Konstantin Zimenko
email: kostyazimenko@gmail.com

| Denis Efimov

Andrey Polyakov

Adaptive Finite-time and Fixed-time Control Design using Output Stability Conditions †

Keywords: Adaptive finite-time control, adaptive fixed-time control, output finite-time stability

The present paper provides a sufficient condition to ensure output finite-time and fixed-time stability. Comparing with analogous researches the proposed result is less restrictive and obtained for a wider class of systems. The presented output stability condition is used for adaptive control design, where the state vector of a plant is extended by adjustable control parameters.

INTRODUCTION

Frequently, the control practice needs regulation algorithms, which ensure output (in particular, a part of states) convergence in a finite time (i.e., the output 𝑌 (𝑡, 𝑥 0 ) = 0 for all 𝑡 ≥ 𝑇 (𝑥 0 ) and some 0 ≤ 𝑇 (𝑥 0 ) < +∞ dependent on the initial conditions 𝑥(0) = 𝑥 0 ) or in a fixed time (i.e., 𝑇 (𝑥 0 ) ≤ 𝑇 max for all initial conditions 𝑥 0 ). Such problem statements usually appear in mechanical and robotic systems, aerospace applications, particle collision systems (see, for example, [START_REF] Vorotnikov | Partial Stability and Control[END_REF][START_REF] Rumyantsev | Stability and stabilization of motion with respect to part of variables[END_REF][START_REF] Fradkov | Pogromsky, Introduction to oscillations and chaos[END_REF][START_REF] Fradkov | Nonlinear and adaptive control of complex systems[END_REF][START_REF] Hai-Ping | On the stability of nonholonomic mechanical systems with respect to partial variables[END_REF][START_REF] Shiriaev | Stabilization of invariant sets for nonlinear non-affine systems[END_REF][START_REF] Haddad | Finite-time partial stability and stabilization, and optimal feedback control[END_REF] ).

The output stabilization is a rather common control issue, certain classes of identification problems and adaptive control systems may be considered in the context of output stability. For example [START_REF] Narendra | Stable adaptive systems[END_REF] , in the case of adaptive control design a closed-loop system has a state vector extended with adjustable control parameters:

[ 𝑥 𝑇 𝜔 𝑇 ] 𝑇 , where 𝑥 ∈ ℝ 𝑛 is the state vector of the plant and 𝜔 ∈ ℝ 𝑟 is the adjustable control parameters vector. Thus, a standard control goal is to guarantee output (partial) stability: the states of the plant should be stabilized at the origin asymptotically or in a finite time, while the adjustable parameters may remain just bounded. A similar output stabilization problem arises in state observer design, where the common dynamics of the system includes the plant and the observer states, 𝑥 and x, respectively, while it is necessary to ensure the convergence of the state estimation error 𝑒 = 𝑥 -x (in nonlinear case the dynamics of 𝑒 may be dependent on 𝑥) [START_REF] Arcak | Nonlinear observers: a circle criterion design and robustness analysis[END_REF][START_REF] Andrieu | LMI Sufficient conditions for contraction and synchronization[END_REF] .

There are a number of results devoted to output finite-time stability (OFTS) analysis. Most of them are about partial stability analysis that is a particular case of output stability (see, for example, [START_REF] Haddad | Finite-time partial stability and stabilization, and optimal feedback control[END_REF][START_REF] Jammazi | A discussion on the Hölder and robust finite-time partial stabilizability of Brockett's integrator[END_REF][START_REF] Jammazi | Continuous and discontinuous homogeneous feedbacks finite-time partially stabilizing controllable multichained systems[END_REF][START_REF] Sun | Fast finite-time stability and its application in adaptive control of high-order nonlinear system[END_REF] ). In [START_REF] Zimenko | On necessary and sufficient conditions for output finite-time stability[END_REF] necessary and sufficient conditions for output finite-time stability are given using Lyapunov functions. However, most of these results are obtained for special classes of systems, and/or particular control problems. The present paper provides a relaxed sufficient condition to analyze OFTS or OFxTS of a wider class of models than in [START_REF] Zimenko | On necessary and sufficient conditions for output finite-time stability[END_REF] .

A finite-time (fixed-time) control (e.g., homogeneity based) can ensure useful properties such as faster convergence, higher accuracy, and better disturbance rejection (see, for example, [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF][START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF][START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF] ). Adaptive finite/fixed-time control is one of the rapidly developing areas of control theory providing an appealing performance for systems with uncertainties (see, for example, [START_REF] Hong | Adaptive finite-time control of nonlinear systems with parametric uncertainty[END_REF][START_REF] Utkin | Adaptive Sliding Mode Control[END_REF][START_REF] Guzman | A new finite-time convergent and robust direct model reference adaptive control for SISO linear time invariant systems[END_REF][START_REF] Basin | Adaptive uniform finite-/fixed-time convergent second-order sliding-mode control[END_REF][START_REF] Sun | Fast finite-time stability and its application in adaptive control of high-order nonlinear system[END_REF] and references therein). Despite a number of available results, the basic problem of adaptive finite/fixed-time regulation still has no solution. In this paper, based on the new OFTS and output fixed-time stability (OFxTS) conditions, a scheme of adaptive finite/fixed-time control design is presented. The proposed scheme allows to combine with an adaptive term different finite-time (fixed-time) control algorithms 𝑢 𝐹 𝑇 𝑆 (𝑢 𝐹 𝑥𝑇 𝑆 ) designed for systems without parametric uncertainties. The developed adaptive control scheme guarantees that for the state of uncertain systems the desired non-asymptotic convergence can be recovered. There is no requirement on persistence of excitation.

The paper is organized in the following way. Notation used in the paper is given in Section 2. Section 3 recalls basics on output global asymptotic stability (oGAS), OFTS, OFxTS and homogeneity property. Section 4 presents the main results on sufficient condition of OFTS/OFxTS and adaptive control design with numerical examples. Finally, concluding remarks are given in Section 5.

NOTATION

• ℝ 𝑛 denotes the 𝑛 dimensional Euclidean space with vector norm | ⋅ |;

• ℝ + = {𝑥 ∈ ℝ ∶ 𝑥 > 0},
where ℝ is the field of real numbers, and ℕ is the set of natural numbers;

• The symbol 1, 𝑚 is used to denote a sequence of integers 1, ..., 𝑚;

• A continuous function 𝜎 ∶ ℝ + ∪ {0} → ℝ + ∪ {0} belongs to class  if it is strictly increasing and 𝜎(0) = 0. It belongs to class  ∞ if it is also unbounded; • A continuous function 𝛽 ∶ ℝ + ∪ {0} × ℝ + ∪ {0} → ℝ + ∪ {0}
𝐷𝑉 (𝑥)𝑓 (𝑥) = lim 𝑡→0 + sup 𝑉 [𝑥 + 𝑡𝑓 ] -𝑉 (𝑥) 𝑡 .

PRELIMINARIES

Consider a system in the form ẋ = 𝑓 (𝑥), 𝑦 = ℎ(𝑥) (1) with states 𝑥 ∈ ℝ 𝑛 and outputs 𝑦 ∈ ℝ 𝑝 . Let the system satisfy the following assumptions:

(A.1) The vector field 𝑓 ∶ ℝ 𝑛 → ℝ 𝑛 ensures forward existence and uniqueness of the system solutions at least locally in time,

𝑓 (0) = 0. (A.2) The function ℎ ∶ ℝ 𝑛 → ℝ 𝑝 is continuously differentiable, ℎ(0) = 0. (A.3) The vector field 𝑓 ∶ ℝ 𝑛 → ℝ 𝑛 is locally Lipschitz continuous on ℝ 𝑛 ⧵ , where  = {𝑥 ∈ ℝ 𝑛 ∶ ℎ(𝑥) = 0}.
For the initial conditions 𝑥 0 ∈ ℝ 𝑛 , let Φ(𝑡, 𝑥 0 ) be a unique maximal solution of the system (1) defined over an interval [0, 𝑇 𝑠 (𝑥 0 )) with some 𝑇 𝑠 (𝑥 0 ) ∈ ℝ + ∪ {+∞} (the solutions are understood in the Filippov sense [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF] ), 𝑌 (𝑡, 𝑥 0 ) = ℎ(Φ(𝑡, 𝑥 0 )).

Below we study only global stability and attractivity properties of the system (1). The local counterparts can be obtained by a direct restriction of the domain of validity for the presented conditions. Note that the preliminaries in this subsection are based on theoretical framework of Input-to-Output Stability and uniform oGAS presented for locally Lipschitz continuous systems in [START_REF] Angeli | Forward completeness, unboundedness observability, and their Lyapunov characterizations[END_REF][START_REF] Dashkovskiy | Input to State Stability and Allied System Properties[END_REF][START_REF] Ingalls | On Input-to-Output Stability for Systems not Uniformly Bounded[END_REF][START_REF] Sontag | Lyapunov Characterizations of Input to Output Stability[END_REF][START_REF] Sontag | A Notion of Input to Output Stability[END_REF][START_REF] Sontag | Notions of Input to Output Stability[END_REF] . In 14 the results on oGAS were extended for a wider class of dynamics, where the Lipschitz continuity may be violated on  (see Assumption A.3).

Definition 1 16, [START_REF] Ingalls | On Input-to-Output Stability for Systems not Uniformly Bounded[END_REF] The system (1) is forward complete if for each 𝑥 0 ∈ ℝ 𝑛 it produces a solution Φ(𝑡, 𝑥 0 ) which is defined on [0, +∞), i.e., 𝑇 𝑠 (𝑥 0 ) = +∞.

Definition 2 [START_REF] Angeli | Forward completeness, unboundedness observability, and their Lyapunov characterizations[END_REF] The system (1) has the unboundedness observability (UO) property if, for each 𝑥 0 such that 𝑇 𝑠 (𝑥 0 ) < +∞, necessarily lim sup

𝑡→𝑇 𝑠 (𝑥 0 ) | | 𝑌 (𝑡, 𝑥 0 ) | | = +∞. (2)
In other words, any unboundedness of the state vector can be observed using the output 𝑦. Hence, if the output is known to be bounded (which is the case under the output stability properties described below), then the UO property is equivalent to forward completeness [START_REF] Ingalls | On Input-to-Output Stability for Systems not Uniformly Bounded[END_REF] . Note, that any system has UO property in the output ℎ(𝑥) = 𝑥. Definition 3 18, [START_REF] Sontag | A Notion of Input to Output Stability[END_REF] A system (1) is oGAS if

• it is forward complete, and

• there exists a -function 𝛽 such that

|𝑌 (𝑡, 𝑥 0 )| ≤ 𝛽(|𝑥 0 |, 𝑡) ∀𝑡 ≥ 0 (3)
holds for all 𝑥 0 ∈ ℝ 𝑛 .

If, in addition, there exists 𝜎 ∈  such that

|𝑌 (𝑡, 𝑥 0 )| ≤ 𝜎(|ℎ(𝑥 0 )|) ∀𝑡 ≥ 0 (4)
holds for all 𝑥 0 ∈ ℝ 𝑛 , then the system is output-Lagrange output globally asymptotically stable (OLoGAS). Finally, if one strengthens (3) to

|𝑌 (𝑡, 𝑥 0 )| ≤ 𝛽(|ℎ(𝑥 0 )|, 𝑡), ∀𝑡 ≥ 0 (5)
for all 𝑥 0 ∈ ℝ 𝑛 , then the system is state-independent output globally asymptotically stable (SIoGAS). Lemma 1 [START_REF] Sontag | Notions of Input to Output Stability[END_REF] For system (1) having the UO property, the following relations are valid:

𝑆𝐼𝑜𝐺𝐴𝑆 ⇒ 𝑂𝐿𝑜𝐺𝐴𝑆 ⇒ 𝑜𝐺𝐴𝑆.
In

for all 𝑡 ≥ 0 and 𝑥 ∈ .

The function 𝑉 is called an OLoGAS-Lyapunov function if it is an oGAS-Lyapunov function, and in addition, inequality (6) can be strengthened to

𝜉 1 (|ℎ(𝑥)|) ≤ 𝑉 (𝑥) ≤ 𝜉 2 (|ℎ(𝑥)|), ∀𝑥 ∈ ℝ 𝑛 . ( 10
)
The function 𝑉 is called the SIoGAS-Lyapunov function if the inequality (10) is satisfied and there exists 𝜉 3 ∈  such that for all 𝑥 ∈ :

𝐷𝑉 (𝑥)𝑓 (𝑥) ≤ -𝜉 3 (𝑉 (𝑥)). (11)
An auxiliary modulus 𝜆 satisfying property (a) is called a strong auxiliary modulus, and one satisfying property (b) is a weak auxiliary modulus [START_REF] Ingalls | On Input-to-Output Stability for Systems not Uniformly Bounded[END_REF] . A strong modulus is a weak modulus with 𝜃(𝑠) = 𝑠 (see [START_REF] Ingalls | On Input-to-Output Stability for Systems not Uniformly Bounded[END_REF] ). Note that in the case of OLoGAS-or SIoGAS-Lyapunov function we have  = ℝ 𝑛 ⧵ . Remark 1 In [START_REF] Angeli | Forward completeness, unboundedness observability, and their Lyapunov characterizations[END_REF][START_REF] Dashkovskiy | Input to State Stability and Allied System Properties[END_REF][START_REF] Ingalls | On Input-to-Output Stability for Systems not Uniformly Bounded[END_REF][START_REF] Sontag | Lyapunov Characterizations of Input to Output Stability[END_REF][START_REF] Sontag | A Notion of Input to Output Stability[END_REF][START_REF] Sontag | Notions of Input to Output Stability[END_REF] all given above definitions are presented in the sense of uniform stability with respect to inputs 𝑢 for the system ẋ = 𝑓 (𝑥, 𝑢), 𝑦 = ℎ(𝑥).

The following theorem gives the necessary and sufficient Lyapunov characterizations of output stability for the system (1).

Theorem 1 [START_REF] Zimenko | On necessary and sufficient conditions for output finite-time stability[END_REF] Suppose the system (1) is UO.

(1) The following claims are equivalent for the system:

(a) it is OLoGAS;

(b) it admits an OLoGAS-Lyapunov function with a weak auxiliary modulus;

(c) it admits an OLoGAS-Lyapunov function with a strong auxiliary modulus.

(2) The following claims are equivalent for the system:

(a) it is SIoGAS;

(b) it admits a SIoGAS-Lyapunov function.

Output Finite-Time Stability

Let us present the definition of the output finite-time stability. Definition 5 The system ( 1) is said to be OFTS if it is oGAS and for any 𝑥 0 ∈ ℝ 𝑛 there exists 0 ≤ 𝑇 0 < +∞ such that 𝑌 (𝑡, 𝑥 0 ) = 0 for all 𝑡 > 𝑇 0 . The function 𝑇 (𝑥 0 ) = inf{𝑇 0 ≥ 0 ∶ 𝑌 (𝑡, 𝑥 0 ) = 0 ∀𝑡 ≥ 𝑇 0 } is called the settling-time function. Definition 6 The system ( 1) is said to be OFxTS if it is OFTS and sup 𝑥 0 ∈ℝ 𝑛 𝑇 (𝑥 0 ) < +∞.

Definition 7

The set 𝑀 is said to be finite-time attractive for (1) if any solution Φ(𝑡, 𝑥 0 ) of (1) reaches 𝑀 in a finite instant of time 𝑡 = 𝑇 𝑀 (𝑥 0 ) and remains there ∀𝑡 ≥ 𝑇 𝑀 (𝑥 0 ). As before,

𝑇 𝑀 ∶ ℝ 𝑛 → ℝ + ∪ {0} is a settling-time function. The set 𝑀 is fixed-time attractive if sup 𝑥 0 ∈ℝ 𝑛 𝑇 (𝑥 0 ) < +∞.
The paper 8 deals with partial finite-time stability that is a particular case of OFTS: Theorem 2 [START_REF] Haddad | Finite-time partial stability and stabilization, and optimal feedback control[END_REF] Consider the system

ẋ1 = 𝑓 1 (𝑥 1 , 𝑥 2 ), 𝑥 1 (0) = 𝑥 10 , ẋ2 = 𝑓 2 (𝑥 1 , 𝑥 2 ), 𝑥 2 (0) = 𝑥 20 , ( 12
)
where

𝑥 1 ∈ ℝ 𝑛 1 , 𝑥 2 ∈ ℝ 𝑛 2 are the states, 𝑓 1 ∶ ℝ 𝑛 1 × ℝ 𝑛 2 → ℝ 𝑛 1 and 𝑓 2 ∶ ℝ 𝑛 1 × ℝ 𝑛 2 → ℝ 𝑛 2 are such that, for every (𝑥 1 , 𝑥 2 ) ∈ ℝ 𝑛 1 × ℝ 𝑛 2 , 𝑓 1 (0, 𝑥 2 ) = 0 and 𝑓 1 (⋅, ⋅), 𝑓 2 (⋅, ⋅) are jointly continuous in 𝑥 1 and 𝑥 2 . If there exist a continuously differentiable function 𝑉 ∶ ℝ 𝑛 1 × ℝ 𝑛 2 → ℝ, a class  functions 𝛼 and 𝛽, a continuous function 𝑘 ∶ ℝ + ∪ {0} → ℝ + , a real number 𝜇 ∈ (0, 1) such that for (𝑥 1 , 𝑥 2 ) ∈ ℝ 𝑛 1 × ℝ 𝑛 2 𝛼(|𝑥 1 |) ≤ 𝑉 (𝑥 1 , 𝑥 2 ) ≤ 𝛽(|𝑥 1 |), (13)
𝐷𝑉 (𝑥 1 , 𝑥 2 ) [ 𝑓 1 (𝑥 1 ,𝑥 2 ) 𝑓 2 (𝑥 1 ,𝑥 2 ) ] ≤ -𝑘(|𝑥 2 |)𝑉 (𝑥 1 , 𝑥 2 ) 𝜇 ( 14
)
then for 𝑦 = 𝑥 1 the system ( 12) is OFTS uniformly in 𝑥 20 . Moreover, there exists a settling-time function

𝑇 ∶ ℝ 𝑛 1 × ℝ 𝑛 2 → [0, ∞) such that 𝑇 (𝑥 10 , 𝑥 20 ) ≤ 𝑞 -1 ( 𝑉 (𝑥 10 , 𝑥 20 ) 1-𝜇 1 -𝜇 ) , (𝑥 10 , 𝑥 20 ) ∈ ℝ 𝑛 1 × ℝ 𝑛 2 ,
where 𝑞 ∶ [0, ∞) → ℝ is continuously differentiable and satisfies

q(𝑡) = 𝑘(|𝑥 2 (𝑡)|), 𝑞(0) = 0, 𝑡 ≥ 0,
and

𝑇 (⋅, ⋅) is jointly continuous on ℝ 𝑛 1 × ℝ 𝑛 2 .
In the paper [START_REF] Zimenko | On necessary and sufficient conditions for output finite-time stability[END_REF] necessary and sufficient Lyapunov characterizations of output finite-time stability are presented for the class of OLoGAS and SIoGAS systems (1). The following lemma on OFTS property is used in the paper.

Lemma 2 [START_REF] Zimenko | On necessary and sufficient conditions for output finite-time stability[END_REF] Consider a forward complete system (1). Let a SIoGAS-Lyapunov function satisfies the inequality

𝐷𝑉 (𝑥)𝑓 (𝑥) ≤ -𝑐𝑉 (𝑥) 𝜇 ( 15
)
for some 𝑐 ∈ ℝ + , 𝜇 ∈ (0, 1) and all 𝑥 ∈ ℝ 𝑛 ⧵ . Then the system ( 1) is SIoGAS and OFTS. Moreover, the settling-time function satisfies 𝑇 (𝑥) ≤ 1 𝑐(1-𝜇) 𝑉 (𝑥) 1-𝜇 . Similarly, the result on fixed-time attractivity of a set can be presented: Lemma 3 Consider a forward complete system (1). Let a SIoGAS-Lyapunov function satisfies the inequality (15) for some 𝑐 ∈ ℝ + , 𝜇 > 1 and all 𝑥 ∈ ℝ 𝑛 ⧵ . Then the system ( 1) is SIoGAS and, for every

𝜀 ∈ ℝ + , the set 𝐵 = {𝑥 ∈ ℝ 𝑛 ∶ 𝑉 (𝑥) < 𝜀} is fixed-time attractive with 𝑇 max = 1 𝑐(𝜇-1)𝜀 𝜇-1 .
Proof. Since 𝑉 (𝑥) is a SIoGAS-Lyapunov function we have that 𝑉 (𝑥) > 0 for ℎ(𝑥) ≠ 0 and 𝑉 (0) = 0. Then the claim is straightforward from (15) with 𝜇 > 1.

Based on Lemmas 2, 3 and 29 the following result extends the output Lyapunov function method providing the background for OFxTS analysis (in 37 partial fixed-time stability is studied using similar arguments):

Corollary 1 Consider a forward complete system (1). Let a SIoGAS-Lyapunov function satisfies the inequality

𝐷𝑉 (𝑥)𝑓 (𝑥) ≤ -𝑘 1 𝑉 (𝑥) 𝜇 -𝑘 2 𝑉 (𝑥) 𝜈 ( 16
)
for some 𝑘 1 , 𝑘 2 ∈ ℝ + , 𝜇 ∈ (0, 1), 𝜈 > 1 and all 𝑥 ∈ ℝ 𝑛 ⧵ . Then the system ( 1) is SIoGAS and OFxTS with

𝑇 (𝑥 0 ) ≤ 1 𝑘 1 (1 -𝜇) + 1 𝑘 2 (𝜈 -1)
.

Proof. Due to (16) we have

V (𝑥) ≤ { -𝑘 1 𝑉 (𝑥) 𝜇 for 𝑉 (𝑥) ≤ 1 -𝑘 2 𝑉 (𝑥) 𝜈 for 𝑉 (𝑥) > 1 .
Hence, for any 𝑥 0 such that 𝑉 (𝑥 0 ) > 1 the last inequality guarantees the set {𝑥 ∈ ℝ 𝑛 ∶ 𝑉 (𝑥) ≤ 1} will be reached in a time [START_REF] Zimenko | On Condition for Output Finite-Time Stability and Adaptive Finite-Time Control Scheme[END_REF] . For 𝑉 (𝑥 0 ) ≤ 1 by Lemma 2 we derive 𝑦(𝑡, 𝑥 0 ) = 0 for 𝑡 ≥ 1 𝑘 1 (1-𝜇) . Therefore, the system (1) is OFxTS and [START_REF] Zimenko | On Condition for Output Finite-Time Stability and Adaptive Finite-Time Control Scheme[END_REF] and ∀𝑥 0 ∈ ℝ 𝑛 .

𝑡 0 ≤ 1 𝑘 2 (𝜈-
𝑦(𝑡, 𝑥 0 ) = 0 for all 𝑡 ≥ 1 𝑘 1 (1-𝜇) + 1 𝑘 2 (𝜈-

Homogeneity

Homogeneity [START_REF] Zubov | On systems of ordinary differential equations with generalized homogeneous right-hand sides[END_REF][START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF] is an intrinsic property of an object, which remains consistent with respect to some scaling. This property provides many advantages to analysis and design of nonlinear control system (including finite-time stability studies).

For

𝑟 𝑖 ∈ ℝ + , 𝑖 = 1, 𝑛, 𝜌 ∈ ℝ + and 𝜆 ∈ ℝ + define vector of weights 𝑟 = [ 𝑟 1 , … , 𝑟 𝑛 ] 𝑇 , dilation matrix 𝐷 𝑟 (𝜆) = diag{𝜆 𝑟 𝑖 } 𝑛 𝑖=1
and homogeneous norm

‖𝑥‖ 𝑟 = ( 𝑛 ∑ 𝑖=1 |𝑥 𝑖 | 𝜌 𝑟 𝑖 ) 1 𝜌 . ( 17
)
Definition 8 [START_REF] Zubov | On systems of ordinary differential equations with generalized homogeneous right-hand sides[END_REF] A function

𝑔 ∶ ℝ 𝑛 → ℝ (vector field 𝑓 ∶ ℝ 𝑛 → ℝ 𝑛 ) is said to be 𝑟-homogeneous of degree 𝑑 ∈ ℝ if 𝑔(𝐷 𝑟 (𝜆)𝑥) = 𝜆 𝑑 𝑔(𝑥) (𝑓 (𝐷 𝑟 (𝜆)𝑥) = 𝜆 𝑑 𝐷 𝑟 (𝜆)𝑓 (𝑥))
for fixed 𝑟, all 𝜆 > 0 and 𝑥 ∈ ℝ 𝑛 .

Introduce the following compact set (homogeneous sphere) 𝕊 𝑟 = {𝑥 ∈ ℝ 𝑛 ∶ ‖𝑥‖ 𝑟 = 1}, then for any 𝑥 ∈ ℝ 𝑛 there is 𝑧 ∈ 𝕊 𝑟 such that 𝑥 = 𝐷 𝑟 (𝜆)𝑧 for 𝜆 = ‖𝑥‖ 𝑟 .

Theorem 3 [START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF][START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF] Let 𝑓 ∶ ℝ 𝑛 → ℝ 𝑛 be defined on ℝ 𝑛 and be a continuous 𝑟-homogeneous vector field with degree 𝜈 (𝜈 < 0). If the origin of the system ẋ = 𝑓 (𝑥) is locally asymptotically stable then it is globally asymptotically stable (globally finite-time stable) and for any 𝜇 > max{0, -𝜈} there exists a continuously differentiable Lyapunov function 𝑉 which is 𝑟-homogeneous with the degree 𝜇. As a direct consequence, the derivative 𝐷𝑉 (𝑥)𝑓 (𝑥) is 𝑟-homogeneous of degree 𝜇 + 𝜈.

According to [START_REF] Zubov | On systems of ordinary differential equations with generalized homogeneous right-hand sides[END_REF][START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF] for such a 𝑉 there exist constants 𝑐 1 , 𝑐 2 , ā, 𝑏 ∈ ℝ + , such that

𝑐 1 ‖𝑥‖ 𝜇 𝑟 ≤ 𝑉 (𝑥) ≤ 𝑐 2 ‖𝑥‖ 𝜇 𝑟 ∀𝑥 ∈ ℝ 𝑛 , ( 18
)
𝜕𝑉 (𝑧) 𝜕𝑧 𝑓 (𝑧) ≤ -ā, | | | | 𝜕𝑉 (𝑧) 𝜕𝑧 | | | | ≤ 𝑏 ∀𝑧 ∈ 𝕊 𝑟 . ( 19
)
A nonlinear system ẋ = 𝑓 (𝑥, 𝑢) is homogeneously stabilizable with degree 𝜈 ∈ ℝ if there exists a feedback 𝑢(𝑥) such that the closed-loop system is homogeneous of degree 𝜈 and globally asymptotically stable. In this case the feedback 𝑢(𝑥) is called homogenizing of degree 𝜈.

MAIN RESULT

On Sufficient Condition for Output Finite-Time Stability

Consider the system in the form ẋ = 𝑓 (𝑥), 𝑦 = ℎ(𝑥) (20) where 𝑥 ∈ ℝ 𝑛 is state vector, 𝑦 ∈ ℝ 𝑝 is output, the vector field 𝑓 ∶ ℝ 𝑛 → ℝ 𝑛 ensures forward existence and uniqueness of the system solutions at least locally in time, the function ℎ ∶ ℝ 𝑛 → ℝ 𝑝 is continuous, 𝑓 (0) = 0 and ℎ(0) = 0. Remark 2 Note that assumptions A.2 and A.3 are relaxed for the system (20). Thus, the system under consideration is of a wider class than in [START_REF] Zimenko | On necessary and sufficient conditions for output finite-time stability[END_REF][START_REF] Angeli | Forward completeness, unboundedness observability, and their Lyapunov characterizations[END_REF][START_REF] Dashkovskiy | Input to State Stability and Allied System Properties[END_REF][START_REF] Ingalls | On Input-to-Output Stability for Systems not Uniformly Bounded[END_REF][START_REF] Sontag | Lyapunov Characterizations of Input to Output Stability[END_REF][START_REF] Sontag | A Notion of Input to Output Stability[END_REF][START_REF] Sontag | Notions of Input to Output Stability[END_REF] .

The following theorem provides a sufficient condition for output finite-time stability of the system (20). In contrast to all previously given results, which utilize OLoGAS and SIoGAS Lyapunov functions, here a more general oGAS scenario is considered.

Theorem 4 Let there exist differentiable on

ℝ 𝑛 ⧵  functions 𝑈 ∶ ℝ 𝑛 → ℝ + ∪ {0} and 𝑊 ∶ ℝ 𝑛 → ℝ + ∪ {0} such that for 𝜉 1 , 𝜉 2 ∈  ∞ the following conditions are satisfied 𝜉 1 (|ℎ(𝑥)|) ≤ 𝑈 (𝑥) ≤ 𝜉 2 (|ℎ(𝑥)|), ∀𝑥 ∈ ℝ 𝑛 , ( 21
)
𝑉 (𝑥) = 𝑈 (𝑥) + 𝑊 (𝑥), (22)
𝐷𝑉 (𝑥)𝑓 (𝑥) ≤ -𝑎𝑈 (𝑥) 𝛼 , ∀𝑥 ∈ ℝ 𝑛 ∖, (23)
|𝐷𝑊 (𝑥)𝑓 (𝑥)| ≤ 𝑁 ∑ 𝑖=1 𝑏 𝑖 𝑈 (𝑥) 𝛽 𝑖 , ∀𝑥 ∈ ℝ 𝑛 ∖, (24)
where 𝑎, 𝑏 𝑖 ∈ ℝ + , 𝛽 𝑖 > 𝛼, 𝛼 ∈ (0, 1), 𝑖 = 1, 𝑁, 𝑁 ∈ ℕ. Then the system (20) is OFTS provided that it is UO.

Proof. Since the inequality ( 23) is satisfied, then 𝐷𝑉 (𝑥)𝑓 (𝑥) ≤ 0 that due to non-negative definiteness of 𝑊 and (21) implies boundedness of the output 𝑦. Hence, by introduced assumptions the trajectories of the system are unique and defined for all 𝑡 ≥ 0.

For some sufficiently small 𝜗 ∈ ℝ + the inequalities ( 22)- (24) imply that

𝐷𝑈 (𝑥)𝑓 (𝑥) ≤ -𝑎𝑈 (𝑥) 𝛼 -𝐷𝑊 (𝑥)𝑓 (𝑥) ≤ -𝑎𝑈 (𝑥) 𝛼 + ∑ 𝑁 𝑖=1 𝑏 𝑖 𝑈 (𝑥) 𝛽 𝑖 < 0 (25)
for all 𝑥 ∈  = {ℝ 𝑛 ∖ ∶ 𝑈 (𝑥) ≤ 𝜗} due to 𝛼 < 𝛽 𝑖 , 𝑖 = 1, 𝑁, and due to (21) the set  is forward invariant. On the other hand, due to (23) we have 𝐷𝑉 (𝑥)𝑓 (𝑥) ≤ -𝑎𝜗 𝛼 for all 𝑥 ∈ ℝ 𝑛 ⧵ ( ∪ ), and due to (22) the set  will be reached in a finite time. Hence, we have that lim 𝑡→+∞ 𝑈 (𝑥(𝑡)) = 0. Moreover, it has been shown that for any 𝜀 > 0 and 𝛿 > 0 there is 𝑇 (𝜀, 𝛿) > 0 such that 𝑈 (𝑥(𝑡)) ≤ 𝜀 for all 𝑡 ≥ 𝑇 (𝜀, 𝛿) provided that |𝑥(0)| ≤ 𝛿 and, consequently, the system (20) is oGAS.

Since lim 𝑡→+∞ 𝑈 (𝑥(𝑡)) = 0, there exists an instant of time 𝜏 > 0 such that

-𝑎𝑈 (𝑥(𝑡)) 𝛼 + 𝑁 ∑ 𝑖=1 𝑏 𝑖 𝑈 (𝑥(𝑡)) 𝛽 𝑖 ≤ -0.5𝑎𝑈 (𝑥(𝑡)) 𝛼 ( 26
)
for all 𝑡 ≥ 𝜏 due to 𝛼 < 𝛽 𝑖 . Therefore, by Lemma 2 the function 𝑈 (𝑥) converges to 0 in a finite time. Then the system ( 20) is OFTS and the settling-time is bounded as follows:

𝑇 (𝑥 0 ) ≤ 𝜏 + 𝑈 1-𝛼 𝜏 0.5𝑎(1 -𝛼) ,
where 𝑈 𝜏 = 𝑈 (𝑥(𝜏)).

Note that choosing 𝑊 (𝑥) = 0 the conditions of Theorem 4 became similar to Theorem 2.

The following theorem provides a sufficient condition for output fixed-time stability of the system (20). Theorem 5 Let the conditions of Theorem 4 are satisfied with (23) replaced by

𝐷𝑉 (𝑥)𝑓 (𝑥) ≤ -𝑎 1 𝑈 (𝑥) 𝛼 1 -𝑎 2 𝑈 (𝑥) 𝛼 2 , ∀𝑥 ∈ ℝ 𝑛 ∖, (27)
and

𝑊 (𝑥) ≤ 𝜎(𝜌 + |ℎ(𝑥)|), (28)
where 𝑎 1 , 𝑎 2 ∈ ℝ + , 𝛼 1 ∈ (0, 1), 𝛼 2 > 1, and 𝛽 𝑖 , 𝑖 = 1, 𝑁 in (24) satisfies 𝛽 𝑖 ∈ (𝛼 1 , 𝛼 2 ), 𝜎 ∈  ∞ and 𝜌 ∈ ℝ + ∪ {0}. Then the system (20) is OFxTS provided that it is UO.

Proof. Analogously to the proof of Theorem 4 one can show that lim 𝑡→+∞ 𝑈 (𝑥(𝑡)) = 0, and the system (20) is oGAS. Then due to (27) we have

𝐷𝑈 (𝑥)𝑓 (𝑥) ≤ -𝑎 1 𝑈 (𝑥) 𝛼 1 -𝑎 2 𝑈 (𝑥) 𝛼 2 -𝐷𝑊 (𝑥)𝑓 (𝑥) ≤ -𝑎 1 𝑈 (𝑥) 𝛼 1 -𝑎 2 𝑈 (𝑥) 𝛼 2 + ∑ 𝑁 𝑖=1 𝑏 𝑖 𝑈 (𝑥) 𝛽 𝑖 ( 29
)
for all 𝑥 ∈ ℝ 𝑛 ∖. Since 𝛼 2 > 𝛽 𝑖 then there exists a constant 𝑈 𝜏 1 > 0 such that

-𝑎 1 𝑈 (𝑥) 𝛼 1 -𝑎 2 𝑈 (𝑥) 𝛼 2 + 𝑁 ∑ 𝑖=1 𝑏 𝑖 𝑈 (𝑥) 𝛽 𝑖 < -0.5𝑎 2 𝑈 (𝑥) 𝛼 2 for all 𝑥 ∈ ℝ 𝑛 ∖{𝑀}, where 𝑀 = {𝑥 ∈ ℝ 𝑛 ∶ 𝑈 (𝑥) ≤ 𝑈 𝜏 1 }. Therefore, 𝐷𝑈 (𝑥)𝑓 (𝑥) ≤ - 𝑎 2 2 𝑈 (𝑥) 𝛼 2 for 𝑥 ∈ ℝ 𝑛 ∖{𝑀}.
Hence, due to Lemma 3 the set 𝑀 is fixed-time attractive with the following settling-time estimate

𝑇 𝑀 (𝑥 0 ) ≤ 1 0.5𝑎 2 (𝛼 2 -1)𝑈 𝛼 2 -1 𝜏 1
.

By the same arguments, there exists 𝑈 𝜏 2 ≤ 𝑈 𝜏 1 such that

-𝑎 1 𝑈 (𝑥) 𝛼 1 -𝑎 2 𝑈 (𝑥) 𝛼 2 + 𝑁 ∑ 𝑖=1 𝑏 𝑖 𝑈 (𝑥) 𝛽 𝑖 ≤ -0.5𝑎 1 𝑈 (𝑥) 𝛼 1 (30)
for all 𝑥 ∈ 𝐵 = {𝑥 ∈ ℝ 𝑛 ∶ 𝑈 (𝑥) ≤ 𝑈 𝜏 2 }. The function 𝑉 is uniformly bounded on 𝑀 ⧵ 𝐵 due to (28) and its time derivative is separated from zero due to (27). On 𝑀 ⧵ 𝐵 we have

𝑈 𝜏 2 ≤ 𝑉 (𝑥) ≤ 𝑘 1 , 𝐷𝑉 (𝑥)𝑓 (𝑥) ≤ -𝑘 2 ,
where

𝑘 1 = 𝑈 𝜏 1 + 𝜎 ( 𝜌 + 𝜉 -1 1 (𝑈 𝜏 1 ) ) , 𝑘 2 = 𝑎 1 𝑈 𝛼 1 𝜏 2 + 𝑎 2 𝑈 𝛼 2 𝜏 2 .
Hence, the set 𝐵 will be reached in a finite time 𝑡 ≤ 𝜏 2 , where

𝜏 2 ≤ 1 0.5𝑎 2 (𝛼 2 -1)𝑈 𝛼 2 -1 𝜏 1 + 𝑘 1 -𝑈 𝜏 2 𝑘 2 .
Finally, due to (30), the system (20) is OFxTS and the settling-time is bounded as follows (note that 𝑈 𝜏 1 , 𝑈 𝜏 2 , 𝑘 1 and 𝑘 2 do not depend on initial conditions, their values are completely predefined by the properties of 𝑉 , 𝑈 , 𝑊 and the system dynamics):

𝑇 (𝑥 0 ) ≤ 1 0.5𝑎 2 (𝛼 2 -1)𝑈 𝛼 2 -1 𝜏 1 + 𝑈 1-𝛼 1 𝜏 2 0.5𝑎 1 (1 -𝛼 1 ) + 𝑘 1 -𝑈 𝜏 2 𝑘 2 .
In Theorem 4, the values of 𝑏 𝑖 and 𝛽 𝑖 for 𝑖 = 1, 𝑁 may depend on 𝑉 (𝑥(0)). In Theorem 5, such a dependence is admitted for 𝛽 𝑖 , 𝑖 = 1, 𝑁 since they belong to a bounded interval (𝛼 1 , 𝛼 2 ), while for 𝑏 𝑖 , 𝑖 = 1, 𝑁 a dependence on 𝑉 (𝑥(0)) or 𝑊 (𝑥(0)) is allowed if there exists a uniform upper bound.

Remark 3 In general, none of the functions 𝑈 (𝑥), 𝑊 (𝑥), 𝑉 (𝑥) is an output Lyapunov function. However, with respect to Definition 4 we have: 26) (( 29) and (30) for Theorem 5) 𝑈 (𝑥) can be considered as local SIoGAS-Lyapunov function for 𝑥 0 ∈ .

• if 𝑉 (𝑥) is smooth and 𝑊 (𝑥) ≤ 𝜉 3 (|ℎ(𝑥)|) for 𝜉 3 ∈ , then 𝑉 (𝑥) is SIoGAS-Lyapunov function; • if 𝑈 (𝑥)
Example 1 Consider the system (20) with

𝑥 = [ 𝑥 1 𝑥 2 ] , 𝑓 (𝑥) = [ -sign(𝑥 1 )|𝑥 1 | 0.5 + 𝑥 2 2 𝑥 1 -|𝑥 1 | 1.5 𝑥 2 ]
, 𝑦 = 𝑥 1 .

The system admits UO property, and for 𝑈 (𝑥) = |𝑥 1 | 1.5 and 𝑊 (𝑥) = 0.75𝑥 2 2 the conditions ( 21)-( 24) are satisfied with 𝐷𝑉 (𝑥)𝑓 (𝑥) ≤ -1.5𝑈 (𝑥) 2∕3 and |𝐷𝑊 (𝑥)𝑓 (𝑥)| ≤ 1.5𝑥 2 (0) 2 𝑈 (𝑥) ≤ 2𝑉 (𝑥(0))𝑈 (𝑥). Then the system is OFTS.

Example 2 Consider the system ẋ1 = -sign(𝑥 1 )|𝑥 1 | 0.5 + 2𝑥 1 sin(𝑥 2 ) -sign(𝑥 1 )𝑥 [START_REF] Vorotnikov | Partial Stability and Control[END_REF] 1 , ẋ2 = |𝑥 1 | 1.5 + sin(𝑥 2 ) sin 2 (𝑥 1 ), 𝑦 = 𝑥 1 .

The system admits UO property (the right-hand side is bounded for a bounded value of 𝑦). For 𝑈 (𝑥) = |𝑥 1 | 1.5 and 𝑊 (𝑥) = 3(1 + cos(𝑥 2 )) the conditions of Theorem 5 are satisfied with 𝑎 1 = 𝑎 2 = 1.5, 𝑏 1 = 6, 𝛼 1 = 2∕3, 𝛼 2 = 5∕3 and 𝛽 1 = 1. Then the system is OFxTS. Note also that for this system there is no Lyapunov function purely dependent on 𝑥 1 guaranteeing a fixed-time convergence for this coordinate as in [START_REF] Sun | Fast finite-time stability and its application in adaptive control of high-order nonlinear system[END_REF] .

Adaptive Control Design

The presented result can be utilized for adaptive finite-time or fixed-time control design. Consider the system

ẋ(𝑡) = 𝑓 (𝑥(𝑡), 𝑢(𝑡), 𝜃), 𝑥(0) = 𝑥 0 , 𝑡 ≥ 0, (31)
where 𝑥(𝑡) ∈ ℝ 𝑛 is the measurable state vector, 𝑢(𝑡) ∈ ℝ 𝑚 is the control input, 𝜃 ∈ ℝ 𝑞 is the vector of unknown parameters and 𝑓 ∶ ℝ 𝑛 × ℝ 𝑚 × ℝ 𝑞 → ℝ 𝑛 . An adaptive control for the system (31) can be presented in the form 9

𝑢(𝑡) = 𝑔(𝑥(𝑡), 𝜔(𝑡)), ω(𝑡) = 𝜅(𝑥(𝑡)),

where 𝜔 ∈ ℝ 𝑞 is the vector of adjustable control parameters, 𝑔 and 𝜅 are mappings defined as 𝑔 ∶ ℝ 𝑛 × ℝ 𝑞 → ℝ 𝑚 and 𝜅 ∶ ℝ 𝑛 → ℝ 𝑞 (in contrast to [START_REF] Hong | Adaptive finite-time control of nonlinear systems with parametric uncertainty[END_REF][START_REF] Sun | Fast finite-time stability and its application in adaptive control of high-order nonlinear system[END_REF] we will not assume that 𝑞 = 1 and 𝜅(𝑥) is a nonnegative funciton). Then the problem of adaptive control design considered in this work is to provide output finite-time stability (finite-time partial stability) of the system (31), (32) with the output 𝑦 = 𝑥 and extended state vector x = [ 𝑥 𝑇 𝜔 𝑇 ] 𝑇 (i.e., we are interested only in convergence of 𝑥 to the origin). To demonstrate how the result of Theorem 4 can be utilized for adaptive finite-time control design let us consider the system (31) in the form

ẋ = 𝐴𝑥 + 𝐵 ( 𝜙(𝑥) 𝑇 𝜃 + 𝑢 ) , ( 33
)
where 𝑥 ∈ ℝ 𝑛 , 𝑢 ∈ ℝ, the pair of system matrix 𝐴 ∈ ℝ 𝑛×𝑛 and control gain matrix 𝐵 ∈ ℝ 𝑛×1 is controllable and 𝜙 ∶ ℝ 𝑛 → ℝ 𝑞 is known. Then following Theorem 4 one may obtain: Theorem 6 Let 𝑢 𝐹 𝑇 𝑆 ∶ ℝ 𝑛 → ℝ be a continuous feedback control such that the system

ẋ = 𝐴𝑥 + 𝐵𝑢 𝐹 𝑇 𝑆 (𝑥), (34)
is finite-time stable and 𝑟-homogeneous of degree 𝜈 < 0. Let 𝑉 𝐹 𝑇 𝑆 ∶ ℝ 𝑛 → ℝ + ∪ {0} be a continuously differentiable 𝑟homogeneous of degree 𝜇 (𝜇 > -𝜈) Lyapunov function for (34). Let |𝜙(𝑥)| < 𝑐‖𝑥‖ 𝛿 𝑟 for some 𝑐 ∈ ℝ + and 𝛿 > 𝜈 + 𝑟 max , 𝑟 max = max 1≤𝑗≤𝑛 𝑟 𝑗 . Then the system (33) with adaptive control

𝑢(𝑥, 𝜔) = 𝑢 𝐹 𝑇 𝑆 (𝑥) -𝜙(𝑥) 𝑇 𝜔 ω = 𝛾𝜙(𝑥) ( 𝜕𝑉 𝐹 𝑇 𝑆 (𝑥) 𝜕𝑥 𝐵 ) 𝑇 ( 35
)
for any 𝛾 ∈ ℝ + is finite-time stable at the origin and the variable 𝜔 remains bounded.

Proof. According to Theorem 3 𝐷𝑉 𝐹 𝑇 𝑆 (𝑥)

( 𝐴𝑥 + 𝐵𝑢 𝐹 𝑇 𝑆 (𝑥) )
is 𝑟-homogeneous of degree 𝜇 + 𝜈. Then, using the homogeneity property and ( 18), (19)

𝐵𝜙(𝑥) 𝑇 (𝜃 -𝜔) -𝛾 -1 ω𝑇 (𝜃 -𝜔) = -𝑎𝑉 𝐹 𝑇 𝑆 (𝑥) 𝛼 .
Thus, the conditions ( 21)-( 23) are satisfied with 𝑈 ( x) = 𝑉 𝐹 𝑇 𝑆 (𝑥), 𝑊 ( x) = 0.5𝛾 -1 (|𝜃 -𝜔|) [START_REF] Vorotnikov | Partial Stability and Control[END_REF] and according to the proof of Theorem 4 the system is oGAS with 𝑦 = 𝑥.

Since 𝐷𝑉 ( x)

( 𝐴𝑥 + 𝐵 ( 𝜙(𝑥) 𝑇 𝜃 + 𝑢( x)
)) ≤ 0 and the amplitudes of 𝑥 and 𝜔 are bounded by the corresponding functions of initial conditions, we have

| | | 𝐷𝑊 ( x) ( 𝐴𝑥 + 𝐵 ( 𝜙(𝑥) 𝑇 𝜃 + 𝑢( x) )) | | | = | | | 𝜕𝑉 𝐹 𝑇 𝑆 (𝑥) 𝜕𝑥 𝐵𝜙(𝑥) 𝑇 (𝜃 -𝜔) | | | ≤ 𝜖 | | | 𝜕𝑉 𝐹 𝑇 𝑆 (𝑥) 𝜕𝑥 𝐵 | | | |𝜙(𝑥)|, (36)
for some 𝜖 ∈ ℝ + dependent on initial conditions. Finally, with the use of [START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF] and ( 18), ( 19) we obtain

| | | 𝐷𝑊 ( x) ( 𝐴𝑥 + 𝐵 ( 𝜙(𝑥) 𝑇 𝜃 + 𝑢( x) )) | | | ≤ 𝑐𝜖 | | | 𝜕𝑉 𝐹 𝑇 𝑆 (𝑥) 𝜕𝑥 𝐵 | | | ‖𝑥‖ 𝛿 𝑟 ≤ 𝑐𝜖 | | | 𝜕𝑉 𝐹 𝑇 𝑆 (𝑧) 𝜕𝑧 | | | |𝐵| | | 𝐷 -1 𝑟 (‖𝑥‖ 𝑟 ) | | ‖𝑥‖ 𝛿+𝜇 𝑟 ≤ 𝑐𝜖𝑏 |𝐵| max { ‖𝑥‖ 𝛿+𝜇-𝑟 min 𝑟 , ‖𝑥‖ 𝛿+𝜇-𝑟 max 𝑟 } ≤ 𝑐𝜖𝑏 |𝐵| max { 𝑐 -𝛽 max 1 , 𝑐 -𝛽 min 1 } max { 𝑉 𝐹 𝑇 𝑆 (𝑥) 𝛽 max , 𝑉 𝐹 𝑇 𝑆 (𝑥) 𝛽 min } = b max{𝑉 𝐹 𝑇 𝑆 (𝑥) 𝛽 max , 𝑉 𝐹 𝑇 𝑆 (𝑥) 𝛽 min },
where

𝑧 ∈ 𝕊 𝑟 , 𝐷 𝑟 (⋅) is the dilation matrix, b = 𝑐𝜖𝑏 |𝐵| max { 𝑐 -𝛽 max 1 , 𝑐 -𝛽 min 1 } and 𝛽 min = 𝜇-𝑟 max +𝛿 𝜇 > 𝛼, 𝛽 max = 𝜇-𝑟 min +𝛿 𝜇
with 𝑟 min = min 1≤𝑗≤𝑛 𝑟 𝑗 . Thus, all conditions of Theorem 4 are verified.

Example 3

To illustrate the application of the proposed adaptive scheme, let us consider a plant defined by

ẋ1 = 𝑥 2 , ẋ2 = 𝜃 1 sin(𝑥 1 𝑥 2 ) + 𝜃 2 𝑥 2 2 + 𝑢, where 𝜃 = [𝜃 1 𝜃 2 ]
𝑇 is the vector of unknown constant parameters and 𝜙(𝑥) = [sin(𝑥 1 𝑥 2 ) 𝑥 [START_REF] Vorotnikov | Partial Stability and Control[END_REF] 2 ] 𝑇 . In this case (34) is the double integrator system. According to [START_REF] Bhat | Continuous finite-time stabilization of the translational and rotational double integrators[END_REF] we choose the finite-time control 𝑢 𝐹 𝑇 𝑆 in the form

𝑢 𝐹 𝑇 𝑆 (𝑥) = -⌈𝑥 2 ⌋ 𝛼 -⌈𝜒 𝛼 ⌋ 𝛼 2-𝛼 ,
where 𝜒 𝛼 = 𝑥 1 + 1 2-𝛼 ⌈𝑥 2 ⌋ 2-𝛼 , 𝛼 ∈ (0, 1) and ⌈𝑥⌋ 𝛽 = |𝑥| 𝛽 sign(𝑥). This finite-time control homogenizes the double integrator system of degree 𝛼 -1 < 0 with the vector of weights 𝑟 = [2 -𝛼, 1] 𝑇 , and the corresponding homogeneous Lyapunov function can be chosen in the form

𝑉 𝐹 𝑇 𝑆 (𝑥) = 2 -𝛼 3 -𝛼 | | 𝜒 𝛼 | | 3-𝛼 2-𝛼 + 𝑠𝑥 2 𝜒 𝛼 + 𝑙 3 -𝛼 |𝑥 2 | 3-𝛼 ,
where 𝑙 and 𝑠 are positive reals. Then for 𝛼 = 0.5, 𝛿 = 𝜌 = 2 and 𝑐 = 1.5 the condition |𝜙(𝑥)| < 𝑐‖𝑥‖ 𝛿 𝑟 is satisfied, and according to Theorem 6 the system is finite-time stable with the use of adaptive control (35) in the form

𝑢(𝑥, 𝜔) = -⌈𝑥 2 ⌋ 𝛼 -⌈𝜒 𝛼 ⌋ 𝛼 2-𝛼 -𝜙(𝑥) 𝑇 𝜔, ω = 𝛾𝜙(𝑥) ( ⌈𝜒 𝛼 ⌋ 1 2-𝛼 |𝑥 2 | 1-𝛼 + 𝑠𝜒 𝛼 + (𝑠 + 𝑟)𝑥 2 |𝑥 2 | 1-𝛼
) .

The results of simulation are shown in Fig. 1 for

𝜃 = [3 -2] 𝑇 , 𝛾 = 𝑙 = 𝑠 = 1.
The results of simulation with using the logarithmic scale are shown in Fig. 2 in order to demonstrate finite-time convergence rate of |𝑥|. The transients for the control 𝑢(𝑥) = 𝑢 𝐹 𝑇 𝑆 (𝑥) are shown in Fig. 3 indicating that the control without adaptive term may not guarantee stability of the system. Remark 4 In 27 it was shown that for a stable homogeneous system ẋ = 𝑓 (𝑥) there exists an implicitly defined homogeneous Lyapunov function 𝑄(𝑉 𝐹 𝑇 𝑆 , 𝑥) = 0, where Ψ is diffeomorphism on ℝ 𝑛 ⧵{0}, the homeomorphism on ℝ 𝑛 , Ψ(0) = 0. In this case by means of the implicit function theorem [START_REF] Courant | Introduction to Calculus and Analysis[END_REF] we have

𝑄(𝑉 𝐹 𝑇 𝑆 , 𝑥) = Ψ(𝑥) 𝑇 𝐷 𝑟 (𝑉 -1 𝐹 𝑇 𝑆 )𝑃 𝐷 𝑟 (𝑉 -1 𝐹 𝑇 𝑆 )Ψ(𝑥) -1,
𝜕𝑉 𝐹 𝑇 𝑆 𝜕𝑥 = - [ 𝜕𝑄 𝜕𝑉 𝐹 𝑇 𝑆 ] -1 𝜕𝑄

𝜕𝑥

for 𝑄(𝑉 𝐹 𝑇 𝑆 , 𝑥) = 0.

The implicit Lyapunov function method is well established for homogenizing finite-time control design. For example, in order to choose homogenizing control for the system (34) the results of [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF][START_REF] Polyakov | Robust stabilization of MIMO systems in finite/fixed time[END_REF] may be used.

The presented results can be extended to linear geometric homogeneous systems using their equivalence to standard homogeneous ones [START_REF] Polyakov | Sliding Mode Control Design Using Canonical Homogeneous Norm[END_REF] .

Remark 5

The control scheme (35) presented in Theorem 6 can be applied with not necessary homogenising control laws 𝑢 𝐹 𝑇 𝑆 . Since for finite-time stable system there exists a Lyapunov function that 𝐷𝑉 𝐹 𝑇 𝑆 (𝑥)(𝐴𝑥 + 𝐵𝑢 𝐹 𝑇 𝑆 (𝑥)) ≤ -𝑎𝑉 𝐹 𝑇 𝑆 (𝑥) 𝛼 , 𝑎 ∈ ℝ + , 𝛼 ∈ (0, 1) then the main condition for applying the scheme (35) is

| | | | 𝜕𝑉 𝐹 𝑇 𝑆 (𝑥) 𝜕𝑥 𝐵 | | | | |𝜙(𝑥)| ≤ 𝑁 ∑ 𝑖=1 𝑏 𝑖 𝑉 𝐹 𝑇 𝑆 (𝑥) 𝛽 𝑖 , ( 37
)
where 𝑏 𝑖 ∈ ℝ + , 𝛽 𝑖 > 𝛼, 𝑖 = 1, 𝑁 and 𝑁 ∈ ℕ.

Let the system (33)

𝐵𝜙(𝑥) 𝑇 θ -𝛾 -1 𝜃 max ω𝑇 diag { 1 1+𝜔 2 𝑖 } 𝑞 𝑖=1 θ = -𝑎 1 𝑉 𝐹 𝑥𝑇 𝑆 (𝑥) 𝛼 1 -𝑎 2 𝑉 𝐹 𝑥𝑇 𝑆 (𝑥) 𝛼 2 .
Thus, the conditions ( 21), ( 22), ( 27) are satisfied with 𝑈 ( x) = 𝑉 𝐹 𝑥𝑇 𝑆 (𝑥), 𝑊 ( x) = 0.5𝛾 - , i.e., the condition ( 27) is satisfied. The rest conditions of Theorem 5 are also valid according to the proof of Theorem 7. Thus, for sufficiently small variations of 𝜃(𝑡) (𝜖 * is sufficiently small) the result of Theorem 7 remains relevant. Similarly, the result of Theorem 6 remains relevant for the case of time-varying parameter 𝜃(𝑡) under additional condition:

𝜖 * | | | | 𝜕𝑉 𝐹 𝑇 𝑆 (𝑥) 𝜕𝑥 𝐵 | | | | |𝜙(𝑥)| ≤ 𝑎 * 𝑉 𝛼 𝐹 𝑇 𝑆 , where 𝑎 * < 𝑎, 𝜖 * ∈ ℝ + ∶ |𝜃(𝑡) -𝜃 *
| ≤ 𝜖 * . The homogeneity property can be used for fixed-time control design. For example, the concept of homogeneity in bi-limit introduced in [START_REF] Andrieu | Homogeneous approximation, recursive observer and output feedback[END_REF] provides that an asymptotically stable system is fixed-time stable if it is homogeneous of negative degree in 0limit and homogeneous of positive degree in ∞-limit. Based on this the fixed-time convergence can be achieved by changing the homogeneity degree in hybrid algorithms (see, for example, [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF][START_REF] Angulo | Robust exact uniformly convergent arbitrary order differentiator[END_REF][START_REF] Zimenko | On simple scheme of finite/fixed-time control design[END_REF]

for 𝑥 ∈ ℝ 𝑛 ⧵ Ω. Let |𝜙(𝑥)| < { 𝜂 1 ‖𝑥‖ 𝛿 1 𝑟 1 for 𝑥 ∈ Ω, 𝜂 2 ‖𝑥‖ 𝛿 2 𝑟 2 for 𝑥 ∈ ℝ 𝑛 ⧵ Ω for some 𝜂 1 , 𝜂 2 ∈ ℝ + and 𝛿 1 > 𝜈 1 + 𝑟 1 max , 𝛿 2 < 𝜈 2 + 𝑟 2 min , 𝑟 1 max = max 1≤𝑗≤𝑛 𝑟 1𝑗 , 𝑟 2 min = min 1≤𝑗≤𝑛 𝑟 2𝑗 .
Then with adaptive control in the form (39) the system ( 33) is fixed-time stable at the origin.

Proof. The proof is a direct consequence of Theorem 7 and the homogeneity property.

Remark 7

As it is shown in the following example the given result can be extended for the case when 𝑉 𝐹 𝑥𝑇 𝑆 is continuously differentiable for 𝑥 ∉ {0} ∪ 𝜕Ω.

Example 4 Consider the system ẋ1 = 𝑥 2 , ẋ2 = 𝜃 1 sin(𝑥 1 𝑥 2 ) + 𝜃 2 𝑥 2 + 𝑢, where 𝜃 = [𝜃 1 𝜃 2 ] 𝑇 is the vector of unknown constant parameters. According to [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF] choose 𝑢 𝐹 𝑥𝑇 𝑆 in the form

𝑢 𝐹 𝑥𝑇 𝑆 = { 𝑉 1+𝜈 1 𝐹 𝑥𝑇 𝑆 𝑘𝐷 𝑟 1 (𝑉 -1 𝐹 𝑥𝑇 𝑆 )𝑥 for 𝑥 𝑇 𝑋 -1 𝑥 < 1 𝑉 1+2𝜈 2 𝐹 𝑥𝑇 𝑆 𝑘𝐷 𝑟 2 (𝑉 -1 𝐹 𝑥𝑇 𝑆 )𝑥 for 𝑥 𝑇 𝑋 -1 𝑥 ≥ 1 with • 𝜈 1 ∈ (-1, 0), 𝜈 2 ∈ ℝ + , 𝑟 1 = [ 1 -𝜈 1 1 ] 𝑇 , 𝑟 2 = [ 1 1 + 𝜈 2 ] 𝑇 ; • 𝑘 = 𝑌 𝑋 -1 , where 𝑌 ∈ ℝ 2×1 , 𝑋 ∈ ℝ 2×2 is a solution of linear matrix inequalities 𝐴𝑋 + 𝑋𝐴 𝑇 + 𝐵𝑌 + 𝑌 𝑇 𝐵 𝑇 + 𝜁 1 𝑋 < 0, 𝑋 > 0, 𝜁 2 𝑋 ≥ 𝑋diag{𝑟 1𝑖 } 2 𝑖=1 + diag{𝑟 1𝑖 } 2 𝑖=1 𝑋 > 0, 𝜁 3 𝑋 ≥ 𝑋diag{𝑟 2𝑖 } 2 𝑖=1 + diag{𝑟 2𝑖 } 2 𝑖=1 𝑋 > 0, for some 𝜁 1 , 𝜁 2 , 𝜁 3 ∈ ℝ + ; • 𝑉 𝐹 𝑥𝑇 𝑆 ∈ ℝ + is defined implicitly by { 𝑄 1 (𝑉 𝐹 𝑥𝑇 𝑆 , 𝑥) = 0 for 𝑥 𝑇 𝑋 -1 𝑥 < 1 𝑄 2 (𝑉 𝐹 𝑥𝑇 𝑆 , 𝑥) = 0 for 𝑥 𝑇 𝑋 -1 𝑥 ≥ 1 , where 𝑄 1 (𝑉 𝐹 𝑥𝑇 𝑆 , 𝑥) = 𝑥 𝑇 𝐷 𝑟 1 (𝑉 -1 𝐹 𝑥𝑇 𝑆 )𝑋 -1 𝐷 𝑟 1 (𝑉 -1 𝐹 𝑥𝑇 𝑆 )𝑥 -1, 𝑄 2 (𝑉 𝐹 𝑥𝑇 𝑆 , 𝑥) = 𝑥 𝑇 𝐷 𝑟 2 (𝑉 -1
𝐹 𝑥𝑇 𝑆 )𝑋 -1 𝐷 𝑟 2 (𝑉 -1 𝐹 𝑥𝑇 𝑆 )𝑥 -1. In order to find 𝑉 𝐹 𝑥𝑇 𝑆 the numerical procedures can be used (for example, the bisection method may be utilized (see, e.g. [START_REF] Polyakov | Robust stabilization of MIMO systems in finite/fixed time[END_REF] ).

The control 𝑢 𝐹 𝑥𝑇 𝑆 homogenizes the system (38) of degree 𝜈 1 < 0 for 𝑥 𝑇 𝑋 -1 𝑥 < 1 and 𝜈 2 > 0 for 𝑥 𝑇 𝑋 -1 𝑥 ≥ 1. Note that 𝑉 𝐹 𝑥𝑇 𝑆 is continuously differentiable for 𝑥 ∉ {0} ∪ {𝑥 ∈ ℝ 𝑛 ∶ 𝑉 (𝑥) = 1}. According to [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF] Remark 8 One of the main advantages of the proposed adaptive scheme is that it allows to combine with an adaptive algorithm different finite-time (fixed-time) controls 𝑢 𝐹 𝑇 𝑆 (𝑢 𝐹 𝑥𝑇 𝑆 ) designed for nominal systems without uncertainties. This variability allows to assign some additional robustness properties to the closed-loop system that are inherent to 𝑢 𝐹 𝑇 𝑆 (𝑢 𝐹 𝑥𝑇 𝑆 ). For example, with the use of homogeneity based algorithms the presented control scheme can cancel a wide class of disturbances, including non-Lipschitz ones (see, for example, [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF] ).

CONCLUSIONS

In the paper a sufficient condition of output finite-time and fixed-time stability is presented. Comparing with existing results the presented approach is less restrictive and/or obtained for a wider class of systems. Based on the provided sufficient condition, a simple scheme of adaptive finite/fixed-time control design is presented. Possible directions for future research include control and observer design based on the use of the presented OFTS/OFxTS condition, and an extension of the approach guaranteeing the parameter convergence also.

  is smooth and 𝐷𝑊 (𝑥)𝑓 (𝑥) ≥ 0, then 𝑈 (𝑥) is SIoGAS-Lyapunov function and the settling-time function in Theorem 4 is bounded by 𝑇 (𝑥 0 ) ≤ 𝑈 1-𝛼 0 𝑎(1-𝛼) , where 𝑈 0 = 𝑈 (𝑥 0 ). • Due to (25) and (

FIGURE 1 FIGURE 2 FIGURE 3

 123 FIGURE 1 System states 𝑥, 𝜔 versus time

  ). The results of simulation are shown in Fig.4for 𝜃 = [3 2] 𝑇 and 𝑥 0 = [0 1] 𝑇 . The results of simulation with using the logarithmic scale are shown in Fig.5for different initial conditions. They show uniformity of the convergence time on the initial conditions. The transients for the control 𝑢(𝑥) = 𝑢 𝐹 𝑥𝑇 𝑆 (𝑥) are shown in Fig.6.

FIGURE 4

 4 FIGURE 4 Simulation plot for 𝑥 0 = [0 1] 𝑇

FIGURE 5 FIGURE 6

 56 FIGURE 5 Simulation plot for different initial conditions 𝑥 0

Definition 4 16,17 For

  the general case, all inverse relations are not satisfied. Let us present definitions for corresponded Lyapunov functions. the system (1), a smooth function 𝑉 and a function 𝜆 ∶ ℝ 𝑛 → ℝ + ∪ {0} are called respectively an oGAS-Lyapunov function and an auxiliary modulus if there exist 𝜉 1 , 𝜉 2 ∈  ∞ such that

	𝜉 1 (|ℎ(𝑥)|) ≤ 𝑉 (𝑥) ≤ 𝜉 2 (|𝑥|) ∀𝑥 ∈ ℝ 𝑛	(6)
	holds and there exists 𝜉 3 ∈  such that
	𝐷𝑉 (𝑥)𝑓 (𝑥) ≤ -𝜉 3 (𝑉 (𝑥), 𝜆(𝑥))	(7)
	𝐷𝜆(𝑥)𝑓 (𝑥) ≤ 0	(8)
	for almost all 𝑥 ∈ ,
	or
	(b) there exists some 𝜃 ∈  such that
	𝜆(Φ(𝑡, 𝑥 0 )) ≤ 𝜃(|𝑥 0 |)

for all 𝑥 ∈ , where  = {𝑥 ∈ ℝ 𝑛 ∶ 𝑉 (𝑥) > 0}, and 𝜆 satisfies the following conditions, either (a) 0 ≤ 𝜆(𝑥) ≤ |𝑥| for all 𝑥 ∈ ℝ 𝑛 , 𝜆 is locally Lipschitz on the set  and satisfies

  𝕊 𝑟 , 0 < 𝛼 = 𝜇+𝜈 𝜇 < 1 and some 𝑎 ∈ ℝ + . Then choosing a candidate Lyapunov function for the system(33),(35) with extended state vector x = [ 𝑥 𝑇 𝜔 𝑇 ] 𝑇 as 𝑉 ( x) = 𝑉 𝐹 𝑇 𝑆 (𝑥) + 0.5𝛾 -1 (|𝜃 -𝜔|) 2

	we obtain	𝐷𝑉 ( x)	( 𝐴𝑥 + 𝐵	( 𝜙(𝑥) 𝑇 𝜃 + 𝑢( x) ))	≤ -𝑎𝑉 𝐹 𝑇 𝑆 (𝑥) 𝛼 +	𝜕𝑉 𝐹 𝑇 𝑆 (𝑥) 𝜕𝑥

one can obtain 𝐷𝑉 𝐹 𝑇 𝑆 (𝑥) ( 𝐴𝑥 + 𝐵𝑢 𝐹 𝑇 𝑆 (𝑥) ) = ‖𝑥‖ 𝜇+𝜈 𝑟 𝜕𝑉 𝐹 𝑇 𝑆 (𝑧) 𝜕𝑧 ( 𝐴𝑧 + 𝐵𝑢 𝐹 𝑇 𝑆 (𝑧) ) ≤ -𝑎𝑉 𝐹 𝑇 𝑆 (𝑥) 𝛼 with 𝑧 ∈

  The unknown parameters are from a compact set, i.e., |𝜃| ≤ 𝜃 max for a known 𝜃 max ∈ ℝ + . Then the results similar to Theorem 6 can be obtained for Theorem 5 by replacing a finite-time stabilizing control 𝑢 𝐹 𝑇 𝑆 with a fixed-time one:Theorem 7 Let assumption A.4 be satisfied and 𝑢 𝐹 𝑥𝑇 𝑆 ∶ ℝ 𝑛 → ℝ be stabilizing in a fixed time control law for the systemẋ = 𝐴𝑥 + 𝐵𝑢 𝐹 𝑥𝑇 𝑆 (𝑥)(38)and 𝑉 𝐹 𝑥𝑇 𝑆 (𝑥) be the corresponding Lyapunov function satisfying𝐷𝑉 𝐹 𝑥𝑇 𝑆 (𝑥)(𝐴𝑥 + 𝐵𝑢 𝐹 𝑥𝑇 𝑆 (𝑥)) ≤ -𝑎 1 𝑉 𝐹 𝑥𝑇 𝑆 (𝑥) 𝛼 1 -𝑎 2 𝑉 𝐹 𝑥𝑇 𝑆 (𝑥) 𝛼 2 , 𝜉 1 (|𝑥|) ≤ 𝑉 𝐹 𝑥𝑇 𝑆 (𝑥) ≤ 𝜉 2 (|𝑥|) with 𝑎 1 , 𝑎 2 ∈ ℝ + , 𝛼 1 ∈ (0, 1), 𝛼 2 > 1 and 𝜉 1 , 𝜉 2 ∈  ∞ . Then the system(33) with adaptive control 𝑢(𝑥, 𝜔) = 𝑢 𝐹 𝑥𝑇 𝑆 (𝑥)-𝜃 max 𝜙(𝑥) 𝑇 [ arctan(𝜔 1 ), ..., arctan(𝜔 𝑞 ) 𝑏 𝑖 ∈ ℝ + , 𝛽 𝑖 ∈ (𝛼 1 , 𝛼 2 ), 𝑖 = 1, 𝑁 and 𝑁 ∈ ℕ. 𝑉 𝐹 𝑥𝑇 𝑆 (𝑥) 𝛼 1 -𝑎 2 𝑉 𝐹 𝑥𝑇 𝑆 (𝑥) 𝛼 2 +

		satisfy the following assumption:
	(A.4) ] 𝑇
			ω = 𝛾𝜃 -1 max diag{1 + 𝜔 2 𝑖 } 𝑞 𝑖=1 𝜙(𝑥)	(	𝜕𝑉 𝐹 𝑥𝑇 𝑆 (𝑥) 𝜕𝑥	𝐵	) 𝑇	(39)
	for any 𝛾 ∈ ℝ + is fixed-time stable at the origin if
				| | | |	𝜕𝑉 𝐹 𝑥𝑇 𝑆 (𝑥) 𝜕𝑥	𝐵	| | | |	|𝜙(𝑥)| ≤	𝑁 𝑖=1 ∑	𝑏 𝑖 𝑉 𝐹 𝑥𝑇 𝑆 (𝑥) 𝛽 𝑖	(40)
	is satisfied for Proof. Choose a candidate Lyapunov function for the system (33), (35) with extended state vector x =	[ 𝑥 𝑇 𝜔 𝑇 ] 𝑇 in the form
					𝑉 ( x) = 𝑉 𝐹 𝑥𝑇 𝑆 (𝑥) + 0.5𝛾 -1	| |	| θ| 2 ,
	where θ = 𝜃 -𝜃 max	[ arctan(𝜔 1 ), ..., arctan(𝜔 𝑞 ) ] 𝑇 . Then we obtain
	𝐷𝑉 ( x)	( 𝐴𝑥 + 𝐵	( 𝜙(𝑥) 𝑇 𝜃 + 𝑢( x) ))	≤ -𝑎 1 𝜕𝑉 𝐹 𝑥𝑇 𝑆 (𝑥) 𝜕𝑥

  The result of Theorem 7 can be applied for the case of time-varying parameter 𝜃(𝑡) under the assumption |𝜃(𝑡)| ≤ 𝜃 max for all 𝑡 ≥ 0. Let 𝜃 * be the mean value of 𝜃(𝑡) and |𝜃(𝑡) -𝜃 * | ≤ 𝜖 * for some 𝜖 * ∈ (0, 𝜃 max ). Let the condition 𝑉 𝐹 𝑥𝑇 𝑆 (𝑥) 𝛼 1 -𝑎 2 𝑉 𝐹 𝑥𝑇 𝑆 (𝑥) 𝛼 2 + 𝜖 * ∑ 𝑁 𝑖=1 𝑏 𝑖 )𝑉 𝐹 𝑥𝑇 𝑆 (𝑥) 𝛼 1 -(𝑎 2 -𝜖 * ∑ 𝑁 𝑖=1 𝑏 𝑖 )𝑉 𝐹 𝑥𝑇 𝑆 (𝑥) 𝛼 2

	assumption A.4 is satisfied, 𝑊 ( x) ≤ 0.5𝛾 -1 𝜃 2 max	( 1 + 0.5 √ 𝑞𝜋	1 ) 2 is globally bounded, i.e., the condition (28) is satisfied. Finally, | | θ| 2 and the system is oGAS. Due to |
	by (40) the inequality (24) holds and all conditions of Theorem 5 are satisfied.
	Remark 6 𝜖 *	𝑁 ∑	𝑏 𝑖 < min{𝑎 1 , 𝑎 2 }
						𝑖=1
	is satisfied, then replacing 𝜃 by 𝜃 * in the expression of 𝑉 and taking into account (40) we obtain
	𝐷𝑉 ( x)	( 𝐴𝑥 + 𝐵	( 𝜙(𝑥) 𝑇 𝜃 + 𝑢( x) ))	≤ -𝑎 1 𝜕𝑉 𝐹 𝑇 𝑆 (𝑥) 𝜕𝑥	𝐵𝜙(𝑥) 𝑇 (𝜃(𝑡) -𝜃 * )
					≤ -(𝑎 1 -] 𝑇	| | | ) 2

for 𝑉 ( x) = 𝑈 ( x) + 𝑊 ( x), 𝑈 ( x) = 𝑉 𝐹 𝑥𝑇 𝑆 (𝑥), 𝑊 ( x) = 0.5𝛾 -1 ( | | | 𝜃 * -

𝜃 max [ arctan(𝜔 1 ), ..., arctan(𝜔 𝑞 )

  ). Corollary 2 Let 𝑢 𝐹 𝑥𝑇 𝑆 ∶ ℝ 𝑛 → ℝ be such that 𝑢 𝐹 𝑥𝑇 𝑆 (𝑥) = 𝑢 1 (𝑥) for 𝑥 ∈ Ω and 𝑢 𝐹 𝑥𝑇 𝑆 (𝑥) = 𝑢 2 (𝑥) for 𝑥 ∈ ℝ 𝑛 ⧵ Ω, where Ω is neighborhood of the origin, 𝑢 𝑖 ∶ ℝ 𝑛 → ℝ are 𝑟 𝑖 -homogeneous functions such that 𝑢 1 (𝑥) = 𝑢 2 (𝑥) for 𝑥 ∈ 𝜕Ω (boundary of Ω). Let the system(38) be fixed-time stable and its continuously differentiable FxTS Lyapunov function 𝑉 𝐹 𝑥𝑇 𝑆 be such that it is 𝑟 1 -homogeneous of degree 𝜇 1 for 𝑥 ∈ Ω and 𝑟 2 -homogeneous of degree 𝜇 2

}

  the inequality 𝐷𝑉 𝐹 𝑥𝑇 𝑆 (𝑥)(𝐴𝑥 + 𝐵𝑢 𝐹 𝑥𝑇 𝑆 (𝑥)) ≤ 𝐹 𝑥𝑇 𝑆 (𝑥) for 𝑉 𝐹 𝑥𝑇 𝑆 (𝑥) < 1, -𝐹 𝑥𝑇 𝑆 (𝑥) for 𝑉 𝐹 𝑥𝑇 𝑆 (𝑥) > 1, -min{ for 𝑉 𝐹 𝑥𝑇 𝑆 (𝑥) = 1, holds for almost all 𝑡 such that 𝑥(𝑡) ≠ 0. Then according to Theorem 7 the system is fixed-time stable with the use of adaptive control in the form (39) 𝑢(𝑥, 𝜔) = 𝑢 𝐹 𝑥𝑇 𝑆 (𝑥) -𝜃 max 𝜙(𝑥)𝑇 [ arctan(𝜔 1 ) arctan(𝜔 2 ) 𝜍 1 (𝑥) for𝑥 𝑇 𝑋 -1 𝑥 < 1 𝜍 2 (𝑥) for 𝑥 𝑇 𝑋 -1 𝑥 ≥ 1 with 𝜍 1 (𝑥) = 𝑥 𝑇 𝐷 𝑟 1 (𝑉 -1 𝐹 𝑥𝑇 𝑆 ) ( diag{𝑟 1𝑖 } 2 𝑖=1 𝑋 -1 + 𝑋 -1 diag{𝑟 1𝑖 } 2 𝑖=1 ) 𝐷 𝑟 1 (𝑉 -1 𝐹 𝑥𝑇 𝑆 )𝑥𝑥 𝑇 𝐷 𝑟 1 (𝑉 -1 𝐹 𝑥𝑇 𝑆 )𝑋 -1 𝐷 𝑟 1 (𝑉 -1 𝐹 𝑥𝑇 𝑆 ), 𝜍 2 (𝑥) = 𝑥 𝑇 𝐷 𝑟 2 (𝑉 -1 𝐹 𝑥𝑇 𝑆 ) ( diag{𝑟 2𝑖 } 2 𝑖=1 𝑋 -1 + 𝑋 -1 diag{𝑟 2𝑖 } 2 𝑖=1 ) 𝐷 𝑟 2 (𝑉 -1 𝐹 𝑥𝑇 𝑆 )𝑥𝑥 𝑇 𝐷 𝑟 2 (𝑉 -1 𝐹 𝑥𝑇 𝑆 )𝑋 -1 𝐷 𝑟 2 (𝑉 -1 𝐹 𝑥𝑇 𝑆

			⎧ ⎪ ⎨ ⎪ ⎩	-	𝜁 1 𝜁 2 𝜁 1 𝜁 3	𝑉 𝑉	1+𝜈 1 1+𝜈 2 𝜁 1 𝜁 2	,	𝜁 1 𝜁 2
								] 𝑇
	ω = 𝛾𝜃 -1 max diag{1 + 𝜔 2 𝑖 } 2 𝑖=1 𝜙(𝑥)	(	𝜕𝑉 𝐹 𝑥𝑇 𝑆 (𝑥) 𝜕𝑥	𝐵	) 𝑇	,
	where according to Remark 4
	𝜕𝑥 𝜕𝑉 𝐹 𝑥𝑇 𝑆 (𝑥)	= -2𝑉 𝐹 𝑥𝑇 𝑆	{

This work is supported by the Russian Science Foundation under grant 21-71-10032.


Liem Binh

Luong Ngyen

Rebecca Bauer

Zineb Lesieur

Florence Galtier

Xavier Duval

Philippe Vanhems

Fabrice Lainé

Pierre Tattevin

Christine Durier

Odile Launay

Binh Luong Liem

Nguyen

L B Luong Nguyen

S M Keita

R Kafi

M Zerrouki

AS. L'Honneur Inserm J Dira

: R Bauer

A Arulananthan

V Foubert

Y Saidi

L Meyer

C Merle

P Geraud

H Chevassus

E Nyiramigisha

Vincent Foulongne

A M Mondain

J Fontanille

V Maugueret

M Ray

F Letois

T Mura

A Bourdin

A Konaté

X Capdevilla

G Du Cailar

A Terminet

MS H Blain

A Leglise

P Le Quellec

L Corne

K Landreau

A Klouche

M Bourgeois

G Sebbane

H Mourad

M Leray

Maarouf

Inserm

: X Duval

Y Costa De Beauregard

N Houhou

B Robin

S Amour

C Del-Signore

B Lina

Hôpital Pontchaillou

Rennes : L Gallais

B Lefeuvre

G Lagathu

Vaccine effectiveness against COVID-19 hospitalization in adults in France: A test negative case control study

Keywords: COVID-19, vaccine, France, vaccine effectiveness, alpha variant

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

Background

France started COVID-19 vaccination on December 27, 2020. Initially restricted to residents of nursing homes and healthcare workers (HCW), it was gradually extended to all adults on May 31, 2021. On July 26, 2021, more than 40 million French residents (almost 60% of the total population) had received at least one dose of one of the four European Medicine Agency (EMA)-approved COVID-19 vaccines: mRNA BNT162b2 (Comirnaty®, Pfizer-BioNTech), mRNA-1273 (Spikevax®, Moderna), ChAdOx1 nCoV-19 (Vaxzevria®, AstraZeneca-Oxford University), and Ad26.COV2-S (Janssen®, Janssen). Measuring vaccine effectiveness (VE) using real-life data is critical to confirm the effectiveness of licensed vaccine, in addition to pivotal clinical trials which could strengthen vaccination adherence, which was particularly low in France during the first months of 2021 [START_REF] Schwarzinger | COVID-19 vaccine hesitancy in a representative working-age population in France: a survey experiment based on vaccine characteristics[END_REF]. The Inserm Fluvac study, conducted since 2013 in five French hospitals (Paris-Cochin, Paris-Bichat, Rennes, Montpellier, Lyon), has been evaluating the effectiveness of influenza vaccines against hospitalization among adults using a test negative design [START_REF] Loubet | Factors associated with poor outcomes among adults hospitalized for influenza in France: A three-year prospective multicenter study[END_REF]. The Fluvac network and study design have been adapted to study COVID-19 VE, and the first results are presented here.

Study population

Between December 23, 2020 and June 15, 2021, 853 patients hospitalized for influenza-like illness ─ defined as the combination of at least one general symptom (fever, malaise, headache, myalgia, or fatigue) and one respiratory symptom (cough, sore throat, shortness of breath, or tachypnoea) ─ were included (Table 1). Patients were prospectively screened by research staff, and detailed clinical data on comorbidities, vaccination status, and vital status were collected at baseline as well as data on hospital stay and vital status at 30 and 90 days.

All patients were tested by PCR technique for SARS-CoV-2 infection within 7 days of symptoms onset, according to local technique guidelines. Patients with a positive SARS-CoV-2 PCR test were classified as "COVID-19 cases", while those with a negative PCR test were classified as "controls". Among the 853 patients included, there were 463 cases and 390 controls. All cases presented with symptoms compatible with SARS-CoV-2 infection. Overall, 308 of 463 cases (67%) were directly diagnosed with SARS-CoV-2 pneumonia at admission. The remaining 155 cases were included for influenza-like illness (n=101, 65%), respiratory symptoms (n=98, 63%), fatigue (n=80, 52%), and myalgia (n=25, 16%). Several controls (n=42, 11%) presented with cardiac disease associated with respiratory symptoms. Among control patients, only 19 respiratory viruses other than SARS-CoV-2 were observed. Cases were younger (median, 66 years vs 77), had a higher body mass index (26.9 kg/m 2 vs 25.4), more often had a professional activity (29.4% vs 10.3%), and had fewer comorbidities (68.0% vs 84.6%). Oxygen therapy was required for 87.3% of cases and 67.4% of controls. Length of stay was 10 days for cases and 8 days for controls. Cases more frequently required intensive ICU admission (26.1% vs 11.3%), with an average ICU length of stay of 6 days for cases and 3 days for controls; 7% of cases required invasive mechanical ventilation versus 1% of controls (Table 1). The in-hospital mortality rate was 5.5% in the case group and 5.6% in the control group.

Vaccinated population

A total of 170 patients received at least one dose of COVID-19 vaccine before symptom onset: 50 in the case group (46 received one dose and four two doses) and 120 in the control group (58 received one dose, 61 two doses, and one three doses). Most vaccinated patients (127/170, 74.7%) had received Comirnaty® (Pfizer vaccine), followed by Vaxzevria® (24/170, 14.1%). Among the four cases hospitalized for COVID-19 after two doses, there were one man admitted less than 7 days after the second dose and three patients more than 7 days after the second dose: two women aged 79 and 42 years with kidney transplants, and one man aged 72 years with multiple myeloma on third-line chemotherapy. All these patients did not require intensive care, and were alive at discharge and one month later (table 2).

Vaccine effectiveness calculation

VE was estimated using a test negative design. We compared the odds of vaccinated patients hospitalized with COVID-19 with the odds of vaccinated patients hospitalized for the same symptoms but with a negative test.

VE and their 95% CIs were estimated using the formula VE = (1-OR) x 100. ORs were calculated by the exponentiated regression coefficients of vaccine status, stratified on medical center in the model using the logistic function in R. We estimated VE using various endpoints, from first dose + 7 days to second dose + 14 days. In univariate analysis, VE J o u r n a l P r e -p r o o f estimates ranged from 79.1% (95% CI=[67.0 ; 86.8]) 7 days after the first dose to 95.5% (95% CI=[84.1 ; 98.8]) 14 days after the second dose (Table 3).

To test result robustness, we performed various sensitivity analyses: one that requalified negative PCR tests to cases if COVID-19 diagnosis was recorded in medical files or if the CT scan showed abnormalities suggestive of COVID-19 infection (N=46 additional cases), and one using the WHO definition of severe acute respiratory infection (SARI) cases (defined by fever 38°C and cough and onset within the past 10 days), which is more stringent (N=323 inclusions). The overall results did not change and VE estimates after the second dose were 6% and 3% lower than in the main analysis, respectively (i.e. when requalifying negative PCR tests to cases in case of COVID-19 diagnosis as mentioned above and when using the WHO definition).

A generalized additional logistic regression model was used to adjust on other covariates: time, age, and stratified on centers. Time was measured in days between January 1, 2021 and hospitalization day. It was included as a spline function to account for bias related to time differences between COVID-19 circulation and vaccine availability. VE was estimated at 96.7% (95% CI=[87.9; 99.9]) 7 days after the second dose.

Discussion

Our results are in line with previous reports showing high COVID-19 VE to prevent hospitalizations, such as in the United-Kingdom, Israel and Qatar which showed >90% effectiveness against severe forms, even against the B.1.1.7 variant (beta), predominant in Qatar [START_REF] Dagan | BNT162b2 mRNA Covid-19 Vaccine in a Nationwide Mass Vaccination Setting[END_REF][START_REF] Abu-Raddad | Effectiveness of the BNT162b2 Covid-19 Vaccine against the B.1.1.7 and B.1.351 Variants[END_REF][START_REF] Hyams | Effectiveness of BNT162b2 and ChAdOx1 nCoV-19 COVID-19 vaccination at preventing hospitalisations in people aged at least 80 years: a test-negative, case-control study[END_REF][START_REF] Bernal | Effectiveness of the Pfizer-BioNTech and Oxford-AstraZeneca vaccines on covid-19 related symptoms, hospital admissions, and mortality in older adults in England: test negative case-control study[END_REF]. Moreover, our results are consistent with a previous cohort from Paris et al. and with a case control study in France from Charmet et al. to assess VE, which reported VE (95% CI) of 76% (54-87) and 84% (75-90) against COVID-19 with the original virus and the B.1.1.7 lineage (beta variant), respectively [START_REF] Paris | Effectiveness of mRNA-BNT162b2, mRNA-1273, and ChAdOx1 nCoV-19 vaccines against COVID-19 in healthcare workers: an observational study using surveillance data[END_REF][START_REF] Charmet | Impact of original, B.1.1.7, and B.1.351/P.1 SARS-CoV-2 lineages on vaccine effectiveness of two doses of COVID-19 mRNA vaccines: Results from a nationwide case-control study in France[END_REF]. However, the latter study had limitations: cases and controls came from two different databases, were not matched based on medical data, and were all declarative [START_REF] Charmet | Impact of original, B.1.1.7, and B.1.351/P.1 SARS-CoV-2 lineages on vaccine effectiveness of two doses of COVID-19 mRNA vaccines: Results from a nationwide case-control study in France[END_REF]. For our study, we prospectively collected clinical data at bedside, with detailed information on patients' hospitalization and careful selection of controls. We documented four hospitalized vaccine breakthrough infections, all in immunocompromised patients, in line with previous reports in kidney transplant recipients [START_REF] Caillard | Occurrence of severe Covid-19 in vaccinated transplant patients[END_REF]. This is in line with data on HCWs reported in Israel [START_REF] Bergwerk | Covid-19 Breakthrough Infections in Vaccinated Health Care Workers[END_REF]. While the epidemics is still ongoing, we carry on data J o u r n a l P r e -p r o o f collection and are generalizing whole-genome sequencing on a random sample of patients and in all vaccine breakthroughs. We believe our study will continue to provide interesting information on COVID-19. To address misclassification and selection biases, we performed sensitivity analyses, with no impact on results. Finally, we were not able at the time of analysis to sequence samples from all patients with documented COVID-19, but national data surveillance shows that the main circulating variant at the time of patient inclusion was the alpha variant [11].

Conclusion

Our results confirm the high effectiveness of COVID-19 vaccine in France to prevent hospitalizations due to the alpha variant during the first semester of 2021. Further data are expected to measure vaccine effectiveness against the delta variant.

Contribution of authors

Liem

  Binh Luong Nguyen: Investigation, Supervision, Project administration, Writingoriginal draft. Rebecca Bauer: Formal analysis, Data curation, Visualization, Methodology, Writing -original draft. Zineb Lesieur: Data curation, Supervision, Project administration, review & editing. Florence Galtier: Investigation, Supervision. Xavier Duval: Investigation, Supervision. Philippe Vanhems: Investigation, Supervision, review & editing. Fabrice Lainé: Investigation, Supervision, review & editing. Pierre Tattevin: Writing -review & editing. Christine Durier: Formal analysis, Methodology, Writing -original draft, review & editing. Odile Launay: Funding acquisition, Methodology, Supervision, Writing -original draft, review & editing.

Table 3 .

 3 Vaccine effectiveness (VE) against COVID-19 hospitalization among adults

	stratified in medical centers
		VE (95% confidence interval)	p value
	VE after 1 dose (D1)
	D1 + 7D	79.1 ([67.0 ; 86.8]	<0.001
	D1 + 14D	84.0 [72.6 ; 90.6]	<0.001
	D1 + 21D	87.0 [76.0 ; 92.9]	<0.001
	D1 + 21D	88.3 [76.3; 94.2]	<0.001
	VE after 2 doses (D2)
	D2 + 7 D	96.2 [86.8 ; 98.9]	<0.001
	D2 + 14D	95.5 [84.1 ; 98.8]	<0.001

J o u r n a l P r e -p r o o f

Funding statement: Inserm

Conflict of interests : None declared

J o u r n a l P r e -p r o o f