consider the system (1), take any x 0 ∈ R n and any

d ∈ L m ∞ , assume that d ∞ ≤ γ -1 • φ( x(t) ) for t ∈ [0, t 1 )
with some (possibly infinite) t 1 ≥ 0, then there is δ ∈ D such that d(t) = γ -1 • φ( x(t) )δ(t) for all t ∈ [0, t 1 ) and, hence, using the properties of (6) we obtain:

x(t, x 0 , d) ≤ β( x 0 , t), ∀t ∈ [0, t 1 ). Next, assume that d ∞ > γ -1 • φ( x(t) ) for t ∈ [t 1 , t 2 ) with some t 2 > t 1 (obviously, always t 1 + t 2 > 0 for any x 0 ∈ R n and d ∈ L m ∞ ), hence, x(t, x 0 , d) ≤ φ -1 • γ( d ∞ ), ∀t ∈ [t 1 , t 2 ). Finally, let again d ∞ ≤ γ -1 • φ( x(t) ) for t ∈ [t 2 , t 3 ) with some t 3 > t 2 , similarly x(t, x 0 , d) ≤ β( x(t 2 ) , t -t 2 ) ≤ β(φ -1 • γ( d ∞ ), 0), ∀t ∈ [t 2 , t 3 ) since in this case x(t 2 ) = φ -1 • γ( d ∞ ) by construction.
Next, all these steps can be repeated iteratively if necessary (they cover all possible scenarios), and it is clear that for all t ≥ 0

x(t, x 0 , d) ≤ β( x 0 , t) + β(φ -1 • γ( d ∞ ), 0),
since by definition β(s, 0) ≥ s for all s ≥ 0, which implies (n)FxT-ISS property of (1).

VI. MAIN RESULTS

For a (n)FxT-ISS dynamical system (1), the result of Proposition 1 guarantees that ( 6) is u(n)FxTS, then the material of the previous sections allows the complete robust accelerated stability notions to be introduced:

Definition 8. A (n)FxT-ISS system (1) is called complete if (6) is complete u(n)FxTS.
The main outcomes of our work are formulated in the theorems below. Proof. Our objective is to construct a nFxT-ISS-LF for a complete nFxT-ISS system (1). In such a case, by Definition 8, the auxiliary system (6) is complete unFxTS with a locally Lipschitz continuous on R n \ {0} escape-time function τ , then by Theorem 1 there is a unFxTS-LF V ∈ C(R n , R + ) for the system (6), locally Lipschitz continuous on R n \ {0}: there exist α 1 , α 2 ∈ K ∞ , α > 1 and c > 0 such that for all z ∈ R n and δ ∈ D:

α 1 ( z ) ≤ V (z) ≤ α 2 ( z ), V (z, γ -1 • φ( z )δ) ≤ -cV α (z).
Returning to the system (1), we obtain:

x ≥ φ -1 • γ( d ) =⇒ V (x, d) ≤ -cV α (x),
hence, V is the required nFxT-ISS-LF.

Theorem 5. If the system (1) is complete FxT-ISS with a locally Lipschitz continuous on R n \ {0} escape-time function τ and settling-time function T , then there exists a FxT-ISS-LF.

Proof. The proof has four main steps. First, using converse arguments, a LF U (x) is constructed that shows local FTS (which is locally equivalent to FxTS) of the unperturbed system (1). Second, it is shown that this LF U (x) is actually an FxT-ISS-LF Lyapunov function if x < ρ, for any ρ > 0 (with the asymptotic gain dependent on ρ). Third, applying the substantiated converse results for nFxT-ISS (which follows from FxT-ISS), another LF W (x) is designed for x ≥ δ for any δ ∈ (0, ρ).

Finally, a desired global FxT-ISS-LF is constructed by uniting U and W .

I Since ( 1) is FxT-ISS, when d = 0 there exists some T (x) such that x(t, x, 0) = 0 for all t ≥ T (x), and T (x) ≤ T max < +∞, for any x ∈ R n . If T (x) is a locally Lipschitz function, then for any ρ > 0 it is possible to define a function U (x) := T (x) 1 1-a , with a ∈ [0, 1), satisfying α 1 ( x ) ≤ U (x) ≤ α 2 ( x ) for some α 1 , α 2 ∈ K ∞ and all x ∈ Ω ρ = {x ∈ R n : x < ρ}, being such that DU (x)f (x, 0) ≤ -c 1 U a (x) [START_REF] Khalil | Nonlinear Systems[END_REF] for some c 1 > 0 and for almost all x ∈ Ω ρ . where s ∈ C(R + , R + ) satisfies

II

s(r) =    1 if r ≥ α 2 (ρ) 0 if r ≤ α 1 (δ)
and ṡ(r) = ∂s(r) ∂r > 0 for all r ∈ (α 1 (δ), α 2 (ρ)). Assume that U (x) ≤ W (x) for all x ∈ {x ∈ R n : α 1 (δ) ≤ U (x) ≤ α 2 (ρ)} (both functions U (x) and W (x) are continuous, positive definite and radially unbounded, then we can adopt such a hypothesis without being restrictive, since multiplying W (x) by a constant as in the proof of Theorem 1 we can always assure its fulfillment), then we have that and gathering all the previous estimates, we arrive to

x ≥ χ( d ) ⇒ ∼ V (x, d) ≤ -α 4 ( x ), (9)
where χ(r) := max{φ -1 • γ(r), ϕ -1 ρ (r)} and α 4 ∈ K ∞ such that

α 4 ( x ) ≥    c [ ∼ V (x)] b , U (x) ≥ α 2 (ρ) c 2 [ ∼ V (x)] a , U (x) ≤ α 1 (δ)
.

Consequently,

∼

V is a FxT-ISS-LF for (1).

VII. CONCLUSION

Several converse Lyapunov theorems are formulated for uFTS, u(n)FxTS and (n)FxTS-ISS properties. The open research directions include relaxation of restrictions given in Assumption 1, avoiding the Lipschitz continuity requirement for τ and T introduced in theorems 4 and 5, investigations of integral ISS property with accelerated convergence rates, or extension of these results to the systems with outputs (to the input-to-output stability concept; note that the necessary and sufficient conditions for output finite-time stability can be found in [START_REF] Zimenko | On necessary and sufficient conditions for output finite-time stability[END_REF]).

  NEARLY FIXED-TIME STABILITY Denote D = {d ∈ L m ∞ : d ∞ ≤ 1} and D = {d ∈ R m : d ≤ 1} in order to analyze the behavior of (1) with a bounded input d. The following uniform stability properties and their necessary Lyapunov characterizations derived below constitute an intermediate step for the main result of this work:

  with τ : R n \ {0} → R + being a locally bounded escape-time function satisfying τ (x 0 ) = sup d∈D τ x0,d .

  due to positive definiteness and radial unboundedness of V and since b > 1.

Theorem 4 .

 4 If the system (1) is complete nFxT-ISS with a locally Lipschitz continuous on R n \ {0} escape-time function τ , then there exists a nFxT-ISS-LF.

  Since T (x) is locally Lipschitz continuous away from the origin, U (x) has the same property and DU (x) ≤ κ+ψ( x )for almost all x ∈ Ω ρ , for some κ ∈ R + and ψ ∈ K. By Assumption 1, f (x, d) -f (x, 0) ≤ L( x ) d for the given L : R + → R + . Thus,DU (x) f (x, d) -f (x, 0) ≤ (κ + ψ( x ))L( x ) d for almost all x ∈ Ω ρ and all d ∈ R m . Let ρ > x ≥ ϕ -1 ρ ( d ),whereϕ ρ ( x ) := c 1 α 1 ( x ) a 2(κ + ψ(ρ)) sup s∈[0,ρ] L(s) (8) is a function from class K ∞ , then DU (x) f (x, d) -f (x, 0) ≤ c1 2 U a (x)and using the inequality[START_REF] Khalil | Nonlinear Systems[END_REF] we have thatDU (x)f (x, d) ≤ -c1 2 U a (x) provided that x ≥ ϕ -1 ρ ( d )for almost all x ∈ Ω ρ and all d ∈ R m . III Since complete FxT-ISS property implies complete nFxT-ISS one, and the local Lipschitz continuity on R n \ {0} of the respective escape-time function τ is assumed, by Theorem 4, there is a nFxT-ISS-LF W ∈ C(R n , R + ), locally Lipschitz continuous on R n \ {0}. Hence, for x / ∈ Ω δ with any δ ∈ (0, ρ) and d ∈ R m :α 1 ( x ) ≤ W (x) ≤ α 2 ( x ), x ≥ φ -1 • γ( d ) =⇒ Ẇ (x, d) ≤ -c W b (x) with α 1 , α 2 ∈ K ∞ ,b > 1 and c > 0. IV Let us define the function ∼ V (x) := s(U (x))W (x) + (1 -s(U (x)))U (x),

∼

  V (x, d) = s Ẇ (x, d) + (1 -s)DU (x)f (x, d) + ṡ(U (x))DU (x)f (x, d)(W (x) -U (x)),


Jingwen Zhu

Hadi Amirpour

Raimund Schatz

Christian Timmerer

Patrick Le Callet

Enhancing Satisfied User Ratio (SUR) Prediction for VMAF Proxy through Video Quality Metrics

Keywords: Video quality, SUR, JND, bitrate ladder, VMAF

In adaptive video streaming, optimizing the selection of representations for the encoding bitrate ladder has a significant impact on the quality and economics of media delivery. An efficient way to select representations for the bitrate ladder of a given clip is to consider the Satisfied User Ratio (SUR) of the perceived quality of consecutive representations. This ensures that only representations with one Just Noticeable Difference (JND) are encoded and streamed by avoiding encoding similar-quality representations. VMAF (Video Multi-method Assessment Fusion) presently stands as the most commonly utilized quality metric for constructing bitrate ladders. Hence, the precise determination of JND-optimal encoding step-sizes for the VMAF proxy holds paramount importance; nevertheless, this task is intricate and can present considerable challenges.

In this paper, we evaluate the effectiveness of different Video Quality Metrics (VQMs) in predicting SUR for the VMAF proxy to better capture content-specific characteristics. Our experimental results provide evidence that incorporating VQMs can improve the precision of the SUR prediction for the VMAF proxy. Compared to a state-of-the-art approach that utilizes video complexity metrics, our proposed approach, which incorporates two quality metrics-specifically, VMAF and SSIM calculated at an optimized quantization parameter (QP)-achieves a substantially reduced Mean Absolute Error (MAE) of 1.67. In contrast, the state-of-the-art approach yields an MAE of 2.01. Hence, we recommend using the above quality metrics to improve the accuracy of the SUR prediction for the VMAF proxy.

I. INTRODUCTION

Video streaming platforms like Netflix, YouTube, and Amazon Prime Video have become an essential part of our daily lives. HTTP Adaptive Streaming (HAS) is the dominant technique utilized in both live and Video-on-Demand (VoD) streaming applications. It relies on Adaptive Bitrate Streaming (ABR) methods [START_REF] Bentaleb | A Survey on Bitrate Adaptation Schemes for Streaming Media Over HTTP[END_REF], where video content is encoded at multiple bitrate-resolution pairs known as representations. These representations are used to construct a bitrate ladder [START_REF] Hadi Amirpour | PSTR: Per-Title Encoding Using Spatio-Temporal Resolutions[END_REF], enabling the dynamic adjustment of video quality according to the viewer's available bandwidth and device type.

Traditionally, a fixed set of representations, such as the HLS bitrate ladder [START_REF] Apple | HTTP Live Streaming (HLS) Authoring Specification for Apple Devices | Apple Developer Documentation[END_REF], is used for all video content. However, this "one-size-fits-all" approach may not be optimal for different types of videos. To address this, the per-title encoding approaches were introduced, where an optimized bitrate ladder is created for each video content, resulting in improved Quality of Experience (QoE). In per-title encoding [START_REF] De Cock | Complexity-Based Consistent-Quality Encoding in the Cloud[END_REF], [START_REF] Hadi Amirpour | PSTR: Per-Title Encoding Using Spatio-Temporal Resolutions[END_REF], [START_REF] Hadi Amirpour | Deep-Stream: Video Streaming Enhancements using Compressed Deep Neural Networks[END_REF], various encoding parameters, including resolution, frame rate, and others, are assessed by encoding the videos using all possible combinations of these parameters. Subsequently, an optimized bitrate ladder is constructed by selecting representations from a convex-hull based on the quality measurements of the encoded representations. Video Multi-method Assessment Fusion (VMAF) [START_REF] Li | Toward A Practical Perceptual Video Quality Metric[END_REF] is widely embraced as an objective quality metric that exhibits a strong correlation with human-perceived quality. It is frequently employed to evaluate the quality of representations and guide the bitrate laddering process [START_REF] Angeliki | VMAF-Based Bitrate Ladder Estimation for Adaptive Streaming[END_REF].

Selecting a subset of representations from the convex-hull is a crucial step in constructing an optimized bitrate ladder. This selection process involves considering various factors, such as available network bandwidth, device capabilities, and perceptual quality metrics like VMAF. While some methods focus on selecting representations based on the probability of clients requesting specific bitrate versions [START_REF] Yuriy | Optimal Design of Encoding Profiles for ABR Streaming[END_REF], [START_REF] Tashtarian | LALISA: Adaptive Bitrate Ladder Optimization in HTTP-based Adaptive Live Streaming[END_REF], other approaches prioritize the selection of representations to minimize perceptual similarity [START_REF] Vignesh V Menon | Perceptually-Aware Per-Title Encoding for Adaptive Video Streaming[END_REF]. These methods aim to avoid including representations in the bitrate ladder that have similar perceptual qualities, as this redundancy may lead to inefficient resource utilization. By diversifying the quality levels of the representations, the bitrate ladder construction aims to provide a wider range of viewing options and enhance the overall streaming experience for users. The human visual system (HVS) has limitations in detecting small distortions in videos due to psychological and physiological mechanisms. HVS can differentiate only a few discrete-scale distortion levels within a wide range of quality levels. The minimum visual difference that can be perceived by the HVS of each individual is defined as the Just Noticeable Difference (JND). The first JND point indicates the transition from perceptually lossless to perceptually lossy coding.

To account for variations in HVS between viewers, a metric called Satisfied User Ratio (SUR) was defined, representing the proportion of the population that could distinguish a given distortion level [START_REF] Zhang | Satisfied-User-Ratio Modeling for Compressed Video[END_REF], [START_REF] Zhu | On the benefit of parameter-driven approaches for the modeling and the prediction of satisfied user ratio for compressed video[END_REF]. SUR is defined with Eq.(1):

SUR(x) = 1 - +∞ -∞ f (x)dx, (1)
where, f (x) corresponds to the Probability Density Function (PDF) that has been appropriately fitted to the individual The variable x serves as a proxy for the SUR curve, encompassing encoding parameters (such as QP, CRF), VMAF scores, bitrates, and other relevant factors tailored to diverse application scenarios [START_REF] Zhu | Elevating Your Streaming Experience with Just Noticeable Difference (JND)-based Encoding[END_REF].

In this paper, we focus on VMAF as the proxy of the SUR curve, because of its codec-independence and superior performance, which is highly correlated with human perception. As shown in Fig. 1, the transition from perceptually lossless to perceptually lossy coding of the VMAF proxy for p% of SUR value is expressed as:

∆VMAF SUR(p%) = VMAF SUR(100%) -VMAF SUR(p%) , (2)
where VMAF SUR(p%) is the VMAF proxy where p% of viewers cannot see the transition and is defined as:

VMAF SUR(p%) = arg min x |SUR(100 -x) -p%| . (3)
This paper focuses on investigating content-dependencies of ∆VMAF SUR(p%) estimation for the VMAF proxy. By leveraging VQMs such as VMAF and SSIM, we aim to explore and leverage the content-specific characteristics that affect ∆VMAF SUR(p%) of a given content clip.

II. DATASET DESCRIPTION

A comprehensive video quality dataset called VideoSet was introduced in a research publication by Wang et al. [START_REF] Wang | VideoSet: A Large-scale Compressed Video Quality Dataset based on JND Measurement[END_REF]. The dataset comprises 220 source video sequences with a duration of 5 seconds, featuring frame rates of either 24 fps or 30 fps. These sequences were encoded using the constant quantization parameter (CQP) rate control mode of the H.264/AVC video coding standard [START_REF] Wiegand | Overview of the H.264/AVC Video Coding Standard[END_REF], with QPs ranging from 1 to 51. The subjective evaluation of JND of individuals was conducted across multiple universities, utilizing 58 dedicated stations for testing.

Although the original JND points were reported for fixed QPs only, we compute the VMAF scores for each encoding and determine the corresponding VMAF values for the first JND. Fig. 2 visually depicts the ∆VMAF SUR(75%) values for all the 220 contents included in the VideoSet, where the first JND occurs for the 100 -p% = 25% (p% = 75%) of viewers. Fig. 2: ∆VMAF SUR(75%) where the first JND occurs for for the 100-p% = 25% (p% = 75%) of viewers in VideoSet [START_REF] Wang | VideoSet: A Large-scale Compressed Video Quality Dataset based on JND Measurement[END_REF] for 1080p.

Please note that p% is selected equal to 75% to align with the literature [START_REF] Zhu | On the benefit of parameter-driven approaches for the modeling and the prediction of satisfied user ratio for compressed video[END_REF], [START_REF] Wang | VideoSet: A Large-scale Compressed Video Quality Dataset based on JND Measurement[END_REF], [START_REF] Hadi Amirpour | Between Two and Six? Towards Correct Estimation of JND Step Sizes for VMAF-Based Bitrate Laddering[END_REF], [START_REF] Zhang | Deep Learning Based Just Noticeable Difference and Perceptual Quality Prediction Models for Compressed Video[END_REF], [START_REF] Wang | Prediction of Satisfied User Ratio for Compressed Video[END_REF]. Fig. 2 illustrates the substantial diversity in ∆VMAF SUR(75%) values, highlighting the content-dependent nature of the perceptual differences. For instance, the range of ∆VMAF SUR(75%) can vary widely, ranging from 0 to 25. This substantial diversity underscores the necessity for content-specific approaches in addressing perceptual differences.

III. STATE-OF-THE-ART METHODS

In the existing literature, two sources provide recommendations for determining the sizing of ∆VMAF SUR(75%) in JND-based bitrate laddering. Jan Ozer [START_REF] Ozer | Finding the Just Noticeable Difference with Netflix VMAF[END_REF], in a blog post, reports an interview with an unnamed Netflix employee who suggests a constant step size of 6 without empirical evidence to support this recommendation. On the other hand, Kah et al. [START_REF] Kah | Fundamental Relationships Between Subjective Quality, User Acceptance, and the VMAF Metric for A Quality-based Bit-rate Ladder Design for Overthe-Top Video Streaming Services[END_REF], in a publication, present the findings of a subjective study campaign on the acceptability and perceived quality of video content and propose a constant value of 2 for ∆VMAF SUR(75%) . To further substantiate these two cases, Amirpour et al. [START_REF] Hadi Amirpour | Between Two and Six? Towards Correct Estimation of JND Step Sizes for VMAF-Based Bitrate Laddering[END_REF] conducted an analysis by applying the two recommendations (∆VMAF SUR(75%) = 2 vs. 6) to VideoSet [START_REF] Wang | VideoSet: A Large-scale Compressed Video Quality Dataset based on JND Measurement[END_REF], comparing the outcomes with the subjective JND ground truth provided in the dataset.

In this study, each recommendation was interpreted as a simple linear model that predicts the ∆VMAF SUR(75%) for each video content in the dataset. The results were subsequently analyzed using common error metrics. It was shown that using the constant value of the 6 rule for ∆VMAF SUR(75%) performs significantly better than the constant value of 2.

However, these estimators still yield high error levels. The main reason for this discrepancy is the substantial variance of ∆VMAF SUR(75%) values across different video contents.

To effectively tackle the challenges in the current works, a content-specific framework [START_REF] Hadi Amirpour | Between Two and Six? Towards Correct Estimation of JND Step Sizes for VMAF-Based Bitrate Laddering[END_REF] was developed to estimate ∆VMAF SUR(75%) by utilizing features extracted from the reference video (see Fig. 3(a)). These features encompass the framewise features, including: (i) Spatial Information (SI) [START_REF]Subjective Video Quality Assessment Methods for Multimedia Applications[END_REF], (ii) Temporal Information (TI) [START_REF]Subjective Video Quality Assessment Methods for Multimedia Applications[END_REF], (iii) Spatial Energy (E) [START_REF] Vignesh V Menon | VCA: Video Complexity Analyzer[END_REF], (iv) Temporal Energy (h) [START_REF] Vignesh V Menon | VCA: Video Complexity Analyzer[END_REF], (v) Brightness (L), (vi) Colourfulness (c) [START_REF] Hasler | Measuring Colorfulness in Natural Images[END_REF], and (vii) Frame rate (fr). By considering these features, the framework provided a more comprehensive estimation of ∆VMAF SUR(75%) , taking into account the spatial and temporal aspects, as well as brightness, colourfulness, and frame rate characteristics of the videos, as:

∆VMAF SUR(75%) = f (SI, T I, E, h, L, f r, c) (4)
However, despite the utilization of this approach, the mean absolute error (MAE) was only reduced from 2.73 down to 2.11, in comparison to using the dataset's ∆VMAF SUR(75%) mean of 6.93 as a constant estimator. Although an improvement was observed, the remaining error levels are still not satisfactory. Therefore, in this paper, we investigate the use of VQMs to improve ∆VMAF SUR(75%) estimation accuracy.

IV. ∆VMAF SUR(75%) PREDICTION PIPELINE USING VQMS

Building upon the work of [START_REF] Hadi Amirpour | Between Two and Six? Towards Correct Estimation of JND Step Sizes for VMAF-Based Bitrate Laddering[END_REF], we propose a new pipeline to further improve the estimation of ∆VMAF SUR(75%) . Illustrated in Fig. 3(b), we leverage VQMs on distorted video that is compressed with a fixed and optimal QP to enhance the estimation process. The rationale behind this idea is that quality metrics not only inherently incorporate video complexity features but may also provide additional insights to further improve the accuracy of the estimation. By integrating VQMs into the ∆VMAF SUR(75%) estimation framework, we aim to achieve more refined and precise predictions of the perceptual differences in video content. The complete pipeline consists of three stages: A) Incorporated VQMs, B) Optimal QP selection, and C) Regression based on VQMs on the optimal QP.

A. Incorporated VQMs

We leverage the following Video Quality Metrics (VQMs) to enhance ∆VMAF SUR(75%) estimation:

• Peak Signal-to-Noise Ratio (PSNR): Measures the difference between an original and distorted video, commonly used for assessing visual quality in compressed or reconstructed videos. • Structural Similarity Index (SSIM) [START_REF] Zhou Wang | Image Quality Assessment: From Error Visibility to Structural Similarity[END_REF]: Evaluates the similarity between reference and distorted images based on luminance, contrast, and structural aspects. • Multi-Scale Structural Similarity Index (MS-SSIM) [START_REF] Wang | Multiscale Structural Similarity for Image Quality Assessment[END_REF]:

Extends SSIM by assessing structural similarity at multiple scales, offering a comprehensive evaluation. • Video Multimethod Assessment Fusion (VMAF) [START_REF] Li | Toward a practical perceptual video quality metric[END_REF]: Considers various visual factors through machine learning and large-scale subjective datasets to predict humanperceived video quality.

• FovVideoVDP [START_REF] Rafał | FovVideoVDP: A Visible Difference Predictor for Wide Field-of-View Video[END_REF]: Simultaneously addresses spatial, temporal, and peripheral perception aspects, integrating contrast sensitivity, cortical magnification, and contrast masking models.

B. Optimal QP selection

Due to the computational cost associated with encoding videos at multiple QPs and measuring the corresponding quality metrics, we propose to encode the video at a single fixed QP (qp) and calculate the VQMs specifically at that optimal QP.

The ultimate optimal QP is determined using a regression model via a brute-force approach. Among the evaluated QPs, we identify the one yielding the lowest Mean Absolute Error (MAE) after regression. As depicted in Fig. 4, illustrating the test set's MAE across various QPs utilizing a linear regression model with only VMAF as input, it becomes evident that the selection of QP notably influences the model's prediction accuracy. The results indicate that the lowest MAE is achieved at a QP value of 29 for this example.

C. Regression

Our prediction model for estimating ∆VMAF SUR(75%) can be represented as:

∆VMAF SUR(75%) = f (VMAFqp, SSIMqp, PSNRqp, MS-SSIMqp, VDPqp), ( 5
)
where qp is the optimal QP previously selected. Initially, a foundational regression model, i.e., ridge regression, with the complexity parameter α = 0.5 was applied. Furthermore, a more optimized machine learning regression model, XGBoost (eXtreme Gradient Boosting) [START_REF] Chen | XGBoost: A Scalable Tree Boosting System[END_REF], was employed using parameters n estimators = 100, max depth = 1, and booster = gbtree.

To evaluate the contribution and dependency of these features in predicting ∆VMAF SUR(75%) , we systematically eliminate the least important feature one by one until only one feature remains. We additionally conducted a comparative analysis of feature importance between the video complexity features outlined in [START_REF] Hadi Amirpour | Between Two and Six? Towards Correct Estimation of JND Step Sizes for VMAF-Based Bitrate Laddering[END_REF] and the optimal VQM features for predicting ∆VMAF SUR(75%) . Further details of the outcomes are provided in Section V.

V. EXPERIMENTAL RESULTS

The evaluation of the proposed method is conducted on VideoSet, which has been described in Section II. Prior to commencing the experiment, a preliminary data cleaning process was undertaken. As shown in Fig. 5 the box plot of the ∆VMAF SUR(75%) of raw data in VideoSet, which shows that there are few data points that deviate significantly from the majority of the dataset. These points have been removed to prevent these extreme values from distorting the overall analysis and interpretation of the data.

The remaining data points were divided into an 80% training set and a 20% test set. Train test split is conducted 20 times using 20 different random seeds. The reported results are the mean values obtained from these multiple splits.

The results are summarized in Table I. When utilizing a fixed ∆VMAF SUR(75%) of 2 [START_REF] Kah | Fundamental Relationships Between Subjective Quality, User Acceptance, and the VMAF Metric for A Quality-based Bit-rate Ladder Design for Overthe-Top Video Streaming Services[END_REF], the MAE is 4.29. Increasing the fixed ∆VMAF SUR(75%) to 6 [START_REF] Ozer | Finding the Just Noticeable Difference with Netflix VMAF[END_REF] results in a reduced MAE of 2.59. The state-of-the-art method [START_REF] Hadi Amirpour | Between Two and Six? Towards Correct Estimation of JND Step Sizes for VMAF-Based Bitrate Laddering[END_REF] achieves an MAE of 2.01. The results indicate that our proposed pipeline both ridge and XGBoost regression leads to lower MAE than SOTA. Utilizing XGBoost with a subset of VMAF 28 , SSIM 28 can further reduce the MAE to 1.67. The overall results presented in Table I demonstrate that incorporating quality metrics of compressed video with optimal QP can significantly reduce the MAE in ∆VMAF SUR(75%) modeling.

Fig 6 portrays the feature importance of both the optimal VQM features and the video complexity features outlined in [START_REF] Hadi Amirpour | Between Two and Six? Towards Correct Estimation of JND Step Sizes for VMAF-Based Bitrate Laddering[END_REF]. This importance is derived from the absolute values of the coefficients associated with each feature within the ridge regression model. The results clearly indicate that VMAF 28 exerts the most substantial influence on the prediction model. Moreover, the VQM features demonstrate notably greater impact on ∆VMAF SUR(75%) prediction compared to the video complexity features. This finding underscores the notion that Fig. 6: Feature importance of optimal VQM features and video complexity features in [START_REF] Hadi Amirpour | Between Two and Six? Towards Correct Estimation of JND Step Sizes for VMAF-Based Bitrate Laddering[END_REF] VQMs not only inherently encompass video complexity features, but also furnish supplementary information, enhancing the predictive accuracy.

VI. CONCLUSION

In this paper, we focused on modeling the ∆VMAF SUR(75%) . The debate surrounding the use of a fixed ∆VMAF SUR(75%) value of 2 or 6 prompted us to explore the content-dependent nature of ∆VMAF SUR(75%) as relying on a fixed value can lead to significant MAE. Additionally, while the state-of-the-art approach considers video complexity, our study delved into the influence of various VQMs, including VMAF, SSIM, VDP, PSNR, and MS-SSIM, on predicting the ∆VMAF SUR(75%) of the first JND. Our findings indicated that calculating these VQMs at a QP of the range 28-30 (depending on the regression model and the selected features) yielded the lowest MAE. We also investigated the interdependency of these features when modeling the ∆VMAF SUR(75%) and discovered that using a subset of VMAF 28 , SSIM 28 resulted in the lowest MAE. This finding suggests that the computation of MS-SSIM, VDP and PSNR may not be necessary to accurately predict the ∆VMAF SUR(75%) . Based on these results, we therefore recommend utilizing VMAF and SSIM quality metrics calculated at a QP of 28 to predict the content-dependent ∆VMAF SUR(75%) .

VII. ACKNOWLEDGE

Fig. 1 :

 1 Fig. 1: Demonstration of SUR curve for the VMAF proxy. distribution of JND obtained from subjective testing. Notably, SUR can be regarded as the inverse of the Cumulative Distribution Function (CDF) associated with the individual JND distribution. In other words, SUR effectively represents the Complementary Cumulative Distribution Function (CCDF).The variable x serves as a proxy for the SUR curve, encompassing encoding parameters (such as QP, CRF), VMAF scores, bitrates, and other relevant factors tailored to diverse application scenarios[START_REF] Zhu | Elevating Your Streaming Experience with Just Noticeable Difference (JND)-based Encoding[END_REF].In this paper, we focus on VMAF as the proxy of the SUR curve, because of its codec-independence and superior performance, which is highly correlated with human perception. As shown in Fig.1, the transition from perceptually lossless to perceptually lossy coding of the VMAF proxy for p% of SUR value is expressed as:

Fig. 3 :

 3 Fig. 3: Comparison of SOTA [16] (a), and our proposed pipelines (b) for ∆VMAF SUR(75%) prediction.

Fig. 4 :

 4 Fig. 4: MAE vs. QP for ridge regression with only VMAF as input features

Fig. 5 :

 5 Fig. 5: The box plot of the ∆VMAF SUR(75%) . Data points falling below the lower threshold (Q 1 -1.5IQR) or exceeding the upper threshold (Q 3 +1.5IQR) are considered outliers and have been excluded, as indicated by the 'x' marker.

TABLE I

 I

	:	Experiment	results	of	different
	∆VMAF SUR(75%) estimation models on VideoSet 1080p.
	Model			Features	qp MAE
	Jan et al. [19]		∆VMAF SUR(75%) = 2	-	4.29
	Kah et al. [20]		∆VMAF SUR(75%) = 6	-	2.59
	Amirpour et al. [16]	SI, TI, E, h, L, c, fr	-	2.01
	Ridge regression		vmaf, ssim, vdp, psnr, ms-ssim	30	1.77
	Ridge regression		vmaf, ssim, vdp, psnr	30	1.77
	Ridge regression		vmaf, ssim, vdp	29	1.78
	Ridge regression		vmaf, vdp	29	1.78
	Ridge regression			vmaf	29	1.80
	XGBoost		vmaf, ssim, vdp, psnr, ms-ssim	28	1.73
	XGBoost		vmaf, ssim, vdp, psnr	28	1.74
	XGBoost		vmaf, ssim, vdp	28	1.73
	XGBoost		vmaf, ssim	28	1.67
	XGBoost			vmaf	28	1.72

We would like to express our sincere gratitude to Amazon Prime, Capacités, and Christian Doppler Laboratory ATHENA for their sponsorship.