Andrey Polyakov
email: andrey.polyakov@inria.fr

Leonid Fridman
email: lfridman@servidor.unam.mx

Stability Notions and Lyapunov Functions for Sliding Mode Control Systems

The paper surveys mathematical tools required for stability and convergence analysis of modern sliding mode control systems. Elements of Filippov theory of differential equations with discontinuous right-hand sides and its recent extensions are discussed. Stability notions (from Lyapunov stability (1982) to fixed-time stability (2012)) are observed. Concepts of generalized derivatives and non-smooth Lyapunov functions are considered. The generalized Lyapunov theorems for stability analysis and convergence time estimation are presented and supported by examples from sliding mode control theory.

Introduction

During whole history of control theory, a special interest of researchers was focused on systems with relay and discontinuous (switching) control elements [START_REF] Flugge-Lotz | Discontinuous automatic control[END_REF][START_REF] Tsypkin | Relay Control Systems[END_REF][START_REF] Utkin | Sliding Modes in Control Optimization[END_REF][START_REF] Boiko | Discontinuous control systems : frequency-domain analysis and design[END_REF]. Relay and variable structure control systems have found applications in many engineering areas. They are simple, effective, cheap and sometimes they have better dynamics than linear systems [START_REF] Tsypkin | Relay Control Systems[END_REF]. In practice both input and output of a system may be of a relay type. For example, automobile engine control sys-tems sometimes use λ -sensor with almost relay output characteristics, i.e only the sign of a controllable output can be measured [START_REF] Choi | Robust throttle control of automotive engines[END_REF]. In the same time, terristors can be considered as relay "actuators" for some power electronic systems [START_REF] Utkin | Sliding Mode Control in Electro-Mechanical Systems[END_REF].

Mathematical backgrounds for a rigorous study of variable structure control systems were presented in the beginning of 1960s by the celebrated Filippov theory of differential equations with discontinuous right-hand sides [START_REF] Filippov | Differential Equations with Discontinuous Righthand Sides[END_REF]. Following this theory, discontinuous differential equations have to be extended to differential inclusions. This extension helps to describe, correctly from a mathematical point of view, such a phenomenon as sliding mode [START_REF] Utkin | Sliding Modes in Control Optimization[END_REF], [START_REF] Edwards | Sliding Mode Control: Theory and Applications[END_REF], [START_REF] Utkin | Sliding Mode Control in Electro-Mechanical Systems[END_REF]. In spite of this, Filippov theory was severely criticized by many authors [START_REF] Neimark | Note on A. Filippov paper[END_REF], [START_REF] Aizerman | Foundations of theory of discontinuous systems i[END_REF], [START_REF] Utkin | Sliding Modes in Control Optimization[END_REF], since it does not describe adequately some discontinuous and relay models. That is why, extensions and specifications of this theory appear rather frequently [START_REF] Aizerman | Foundations of theory of discontinuous systems i[END_REF], [START_REF] Bartolini | Variable structure systems nonlinear in the control law[END_REF]. Recently, in [START_REF] Heemels | Input-to-state stability and interconnections of discontinuous dynamical systems[END_REF] an extension of Filippov theory was presented in order to study Input-to-State Stability (ISS) and some other robustness properties of discontinuous models.

Analysis of sliding mode systems is usually related to a specific property, which is called finite-time stability [START_REF] Roxin | On finite stability in control systems[END_REF], [START_REF] Utkin | Sliding Modes in Control Optimization[END_REF], [START_REF] Bhat | Finite time stability of continuous autonomous systems[END_REF], [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF], [START_REF] Moulay | Finite-time stability conditions for nonautonomous continuous systems[END_REF]. Indeed, the simplest example of a finite-time stable system is the relay sliding mode system: ẋ = -sign[x], x ∈ R, x(0) = x 0 . Any solution of this system reaches the origin in a finite time T (x 0 ) = |x 0 | and remains there for all later time instants.

Sometimes, this conceptually very simple property is hard to prove theoretically. From a practical point of view, it is also important to estimate a time of stabilization (settling time). Both these problems can be tackled by Lyapunov Function Method [START_REF] Lyapunov | The general problem of the stability of motion[END_REF][START_REF] Barbashin | Lyapunov Functions[END_REF][START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF]. However, designing a finite-time Lyapunov function of a rather simple form is a difficult problem for many sliding mode systems. In particular, appropriate Lyapunov functions for second order sliding mode systems are non-smooth [START_REF] Polyakov | Reaching time estimation for "super-twisting" second order sliding mode controller via Lyapunov function designing[END_REF][START_REF] Polyakov | Lyapunov function design for finite-time convergence analysis: "twisting" controller for second order sliding mode realization[END_REF][START_REF] Poznyak | Analysis of finite-time convergence by the method of Lyapunov functions in systems with second order sliding modes[END_REF] or even non-Lipschitz [START_REF] Moreno | A Lyapunov approach to analysis and design of second order sliding mode algorithms[END_REF][START_REF] Moreno | Strict Lyapunov functions for the super-twisting algorithm[END_REF][START_REF] Gonzalez | Variable gain super-twisting sliding mode control[END_REF]. Some problems of a stability analysis using generalized Lyapunov functions are studied in [START_REF] Zubov | Methods of A.M. Lyapunov and Their Applications[END_REF][START_REF] Roxin | On stability in control systems[END_REF][START_REF] Clarke | Qualitative properties of trajectories of control systems: A survey[END_REF][START_REF] Chellaboina | Generalized Lyapunov and invariant set theorems for nonlinear dynamical systems[END_REF].

One more extension of a conventional stability property is called fixed-time stability [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF]. In addition to finite-time stability it assumes uniform boundedness of a settling time on a set of admissible initial conditions (attraction domain). This phenomenon was initially discovered in the context of systems that are homogeneous in the bi-limit [START_REF] Andrieu | Homogeneous approximation, recursive obsertver and output feedback[END_REF]. In particular, if an asymptotically stable system has an asymptotically stable homogeneous approximation at the 0-limit with negative degree and an asymptotically stable homogeneous approximation at the +∞-limit with positive degree, then it is fixed-time stable. An important application of this concepts was considered in the paper [START_REF] Cruz-Zavala | Uniform robust exact differentiator[END_REF], which designs a uniform (fixed-time) exact differentiator basing on the second order sliding mode technique. Analysis of fixed-time stable sliding mode system requires applying generalized Lyapunov functions [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF], [START_REF] Cruz-Zavala | Uniform robust exact differentiator[END_REF].

The main goal of this paper is to survey mathematical tools required for stability analysis of modern sliding mode control systems. The paper is organized as follows. The next section presents notations, which are used in the paper. Section 3 considers elements of the theory of differential equations with discontinuous right-hand sides, which are required for a correct description of sliding modes. Stability notions, which frequently appear in sliding mode control systems, are discussed in Section 4. Concepts of generalized derivatives are studied in Section 5 in order to present a generalized Lyapunov function method in Section 6. Finally, some concluding remarks are given.

Notations

• R is the set of real numbers and R = R ∪ {-∞} ∪ {+∞}, R + = {x ∈ R :

x > 0} and R + = R + ∪ {+∞}.

• I denotes one of the following intervals: • The inner product of x, y ∈ R n is denoted by x, y and x =

x, x .

• The set consisting of elements x 1 , x 2 , ..., x n is denoted by {x 1 , x 2 , ..., x n }.

• The set of all subsets of a set M ⊆ R n is denoted by 2 M .

• The sign function is defined by

sign σ [ρ] =          1 if ρ > 0, -1 if ρ < 0, σ if ρ = 0, (1)
where σ ∈ R : -1 ≤ σ ≤ 1. If σ = 0 we use the notation sign [ρ].

• The set-valued modification of the sign function is given by sign

[ρ] =          {1} if ρ > 0, {-1} if ρ < 0, [-1, 1] if ρ = 0.
(2)

• x [α] = |x| α sign[x] is a power operation, which preserves the sign of a number x ∈ R.

• The geometric sum of two sets is denoted by " +", i.e.

M 1 +M 2 = x1∈M1,x2∈M2 {x 1 + x 2 }, (3)
where

M 1 ⊆ R n , M 2 ⊆ R n .
• The Cartesian product of sets is denoted by ×.

• The product of a scalar y ∈ R and a set M ⊆ R n is denoted by "•" :

y • M = M • y = x∈M {yx}. (4)
• The product of a matrix A ∈ R m×n to a set M ⊆ R n is also denoted by "•":

A • M = x∈M {Ax}. ( 5
)
• ∂Ω is the boundary set of Ω ⊆ R n .

• B(r) = {x ∈ R n : x < r} is an open ball of the radius r ∈ R + with the center at the origin. Under introduced notations, {y} +B(ε) is an open ball of the radius ε > 0 with the center at y ∈ R n .

• int(Ω) is the interior of a set Ω ⊆ R n , i.e. x ∈ int(Ω) iff ∃r ∈ R + :

{x} + B(r) ⊆ Ω.

• Let k be a given natural number. C k (Ω) is the set of continuous functions defined on a set Ω ⊆ R n , which are continuously differentiable up to the order k.

• If V (•) ∈ C 1 then ∇V (x) = ∂V ∂x1 , ..., ∂V ∂xn T . If s : R n → R m , s(•) = (s 1 (•), ..., s m (•)) T , s i (•) ∈ C 1 then ∇s(x) is the matrix R n×m of the partial derivatives ∂sj ∂xi .
• W n I is the set of vector-valued, componentwise locally absolutely continuous functions, which map I to R n .

Discontinuous systems, sliding modes and disturbances

Systems with discontinuous right-hand sides

The classical theory of differential equations [START_REF] Coddington | Theory of Ordinary Differential Equations[END_REF] introduces a solution of the ordinary differential equation (ODE)

ẋ = f (t, x), f : R × R n → R n , (6)
as a differentiable function x : R → R n , which satisfies (6) on some segment (or interval) I ⊆ R. The modern control theory frequently deals with dynamic systems, which are modeled by ODE with discontinuous right-hand sides [START_REF] Utkin | Sliding Mode Control in Electro-Mechanical Systems[END_REF][START_REF] Liberzon | Switchings in Systems and Control[END_REF][START_REF] Gelig | Stability Of Stationary Sets In Control Systems With Discontinuous Nonlinearities[END_REF]. The classical definition is not applicable to such ODE. This section observes definitions of solutions for systems with piecewise continuous right-hand sides, which are useful for sliding mode control theory.

Recall that a function f :

R n+1 → R n is piece-wise continuous iff R n+1 consists of a finite number of domains (open connected sets) G j ⊂ R n+1 , j = 1, 2, ..., N ; G i G j = ∅ for i = j and the boundary set S = N i=1
∂G j of measure zero such that f (t, x) is continuous in each G j and for each (t * , x * ) ∈ ∂G j there exists a vector f j (t * , x * ), possible depended on j, such that for any sequence

(t k , x k ) ∈ G j : (t k , x k ) → (t * , x * ) we have f (t k , x k ) → f j (t * , x * ).
Let functions f j : R n+1 → R n be defined on ∂G j according to this limiting process, i.e.

f j (t, x) = lim (t k ,x k )→(t,x) f (t k , x k ), (t k , x k ) ∈ G j , (t, x) ∈ ∂G j .

Filippov definition

Introduce the following differential inclusion

ẋ ∈ K[f ](t, x), t ∈ R, (7)
K[f ](t, x) =        {f (t, x)} if (t, x) ∈ R n+1 \S, co j∈N (t,x) f j (t, x) if (t, x) ∈ S, (8)
where co(M ) is the convex closure of a set M and the set-valued index function N : R n+1 → 2 {1,2,...,N } defined on S indicates domains G j , which have a common boundary point (t, x) ∈ S, i.e.

N (t, x) = {j ∈ {1, 2, ..., N } : (t, x) ∈ ∂G j } . For (t, x) ∈ S the set K[f ](t, x) is a convex polyhedron.
Definition 1 ( [START_REF] Filippov | Differential Equations with Discontinuous Righthand Sides[END_REF], page 50). An absolutely continuous function x : I → R n defined on some interval or segment I is called a solution of (6) if it satisfies the differential inclusion [START_REF] Filippov | Differential Equations with Discontinuous Righthand Sides[END_REF] almost everywhere on I.

Consider the simplest case when the function f (t, x) has discontinuities on a smooth surface S = {x ∈ R n : s(x) = 0}, which separates R n on two domains

G + = {x ∈ R n : s(x) > 0} and G -= {x ∈ R n : s(x) < 0}.
Let P (x) be the tangential plane to the surface S at a point x ∈ S and

f + (t, x) = lim xi→x,xi∈G + f (t, x i ) and f -(t, x) = lim xi→x,xi∈G -f (t, x i ) For x ∈ S the set K[f ](t, x) defines a segment connecting the vectors f + (t, x)
and f -(t, x) (see Fig. 1(a), 1(b)). If this segment crosses P (x) then the cross point is the end of the velocity vector, which defines the system motion on the surface S (see Fig. 1(b)). In this case the system (7) has trajectories, which start to slide on the surface S according to the sliding motion equation

ẋ = f 0 (t, x), (9)
where the function

f 0 (t, x) = ∇s(x), f -(t, x) f + (t, x) + ∇s(x), f + (t, x) f -(t, x) ∇s(x), f + (t, x) -f -(t, x) (10)
is the velocity vector defined by a cross-point of the segment and the plane

P (x), i.e. f 0 (t, x) = µf + (t, x) + (1 -µ)f -(t, x) with µ ∈ [0, 1] such that ∇s(x), µf + (t, x) + (1 -µ)f -(t, x) = 0. If ∇s(x) ⊥ µf -(t, x)+(1-µ)f + (t, x)
for every µ ∈ [0, 1] then any trajectory of ( 7) comes through the surface (see Fig. 1(a)) resulting an isolated "switching" in the right-hand side of ( 6). Seemingly, Filippov definition is the most simple and widespread definition of solutions for ODE with discontinuous by x right-hand sides. However, this definition was severely criticized by many authors [START_REF] Neimark | Note on A. Filippov paper[END_REF], [START_REF] Utkin | Sliding Modes in Control Optimization[END_REF], [START_REF] Aizerman | Foundations of theory of discontinuous systems i[END_REF] since its appearance in 1960s. In fact, it does not cover correctly many real-life systems, which have discontinuous models. Definitely, contradictions to reality usually are provoked by model inadequacies, but some problems can be avoided by modifications of Filippov definition.

Example 1. Consider the discontinuous control system

   ẋ1 = u, ẋ2 = (εu 2 + ε 2 |u| -ε)x 2 , u = -sign[x 1 ], (11)
where x 1 , x 2 ∈ R are system states, ε ∈ R + is some small parameter 0 < ε 1, u ∈ R is the relay control with the sign function defined by (1).

If we apply Filippov definition only to the first equation of [START_REF] Bartolini | Variable structure systems nonlinear in the control law[END_REF], we obtain the following sliding motion equation ẋ1 = 0 for x 1 = 0, which implicitly implies u = 0 for x 1 = 0. So, the expectable sliding motion equation for [START_REF] Bartolini | Variable structure systems nonlinear in the control law[END_REF] is

   ẋ1 = 0, ẋ2 = -εx 2 , for x 1 = 0. ( 12
)
However, considering Filippov definition for the whole system (11) we derive

f + (x 1 , x 2 ) =   -1 ε 2 x 2   for x 1 → +0 f -(x 1 , x 2 ) =   1 ε 2 x 2   for x 1 → -0
and the formula (10) for s(x) = x 1 gives another sliding motion equation:

  ẋ1 ẋ2   = ∇s(x), f -(t, x) f + (t, x) + ∇s(x), f + (t, x) f -(t, x) ∇s(x), f + (t, x) -f -(t, x) =   0 ε 2 x 2  
From the practical point of view the sliding motion equation (12) looks more realistic. Indeed, in practice we usually do not have ideal relays, so the model of switchings like (1) is just a "comfortable" approximation of real "relay" elements, which are continuous functions (or singular outputs of additional dynamics [START_REF] Fridman | An averaging approach to chattering[END_REF]) probably with hysteresis or delay effects. In this case, a "real" sliding mode is, in fact, a switching regime of bounded frequency. An average value of the control input

|u| average = 1 t -t 0 t t0 |u(τ )|dτ, t > t 0 : x 1 (t 0 ) = 0
in the "real" sliding mode is less than 1, particulary |u| average ≤ 1 -ε (see [START_REF] Fridman | An averaging approach to chattering[END_REF] for details). Hence, ε|u| 2 average + ε 2 |u| average -ε ≤ -ε 2 and the system (11) has asymptotically stable equilibrium point (x 1 , x 2 ) = 0 ∈ R 2 , but Filippov definition quite the contrary provides instability of the system.

Such problems with Filippov definition may appear if the control input u is incorporated to the system (11) in nonlinear way. More detailed study of such discontinuous models is presented in [START_REF] Bartolini | Variable structure systems nonlinear in the control law[END_REF].

This example demonstrates two important things:

• Filippov defintion is not appropriate for some discontinuous models, since it does not describe a real system motion.

• Stability properties of a system with discontinuous right-hand side may depend on a definition of solutions.

Remark 1 (On Filippov regularization). The regularization of the ODE system with discontinuous right-hand side can be also done even if the function f (t, x) in (6) is not piecewise continuous, but locally measurable. In this case the differential inclusion (7) has the following right-hand side [START_REF] Filippov | Differential Equations with Discontinuous Righthand Sides[END_REF]:

K[f ](t, x) = δ>0 µ(N )=0 co f (t, {x} +B(δ)\N ),
where the intersections are taken over all sets N ⊂ R n of measure zero (µ(N ) = 0) and all δ > 0, co(M ) denotes the convex closure of the set M .

Utkin definition (equivalent control method)

The modification of Filippov definition, which delivers an important impact to the sliding mode control theory, is called the equivalent control method [START_REF] Utkin | Sliding Modes in Control Optimization[END_REF].

Consider the system ẋ = f (t, x, u(t, x)), t ∈ R, (13)
where f : R × R n × R m → R n is a continuous vector-valued function and a piecewise continuous function

u : R × R n → R m , u(t, x) = (u 1 (t, x), u 2 (t, x), ..., u m (t, x)) T
has a sense of a feedback control.

Assumption 1. Each component u i (t, x) is discontinuous only on a surface

S i = {(t, x) ∈ R n : s i (t, x) = 0},
where functions s i : R n+1 → R are smooth, i.e.

s i ∈ C 1 (R n+1 ).
Introduce the following differential inclusion

ẋ ∈ f (t, x, K[u](t, x)), t ∈ R, (14)
where

K[u](t, x) = (K[u 1 ](t, x), ..., K[u m ](t, x)) T , K[u i ](t,x)=            {u i (t, x)}, s i (t, x) = 0, co      lim (t j ,x j )→(t,x) si(tj ,xj )>0 u i (t j ,x j ), lim (t j ,x j )→(t,x) si(tj ,xj )<0 u i (t j ,x j )      , s i (t, x) = 0. (15)
The set f (t, x, K[u 1 ](t, x), ..., K[u m ](t, x)) is non-convex in general case [START_REF] Bartolini | Variable structure systems nonlinear in the control law[END_REF].

Definition 2. An absolutely continuous function x : I → R n defined on some interval or segment I is called a solution of (13) if there exists a measurable function u eq : I → R m such that u eq (t) ∈ K[u](t, x(t)) and ẋ(t) = f (t, x(t), u eq (t)) almost everywhere on I.

The given definition introduces a solution of the differential equation ( 13), which we call Utkin solution, since it follows the basic idea of the equivalent control method introduced by V.I. Utkin [START_REF] Utkin | Sliding Modes in Control Optimization[END_REF], page 14 (see also [START_REF] Filippov | Differential Equations with Discontinuous Righthand Sides[END_REF], page 54).

Obviously, for (t, x(t)) / ∈ S we have u eq (t) = u(t, x(t)). So, the only question is how to define u eq (t) on a switching surface. The scheme presented in [START_REF] Utkin | Sliding Modes in Control Optimization[END_REF] is based on resolving of the equation ṡ(t, x) = ∂s ∂t + ∇ T s(x)f (t, x, u eq ) = 0 in algebraic way. The obtained solution u eq (t, x) is called equivalent control [START_REF] Utkin | Sliding Modes in Control Optimization[END_REF].

In order to show a difference between Utkin and Filippov definitions we consider the system (13) with u ∈ R (m = 1) and a time-invariant switching

surface S = {x ∈ R n : s(x) = 0}. Denote u + (t, x) = lim xj →x,s(xj )>0 u(t, x j ) and u -(t, x) = lim xj →x,s(xj )<0 u(t, x j ), f + (t, x) = f (t, x, u + (t, x)) and f -(t, x) = f (t, x, u -(t, x)).
The sliding mode existence condition

∃µ ∈ [0, 1] : ∇s(x) ⊥ µf -(t, x) + (1 -µ)f + (t, x)
is the same for both definitions.

The sliding motion equation obtained by Filippov definition has the form (9) recalled here by

ẋ = f 0 (t, x), f 0 (t, x) = ∇s(x), f -(t, x) f + (t, x) + ∇s(x), f + (t, x) f -(t, x) ∇s(x), f + (t, x) -f -(t, x) .
The corresponding vector f 0 (t, x) is defined by a cross-point of the tangential plane at the point x ∈ S and a segment connecting the ends of the vectors f + (t, x) and f -(t, x) (see Fig. 3(a)).

Utkin definition considers a set K[u](t, x), which is the convex closure of a set of limit values of a discontinuous control function u(t, x). For different u 1 , u 2 , u 3 , ... ∈ K[u](t, x) the vectors f (t, x, u 1 ), f (t, x, u 2 ), f (t, x, u 3 ), ... end on an arc connecting the ends of the vectors f + (t, x) and f -(t, x) (see Fig. 3(b)).

In this case the vector f (t, x, u eq ) defining the right-hand side of the sliding motion equation is derived by a cross-point of this arc and a tangential plane at the point x ∈ S (see Fig. 3

(b)), i.e. ẋ = f (t, x, u eq (t, x)), x ∈ S, (16)
where u eq (t, x) ∈ K[u](t, x) : ∇s(x) ⊥ f (t, x, u eq (t, x)).

Sometimes Utkin definition gives quite strange, from mathematical point of view, results, but they are very consistent with real-life applications.

Example 2. ([7]) Consider the system

ẋ = Ax + bu 1 + cu 2 , u 1 = sign[x 1 ], u 2 = sign[x 1 ], (17)
where

x = (x 1 , x 2 , ..., x n ) T ∈ R n , A ∈ R n×n , c, b ∈ R n , c = b. Filippov definition provides the inclusion ẋ ∈ {Ax} +(b + c) • sign[x 1 ], ( 18
)
where + is the geometric (Minkovski) sum of sets (see (3)), sign is the set-valued modification of the sign function (see [START_REF] Tsypkin | Relay Control Systems[END_REF]) and the product of a vector to a set is defined by [START_REF] Choi | Robust throttle control of automotive engines[END_REF].

If the functions u 1 and u 2 are independent control inputs, then Utkin definition gives

ẋ ∈ {Ax} +b • sign[x 1 ] +c • sign[x 1 ]. ( 19
)
The right-hand sides of ( 18) and ( 19) coincide if the vectors c and b are collinear, otherwise Filippov and Utkin definitions generate different set-valued mappings.

For example, if

x = (x 1 , x 2 ) T ∈ R 2 , A = 0, b = (-1, 0) T and c = (0, -1) T , then a) Filippov definition gives K[f ](x) = [-1, 1]•   1 1   for x 1 = 0, i.e.K[f ](x)
is a segment connecting the points (-1, -1) and (1, 1) (see Fig. 2(a)); the corresponding sliding motion equation is

ẋ = 0 for x 1 = 0; b)Utkin definition generates the square box, i.e. K[f ](x) = [-1, 1] × [-1, 1]
for x 1 = 0 (see Fig. 2(b)), so sliding motion equation has the form

ẋ =   0 u eq (t)   for x 1 = 0,
where u eq : R → R is an arbitrary locally measurable function such that |u eq (t)| ≤ 1 for every t ∈ R.

Control inputs u 1 and u 2 are independent and relay elements are not identical in practice. They can not switch absolutely synchronously. This admits a motion of the system along the switching line x 1 = 0. In this case, Utkin definition is more adequate to reality than Filippov one.

Aizerman-Pyatnitskii definition

The Aizerman-Pyatnitskii definition covers solutions of both definitions considered above by means of introduction of the following differential inclusion

ẋ ∈ co f (t, x, K[u](t, x)), t ∈ R, (20)
for the system [START_REF] Roxin | On finite stability in control systems[END_REF].

Definition 3 (Aizerman-Pyatnitskii definition ([10] and [START_REF] Filippov | Differential Equations with Discontinuous Righthand Sides[END_REF], page 55)).

An absolutely continuous function x : I → R n defined on some interval or segment I is called a solution of (6) if it satisfies the differential inclusion [START_REF] Polyakov | Reaching time estimation for "super-twisting" second order sliding mode controller via Lyapunov function designing[END_REF] almost everywhere on I.

Returning to the example considered above for u ∈ R (m = 1) Aizerman-Pyatnitskii definition gives the inclusion

ẋ ∈ F SM (t, x) = co{f 0 (t, x), f (t, x, u eq (t, x))},
which describes the motion of the discontinuous system (13) in a sliding mode

(see Fig. 3(c) with f α ∈ F SM (t, x)).
A criticism of Aizerman-Pyatnitskii definition is related to nonuniqueness of solutions even for simple nonlinear cases. However, if some stability property is proven for Aizerman-Pyatnickii definition, then the same property holds for both Filippov and Utkin solutions. The affine control system is the case when all definitions may be equivalent.

Theorem 1 ([37], Theorem 14, page 44). Let a right-hand side of the system (6) be affine with respect to control:

f (t, x) = a(t, x) + b(t, x)u(t, x),
where a : R n+1 → R n is a continuous vector-valued function, b : R n+1 → R n×m is a continuous matrix-valued function and u : R n+1 → R m is a piecewise continuous function u(t, x) = (u 1 (t, x), ..., u m (t, x)) T , such that u i has a unique time-invariant switching surface s i (x) = 0,

s i ∈ C 1 (R n ).
Definitions of Filippov, Utkin and Aizerman-Pyatnitskii are equivalent iff

det ∇ T s(x)b(t, x) = 0 if (t, x) ∈ S, (21)
where s(x) = (s 1 (x), s 2 (x), ..., s m (x)) T , ∇s(x) ∈ R n×m is the matrix of partial derivatives ∂sj ∂xi and S is a discontinuity set of u(t, x).

The present theorem has the simple geometric interpretation for the single input system. The affine control system is linear with respect to the control input, which is the only discontinuous term of the right-hand side of the system [START_REF] Utkin | Sliding Mode Control in Electro-Mechanical Systems[END_REF]. In this case all regularization procedures provide the set-valued extension depicted on Fig. 3(a). The condition [START_REF] Polyakov | Lyapunov function design for finite-time convergence analysis: "twisting" controller for second order sliding mode realization[END_REF] excludes non-uniqueness of this setvalued extension for multi-input case. For example, the system considered in Example 17 is affine, but it does not satisfy the condition (21).

System disturbances and extended differential inclusion

Some modifications of presented definitions of solutions are required again if a model of a dynamic system includes disturbances into considerations. For example, the paper [START_REF] Heemels | Input-to-state stability and interconnections of discontinuous dynamical systems[END_REF] extends Filippov definition to discontinuous disturbed systems. It demonstrates that the presented extension is useful for ISS analysis.

The present survey is mostly oriented on sliding mode control systems. The robustness of sliding mode control systems (at least theoretically) is related to invariance of qualitative behavior of closed-loop system on matched disturbances with some a priori known maximum magnitude [START_REF] Utkin | Sliding Modes in Control Optimization[END_REF], [START_REF] Edwards | Sliding Mode Control: Theory and Applications[END_REF], [START_REF] Utkin | Sliding Mode Control in Electro-Mechanical Systems[END_REF]. This property usually allows reducing a problem of stability analysis of a disturbed discontinuous sliding mode control system to a similar problem presented for an extended differential inclusion. The idea explained in the next example was also used in papers [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF], [START_REF] Levant | Quasi-continuous high-order sliding-mode controllers[END_REF].

Example 3. Consider the simplest disturbed sliding mode system

ẋ = -d 1 (t) sign[x] + d 2 (t), (22)
where x ∈ R, unknown functions d i : R → R are bounded by

d min i ≤ d i (t) ≤ d max i , i = 1, 2, (23)
and the function sign[x] is defined by [START_REF] Flugge-Lotz | Discontinuous automatic control[END_REF].

Obviously, all solutions of the system (22) belong to a solution set of the following extended differential inclusion

ẋ ∈ -[d min 1 , d max 1 ] • sign[x] + [d min 2 , d max 2 ]. ( 24
)
Stability of the system (24) implies the same property for [START_REF] Poznyak | Analysis of finite-time convergence by the method of Lyapunov functions in systems with second order sliding modes[END_REF]. In particular,

for d min 1 > max{|d min 2 |, |d max
for original discontinuous differential equations. That is why, in this paper we do not discuss "robust" modifications of stability notions for differential inclusions.

Models of sliding mode control systems usually have the form

ẋ = f (t, x, u(t, x), d(t)), t ∈ R, (25)
where x ∈ R n is the vector of system states, u ∈ R m is the vector of control inputs, d ∈ R k is the vector of disturbances, the function f : R n+m+k+1 → R n is assumed to be continuous, the control function u : R n+1 → R m is piecewise continuous, the vector-valued function d : R → R k is assumed to be locally measurable and bounded as follows:

d min i ≤ d i (t) ≤ d max i , (26)
where

d(t) = (d 1 (t), d 2 (t), ..., d k (t)) T , t ∈ R.
All further considerations deal with the extended differential inclusion

ẋ ∈ F (t, x), t ∈ R, (27)
where

F (t, x) = co{f (t, x, K[u](t, x), D)}, the set-valued function K[u](t, x) is
defined by [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF] and

D = (d 1 , d 2 , ..., d k ) T ∈ R k : d i ∈ [d min i , d max i ], i = 1, 2, ..., k . (28)
The same extended differential inclusion can be used if the vector d (or its part) has a sense of parametric uncertainties.

Existence of solutions

Let us recall initially the classical result of Caratheodory about existence of solutions for ODEs with right-hand sides, which are discontinuous on time.

Theorem 2 ([33], Theorem 1.1, Chapter 2). Let the function and a number b ∈ (0, a] such that x(t 0 ) = x 0 and the equality

g : R × R n → R n (t, x) → g(t, x) be continuous by x in Ω = {x 0 } + B(r), r ∈ R + , x 0 ∈ R n for any fixed t ∈ I = [t 0 -a, t 0 + a], a ∈ R + , t 0 ∈ R
ẋ(t) = g(t, x(t))
hold almost everywhere on

[t 0 -b, t 0 + b].
Introduce the following distances

ρ(x, M) = inf y∈M x -y , x ∈ R n , M ⊆ R n , ρ(M 1 , M 2 ) = sup x∈M1 ρ(x, M 2 ), M 1 ⊆ R n , M 2 ⊆ R n . (29)
Remark, the distance

ρ(M 1 , M 2 ) is not symmetric, i.e. ρ(M 1 , M 2 ) = ρ(M 2 , M 1 )
in the general case.

Definition 4. A set-valued function F : R n+1 → 2 R n+1 is said to be upper semi-continuous at a point (t * , x * ) ∈ R n+1 if (t, x) → (t * , x * ) implies ρ(F (t, x), F (t * , x * )) → 0.
For instance, the function sign[x] defined by ( 2) is upper semi-continuous.

G = {(t, x) ∈ R n+1 : |t -t 0 | ≤ a and x -x 0 ≤ b}, (30)
where

a, b ∈ R + , t 0 ∈ R, x 0 ∈ R n . Let F (t, x) be nonempty, compact and convex for (t, x) ∈ G.
If there exists K > 0 such that ρ(0, F (t, x)) < K for (t, x) ∈ G then there exists at least one absolutely continuous function x : R → R n defined at least on the segment [t 0 -α, t 0 + α], α = min{a, b/K}, such that x(t 0 ) = x 0 and the inclusion ẋ(t) ∈ F (t, x(t)) holds almost everywhere on [t 0 -α, t 0 + α].

Filippov and Aizerman-Pyatnickii set-valued extensions of the discontinuous ODE (see formulas [START_REF] Filippov | Differential Equations with Discontinuous Righthand Sides[END_REF] and ( 20)) and the extended differential inclusion [START_REF] Roxin | On stability in control systems[END_REF] satisfy all conditions of Theorem 3 implying local existence of the corresponding solutions.

The existence analysis of Utkin solutions is more complicated in general case.

Since the function f (t, x, u) is continuous, then for any measurable bounded function u 0 : I → R m the composition f (t, x, u 0 (t)) satisfies all conditions of Theorem 2 and the equation ẋ = f (t, x, u 0 (t)) has an absolutely continuous solution x 0 (t), but u 0 (t) may not belong to the set K[u](t, x 0 (t)).

In some cases, the existence of Utkin solution can be proven using the celebrated Filippov's lemma. Then there exists a measurable function u eq : R → R m such that u eq (t) ∈ U (t, x(t)) and ẋ(t) = f (t, x(t), u eq (t)) almost everywhere on I.

If the differential inclusion ( 14) has a convex right-hand side then Theorem 3 together with Lemma 1 results local existence of Utkin solutions. If the setvalued function f (t, x, K[u](t, x)) is non-convex, the existence analysis of Utkin solutions becomes very difficult (see [START_REF] Bartolini | Variable structure systems nonlinear in the control law[END_REF] for the details).

Some additional restrictions to right-hand sides are required for a prolongation of solutions. In particular, the famous Winter's theorem (see, for example, [START_REF] Poznyak | Advanced Mathematical Tools for Automatic Control Engineers[END_REF], page 515) about a non-local existence of solutions of ODE can be expanded to differential inclusions.

Theorem 4 ( [START_REF] Gelig | Stability of Nonlinear Systems with Nonunique Equilibrium Position[END_REF], page 169). Let a set-valued function F : R n+1 → R n+1 be defined and upper-semicontinuous in R n+1 . Let F (t, x) be nonempty, compact and convex for any (t, x) ∈ R n+1 .

If there exists a real valued function L : R

+ ∪ {0} → R + ∪ {0} such that ρ(0, F (t, x)) ≤ L( x ) and +∞ 0 1 L(r) dr = +∞,
then for any (t 0 , x 0 ) ∈ R n+1 the system (27) has a solution x(t) : x(t 0 ) = x 0 defined for all t ∈ R.

Based on Lyapunov function method, the less conservative conditions for prolongation of solutions are given below.

Stability and convergence rate

Consider the differential inclusion [START_REF] Roxin | On stability in control systems[END_REF] for t > t 0 with an initial condition

x(t 0 ) = x 0 , (31)
where x 0 ∈ R n is given.

Cauchy problem [START_REF] Roxin | On stability in control systems[END_REF], [START_REF] Andrieu | Homogeneous approximation, recursive obsertver and output feedback[END_REF] obviously may not have a unique solution for a given t 0 ∈ R and a given x 0 ∈ R n . Let us denote the set of all solutions of Cauchy problem [START_REF] Roxin | On stability in control systems[END_REF], [START_REF] Andrieu | Homogeneous approximation, recursive obsertver and output feedback[END_REF] by Φ(t 0 , x 0 ) and a solution of (27), (31) by x(t, t 0 , x 0 ) ∈ Φ(t 0 , x 0 ).

Nonuniqueness of solutions implies two types of stability for differential inclusions [START_REF] Roxin | On stability in control systems[END_REF]: weak stability(a property holds for a solution) and strong stability (a property holds for all solutions) (see, for example, [START_REF] Roxin | On stability in control systems[END_REF], [START_REF] Roxin | On finite stability in control systems[END_REF], [START_REF] Filippov | Differential Equations with Discontinuous Righthand Sides[END_REF]). Weak stability usually is not enough for robust control purposes. This section observes only strong stability properties of the system [START_REF] Roxin | On stability in control systems[END_REF]. All conditions presented in definitions below are assumed to be held for all solutions x(t, t 0 , x 0 ) ∈ Φ(t 0 , x 0 ).

Lyapunov, asymptotic and exponential stability

The concept of stability introduced in the famous thesis of A.M. Lyapunov [START_REF] Lyapunov | The general problem of the stability of motion[END_REF] is one of central notions of the modern stability theory. It considers some nominal motion x * (t, t 0 , x 0 ) of a dynamic system and studies small perturbations of the initial condition x 0 . If they imply small deviations of perturbed motions from x * (t, t 0 , x 0 ) then the nominal motion is called stable. We study different stability forms of the zero solution (or, equivalently, the origin) of the system [START_REF] Roxin | On stability in control systems[END_REF], since making the change of variables y = x -x * we transform any problem of stability analysis for some nontrivial solution x * (t, t * , x * 0 ) to the same problem for the zero solution.

Assume that 0 ∈ F (t, 0) for t ∈ R, where F (t, x) is defined by [START_REF] Roxin | On stability in control systems[END_REF]. Then the function x 0 (t) = 0 belongs to a solution set Φ(t, t 0 , 0) for any t 0 ∈ R.

Definition 5 (Lyapunov stability). The origin of the system (27) is said to be Lyapunov stable if for ∀ε ∈ R + and ∀t 0 ∈ R there exists δ = δ(ε, t 0 ) ∈ R + such that for ∀x 0 ∈ B(δ) 1) any solution x(t, t 0 , x 0 ) of Cauchy problem [START_REF] Roxin | On stability in control systems[END_REF], [START_REF] Andrieu | Homogeneous approximation, recursive obsertver and output feedback[END_REF]

exists for t > t 0 ; 2) x(t, t 0 , x 0 ) ∈ B(ε) for t > t 0 .
If the function δ does not depend on t 0 then the origin is called uniformly Lyapunov stable. For instance, if F (t, x) is independent of t (time-invariant case) and the zero solution of ( 27) is Lyapunov stable, then it is uniformly Lyapunov stable.

Proposition 1. If the origin of the system (27) is Lyapunov stable then x(t) = 0 is the unique solution of Cauchy problem [START_REF] Roxin | On stability in control systems[END_REF], [START_REF] Andrieu | Homogeneous approximation, recursive obsertver and output feedback[END_REF] with x 0 = 0 and t 0 ∈ R.

The origin, which does not satisfy any condition from Definition 5, is called unstable.

Definition 6 (Asymptotic attractivity). The origin of the system (27) is said to be asymptotically attractive if for ∀t 0 ∈ R there exists a set

U(t 0 ) ⊆ R n : 0 ∈ int(U(t 0 )) such that ∀x 0 ∈ U(t 0 )
• any solution x(t, t 0 , x 0 ) of Cauchy problem [START_REF] Roxin | On stability in control systems[END_REF], [START_REF] Andrieu | Homogeneous approximation, recursive obsertver and output feedback[END_REF]

exists for t > t 0 ; • lim t→+∞ x(t, t 0 , x 0 ) = 0.
The set U(t 0 ) is called attraction domain.

Finding the maximum attraction domain is an important problem for many practical control applications.

Definition 7 (Asymptotic stability). The origin of the system (27) is said to be asymptotically stable if it is Lyapunov stable and asymptotically attractive.

If U(t 0 ) = R n then the asymptotically stable (attractive) origin of the system ( 27) is called globally asymptotically stable (attractive).

Requirement of Lyapunov stability is very important in Definition 7, since even global asymptotic attractivity does not imply Lyapunov stability.

Example 4 ( [START_REF] Vinograd | The inadequacy of the method of characteristic exponents for the study of nonlinear differential equations[END_REF], page 433 or [START_REF] Hahn | Stability of Motion[END_REF], page 191). The system

ẋ1 = x 2 1 (x 2 -x 1 ) + x 5 2 (x 2 1 + x 2 2 ) 1 + (x 2 1 + x 2 2 ) 2 and ẋ2 = x 2 2 (x 2 -2x 1 ) (x 2 1 + x 2 2 ) 1 + (x 2 1 + x 2 2 )
2 has the globally asymptotically attractive origin. However, it is not Lyapunov stable, since this system has trajectories (see Fig. 4), which start in arbitrary small ball with the center at the origin and always leave the ball B(ε 0 ) of a fixed radius ε 0 ∈ R + (i.e. Condition 2 of Definition 5 does not hold for ε ∈ (0, ε 0 )). is said to be uniformly asymptotically stable if it is uniformly Lyapunov stable and uniformly asymptotically attractive.

If U = R n then
ẋ ∈ F (x)
is asymptotically stable then it is uniformly asymptotically stable.

Frequently, an asymptotic stability of a closed-loop system is not enough for a "good" quality of control. A rate of transition processes also has to be adjusted in order to provide a better performance to a control system. For this purpose some concepts of "rated" stability can be used such as exponential, finite-time or fixed-time stability.

Definition 10 (Exponential stability). The origin of the system (27) is said to be exponentially stable if there exist an attraction domain U ⊆ R n : 0 ∈ int(U)

and numbers C, r ∈ R + such that x(t, t 0 , x 0 ) ≤ C x 0 e -r(t-t0) , t > t 0 . ( 32
)
for t 0 ∈ R and x 0 ∈ U.

The inequality [START_REF] Cruz-Zavala | Uniform robust exact differentiator[END_REF] expresses the so-called exponential convergence (attractivity) property. The linear control theory usually deals with this property [START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF].

Exponential stability obviously implies both Lyapunov stability and asymptotic stability.

Finite-time Stability

Introduce the functional T 0 : W n [t0,+∞) → R + ∪ {0} by the following formula

T 0 (y(•)) = inf τ ≥t0:y(τ )=0
τ.

If y(τ ) = 0 for all t ∈ [t 0 , +∞) then T 0 (y(•)) = +∞.

Let us define the settling-time function of the system (27) as follows

T (t 0 , x 0 ) = sup x(t,t0,x0)∈Φ(t0,x0) T 0 (x(t, t 0 , x 0 )) -t 0 , (33)
where Φ(t 0 , x 0 ) is the set of all solutions of the Cauchy problem ( 27), [START_REF] Andrieu | Homogeneous approximation, recursive obsertver and output feedback[END_REF].

Definition 11 (Finite-time attractivity). The origin of the system (27) is said to be finite-time attractive if for ∀t 0 ∈ R there exists a set

V(t 0 ) ⊆ R n : 0 ∈ int(V(t 0 )) such that ∀x 0 ∈ V(t 0 )
• any solution x(t, t 0 , x 0 ) of Cauchy problem (27), ( 31) exists for t > t 0 ;

• T (t 0 , x 0 ) < +∞ for x 0 ∈ V(t 0 ) and for t 0 ∈ R.

The set V(t 0 ) is called finite-time attraction domain.

It is worth to stress that the finite-time attractivity property, introduced originally in [START_REF] Bhat | Finite time stability of continuous autonomous systems[END_REF], does not imply asymptotic attractivity. However, it is important for many control applications. For example, antimissile control problem has to be studied only on a finite interval of time, since there is nothing to control after missile explosion. In practice, Lyapunov stability is additionally required in order to guarantee a robustness of a control system.

Definition 12 (Finite-time stability ( [START_REF] Roxin | On finite stability in control systems[END_REF], [START_REF] Bhat | Finite time stability of continuous autonomous systems[END_REF])). The origin of the system (27) is said to be finite-time stable if it is Lyapunov stable and finite-time attractive.

If V(t 0 ) = R n then the origin of ( 27) is called globally finite-time stable.

Example 5. Consider the sliding mode system

ẋ = - 2 √ π sign[x] + |2tx|, t > t 0 , x ∈ R,
which, according to Filippov definition, is extended to the differential inclusion

ẋ ∈ - 2 √ π • sign[x] +{|2tx|}, t > t 0 , x ∈ R, (34)
where t 0 ∈ R. It can be shown that the origin of this system is finite-time attractive with an attraction domain

V(t 0 ) = B e t 2 0 (1 -erf(|t 0 |))
, where

erf(z) = 2 √ π z 0 e -τ 2 dτ, z ∈ R
is the so-called Gauss error function. Moreover, the origin of the considered system is Lyapunov stable (for ∀ε > 0 and for ∀t 0 ∈ R we can select δ =

δ(t 0 ) = min ε, e t 2 0 (1 -erf(|t 0 |))
), so it is finite-time stable. In particular, for t 0 > 0 the settling-time function has the form

T (t 0 , x 0 ) = erf -1 |x 0 |e -t 2 0 + erf(t 0 ) -t 0 ,
where erf -1 (•) denotes the inverse function to erf(•).

The proposition 1 implies the following property of a finite-time stable system.

Proposition 3 ([14], Proposition 2.3). If the origin of the system (27) is finite-time stable then it is asymptotically stable and x(t, t 0 , x 0 ) = 0 for t > t 0 + T 0 (t 0 , x 0 ).

A uniform finite-time attractivity requests an additional property for the system [START_REF] Roxin | On stability in control systems[END_REF].

Definition 13 (Uniform finite-time attractivity). The origin of the system ( 27) is said to be uniformly finite-time attractive if it is finite-time attractive with a time-invariant attraction domain V ⊆ R n such that the settling time function T (t 0 , x 0 ) is locally bounded on R × V uniformly on t 0 ∈ R, i.e. for any y ∈ V there exists ε ∈ R + such that {y} +B(ε) ⊆ V and sup t0∈R, x0∈{y} +B(ε)

T (t 0 , x 0 ) < +∞.

Definition 14 (Uniform finite-time stability, [START_REF] Roxin | On finite stability in control systems[END_REF], [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF]). The origin of the system (27) is said to be uniformly finite-time stable if it is uniformly Lyapunov stable and uniformly finite-time attractive.

The origin of ( 27) is called globally uniformly finite-time stable if V = R n .

Obviously, a settling-time function of time-invariant finite-time stable system ( 27) is independent of t 0 , i.e. T = T (x 0 ). However, in contrast to asymptotic and Lyapunov stability, finite-time stability of a time-invariant system does not imply its uniform finite-time stability in general case.

Example 6 ([14], page 756). Let a vector field f : R 2 → R 2 of a timeinvariant system be defined on the quadrants

Q I = x ∈ R 2 \{0} : x 1 ≥ 0, x 2 ≥ 0 Q II = x ∈ R 2 : x 1 < 0, x 2 ≥ 0 Q III = x ∈ R 2 : x 1 ≤ 0, x 2 < 0 Q IV = x ∈ R 2 : x 1 > 0, x 2 < 0
as show in Fig. 5. The vector field f is continuous, f (0) = 0 and x = (x 1 , x 2 ) T = (r cos(θ), r sin(θ)) T , r > 0, θ ∈ [0, 2π). In [START_REF] Bhat | Finite time stability of continuous autonomous systems[END_REF] it was shown that this system is finite-time stable. Moreover, it is uniformly asymptotically stable, but it is not uniformly finite-time stable. For the sequence of the initial conditions x i 0 = (0, -1/i) T , i = 1, 2, ... we have (see [START_REF] Bhat | Finite time stability of continuous autonomous systems[END_REF] for the details)

x i 0 → 0 and T (x i 0 ) → +∞.

So, for any open ball B(r), r > 0 with the center at the origin we have

sup x0∈B(r) T (x 0 ) = +∞.
Uniform finite-time stability is the usual property for sliding mode systems [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF], [START_REF] Levant | Quasi-continuous high-order sliding-mode controllers[END_REF]. The further considerations deals mainly with this property and its modifications.

Fixed-time Stability

This subsection discusses a recent extension of the uniform finite-time stability concept, which is called fixed-time stability [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF]. Fixed-time stability asks more strong uniform attractivity property for the system [START_REF] Roxin | On stability in control systems[END_REF]. As it was demonstrated in [START_REF] Cruz-Zavala | Uniform robust exact differentiator[END_REF], [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF], this property is very important for some applications, such as control and observation with predefined convergence time.

In order to demonstrate the necessity of more detailed elaboration of uniformity properties of finite-time stable systems let us consider the following motivating example.

Example 7. Consider two systems

(I) ẋ = -x [ 1 2 ] (1 -|x|) , (II) ẋ =    -x [ 1 2 ] for x < 1, 0 for x ≥ 1,
which are uniformly finite-time stable with the finite-time attraction domain

V = B(1)
. Indeed, the settling-time functions of these systems are continuous on V:

T (I) (x 0 ) = ln 1 + |x 0 | 1 2 1 -|x 0 | 1 2 , T (II) (x 0 ) = 2|x 0 | 1 2 .
So, for any y ∈ V we can select the ball {y} +B(ε) ⊆ V, where ε=(1 -|y|)/2, such that sup x0∈{y} +B(ε)

T (I) (x 0 ) <+∞ and sup x0∈{y} +B(ε) T (II) (x 0 ) <+∞.
On the other hand,

T (I) (x 0 ) → +∞ if x 0 → ±1, but T (II) (x 0 ) → 2 if
x 0 → ±1. Therefore, these systems have different uniformity properties of finitetime attractivity with respect to domain of initial conditions. Definition 15 (Fixed-time attractivity). The origin of the system (27) is said to be fixed-time attractive if it is uniformly finite-time attractive with an attraction domain V and the settling time function T (t 0 , x 0 ) is bounded on R × V, i.e. there exists a number

T max ∈ R + such that T (t 0 , x 0 ) ≤ T max if t 0 ∈ R and x 0 ∈ V.
Systems (I) and (II) from Example 7 are both fixed-time attractive with respect to attraction domain B(r) if r ∈ (0, 1), but the system (I) loses this property for the maximum attraction domain B(1).

Definition 16 (Fixed-time stability, [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF]). The origin of the system ( 27) is said to be fixed-time stable if it is Lyapunov stable and fixed-time attractive.

If V = R n then the origin of the system ( 27) is called globally fixed-time stable. Locally differences between finite-time and fixed-time stability are questionable. Fixed-time stability definitely provides more advantages to a control system in a global case [START_REF] Cruz-Zavala | Uniform robust exact differentiator[END_REF], [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF].

Example 8. Consider the system

ẋ = -x [ 1 2 ] -x [ 3 2 ] , x ∈ R, t > t 0 ,
which has solutions defined for all t ≥ t 0 :

x(t, t 0 , x 0 )=    sign(x 0 ) tan 2 arctan(|x 0 | 1 2 ) -t-t0 2 , t ≤ t 0 +2 arctan(|x 0 | 1 2 ), 0, t > t 0 +2 arctan(|x 0 | 1 2 ).
Any solution x(t, t 0 , x 0 ) of this system converges to the origin in a finite time.

Moreover, for any x 0 ∈ R, t 0 ∈ R the equality x(t, t 0 , x 0 ) = 0 holds for all t ≥ t 0 + π, i.e. the system is globally fixed-time stable with T max = π.

Generalized derivatives

The celebrated Second Lyapunov Method is founded on the so-called energetic approach to stability analysis. It considers any positive definite function as an possible energetic characteristic ("energy") of a dynamic system and studies evolution of this "energy" in time. If a dynamic system has an energetic function, which is decreasing (strongly decreasing or bounded) along any trajectory of the system, then this system has a stability property and the corresponding energetic function is called Lyapunov function.

For example, to analyze asymptotic stability of the origin of the system

ẋ = f (t, x), f ∈ C(R n+1 ), t ∈ R + , x ∈ R n (35)
it is sufficient to find a continuous positive definite function V (•) such that for any solution x(t) of the system (35) the function V (x(t)) is decreasing and tending to zero for t → +∞. The existence of such function guarantees asymptotic stability of the origin of the system (35) due to Zubov's theorem (see [26] [40]).

If the function V (x) is continuously differentiable then the required monotonicity property can be rewritten in the form of the classical condition [START_REF] Lyapunov | The general problem of the stability of motion[END_REF]:

V (x) = ∇ T V (x)f (t, x) < 0. ( 36
)
The inequality ( 36) is very usable, since it does not require knowing the solutions of ( 35) in order to check the asymptotic stability. From the practical point of view, it is important to represent monotonicity conditions in the form of differential or algebraic inequalities like [START_REF] Fridman | An averaging approach to chattering[END_REF].

Analysis of sliding mode systems is frequently based on non-smooth or even discontinuous Lyapunov functions [START_REF] Roxin | On finite stability in control systems[END_REF][START_REF] Roxin | On stability in control systems[END_REF][START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF][START_REF] Polyakov | Reaching time estimation for "super-twisting" second order sliding mode controller via Lyapunov function designing[END_REF][START_REF] Moreno | Strict Lyapunov functions for the super-twisting algorithm[END_REF], which require consideration of generalized derivatives and generalized gradients in order to verify stability conditions. This section presents all necessary backgrounds for the corresponding non-smooth analysis.

Derivative Numbers and Monotonicity

Let I be one of the following intervals: The function ϕ : R → R is called decreasing on I iff

∀t 1 , t 2 ∈ I : t 1 ≤ t 2 ⇒ ϕ(t 1 ) ≥ ϕ(t 2 ).
Let K be a set of all sequences of real numbers converging to zero, i.e.

{h n } ∈ K ⇔ h n → 0, h n = 0.
Let a real-valued function ϕ : R → R be defined on I.

Definition 17 ([46], page 207). A number

D {hn} ϕ(t) = lim n→+∞ ϕ(t + h n ) -ϕ(t) h n , {h n } ∈ K : t + h n ∈ I
is called derivative number of the function ϕ(t) at a point t ∈ I, if finite or infinite limit exists.

The set of all derivative numbers of the function ϕ(t) at a point t ∈ I is called contingent derivative:

D K ϕ(t) = {hn}∈K {D {hn} ϕ(t)} ⊆ R.
A contingent derivative of a vector-valued function ϕ : R → R n can be defined in the same way. If a function ϕ(t) is differentiable at a point t ∈ I then

D K ϕ(t) = { φ(t)}.
Lemma 2 ( [START_REF] Natanson | Theory of functions of a real variable[END_REF], page 208). If a function ϕ : R → R is defined on I then 1) the set D K ϕ(t) ⊆ R is nonempty for any t ∈ I;

2) for any t ∈ I and for any sequence {h n } ∈ K : t + {h n } ∈ I there exists a subsequence {h n } ⊆ {h n } such that finite or infinite derivative number

D {h n } ϕ(t) exists.
Remark, Lemma 2 remains true for a vector-valued function ϕ : R → R n .

Inequalities y < 0, y ≤ 0, y > 0, y ≥ 0 for y ∈ R n are understood in a componentwise sense. If for ∀y ∈ D K ϕ(t) we have y < 0 then we write D K ϕ(t) < 0. Other ordering relations ≤, >, ≥ for contingent derivatives are interpreted analogously.

The contingent derivative also helps to prove monotonicity of a non-differentiable function. Example 9. The function ϕ(t) = -t-sign σ [t] has a negative contingent derivative for all t ∈ R and for any σ ∈ [-1, 1], where the function sign σ is defined by (1). Indeed,

D K ϕ(t) = {-1} for t = 0, D K ϕ(0) = {-∞} if σ ∈ (-1, 1)
and

D K ϕ(0) = {-∞, -1} if σ ∈ {-1, 1}.
The next lemma simplifies the monotonicity analysis of nonnegative functions.

Lemma 4. If 1) the function ϕ : R → R is nonnegative on I;

2) the inequality D K ϕ(t) ≤ 0 holds for t ∈ I : ϕ(t) = 0;

3) the function ϕ(t) is continuous at any t ∈ I : ϕ(t) = 0; then ϕ(t) is decreasing function on I and differentiable almost everywhere on I.

Proof. Suppose the contrary: ∃t 1 , t 2 ∈ I :

t 1 < t 2 and 0 ≤ ϕ(t 1 ) < ϕ(t 2 ). If ϕ(t 0 ) = 0 for all t ∈ [t 1 , t 2 ] then Lemma 3 implies that the function ϕ(t) is decreasing on [t 1 , t 2 ] and ϕ(t 1 ) ≥ ϕ(t 2 ).
If there exists t 0 ∈ [t 1 , t 2 ] such that ϕ(t 0 ) = 0 and ϕ(t) > 0 for all t ∈ (t 0 , t 2 ] then Lemma 3 guarantees that the function ϕ(t) is decreasing on (t 0 , t 2 ]. Taking into account the condition 3) we obtain the contradiction ϕ(t 2 ) ≤ ϕ(t 0 ) = 0.

Finally, let there exists a point t * ∈ (t 1 , t 2 ] such that ϕ(t * ) > 0 and any neighborhood of the point t * contains a point t 0 ∈ [t 1 , t * ] : ϕ(t 0 ) = 0. In this case, let us select the sequence h n = t n -t * < 0 such that ϕ(t n ) = 0 and t n → t * as n → ∞. For this sequence we obviously have

D {hn} ϕ(t 1 )= lim n→∞ ϕ(t * +h n ) -ϕ(t * ) h n = lim n→∞ -ϕ(t * ) h n = + ∞.
This contradicts to the condition 2).

Absolutely continuous functions are differentiable almost everywhere. Monotonicity conditions for them are less restrictive. Lemma below shows relations between solutions of a differential inclusion [START_REF] Roxin | On stability in control systems[END_REF] and its contingent derivatives.

Lemma 6 ( [START_REF] Filippov | Differential Equations with Discontinuous Righthand Sides[END_REF], page 70). Let a set-valued function F : R n+1 → 2 R n be defined, upper-semicontinuous on a closed nonempty set Ω ∈ R n+1 and the set F (t, x) be nonempty, compact and convex for all (t, x) ∈ Ω.

Let an absolutely continuous function x : R → R n be defined on I and

(t, x(t)) ∈ Ω if t ∈ I. Then ẋ(t) ∈ F (t, x(t))
almost everywhere on

I    ⇔ D K x(t) ⊆ F (t, x(t))
everywhere on I.

Dini derivatives and comparison systems

The generalized derivatives presented above are closely related with wellknown Dini derivatives (see, for example, [START_REF] Szarski | Differential Inequalities[END_REF]).

• Right-hand Upper Dini derivative:

D + ϕ(t) = lim sup h→0 + ϕ(t + h) -ϕ(t) h .
• Right-hand Lower Dini derivative:

D + ϕ(t) = lim inf h→0 + ϕ(t + h) -ϕ(t) h .
• Left-hand Upper Dini derivative:

D -ϕ(t) = lim sup h→0 - ϕ(t + h) -ϕ(t) h .
• Left-hand Lower Dini derivative:

D -ϕ(t) = lim inf h→0 - ϕ(t + h) -ϕ(t) h .
Obviously, D + ϕ(t) ≤ D + ϕ(t) and D -ϕ(t) ≤ D -ϕ(t). Moreover, definitions of lim sup and lim inf directly imply that all Dini derivatives belong to the set

D K ϕ(t) and D K ϕ(t) ≤ 0 ⇔    D -ϕ(t) ≤ 0, D + ϕ(t) ≤ 0. D K ϕ(t) ≥ 0 ⇔    D -ϕ(t) ≥ 0, D + ϕ(t) ≥ 0.
Therefore, all further results for contingent derivative can be rewritten in terms of Dini derivatives.

Theorem 5 (Denjoy-Young-Saks Theorem, [START_REF] Bruckner | Differentiation of Real Functions[END_REF], page 65). If ϕ : R → R is a function defined on an interval I, then for almost all t ∈ I Dini derivatives of ϕ(t) satisfy one of the following four conditions:

• ϕ(t) has a finite derivative;

• D + ϕ(t) = D -ϕ(t) is finite and D -ϕ(t) = +∞, D + ϕ(t) = -∞; • D -ϕ(t) = D + ϕ(t) is finite and D + ϕ(t) = +∞, D -ϕ(t) = -∞; • D -ϕ(t) = D + ϕ(t) = +∞, D -ϕ(t) = D + ϕ(t) = -∞.
This theorem has the following simple corollary, which is important for some further considerations.

Corollary 1. If ϕ : R → R is a function defined on I, then the equality

D K ϕ(t) = {-∞} (D K ϕ(t) = {+∞}
) may hold only on a set ∆ ⊆ I of measure zero.

Consider the system ẏ = g(t, y), (t, y) ∈ R 2 , g : R 2 → R,

where a function g(t, y) is continuous and defined on a set G = (a, b) × (y 1 , y 2 ), a, b, y 1 , y 2 ∈ R : a < b, y 1 < y 2 . In this case the system (37) has the so-called right-hand maximum solutions for any initial condition y(t 0 ) = y 0 , (t 0 , y 0 ) ∈ G (see [START_REF] Szarski | Differential Inequalities[END_REF], Remark 9.1, page 25).

Definition 18. A solution y * (t, t 0 , y 0 ) of the system (37) with initial conditions y(t 0 ) = y 0 , (t 0 , y 0 ) ∈ G is said to be right-hand maximum if any other solution y(t, t 0 , y 0 ) of the system (37) with the same initial condition satisfies the inequality y(t, t 0 , y 0 ) ≤ y * (t, t 0 , y 0 ) for all t ∈ I, where I is a time interval on which all solutions exist. Now we can formulate the following comparison theorem.

Theorem 6 ([47], page 25). Let 1) the right-hand side of the equation (37) be continuous in a region G;

2) y * (t, t 0 , y 0 ) be the right-hand maximum solution of (37) with the initial condition y(t 0 ) = y 0 , (t 0 , y 0 ) ∈ G, which is defined on [t 0 , t 0 + α), α ∈ R + ;

3) a function V : R → R be defined and continuous on

[t 0 , t 0 + β), β ∈ R + , (t, V (t)) ∈ G for t ∈ [t 0 , t 0 + β) and V (t 0 ) ≤ y 0 , D + V (t) ≤ g(t, V (t)) for t ∈ (t 0 , t 0 + β), then V (t) ≤ y * (t, t 0 , y 0 ) for t ∈ [t 0 , t 0 + min{α, β}).
Theorem 6 remains true if Dini derivative D + is replaced with some other derivative D + , D -, D -or D K (see [START_REF] Szarski | Differential Inequalities[END_REF], Remark 2.2, page 11).

Generalized directional derivatives of continuous and discontinuous functions

Stability analysis based on Lyapunov functions requires calculation of derivatives of positive definite functions along trajectories of a dynamic system. If

Lyapunov function is non-differentiable, a concept of generalized directional derivatives (see, for example, [START_REF] Clarke | Qualitative properties of trajectories of control systems: A survey[END_REF][START_REF] Aubin | Applied Nonlinear Analysis[END_REF][START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF]) can be used for this analysis. This survey introduces generalized directional derivatives by analogy with contingent derivatives for scalar functions.

Let M(d) be a set of all sequences of real vectors converging to d ∈ R n , i.e.

{v n } ∈ M(d) ⇔ v n → d, v n ∈ R n .
D {hn},{vn} V (x, d) = lim n→+∞ V (x+hnvn)-V (x) hn , {h n } ∈ K, {v n } ∈ M(d) : x + h n v n ∈ Ω
is called directional derivative number of the function V (x) at the point x ∈ Ω on the direction d ∈ R n , if finite or infinite limit exists.

The set of all directional derivative numbers of the function V (x) at the point x ∈ Ω on the direction d ∈ R n is called directional contingent derivative:

D K,M(d) V (x) = {hn}∈K,{vn}∈M(d) {D {hn},{vn} V (x, d)}.
Similarly to Lemma 2 it can be shown that if x ∈ Ω then the set

D K,M(d) V (x)
is nonempty for any function V defined on an open nonempty set Ω ⊆ R n and any d ∈ R n . A chain rule for the introduced contingent derivative is described by the following lemma.

Lemma 7. Let a function V : R n → R be defined on an open nonempty set Ω ⊆ R n and a function x : R → R n be defined on I, such that x(t) ∈ Ω if t ∈ I and the contingent derivative D K x(t) ⊆ R n is bounded for all t ∈ I.

Then the inclusion

D K V (x(t)) ⊆ d∈D K x(t) D K,M(d) V (x)
holds for all t ∈ I.

Proof. Since x(t) ∈ Ω for t ∈ I then Lemma 2 implies that D K V (x(t)) is nonempty for any t ∈ I. Let D {hn} V (x(t)) ∈ D K V (x(t)) be an arbitrary derivative number, i.e. by Definition 17 the finite or infinite limit

lim n→∞ V (x(t + h n )) -V (x(t)) h n , {h n } ∈ K : t + h n ∈ I exists.
Consider now the sequence:

v n = x(t + h n ) -x(t) h n .
Lemma 2 and inequality |D K x(t)| < +∞ implies that there exist finite d ∈ D K x(t) and a subsequence {h n } of the sequence {h n } such that v n → d.

Hence,

D {hn} V (x(t))= lim n→∞ V (x(t + h n )) -V (x(t)) h n = lim n →∞ V (x(t + h n )) -V (x(t)) h n = lim n →∞ V (x(t) + h n v n ) -V (x(t)) h n = D {h n },{v n } V (x).
The proven lemma together with Lemmas 6 and 4 imply the following corollary, which is useful for a non-smooth Lyapunov analysis.

Corollary 2. Let a set-valued function F : R n+1 → 2 R n be defined and uppersemicontinuous on I × Ω and the set F (t, x) be nonempty, compact and convex for any (t, x) ∈ I × Ω, where Ω ⊆ R n is an open nonempty set.

Let x(t, t 0 , x 0 ) be an arbitrary solution of Cauchy problem [START_REF] Roxin | On stability in control systems[END_REF], [START_REF] Andrieu | Homogeneous approximation, recursive obsertver and output feedback[END_REF] defined on [t 0 , t 0 + α), where t 0 ∈ I, x 0 ∈ Ω and α ∈ R + . Let a function V : R n → R be nonnegative on Ω.

If the inequality D F (t,x) V (x) ≤ 0 holds for every t ∈ I and every x ∈ Ω :

V (x) = 0 then the function of time V (x(t, t 0 , x 0 )) is decreasing on [t 0 , t 0 + α),
where

D F (t,x) V (x) = d∈F (t,x) D K,M(d) V (x). (38)

Clarke's gradient of Lipschitz continuous functions

Let a function V : R n → R be defined and Lipschitz continuous on an open nonempty set. Then, by Rademacher theorem [START_REF] Rademacher | Uber partielle und totale differenzierbarkeit von funktionen mehrerer variabeln und uber die transformation der doppelintegrale[END_REF], its gradient exists almost everywhere on Ω and for each x ∈ Ω the following set can be constructed:

∇ C V (x)= co {x k }∈M(x):∃∇V (x k ) lim x k →x ∇V (x k ) , (39)
which is called the Clarke's generalized gradient of the function V (x) at the point x ∈ Ω. The set ∇ C V (x) is nonempty, convex and compact for any

x ∈ Ω and the set-valued mapping ∇ C V : R n → 2 R n is upper-semicontinuous on Ω (see [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF], Proposition 2.6.2, page 70).

The formula [START_REF] Filippov | On certain questions in the theory of optimal control[END_REF] gives a procedure for calculation of the generalized gradient of a function. where t 0 ∈ I, x 0 ∈ Ω and α ∈ R + . Let a function V : R n → R be defined and Lipschitz continuous on Ω.

If the inequality D C F (t,x) V (x) ≤ 0 holds almost everywhere on I for every x ∈ Ω then the function of time V (x(t, t 0 , x 0 )) is decreasing on [t 0 , t 0 + α), where

D C F (t,x) V (x) = d∈F (t,x) p∈∇ C V (x) p T d (40)
If the function V : R n → R is continuously differentiable then the usual total derivative

VF (t,x) (x) = d∈F (t,x) ∇ T V (x)d (41)
can be used for monotonicity analysis instead of Clarke's or contingent derivative. In this case we have

D F (t,x) V (x) = D C F (t,x) V (x) = VF (t,x) (x).

Lyapunov function method and convergence rate

Lyapunov function method is a very effective tool for analysis and design of both linear and nonlinear control systems [START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF]. Initially, the method was presented for "unrated" (Lyapunov and asymptotic) stability analysis [START_REF] Lyapunov | The general problem of the stability of motion[END_REF]. A development of control theory had required to study a convergence rate together with a stability properties of a control system. This section observes the most important achievements of the Lyapunov function method related to a convergence rate estimation of sliding mode systems. 2) there exists a continuous positive definite function V : R n → R such that

V (x) ≤ V (x) for x ∈ Ω. A positive definite function W : R n → R is called radially unbounded if W (x) → +∞ for x → +∞.
Ω ⊆ R n : 0 ∈ int(Ω) and D F (t,x) V (x) ≤ 0 for t ∈ R and x ∈ Ω\{0}. ( 42
)
Then the origin of the system (27) is Lyapunov stable.

Proof. Since V (x) is proper, then there exist continuous positive definite function V (x) such that V (x) ≤ V (x) for all x ∈ Ω.

Let h = sup r∈R+:B(r)⊆Ω r and λ(ε) = inf

x∈R n : x =ε V (x) > 0, where ε ∈ (0, h].
The function V (x) is continuous at the origin, so ∃δ ∈ (0, ε) :

V (x) < λ(ε) if x ∈ B(δ). Moreover, B(δ) ⊆ U(ε) = Π(V, λ(ε)) ∩ B(ε).
Let t 0 ∈ R and x 0 ∈ U(ε) (in partial case x 0 ∈ B(δ)). The system [START_REF] Roxin | On stability in control systems[END_REF] satisfies Theorem 3 and it has solutions, which can be continued up to the boundary of Ω. Consider an arbitrary solution x(t, t 0 , x 0 ) of ( 27). The inequality [START_REF] Vinograd | The inadequacy of the method of characteristic exponents for the study of nonlinear differential equations[END_REF] and Corollary 2 implies that the function of time V (x(t, t 0 , x 0 )) is decreasing for t > t 0 , i.e. V (x(t, t 0 , x 0 )) ≤ V (x 0 ) < λ(ε).

In this case, x(t, t 0 , x 0 ) ∈ B(ε) for t > t 0 . Indeed, otherwise there exists t * > t 0 : x(t * , t 0 , x 0 ) = ε, so V (x(t * , t 0 , x 0 )) ≥ V (x(t * , t 0 , x 0 )) ≥ λ(ε).

The proven property also implies that even if a solution of (27) with t 0 ∈ R and x 0 ∈ U(ε) was initially defined on finite interval [t 0 , t 0 + α), α ∈ R + , it can be prolonged for all t > t 0 .

Asymptotic stability requires analysis of an attraction set. Lyapunov function approach may provide an estimate of this set. Then the origin of the system (27) is asymptotically stable with an attraction domain

U = Π(V, λ(h)) ∩ B(h), (43)
This theorem can be proven by analogy to a classical theorem on exponential stability (see, for example, [START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF], page 171) using Lemma 6.

The presented theorems shows that discontinuous and non-Lipschitzian Lyapunov functions can also be used for stability analysis. If V (x) is Lipschitz continuous then all theorems on stability can be reformulated using Clarke's gradient.

The following important theorem declares that a smooth Lyapunov function always exists for a time-invariant asymptotically stable differential inclusion [START_REF] Roxin | On stability in control systems[END_REF].

Theorem 10 ([44], Theorem 1.2). Let a set-valued function F : R n → R n be defined and upper-semicontinuous in R n . Let F (x) be nonempty, compact and convex for any x ∈ R n . If the origin of the system

ẋ ∈ F (x)
is globally uniformly asymptotically stable iff there exists a globally proper func-

tion V (•) ∈ C ∞ (R n ) and a function W (•) ∈ C ∞ (R n ) : W (x) > 0 for x = 0 such that max y∈F (x) ∇ T V (x)y ≤ -W (x), x ∈ R n \{0}.
However, the practice shows that designing of a Lyapunov function for nonlinear and/or discontinuous system is a nontrivial problem even for a two dimensional case. Frequently, in order to analyze stability of a sliding mode control system it is simpler to design a non-smooth Lyapunov function (see, for example, [START_REF] Utkin | Sliding Modes in Control Optimization[END_REF], [START_REF] Polyakov | Reaching time estimation for "super-twisting" second order sliding mode controller via Lyapunov function designing[END_REF], [START_REF] Moreno | Strict Lyapunov functions for the super-twisting algorithm[END_REF]).

Lyapunov analysis of finite-time stability

Analysis of finite-time stability using the Lyapunov function method allows us to estimate of a settling time a priori. The proof of the next theorem follows the ideas introduced in [START_REF] Roxin | On finite stability in control systems[END_REF] and [START_REF] Polyakov | Unified Lyapunov function for a finite-time stability analysis of relay second-order sliding mode control systems[END_REF].

Example 10. Consider again the uniformly finite-time stable system

ẋ = -x [ 1 2 ] (1 -|x|) , x ∈ R,
and show that its settling-time function

T (x) = ln 1 + |x| 1 2 1 -|x| 1 2
satisfies all conditions of Theorem 11. Indeed, it is continuous and proper on B(1). Finally, it is differentiable for x ∈ B(1)\{0} and

Ṫ (x) = ∂T ∂x ẋ = 1 x [ 1 2 ] (1 -|x|) ẋ = -1 for x = 0.
The last example shows that a settling-time function of finite-time stable system is a Lyapunov function in a generalized sense. Theorem 11 operates with a very large class Lyapunov functions. However, its conditions are still rather conservative. For example, the settling-time function from Example 6 can not be considered as a Lyapunov function candidate, since it is discontinuous at the origin, so it is not proper. However, even proper settling-time functions of sliding mode systems may not satisfy the condition [START_REF] Clarke | Asymptotic stability and smooth Lyapunov functions[END_REF].

Example 11. Consider the twisting second order sliding mode system [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF]   ẋ1 ẋ2

  ∈ F (x 1 , x 2 ) =   y -2sign[x 1 ] -sign[x 2 ]   , (46)
which is uniformly finite-time stable with the settling-time function [START_REF] Polyakov | Unified Lyapunov function for a finite-time stability analysis of relay second-order sliding mode control systems[END_REF] :

T tw (x) = p |x 1 | + x 2 2 2(2 + sign[x 1 x 2 ]) + |x 2 | sign[x 1 x 2 ] 2 + sign[x 1 x 2 ] , p = 4 √ 2 3 - √ 3
The function T tw is globally proper, Lipschtz continuous outside the origin and continuously differentiable for xy = 0

D F (x1,x2) T tw (x 1 , x 2 ) = ∂T tw ∂x 1 x 2 + ∂T tw ∂x 2 (-2 sign[x 1 ]-sign[x 2 ]) = -1 for x 1 x 2 = 0.
However, D F (x1,x2) T tw (x 1 , x 2 ) R + = ∅ for x 1 = 0. So, T tw (x, y) does not satisfy [START_REF] Clarke | Asymptotic stability and smooth Lyapunov functions[END_REF]. Applying Clarke's gradient does not help to avoid this problem.

In the same time, if x(t, t 0 , x 0 ) is an arbitrary solution of the system (46), then D K T tw (x(t, t 0 , x 0 )) ≤ -1 for ∀t > t 0 : x(t, t 0 , x 0 ) = 0 (see [START_REF] Polyakov | Unified Lyapunov function for a finite-time stability analysis of relay second-order sliding mode control systems[END_REF] for the details).

Remark, if p > 4 Sometimes the less restrictive finite-time stability condition D K V (x(t, t 0 , x 0 )) ≤ -1, t ≥ t 0 : x(t, t 0 , x 0 ) = 0, x(t, t 0 , x 0 ) ∈ Φ(t 0 , x 0 ), t 0 ∈ R, x 0 ∈ U. [START_REF] Szarski | Differential Inequalities[END_REF] has to be considered instead of [START_REF] Clarke | Asymptotic stability and smooth Lyapunov functions[END_REF]. Examples of applying the condition [START_REF] Szarski | Differential Inequalities[END_REF] for analysis of second order sliding mode systems can be found in [START_REF] Polyakov | Unified Lyapunov function for a finite-time stability analysis of relay second-order sliding mode control systems[END_REF], [START_REF] Poznyak | Analysis of finite-time convergence by the method of Lyapunov functions in systems with second order sliding modes[END_REF].

They demonstrate that frequently we do not need to know a solution x(t, t 0 , x 0 ) of ( 27) in order to check the condition [START_REF] Szarski | Differential Inequalities[END_REF].

T (x 0 ) ≤ V 1-ρ (x 0 )
r(1 -ρ) for x 0 ∈ U.

Proof. Let x(t, t 0 , x 0 ), x 0 ∈ U be any solution of [START_REF] Roxin | On stability in control systems[END_REF] and Ṽ (t) = V (x(t, t 0 , x 0 )).

Since D K Ṽ (t) ≤ D F (t,x) V (x(t, t 0 , x 0 )) ≤ -r Ṽ ρ (t) (see, Lemma 7) then Lemma 6 implies that Ṽ (t) ≤ y(t), t > t 0 , where y(t) is a right-hand maximum solution of the following Cauchy problem ẏ(t) = -ry ρ (t), y(t 0 ) = V (x 0 ), i.e.

y(t) =    V (x 0 ) 1-ρ -r(1 -ρ)(t -t 0 ) 1 1-ρ for t ∈ [t 0 , t 0 + V 1-ρ (x0) r(1-ρ) ], 0 for t > V 1-ρ (x0) r(1-ρ) .
This implies V (x(t, t 0 , x 0 )) = 0 for ∀t > The function [START_REF] Moreno | Strict Lyapunov functions for the super-twisting algorithm[END_REF] V (x, y) = (2β + α 2 /2)|x|

+ y 2 -αyx [ 1 2 ]
is the generalized Lyapunov function for the system [START_REF] Bruckner | Differentiation of Real Functions[END_REF]. Indeed, this function is globally proper and continuous (but not Lipschitz continuous on the line x = 0).

For x = 0 this function is differentiable and DV F (x,y) (x, y) ≤ -γ V (x, y)

Consider the function V (s) = |s| and calculate its generalized derivative along trajectories of the last system

D F V (s) ≤ -α 2 V (s) + β 2 V 3 (s) 1 2
for s = 0

(see [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF] for the details). This implies that the sliding surface s = 0 is fixed-time attractive with the estimate of a reaching time:

T s ≤ 2 √ α 2 + 2 √ β 2 .
The sliding motion equation for s = 0 has the form

ẋ = - α 1 2 x + β 1 2 x 3 [ 1 2 ] .
This system is fixed-time stable and a global estimate of the settling-time function T (x, y) for the original system is

T (x, y) ≤ T max ≤ 2 √ 2 √ α 1 + 2 √ 2 √ β 1 + 2 √ α 2 + 2 √ β 2 .

Conclusions

The paper surveys mathematical tools required for stability analysis of sliding mode systems. It discusses definitions of solutions for systems with discontinuous right-hand sides, which effectively describe sliding mode systems. It observes an evolution of stability notions, convergence rate properties and underlines differences between finite-time and fixed-time stable systems in local and global cases. The paper considers elements of the theory of generalized derivatives and presents a generalized Lyapunov function method for asymptotic, exponential, finite-time and fixed-time stability analysis of discontinuous systems.

Theorems on finite-time and fixed-time stability provide rigorous mathematical justifications of formal applying non-Lipschitz Lyapunov functions presented in [START_REF] Moreno | A Lyapunov approach to analysis and design of second order sliding mode algorithms[END_REF][START_REF] Moreno | Strict Lyapunov functions for the super-twisting algorithm[END_REF][START_REF] Gonzalez | Variable gain super-twisting sliding mode control[END_REF] for stability analysis of second order sliding mode systems.

It is worth to stress that the presented tutorial summarize methods required for stability analysis of the so-called "ideal" sliding modes. The practical realization of sliding mode control requires extended analysis, which takes into account sampling, hysteresis and delay effects, measurement errors, discretization, etc. Robustness analysis of "real" sliding modes goes out of the scope of this paper. Practical stability analysis of sliding mode systems based on two Lyapunov functions was presented in [START_REF] Clarke | Stability analysis of sliding-mode feedback control[END_REF]. Stability of the real coordinates in the sliding mode was studied in [START_REF] Zolezzi | Real states of stable sliding mode control systems[END_REF]. More general approach to robustness analysis of "real" sliding modes based on ISS theory of homogeneous systems can be found in [START_REF] Bernuau | Robustness of finitetime stability property for sliding modes[END_REF].

Acknowledgements

We would like to thank the editor and the reviewers, who provided valuable comments helping us to improve the presentation of the paper.

  [a, b], (a, b), [a, b) or (a, b], where a, b ∈ R, a < b.

  (a) Switching case.(b) Sliding mode case.

Figure 1 :

 1 Figure 1: Geometrical illustration of Filippov definition.

  (a) Filippov definition. (b) Utkin definition.

Figure 2 :

 2 Figure 2: Example of Filippov's and Utkin's sets.

Figure 3 :

 3 Figure 3: The sliding motion for different definitions.

Theorem 3 (

 3 [START_REF] Filippov | Differential Equations with Discontinuous Righthand Sides[END_REF], page 77). Let a set-valued function F : G → 2 R n be defined and upper semi-continuous at each point of the set

Figure 4 :Definition 9 (

 49 Figure 4: Example of R.E. Vinograd [42].

Figure 5 :

 5 Figure 5: Example of S.P. Bhat and D. Bernstein [14].

  [a, b], (a, b), [a, b) or (a, b], where a, b ∈ R, a < b.

Lemma 3 (

 3 [START_REF] Natanson | Theory of functions of a real variable[END_REF], page 266). If a function ϕ : R → R is defined on I and the inequality D K ϕ(t) ≤ 0 holds for all t ∈ I, then ϕ(t) is decreasing function on I and differentiable almost everywhere on I. Lemma 3 require neither the continuity of the function ϕ(t) nor the finiteness of its derivative numbers. It gives a background for the discontinuous Lyapunov function method.

Lemma 5 (

 5 [START_REF] Szarski | Differential Inequalities[END_REF], page 13). If a function ϕ : R → R defined on I is absolutely continuous and φ(t) ≤ 0 almost everywhere on I then ϕ(t) is decreasing function on I.

  Let a function V : R n → R be defined on an open nonempty set Ω ⊆ R n and d ∈ R n . Definition 19. A number

6. 1 .

 1 Analysis of Lyapunov, asymptotic and exponential stability The continuous function W : R n → R defined on R n is said to be positive definite iff W (0) = 0 and W (x) > 0 for x ∈ R n \{0}. Definition 20. A function V : R n → R is said to be proper on an open nonempty set Ω ⊆ R n : 0 ∈ int(Ω) iff 1) it is defined on Ω and continuous at the origin;

Definition 21 . 2 . 7 .

 2127 A function V : R n → R is said to be globally proper iff it is proper on R n and the positive definite function V : R n → R is radially unbounded.If V is continuous on Ω, then V (x) = V (x) for x ∈ Ω and Definition 21 corresponds to the usual notion of proper positive definite function (see, for example,[START_REF] Clarke | Asymptotic stability and smooth Lyapunov functions[END_REF]). For a given number r ∈ R and a given positive definite function W : R n → R defined on Ω let us introduce the set Π(W, r) = {x ∈ Ω : W (x) < r} which is called the level set of the function W . Theorems on Lyapunov and asymptotic stability given below are obtained by a combination of Zubov's theorems (see, for example, [40], pages 566-568) with Corollary Theorem Let a function V : R n → R be proper on an open nonempty set

Theorem 8 .

 8 Let a function V : R n → R be proper on an open nonempty set Ω ⊆ R n : 0 ∈ int(Ω), a function W : R n → R be a continuous positive definite andD F (t,x) V (x) ≤ -W (x)for t ∈ R and x ∈ Ω\{0}.

3

 3 then the function T tw (x) satisfies the conditions of Theorem 11 and D F (x1,x2) T tw (x) = {-∞} for x 1 = 0.

Example 12 . 2 .for x 1 x 2 < 0 2 .

 12202 Consider the system ẋ = -(2 -sign[x 1 x 2 ]) x x, x = (x 1 , x 2 ) T ∈ R It is uniformly finite-time stable. Its settling time function is discontinuous However, the function T (x) is the generalized Lyapunov function, since it is globally proper andD K T (x(t, t 0 , x 0 )) = -1 for t > t 0 : x(t, t 0 , x 0 ) = 0, where x(t, t 0 , x 0 ) ∈ Φ(t 0 , x 0 ), t 0 ∈ Rand x 0 ∈ R Theorem 12 ([14], Theorem 4.2). Let a continuous function V : R n → R be proper on an open nonempty set Ω ⊆ R n : 0 ∈ int(Ω) andD F (t,x) V (x) ≤ -rV ρ (x), t > t 0 , x ∈ Ω,where r ∈ R + , 0 < ρ < 1. Then the origin of the system (27) is uniformly finite-time stable with an attraction domain U defined by[START_REF] Hahn | Stability of Motion[END_REF] and the settling time function T (•) is estimated as follows

  V 1-ρ (x0) r(1-ρ) .A global finite-time stability can be analyzed using globally proper Lyapunov functions in Theorems 11 and 12.Example 13. Consider the so-called super-twisting system[START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF] wherex ∈ R, y ∈ R, α > 0, β > 0. Recall, x [µ] = |x| µ sign[x], µ ∈ R + .

  a uniformly asymptotically stable (attractive) origin of the system (27) is called globally uniformly asymptotically stable (attractive). Uniform asymptotic stability always implies asymptotic stability. The converse proposition also holds for time-invariant systems.

Proposition 2

([44]

, Proposition 2.2, page 78). Let a set-valued function F : R n → R n be defined and upper-semicontinuous in R n . Let F (x) be nonempty, compact and convex for any x ∈ R n . If the origin of the system

  The next lemma presents a chain rule for the Clarke's generalized gradient.Lemma 8([52], Theorem 2, page 336). Let a Lipschitz continuous function V : R n → R be defined in an open nonempty set Ω ⊆ R n and a absolutely continuous function x : R → R n be defined on I such that x(t) ∈ Ω for everyt ∈ I.Then there exists a function p : R → R n defined on I such that p(t) ∈ ∇ C V (x(t)) and V (x(t)) = p T (t) ẋ(t) almost everywhere on I. Let a set-valued function F : R n+1 → 2 R n be defined and uppersemicontinuous on I × Ω and a set F (t, x) be nonempty, compact and convex for any (t, x) ∈ I × Ω, where Ω ⊆ R n is an open nonempty set. Let x(t, t 0 , x 0 ) be an arbitrary solution of Cauchy problem[START_REF] Roxin | On stability in control systems[END_REF],[START_REF] Andrieu | Homogeneous approximation, recursive obsertver and output feedback[END_REF] defined on [t 0 , t 0 + α),

	Lemmas 8 and 5 implies the following corollary.
	Corollary 3.

This work is supported by ANR Grant CHASLIM (ANR-11-BS03-0007).

Grant 190776 Bilateral Cooperation CONACyT(Mexico) -CNRS(France), Grant CONA-CyT 132125 y Programa de Apoyo a Proyectos de Investiagacion e Inovacion Tecnologico (PAPIIT) 113613 of UNAM. Preprint submitted to Elsevier December 2, 2013

|} both these systems have asymptotically stable origins.This example shows that the conventional properties, like asymptotic or finite stability, discovered for differential inclusions may provide "robust" stability

where λ(h) = inf x∈R n : x =h V (x) and h ≤ sup r∈R+:B(r)⊆Ω r.

If V is globally proper and Ω = R n then the origin of the system (27) is globally asymptotically stable (U = R n ).

Proof. Theorem 7 implies that an arbitrary solution x(t, t 0 , x 0 ) of ( 27) with t 0 ∈ R and x 0 ∈ U(ε) is defined for all t > t 0 and x(t, t 0 , x 0 ) ∈ B(ε), where ε ∈ (0, h] and U(ε) = Π(V, λ(ε)) ∩ B(ε). Moreover, the function of time Ṽ (t) =

V (x(t, t 0 , x 0 )) is decreasing for all t > t 0 . So, in order to prove asymptotic stability we just need to show that µ = 0, where µ = inf t>t0 Ṽ (t).

Suppose a contradiction, i.e. µ > 0.

The function V (x) is continuous at the origin, so there exists r > 0 such that V (x) < µ for all x ∈ B(r). Since µ > 0 then x(t, t 0 , x 0 ) / ∈ B(r) for all t > t 0 .

Introduce the following compact set Θ = {x ∈ R n : r ≤ x ≤ ε}. Since W (x) is continuous and positive definite, then we have

The inequality D F (t,x) V (x) ≤ -W (x) and the exclusion x(t, t 0 , x 0 ) / ∈ B(r)

Since Ṽ (t) is decreasing then it is differentiable almost everywhere on [t 0 , t 0 + ∆], where ∆ = V (x 0 )/W 0 . Hence (see, for example, [START_REF] Wheeden | Measure and Integral: An Introduction to Real Analysis[END_REF], page 111),

i.e. V (t 0 +∆) ≤ 0 < µ. This contradicts our supposition. So, V (x(t, t 0 , x 0 )) → 0 or equivalently x(t, t 0 , x 0 ) → 0 if t → +∞.

If the function V is globally proper then global asymptotic attractiveness follows from lim ε→+∞ λ(ε) = +∞ due to radial unboundedness of V .

Exponential convergence asks for additional properties of Lyapunov functions.

Theorem 9. Let conditions of Theorem 8 hold, the function V (x) is continuous on an open nonempty set Ω ⊂ R n : 0 ∈ int(Ω) and there exist α, r 1 , r 2 ∈ R + :

then the origin of the system (27) is exponentially stable with a rate α ∈ R + .

Theorem 11. Let a function V : R n → R be proper on an open nonempty set

Then the origin of the system (27) is finite-time stable with an attraction domain U defined by [START_REF] Hahn | Stability of Motion[END_REF] and

where T (•) is a settling-time function.

If a function V is globally proper on Ω = R n then the inequality (44) implies global finite-time stability of the system [START_REF] Roxin | On stability in control systems[END_REF].

Proof. Theorem 8 implies that the origin of the system ( 27) is asymptotically stable with the attraction domain U. This means that any solution

x(t, t 0 , x 0 ), x 0 ∈ U of the system (27) exists for ∀t > t 0 . Therefore, we need to show finite-time attractivity. Consider the interval [t 0 , t 1 ],

Suppose a contradiction: x(t, t 0 , x 0 ) = 0 for ∀t ∈ [t 0 , t 1 ]. Denote Ṽ (t) =

V (x(t, t 0 , x 0 )). Lemma 7 implies

Hence, by Lemma 3 the function Ṽ (t) is decreasing on [t 0 , t 1 ] and differentiable almost everywhere on [t 0 , t 1 ]. Then

. the origin of the system ( 27) is finite-time attractive with the settling time estimate [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF].

Evidently, if under conditions of Theorem 11 there exists a continuous function V : R n → R such that V (x) ≤ V (x) for ∀x ∈ Ω then the origin of the system ( 27) is uniformly finite-time stable.

where γ = γ(α, β) > 0 is a positive number (see [START_REF] Moreno | Strict Lyapunov functions for the super-twisting algorithm[END_REF] for details).

For x = 0 and y = 0 we need to calculate a generalized directional derivative.

So, consider the limit

Obviously, D {hn},{un} V (0, y) = -∞. Therefore,

and the super-twisting system is uniformly finite-time stable with the settling time estimate T (x, y) ≤ 2 V (x, y)/γ.

By Corollary 1, the set of time instants t > t 0 : D K V (x(t), y(t)) = {-∞} may have only the measure zero. This means that the line x = 0 for y = 0 can not be sliding set of the system [START_REF] Bruckner | Differentiation of Real Functions[END_REF]. The sliding mode may appear only at the origin.

Fixed-time stability analysis

Locally fixed-time stability property is very close to finite-time stability, so it can be established using Theorem 11 just including additional condition :

V (x) ≤ T max for ∀x ∈ Ω , where T max ∈ R + . An alternative Lyapunov characterization of fixed-time stability can be obtained using the ideas introduced in the proof of Corollary 2.24 from [START_REF] Andrieu | Homogeneous approximation, recursive obsertver and output feedback[END_REF].

Theorem 13 ([30], page 2106). Let a continuous function V : R n → R be

proper on an open connected set Ω : 0 ∈ int(Ω). If for some numbers µ ∈

holds, then the origin of the system (27) is fixed-time stable with the attraction domain U defined by [START_REF] Hahn | Stability of Motion[END_REF] and the maximum settling time is estimated by

If Ω = R n and a function V is radially unbounded then the origin of the system (27) is globally fixed-time stable.

Proof. Theorem 8 implies that the origin of the system ( 27) is asymptotically stable with the attraction domain U. This means that any solution

x(t, t 0 , x 0 ), x 0 ∈ U of the system (27) exists for ∀t > t 0 . We just need to proof that the estimate (49) implies fixed-time attractivity.

Indeed, for any trajectory x(t, t 0 , x 0 ) of the system (6) with V (x 0 ) > 1, there exists a time instant

On the other hand, for any trajectory x(t, t 1 , x 1 ) with V (x 1 ) ≤ 1, there exists a time instant

)) → 0 for t → T 2 . These facts can be easily proven analogously to Theorem 12. This result also can be used for fixed-time stability analysis of high-order sliding mode control systems.