Jian Wang

Jesus Mendoza Avila

Denis Efimov

Alexander Aleksandrov

Leonid Fridman

On existence of oscillations in Persidskii systems

The conditions of existence of oscillations in the sense of Yakubovich are considered for a class of generalized nonlinear Persidskii systems. To this end, the conditions of local instability at the origin and global boundedness of solutions are presented in the form of linear matrix inequalities. The proposed theory is applied for robustness analysis of nonlinear feedback controls in linear systems with respect to unmodeled dynamics.

INTRODUCTION

Stability analysis is a complex issue, therefore, in the theory of control and dynamical systems there are various developments devoted to this problem [START_REF] Khalil | Nonlinear Systems[END_REF]. The main approach to check the stability in the nonlinear framework is based on the Lyapunov function method [START_REF] Lyapunov | Stability of motion: General problem[END_REF]; [START_REF] Malkin | Theory of Stability of Motion[END_REF]; [START_REF] Hahn | Stability of Motion[END_REF] and its extensions [START_REF] Lasalle | Stability by Liapunov's Direct Method with Aplications[END_REF]; [START_REF] Van Der Schaft | L 2 -gain and passivity techniques in nonlinear control[END_REF]; [START_REF] Lin | A smooth converse Lyapunov theorem for robust stability[END_REF]. The shortage of this procedure is the lack of constructive tools assigning a corresponding Lyapunov function to a generic nonlinear system.

In line with stability analysis, studying more complex modes of dynamical systems, including oscillations (periodical or chaotic), attracts the researchers nowadays [START_REF] Fradkov | Introduction to oscillations and chaos[END_REF]; [START_REF] Leonov | Frequency Methods in Oscillation Theory[END_REF]; [START_REF] Martinez | Analysis and design of oscillatory control systems[END_REF]; [START_REF] Meglio | A first principle model for multiphase slugging flow in vertical risers[END_REF]. A generic and useful theory for studying irregular oscillations has been proposed by V.A. [START_REF] Yakubovich | Frequency oscillations conditions in nonlinear systems with stationary single nonlinearity[END_REF][START_REF] Yakubovich | Oscillations in systems with discontinuous and hysteresis nonlinearities[END_REF]; [START_REF] Yakubovich | Conditions for self-induced oscillations in nonlinear systems[END_REF]. The conditions of oscillations in the sense of Yakubovich are based on existence of two Lyapunov functions [START_REF] Efimov | Oscillatority of nonlinear systems with static feedback[END_REF]; [START_REF] Efimov | On conditions of oscillations and multi-homogeneity[END_REF]. The first Lyapunov function ensures local instability of an equilibrium, while the second one provides global boundedness of the system trajectories.

In this work a class of Persidskii systems is studied, which have been first introduced in [START_REF] Barbashin | On construction of Lyapunov functions for nonlinear systems[END_REF] with the Lyapunov functions being a linear combination of the integrals of the nonlinearities, and next by [START_REF] Persidskii | Problem of absolute stability[END_REF], where a combination of the absolute values of the states was used in the Lyapunov function. This class of systems has been widely investigated in the context of diagonal stability [START_REF] Kazkurewicz | Matrix Diagonal Stability in Systems and Computation[END_REF], neural networks [START_REF] Hopfield | Computing with neural circuits: a model[END_REF]; [START_REF] Sontag | Recurrent networks[END_REF], sliding mode controls [START_REF] Aparicio | Stabilization of systems with switchings on the axis of their coordinates and its input-to-state properties[END_REF] and in other applications [START_REF] Erickson | Stability analysis of fixed-point digital filters using computer generated Lyapunov functions -Part I: Direct form and coupled form filters[END_REF]. In a recent work [START_REF] Efimov | Robust stability analysis and implementation of Persidskii systems[END_REF], input-tostate stability conditions have been proposed.

The goal of this work is to introduce the conditions of existence of oscillations in the sense of Yakubovich for a class of generalized Persidskii systems (Section 2). For this purpose, the conditions of local instability at the origin and global boundedness of trajectories are obtained in Section 4 (the latter extends the result obtained in [START_REF] Efimov | Robust stability analysis and implementation of Persidskii systems[END_REF]), which can be verified by solving linear matrix inequalities (LMIs). In Section 5 these results are used to investigate robustness of nonlinear control algorithms applied to linear systems, and when an unmodeled dynamics appears in the control channel (such a dynamics, usually, does not allow the system to be stabilized at the origin, then appearance of oscillations implies that at least boundedness of trajectories is preserved).

Notation • R + = {x ∈ R : x ≥ 0},
ẋ(t) = A 0 x(t) + M j=1 A j f j (x(t)) + d(t), t ≥ 0, (1)
where

x(t) = [x 1 (t) . . . x n (t)] ∈ R n is the state vector, x(0) ∈ R n ; and d(t) ∈ R n is the external perturbation, d ∈ L n ∞ ; f j (x) = [f j 1 (x 1 ) . . . f j n (x n )] ,
f j (0) = 0, j = 1, M are continuous functions (the solutions of the system (1) exist in the forward time at least locally), the matrices A k ∈ R n×n for k = 0, M . Assumption 1. For any i = 1, n, j = 1, M :

sf j i (s) > 0 ∀s ∈ R \ {0}.
In the assumption above, it is stated that all nonlinearities belong to a sector and may take zero values at the origin only. Under this hypothesis, after a proper re-indexing and decomposition of f j in (1), there exists m ∈ {0, . . . , M } such that for all i = 1, n, z = 1, m:

lim s→±∞ f z i (s) = ±∞;
and there exists µ ∈ {m, . . . , M } such that for all i = 1, n, z = 1, µ:

lim s→±∞ s 0 f z i (σ)dσ = +∞.
Thus, it is supposed that some of the nonlinearities are radially unbounded, and m = 0 corresponds to the case when all nonlinearities are bounded (at least for negative or positive argument). It is also claimed that other part of these nonlinearities may have unbounded integral (clearly if m > 0, then all radially unbounded nonlinearities also have unbounded integrals and µ ≥ m due to the introduced sector condition).

If A r = 0 for all r = 0, M -1 and µ = M , then we recover the system studied by Persidskii in the conventional framework [START_REF] Persidskii | Problem of absolute stability[END_REF].

Our goal is to propose conditions establishing a possible presence of oscillating trajectories in (1) (in the sense of Yakubovich, given below).

PRELIMINARIES

Consider a nonlinear system:

ẋ(t) = f (x(t), d(t)), t ≥ 0, (2) where x(t) ∈ R n is the state, d(t) ∈ R m is the external input, d ∈ L m
∞ , and f : R n+m → R n is a locally Lipschitz (or Hölder) continuous function, f (0, 0) = 0. For an initial condition x 0 ∈ R n and input d ∈ L m ∞ , define the corresponding solutions by x(t, x 0 , d) for any t ≥ 0 for which the solution exists.

Basic properties

In this work we will be interested in the following stability properties [START_REF] Khalil | Nonlinear Systems[END_REF]; [START_REF] Krasovskii | Stability of Motion[END_REF]; [START_REF] Hahn | Stability of Motion[END_REF]; [START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF]: Definition 1. The system ( 2) is called input-to-state practically stable (ISpS ), if there are some functions β ∈ KL, γ ∈ K and a constant c ≥ 0 such that

x(t, x 0 , d) ≤ β( x 0 , t) + γ(||d|| [0,t) ) + c ∀t ≥ 0 for any input d ∈ L m
∞ and any x 0 ∈ R n . The function γ is called nonlinear asymptotic gain. The system is called ISS if c = 0. Definition 2. The system (2) is called globally practically stable if there are functions σ, γ ∈ K and a constant c ≥ 0 such that

x(t, x 0 , d) ≤ σ( x 0 ) + γ(||d|| [0,t) ) + c ∀t ≥ 0 for any x 0 ∈ R n and d ∈ L m ∞ . For d = 0, (2) is called Lagrange stable, and it is called just stable if c = 0.
It is straightforward to conclude that any ISpS (ISS) system (2) is practically stable (stable) with σ(s) = β(s, 0). Definition 3. The system (2) is called unstable at the origin for d = 0, if there is an > 0 such that for any δ > 0, x(t , x 0 , 0) / ∈ B( ) for some x 0 ∈ B(δ) and some t ≥ 0.

These properties have the following Lyapunov/Chetaev function characterizations:

Definition 4. A smooth function V : R n → R + is called ISpS Lyapunov function for the system (2) if for all x ∈ R n , d ∈ R m and some r ≥ 0, α 1 , α 2 , α 3 ∈ K ∞ and θ ∈ K: α 1 ( x ) ≤ V (x) ≤ α 2 ( x ), DV (x)f (x, d) ≤ r + θ( d ) -α 3 ( x ). Such a function V is called ISS Lyapunov function if r = 0.
For global practical stability it is enough that an ISpS Lyapunov function exists for x > X with some X ∈ R + . Definition 5. A smooth function V : R n → R is called Chetaev function for the system (2) with d = 0 if V (0) = 0, and there exist 0 > 0 such that V + ∩ B( ) = ∅ for any ∈ (0, 0 ], where V + = {x ∈ B( 0 ) : V (x) > 0}, and DV (x)f (x, 0) > 0 ∀x ∈ V + . Theorem 6. The system (2) is ISpS (ISS) iff it admits an ISpS (ISS) Lyapunov function. Theorem 7. The system (2) with d = 0 is unstable if it admits a Chetaev function.

Under additional mild restrictions, the existence of a Chetaev function is also necessary for instability [START_REF] Krasovskii | Stability of Motion[END_REF]; [START_REF] Efimov | On conditions of oscillations and multi-homogeneity[END_REF].

Oscillations in the sense of Yakubovich

A function h : R n → R is called monotone if the condition x 1 ≤ x 1 , . . . , x n ≤ x n implies that either h(x 1 , . . . , x n ) ≤ h(x 1 , . . . , x n ) or h(x 1 , . . . , x n ) ≥ h(x 1 , . . . , x n ). Definition 8. [START_REF] Yakubovich | Frequency oscillations conditions in nonlinear systems with stationary single nonlinearity[END_REF]; [START_REF] Efimov | Oscillatority of nonlinear systems with static feedback[END_REF] For -∞ < π -< π + < +∞ the solution x(t, x 0 , 0) of the system (2) with x 0 ∈ R n and d = 0 is called [π -, π + ]-oscillation with respect to an output h(x) (where h : R n → R is a continuous monotone function) if the solution is defined for all t ≥ 0 and lim inf

t→+∞ h(x(t, x 0 , 0)) = π -, lim sup t→+∞ h(x(t, x 0 , 0)) = π + .
The solution x(t, x 0 , 0) of the system (2) is called oscillating, if there exist some output h and constants π -, π + such that solution x(t, x 0 , 0) is [π -, π + ]-oscillation with respect to the output h. A forward complete system (2) with d = 0 is called oscillatory, if for almost all x 0 ∈ R n the solutions x(t, x 0 , 0) of the system are oscillating. The oscillatory system (2) is called uniformly oscillatory, if for almost all x 0 ∈ R n for the corresponding solutions x(t, x 0 , 0) there exist an output h and constants π -, π + independently on initial conditions.

In other words the solution x(t, x 0 , 0) is oscillating if the output h(x(t, x 0 , 0)) is asymptotically bounded and there is no limit value of h(x(t, x 0 , 0)) for t → +∞. Note that the term "almost all solutions" is used to emphasize that generally the system (2) has a nonempty set of equilibria.

The notion of oscillations in the sense of Yakubovich is rather generic, it includes periodical oscillations (limit cycles), quasi-periodical, recurrent and chaotic trajectories. An oscillating trajectory in the sense of Yakubovich could be repelling. The trajectories could also be unbounded, it is required to find a function of the state vector, which is bounded and admits certain requirements introduced in Definition 8. Despite its complexity this notion has a simple Lyapunov characterization. Theorem 9. [START_REF] Efimov | Oscillatority of nonlinear systems with static feedback[END_REF] Let the system (2) have two continuously differentiable Lyapunov functions V 1 : R n → R + and V 2 : R n → R + fulfilling the following inequalities for all x ∈ R n :

υ 1 ( x ) ≤ V 1 (x) ≤ υ 2 ( x ), υ 3 ( x ) ≤ V 2 (x) ≤ υ 4 ( x ), υ 1 , υ 2 , υ 3 , υ 4 ∈ K ∞ ,
and for some 0

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

Ω ∩ Π = ∅, Ω = {x ∈ R n : υ -1 2 • υ 1 (X 1 ) ≤ x ≤ υ -1 3 • υ 4 (X 2 )}. Then the system (2) is oscillatory.
The Lyapunov function for the linearized system (2) at the origin is a candidate for the function V 1 Yakubovich (1975) (it can be replaced with Chetaev one [START_REF] Efimov | On conditions of oscillations and multi-homogeneity[END_REF], as in this work). Instead of existence of the function V 2 one can require just boundedness of solutions of the system (2) (if this fact can be verified using another approach not dealing with a Lyapunov function analysis). In [START_REF] Efimov | Oscillatority of nonlinear systems with static feedback[END_REF], for a class of uniformly oscillatory systems (2) it is shown that these conditions are also necessary.

Stability of extended Persidskii systems

In the expressions below, it is assumed that if the upper limit of a summation or a sequence is smaller than the lower one, then the corresponding terms have to be omitted and the conditions avoided. Theorem 10. [START_REF] Efimov | Robust stability analysis and implementation of Persidskii systems[END_REF] Let Assumption 1 be satisfied and there exist

P = P ∈ R n×n ; Ξ k = diag{ξ k } with ξ k = [ξ k 1 . . . ξ k n ] ∈ R n for k = 0, M ; Λ j = diag{λ j } with λ j = [λ j 1 . . . λ j n ] ∈ R n for j = 1, M ; Υ s,j = diag{υ s,j } with υ s,j = [υ s,j 1 . . . υ s,j n ] ∈ R n for s = 0, M and j = s + 1, M , and Γ = Γ > 0 such that Λ j ≥ 0, j = 1, M ; Ξ k ≥ 0, k = 0, M ; P > 0 or P ≥ 0, µ z=1 Λ z > 0; Υ s,j ≥ 0, s = 0, M , j = s + 1, M ; (3)      Ξ 0 > 0 m = 0 m k=0 Ξ k + 2 m s=0 m j=s+1 Υ s,j > 0 m > 0 ; Q ≤ 0,
where

Q =        Q1,1 Q1,2 Q1,3 • • • Q 1,M +1 P Q 1,2 Q2,2 Q2,3 • • • Q 2,M +1 Λ 1 Q 1,3 Q 2,3 Q3,3 • • • Q 3,M +1 Λ 2 . . . . . . . . . . . . . . . . . . Q 1,M +1 Q 2,M +1 Q 3,M +1 • • • Q M +1,M +1 Λ M P Λ 1 Λ 2 • • • Λ M -Γ        Q1,1 = A 0 P + P A0 + Ξ 0 ;
Qj+1,j+1 = A j Λ j + Λ j Aj + Ξ j , j = 1, M ;

Q1,j+1 = P Aj + A 0 Λ j + Υ0,j , j = 1, M ;

Qs+1,j+1 = A s Λ j + Λ s Aj + Υs,j , s = 1, M , j = s + 1, M .

Then the system (1) is ISS.

The proof of this result is based on the following Lyapunov function:

V (x) = x P x + 2 M j=1 n i=1 λ j i xi 0 f j i (s)ds, (4)
whose time derivative with respect to (1) admits an estimate for Q ≤ 0:

DV (x)f (x, d) ≤ -x Ξ 0 x - M j=1 f j (x) Ξ j f j (x) -2 M j=1 x Υ 0,j f j (x) -2 M s=1 M j=s+1 f s (x) Υ s,j f j (x)
+d Γd.

(5) The feature of the LMI (3) consists in a fine treatment of all diagonal items in the off-diagonal blocks of Q represented by the matrices Υ s,j ≥ 0 for s = 0, M and j = s + 1, M . Without introduction of Υ s,j all corresponding elements would be considered as perturbations, but actually due to the sector condition imposed on nonlinearities f j (x) in Assumption 1, these terms can be sign-definite and, hence, helpful.

Remark 11. The condition Ξ 0 > 0 of Theorem 10 for the case m = 0 can be relaxed to

Ξ 0 + 2 M j=1
Υ 0,j > 0 under an additional hypothesis that the functions x i f j i (x i ) are radially unbounded for all i = 1, n and j = 1, M .

CONDITIONS OF OSCILLATIONS

In this section, following the ideas given in Theorem 9, first, the conditions of ISpS will be presented extending the result of Theorem 10 by using Theorem 6 and the Lyapunov function (4). Second, the conditions of instability will be given based on Theorem 7. Finally, the conditions of oscillatory behavior will be proposed combining the previously obtained results.

Global practical stability

To establish global practical stability property we can use the fact that it is related with ISS, whose conditions for (1) are given in Theorem 10. Note that for this property (since it is simpler than ISS) the derivative of the Lyapunov function (4) should be negative definite in x only for sufficiently big values of the state, and even its positive definiteness has to be ensured only outside of a ball. These observations provide an insight how we are going to develop the result of Theorem 10, but before we need to introduce an additional conventions on the shale and ordering of the functions f j (x), j = 1, M : Assumption 2. Let

lim s→±∞ |f j i (s)| |s| ∈ {0, +∞}, lim s→±∞ |f j i (s)| |f k i (s)| ∈ {0, +∞}
for all i = 1, n and all j = k = 1, M .

The last condition implies that all f j i (x), j = 1, M have nonlinear and different asymptotic growth (this property also can be assured by putting linear components of f j (x) in the linear part explicitly, the same for the functions of a common rate). Assumption 3. For any i = 1, n there exists Z i > 0 such that for all |s| ≥ Z i :

sf j i (s) > 0, ∀j = 1, M ; |f 1 i (s)| ≥ |f 2 i (s)| ≥ • • • ≥ |f Ri i (s)| ≥ |s| ≥ |f Ri+1 i (s)| ≥ • • • ≥ |f M i (s)|, where R i ∈ {0, . . . , M } is an index.
In the above inequality, the limit terms f 0 i and f M +1 i in the cases R i = 0 and R i = M , respectively, have to be omitted. This hypothesis is not restrictive, and in the most cases it can be respected by rearranging and reordering the nonlinearities in (1). For brevity of formulation of the result below define R = sign(min 1≤i≤n R i ), the indices m and µ save their meaning under Assumption 3. Theorem 12. Let assumptions 2, 3 be satisfied and there exist indices κ ∈ {1, . . . , µ} and ι ∈ {R, . . . , m}, the matrices

P = P ∈ R n×n ; Ξ k = diag{ξ k } with ξ k = [ξ k 1 . . . ξ k n ] ∈ R n for k = 0, M ; Λ j = diag{λ j } with λ j = [λ j 1 . . . λ j n ] ∈ R n for j = 1
, M ; Υ s,j = diag{υ s,j } with υ s,j = [υ s,j 1 . . . υ s,j n ] ∈ R n for s = 0, M and j = s + 1, M , and Γ = Γ > 0 such that Λ j ≥ 0, j = 1, κ;

     P > 0 µ = 0 P ≥ 0, κ z=1 Λ z > 0 µ ≥ 1 ; Ξ s ≥ 0, Υ s,j ≥ 0, s = R, ι, j = s + 1, ι,      Ξ 0 > 0 m = 0 ι s=R Ξ s + 2 ι s=R ι j=s+1 Υ s,j > 0 m ≥ 1 ; Q ≤ 0,
where the matrix Q is given in Theorem 10. Then the system (1) is globally practically stable. If additionally Assumption 1 holds and Λ j ≥ 0, j = 1, M , then the system (1) is ISpS.

Proof. The proof is only sketched below due to space limitations, and it is based on the Lyapunov function given in (4). It can be shown that under the conditions of the theorem, the function V outside a ball around the origin is positive and radially unbounded, and its derivative with respect to (1) admits an estimate given in (5), which is negative and radially unbounded with respect to the state x, positive definite with respect to d and has a constant bias (as in Theorem 6 for ISpS).

The results of this subsection introduce a series of tools for analysis of boundedness of trajectories in the extended Persidskii system (1), these tools can be combined and also relaxed considering more particular scenarios and applications.

Local instability at the origin

For local instability it is necessary to ensure local positive definiteness of V in some directions at the origin and positive definiteness of the derivative V also locally in the domain where V (x) > 0. To formulate these restrictions let us introduce a reordering of the nonlinearities:

Assumption 4. Let lim s→0 |f j i (s)| |s| ∈ {0, +∞}, lim s→0 |f j i (s)| |f k i (s)| ∈ {0, +∞} for all i = 1, n and all j = k = 1, M .
This condition reads that all f j i (x), j = 1, M have nonlinear and different local growth at the origin. Assumption 5. There exist Z > 0, η > 0 such that for all s ∈ (-Z, 0) ∪ (0, Z):

ηsf j i (s) ≥ s 0 f j i (σ)dσ > 0, ∀j = 1, M ; |f 1 i (s)| ≥ |f 2 i (s)| ≥ • • • ≥ |f Ri i (s)| ≥ |s| ≥ |f Ri+1 i (s)| ≥ • • • ≥ |f M i (s)|
, where R i ∈ {0, . . . , M } is an index for all i = 1, n.

As we can see, all restrictions are imposed for x < Z and there is no requirement on radial unboundedness of functions and integrals for a local investigation. It is straightforward to verify that always there exists a value of ϑ 1 sufficiently big such that the matrix Ã0 is unstable in the direction of the variable z. Since α ∈ [0, 1), then also for any value of ϑ 1 the function f j 1 (x) = f j 1 (x 1 )ϑ 1 x 1 stays in the required sector onto a vicinity of the origin. Take P = P > 0, λ 1 = [λ 1 1 , 0, 0] ∈ R 3 with λ 1 1 > 0, ξ 0 ∈ R 3 with ξ 0 1 < 0, and the matrix Υ 0,1 with requirement that (Υ 0,1 ) 1,1 < 0, then the conditions of the theorem are satisfied with ι = 1.

By Theorem 14 the system oscillatory. The results of simulation for δ = 3, K 1 = 1 and α = 0.5 are shown in Fig. 1.

CONCLUSIONS

For a class of extended Persidskii systems new conditions of ISpS and instability are proposed, which are formulated in terms of LMIs. The proposed theory is combined to predict presence of oscillations in this class of models. These results are used to investigate robustness of nonlinear controls for linear systems with respect to unmodeled dynamics.

  This work is partially supported by 111 project No. D17019 (China), by the Government of Russian Federation (Grant 08-08), by the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898 and by the Russian Foundation for Basic Research (grant no. 19-01-00146-a).

Fig. 1 .

 1 Fig. 1. The results of simulation for δ = 3, K 1 = 1 and α = 0.5 To use Theorem 13, take any ϑ 1 > 0 and shift the nonlinearity f 1 by linear feedback, then we obtain the new matrixÃ0 = A 0 + ϑ 1 A 1 = 0 1 0 0 -δ δ -ϑ 1 δK 1 0 -δ .

  where R is the set of real numbers. • | • | and • denote the absolute value in R and the Euclidean norm on R n , respectively; for any > 0 define an open ball around the origin by B( ) = {x ∈ R + → R m define the norm ||d|| [t0,t1) = ess sup t∈[t0,t1) d(t) , then ||d|| ∞ = ||d|| [0,+∞) and the set of d with the property ||d|| ∞ < +∞ we further denote as L m ∞ (the set of essentially bounded measurable functions).• A continuous function α : R + → R + belongs to the class K if α(0) = 0 and the function is strictly increasing. The function α : R + → R + belongs to the class K ∞ if α ∈ K and it is increasing to infinity. A continuous function β : R + × R + → R + belongs to the class KL if β(•, s) ∈ K ∞ and β(s, •) is a nonincreasing function with lim t→+∞ β(s, t) = 0 for each fixed s ∈ R + .

	• The notation DV (x)v stands for the directional
	derivative of a continuously differentiable function V
	evaluated at point x with respect to a vector v ∈ R n .
	• Denote the identity matrix of dimension n × n by I n ,
	the vector of dimension n with all elements equal 1
	by 1 n , and diag{g} represents a diagonal matrix of dimension n × n with a vector g ∈ R n on the main
	diagonal.
	• A series of integers 1, 2, ..., n is denoted by 1, n.
	2. PROBLEM STATEMENT
	Consider the following class of extended Persidskii sys-
	tems:

n : x < }.

• For a (Lebesgue) measurable function d : R

Theorem 13. Let assumptions 4, 5 be satisfied and there exist indices κ ∈ {1, . . . , M } and ι ∈ {1, . . . , n}, the matrices P = P ∈ R n×n ; Ξ k = diag{ξ k } with ξ k = [ξ k 1 . . . ξ k n ] ∈ R n for k = 0, M ; Λ j = diag{λ j } with λ j = [λ j 1 . . . λ j n ] ∈ R n for j = 1, M ; Υ s,j = diag{υ s,j } with υ s,j = [υ s,j 1 . . . υ s,j n ] ∈ R n for s = 0, M and j = s + 1, M such that

where ρ > 0 is a parameter and the matrix Q is given in Theorem 10. Then the system (1) is unstable.

Proof. The proof is again only sketched. By considering the function V (x) given in (4) and calculating a lower estimate for its derivative with respect to (1) for the case d = 0, the conditions of Theorem 7 can be verified under the restrictions of the theorem, which implies that the origin is unstable.

Existence of oscillations

Following the result of Theorem 9, it is straightforward to formulate the required conditions: Theorem 14. Let the requirements of theorems 12, 13 be satisfied and the system (1) with d = 0 admit the only equilibrium at the origin. Then (1) with d = 0 is oscillatory.

Proof. Under these restrictions, the set Π from Theorem 9 contains just the origin. Fulfillment of the conditions of Theorem 12 ensures the global boundedness of all trajectories (it gives function V 2 from Theorem 9), while Theorem 13 guarantees instability of the only steady state (it provides a kind of function V 1 from Theorem 9). Hence, there is a compact forward invariant set Ω ⊂ R n attracting asymptotically almost all trajectories of the system, and obviously Ω ∩ Π = ∅.

Let us demonstrate the efficiency of the developed theory in an application.

STABILITY ROBUSTNESS AGAINST

UNMODELED DYNAMICS Consider a linear system: ż(t) = Az(t) + Bu(t), (6) where z(t) ∈ R ν and u(t) ∈ R µ are the state and the control vectors, ν > 0 and µ > 0 are some integers, A ∈ R ν×ν and B ∈ R ν×µ . Assume that there is a control input for this system:

where M > 0 is an integer and φ j : R ν → R ν are some nonlinear functions, φ j (z) = [φ j 1 (z 1 ) . . . φ j ν (z ν )] , φ j (0) = 0, j = 1, M ; K j ∈ R µ×ν for j = 1, M ; and the closed-loop system is stable and possesses a desired performance (clearly it is a system in the form (1) with d = 0). Next, assume that in the control channel there is an unmodeled dynamics (actuator) represented by a linear stable filter:

where ζ(t) ∈ R is the state of the filter for an integer > 0, υ(t) ∈ R µ is the applied to the actuator control designed for the system (6), and u(t) ∈ R µ is the output, which is the control for the system (6); C ∈ R µ× , D ∈ R × and G ∈ R ×µ . The closed-loop dynamics of ( 6) and ( 7) can be rewritten in the form of ( 1) with d = 0:

x

Therefore, using the result of Theorem 14 we can verify the oscillatory property of the closed-loop system under such an unmodeled dynamics of the actuator.

Let ν = 1, = 2 and

where α ∈ [0, 1) and δ > 0 are tuning parameters, K 1 > 0. Such a control can be used for finite-time stabilization [START_REF] Bernuau | On homogeneity and its application in sliding mode[END_REF]. Hence,

, A 1 = 0 0 0 0 0 0 -δK 1 0 0 , and the matrix A 0 is not Hurwitz, it has one zero eigenvalue and two equal -δ.

To check the conditions of Theorem 12 select P = P > 0,

2 > 0, ξ 0 3 > 0, and the matrix Υ 0,1 in a generic form (full matrix) with requirement that (Υ 0,1 ) 1,1 > 0,

i.e. the first element of this matrix corresponding to the term |z| α+1 is positive. It is obvious that the Lyapunov function V (x) given in ( 4) is positive definite under introduced restrictions. It is also straightforward to verify that there exist the matrices demanded above such that Q ≤ 0 and in (5) DV (x)f (x, 0) ≤ -(Υ 0,1 ) 1,1 |z| α+1 -ξ 0 2 ζ 2 1 -ξ 0 3 ζ 2 2 + 2 (Υ 0,1 ) 2,1 |ζ 1 ||z| α + (Υ 0,1 ) 3,1 |ζ 2 ||z| α , then the required global boundedness of trajectories is proven. Note that in the linear case, i.e. for α = 1, some additional restrictions on δ and K 1 have to be imposed to prove the boundedness of solutions.


Youness Braidiz
email: youness.braidiz@centralelille.fr

Denis Efimov
email: denis.efimov@inria.fr

Andrey Polyakov
email: andrey.polyakov@inria.fr

Wilfrid Perruquetti
email: wilfrid.perruquetti@centralelille.fr

On finite-time stability of sub-homogeneous differential inclusions

Keywords: Homogeneity, Finite time stability, Nonlinear systems

Sub-homogeneity property is introduced and is related to a differential inclusion (DI). It is shown that a nonlinear ordinary differential equation (ODE), which may not admit a homogeneous approximation, can be transformed into a sub-homogeneous DI (which is a homogeneous extension of the original ODE). Using this homogeneous extension, one can directly recover finite-time stability property for some particular classes of nonlinear systems. In the last section, such a sub homogeneity property is used to design a nonlinear finite-time observer.

INTRODUCTION

DIs arise naturally in various branches of modern mathematics, such as mathematical economics, theory of games, convex analysis, etc. The theory of DIs was initiated in [START_REF] Marchaud | Sur les champs de demi-droites et les équations différentielles du premier ordre[END_REF][START_REF] Marchaud | Sur les champs continus de demicônes convexes et leurs intégrales[END_REF]; [START_REF] Zaremba | Sur les équations au paratingent[END_REF], where authors showed that it plays a significant role in the description of processes in control theory. Hence, DIs appear as a model of dynamical systems which do not satisfy the classical assumptions of regularity [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF]. Furthermore, they also apear as a homogeneous extension of nonhomogeneous systems to study their finite-time stability (FTS) (see Braidiz et al. (2019a)).

Homogeneity was introduced for ordinary differential equations (in [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF]; [START_REF] Khomenyuk | Systems of ordinary differential equations with generalized homogeneous right-hand sides[END_REF]; [START_REF] Hermes | Nilpotent approximations of control systems and distributions[END_REF]; [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF]; [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF]), time delay systems [START_REF] Efimov | Weighted homogeneity for time-delay systems: Finite-time and independent of delay stability[END_REF], discrete-time systems [START_REF] Sanchez | On homogeneity of discretetime systems: stability and convergence rates[END_REF] and partial differential equations (in [START_REF] Polyakov | On homogeneous distributed parameter systems[END_REF][START_REF] Polyakov | On homogeneous finite-time control for linear evolution equation in Hilbert space[END_REF]). The notion of homogeneity was also used to prove FTS of systems with disturbances in Braidiz et al. (2019) and weak homogeneous systems in [START_REF] Braidiz | On finite-time stability of weak homogeneous systems[END_REF]. Nevertherless, requiring homogeneity for finite-time design is a restrictive constraint. However, many useful properties can be guaranteed for locally homogeneous systems, and the class of such dynamics is very large (see also [START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF]; [START_REF] Bernuau | On homogeneity and its application in sliding mode control[END_REF]), but still limited.

The goal of this note is to extend the homogeneity theory to the class of nonlinear dynamical systems, which do not This work was partially supported by ANR 15 CE23 0007 (Project Finite4SoS), by the Government of Russian Federation (Grant 08-08) and the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898.

admit a homogeneous approximation. To this end, a notion of homogeneous set-valued extension is defined, and its properties are investigated using the introduced concept of sub-homogeneity. Sub-homogeneity is a sort of symmetry, which appears as a general definition of homogeneity for some classes of DIs. It will be demonstrated that this concept, as the homogeneity [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], is a useful tool for analysis of nonlinear dynamical systems. In particular, it allows us to investigate their FTS.

The main contribution of this work is proving the FTS of a class of DIs which are sub-homogeneous. The notion of sub-homogeneity which will be proposed in this paper can be used to study the FTS of globally asymptotically stable (GAS) systems when their homogeneous approximation does not exist. Hence, this allows us to enlarge a lot the class of considered systems.

The paper is organized as follows. After introducing notations and definitions about asymptotic/finite-time stability and homogeneity in Section 2, we investigate the FTS of sub-homogeneous DIs in Section 3. In the last section, to illustrate the obtained results, we present an example of finite-time observer design by using the sub-homogeneous DIs. • The notation

PRELIMINARIES

Notations

• R + = {x ∈ R : x ≥ 0} where R is the set of real numbers. | • | denotes the absolute value in R, • denotes the Euclidean norm in R n and x α = |x| α sign(x) for x ∈ R and α ∈ R + . • S = {x ∈ R n : x = 1} denotes the unit sphere in R n . B(x 0 , r) = {x ∈ R n : x -x 0 < r} denotes the open ball of radius r > 0 centered at a point x 0 . We define B(K, r) = y ∈ R n : inf x∈K y -x < r for all compact K ⊂ R n . • A continuous function α : R + → R + belongs to the class K if α(0) = 0
∇V (x) = ∂V ∂x1 (x), • • • , ∂V ∂xn (x) stands for the first derivative of a continuously differentiable function V at point x ∈ R n . For f : x ∈ R n → f (x) ∈ R n , the notation ∇V (x), f (x) = n i=1 ∂V ∂xi (x)f i (x)
stands for the directional derivative of a continuously differentiable function V with respect to the vector field f evaluated at point x ∈ R n .

• C p (R n , R) denotes the space of functions f : R n → R which have p continuous derivatives and C ∞ (R n , R) denotes the space of functions f : R n → R which are smooth. We denote by CL 0 (R n , R) (respectively CL k (R n , R)) the set of continuous functions on R n , locally Lipschitz on R n \ {0} with value in R n (respectively the set of continuous functions on R n , C k on R n \ {0} with value in R). • For F : R n ⇒ R n , we denote by D(F ) the domain of definition of F. • coA denotes the closed convex hull of A ⊂ R n .

Stability properties and generalized homogeneity

In this subsection we give some definitions of homogeneity and different kinds of stability for a system:

ẋ(t) = f (x(t)), t ≥ 0 x(0) = x 0 ∈ R n .
(1)

where f : R n → R n , f (0) = 0 is considered such that (1) possesses a unique solution in forward time for all initial conditions x 0 ∈ R n except the origin, denoted as x(t, x 0 ).

Asymptotic, finite-time stability:

The following definitions are inspired from [START_REF] Zubov | Methods of AM Lyapunov and their Application[END_REF]; [START_REF] Clarke | Asymptotic stability and smooth Lyapunov functions[END_REF]; [START_REF] Bernuau | Finite-time stability of continuous autonomous systems[END_REF] and [START_REF] Polyakov | Stability notions and Lyapunov functions for sliding mode control systems[END_REF].

Definition 2.1. The origin of system (1) is

• Lyapunov stable: if there exists a nonempty open neighborhood of the origin V ⊂ R n and a function α ∈ K ∞ such that for all x 0 ∈ V we have x(t, x 0 ) ≤ α( x 0 ), ∀t ≥ 0. • Asymptotically stable: if it is Lyapunov stable and for all x 0 ∈ V, lim t→+∞ x(t, x 0 ) = 0.
• FTS: if it is Lyapunov stable and there exists a function T :

V → R + such that ∀x 0 ∈ V, x(t, x 0 ) = 0, ∀t ≥ T (x 0 ).
T is called a settling-time function of (1).

In addition, if V = R n , then all these properties hold globally.

The following proposition shows how to investigate the FTS by using Lyapunov theory. Proposition 2.1. [START_REF] Moulay | Finite time stability and stabilization of a class of continuous systems[END_REF] Assume that (1) has a unique solution in forward time outside the origin. The origin of this system is FTS with a continuous settling-time function at the origin if and only if there exist a real number c > 0, α ∈ (0, 1) and a class

CL ∞ Lyapunov function V : V ⊂ R n → R + , 0 ∈ V satisfying ∇V (x), f (x) ≤ -c[V (x)] α , ∀x ∈ V.
Now we define the stability properties for DIs. Let us consider the following differential inclusion

ẋ(t) ∈ F (x(t)), ∀t ≥ 0, x(0) = x 0 ∈ R n , (2)
where

F : R n ⇒ R n . Definition 2.2. A set-valued map F is called upper semi- continuous at x ∈ D(F ) if ∀ε > 0, ∃δ > 0 such that ∀x ∈ B(x, δ), F (x) ⊂ B(F (x), ε). For each Ω ⊆ R n , we let S(Ω) ⊂ C(R + × R n , R n ) denote the set of solutions of (2) satisfying x(0) ∈ Ω.
If Ω is a singleton {x 0 }, we will use the shorthand S(x 0 ). We denote S = S(R n ) as the set of all solutions. The domain of a solution x(•) will be denoted by R + if it is forward complete. The next definition concerns GAS and FTS properties of DIs. Definition 2.3. The origin of the system (2) is

• strongly GAS (resp., FTS) if all solutions x(•) ∈ S of (2) are GAS (resp., FTS) (Definition 2.1). • weakly GAS/FTS if there exists a subset Ŝ of the set S of solutions of (2), such that all x(•) ∈ Ŝ are uniformly GAS/FTS.

By the definition of weak GAS (resp., FTS) ∃β ∈ KL (resp., β ∈ GKL) such that x(t) ≤ β( x 0 , t), for all x 0 ∈ R n , x(•) ∈ Ŝ(x 0 ). Notice that, in the case of the weak uniform GAS or FTS, the set Ŝ may depend on the function β:

Ŝ = Ŝβ .
In this case, for σ ∈ K ∞ and R > 0 we denote

C β σ (R) =    y ∈ C ([0, 1], R n ) ; y(1) = 0, y σ(t) 1 + σ(t) = x(t), t ≥ 0, x(•) ∈ Ŝβ (B(0, R))    .
The following lemma proves that the set Ŝβ of weakly uniformly stable solutions of ( 2)

is homeomorphic to a compact in C([0, 1], R n ). Lemma 2.1. If F is upper semicontinuous and Ŝβ is the set of all GAS solutions of (2), R > 0. Then, C σ (R) is a compactum in (C([0, 1], R n ), • C ) , with f C = sup x∈[0,1] |f (x)| denotes the uniform norm.
The proofs of several results are skipped or shortened due to space limitations.

Homogeneity: In control theory, homogeneity simplifies qualitative analysis of nonlinear dynamic systems. So that, it allows a local properties (e.g., local stability) to be extended globally using a property of the solutions (homogeneity is a Lie symmetry). In order to define this concept, let us introduce the notion of dilation group. Definition 2.4. A map d : R → R n×n is called a linear dilation in R n iff it satisfies and satisfies the following properties:

• Group property: d(0) = I n , d(t + s) = d(t)d(s), t, s ∈ R. • Limit property: lim
d ds d(s) = G d d(s) = d(s)G d and d(s) = e G d s = +∞ i=0 s i G i d i! . Definition 2.5. The dilation d is monotone in R n , if d(s) R n×n < 1, ∀s < 0, with d(s) R n×n = sup x∈S d(s)x .
Now we will define the canonical homogeneous norm. Definition 2.6. If the dilation d is monotone, the continuous function

• d : x ∈ R n → x d = e sx ∈ R + , where s x ∈ R : d(-s x )x = 1, x d → 0 as x → 0 and d(s)x d = e s x d > 0, is called the canonical homogeneous norm.
Note that in the sequel of this paper, we will use only monotone dilations. The existence of a homogeneous Lyapunov function for a GAS homogeneous system was provided by Zubov in 1957[START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF] and [START_REF] Polyakov | Sliding mode control design using canonical homogeneous norm[END_REF] by using weighted and generalized homogeneity.

SUB-HOMOGENEOUS DIFFERENTIAL INCLUSIONS (DI)

In this section, we define sub-homogeneous DI of (1) and then investigate its FTS, a way to obtain a subhomogeneous DI for (1) is also indicated. Let us introduce the following assumptions for (2):

-H F gas : (2) is strongly GAS, -H F sub : for all x ∈ R n , e νs d(s)F (x) ⊆ F (d(s)x), ∀s ≥ 0, where d is a generalized linear dilation and ν ∈ R. -H F exi : F (x) is nonempty, convex, compact and upper semicontinuous. Definition 3.1. F is d-sub-homogeneous with degree of homogeneity ν ∈ R n , iff H F sub holds.

Defining the notion of sub-homogeneity comes from the fact that there are some DIs which satisfy this property (with inclusion) and not with the equality (when additionally F (d(s)x) ⊆ e νs d(s)F (x), ∀s ≤ 0). This class of DIs appears sometimes as extension of dynamical systems, which do not have homogeneous approximation and which are not easy to be studied. As an example of this extension we have

F (x) = co s≤0 e -νs d(-s)f (d(s)x) ;
(3) with f is defined in (1). Such a set-valued map F is dsub-homogeneous and f (x) ∈ F (x), ∀x ∈ R n . Hence, this extension will be used in the last section to design finitetime observer for non-homogenoeus systems. The following proposition ensures symmetry of solutions of (2) when it satisfies the assumption H F sub . Proposition 3.1. Let F be a d-sub-homogeneous set-valued map of degree ν ∈ R. If for x 0 ∈ R n , the trajectory x(•, x 0 ) is a solution of (2) starting at x 0 , then for all s ≥ 0, the absolutely continuous curve t -→ d(s)x(e νs t, x 0 ) is a trajectory of (2) starting at d(s)x 0 , (i.e., if x(•, x 0 ) ∈ S(x 0 ), then d(s)x(e νs •, x 0 ) ∈ S(d(s)x 0 ) for all s ≥ 0, ( and d(s)S(x 0 ) ⊆ S(d(s)x 0 )) for all s ≥ 0).

Proof. We consider a trajectory x(•, x 0 ) of (2) starting at x 0 . The curve t -→ d(s)x(e νs t, x 0 ) is clearly a continuous curve for all s ≥ 0. Moreover, for almost all t ∈ R we have: )x(e νs t, x 0 )] = e νs d(s) ẋ(e νs t, x 0 ) ∈ e νs d(s)F (x(e νs t, x 0 )).

d dt [d(s
Since F is d-sub-homogeneous with degree ν, one gets

d dt [d(s)x(e νs t, x 0 )] ∈ F (d(s)x(e νs t, x 0 )), (4)
then, thus t -→ d(s)x(e νs t, x 0 ) is a solution of (2) for all s ≥ 0. Theorem 3.1. If the set-valued map F satisfies H F sub , H F gas and H F exi , then, for k > max{-ν, 0}, there exists a pair (V, W ) of continuous functions and p ∈ N \ {0}, such that all the eigenvalues of (kI n -pG d ) have a positive real parts, and ∇V (x), h ≤ -W (x) for all x = 0.

1) V ∈ C p (R n , R + ), V is d-
Proof. The sketch of the proof is as follows: under the introduced hypotheses and using the result from [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF] we construct a d-homogeneous Lyapunov function V, which satisfies 1), 2) and 3). Theorem 3.1 asserts that a strongly GAS sub-homogeneous DI admits a homogeneous Lyapunov function. This result is a generalization of the theorem of [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF]. The next theorem proves the FTS of the GAS subhomogeneous DI (2) with negative degree of homogeneity. Theorem 3.2. If the set-valued map F satisfies H F exi , H F gas , H F sub and its degree of homogeneity ν < 0, then (2) is globally FTS.

Proof. According to Theorem 3.1, there exists a continuous pair (V, W ) satisfying the conditions 1), 2) and 3) above. Let τ ≥ 0, using the homogeneity of V and the sub-homogeneity of F, we get

e (k+ν)τ max h∈F (x) ∇V (x), h ≤ e kτ max h∈e ντ d(τ )F (x) ∇V (x), d(-τ )h ≤ max h∈F (d(τ )x) ∇V (d(τ )x), h ≤ -W (d(τ )x).
(5)

Let R > 0, for τ = -ln x(t) d R ≥ 0, and from ( 5) we obtain

V (x(t)) = max h∈F (x(t)) ∇V (x(t)), h ≤ -e -(k+ν)τ min y d =R W (y) ≤ - W R k+ν x(t) k+ν d ≤ - W aR k+ν V (x(t)) k+ν k , (6)
where a = max

y d =1 V (y)
k+ν k

and W = min

y d =R W (y). If
ν < 0 and by using Proposition 2.1 we conclude that the origin of (2) is FTS ( settling time function is estimated by

T (x 0 ) ≤ [V (x0)] 1-α (1-α)c
for all initial conditions x 0 ∈ B(0, R)

and for all R > 0. This proves the global FTS of (2).

This result demonstrates how useful the notion of subhomogeneity for analysis of FTS. In many papers, researchers study DIs which are strongly GAS. However, in this work, we will see that DIs which are weakly GAS appear as a sub-homogeneous extensions of nonhomogeneous dynamical systems. Therefore, the next theorem will study the FTS of this class of DIs. Theorem 3.3. If the set-valued map F satisfies H F exi , H F sub , there exists a set Ŝβ of GAS solutions of DI (2) ( (2) is weakly GAS) and ν < 0, then all the solutions in Ŝβ are globally FTS (origin of DI (2) is weakly globally FTS).

Proof. Let Ŝβ be the set of solutions of the DIs (2), which are GAS. Let R > 0,

x 0 ∈ B 2R , x(•, x 0 ) ∈ Ŝβ (x 0 ), B R = {x ∈ R n : d(-ln(R))
x ≤ 1} and let us define the time τ (x 0 , B R ) for GAS solutions of (2) starting from x 0 , to reach and stay in B R by:

τ (x 0 , B R ) = sup x(•,x0)∈ Ŝβ (x0) inf {T > 0 : x(t, x 0 ) ∈ B R , ∀t ≥ T } .
Using Lemma 2.1, one can prove that τ is finite for every x 0 ∈ B R . The fact that ν < 0 implies the weak finitetime stability of the sub-homogeneous extension (2). This property holds for every R > 0, which implies the global weak FTS of (2).

Theorem 3.3 generalizes the results from [START_REF] Bernuau | Finite-time stability of continuous autonomous systems[END_REF] about homogeneous DI. Also, it shows that if a DI is weakly GAS and the degree of sub-homogeneity is negative, we can guarantee its weak FTS.

FINITE-TIME OBSERVER DESIGN USING SUB-HOMOGENEITY

The notion of sub-homogeneity is instrumental to introduce a new observer design method. Throughout this section we consider the following system

ẋ(t) = A(y)x(t) + φ(y, u); x(t) ∈ R n , y(t) = Cx(t), t ∈ [0, +∞). (7)
where

A(y) =          0 a 1 (y) 0 • • • 0 . . . . . . . . . . . . . . . . . . . . . 0 . . . . . . a n-1 (y) 0 • • • • • • 0          , C = (1, 0, • • • , 0) .
(8) The functions a i : R → R and 0 < a i ≤ a(y) ≤ āi < 1, ∀y ∈ R, i = 1, • • • , n -1 are continuous and they do not have a homogeneous approximation. The goal of this section is to introduce an observer for the system (7). The observer is given by the system ẋ(t) = A(y)x(t) + φ(y, u) -L(E 1 ), t ∈ [0, +∞),

where x(t) ∈ R n , is the state estimate L(E 1 ) = (k 1 E 1 α1 , • • • , k n E 1 αn ) , k i > 0, α i = n-i n , ∀i = 1, • • • , n; and E = x-x = (E 1 , • • • , E n ) is the estimation error vector.
Here is the observer error dynamics Ė = A(y)E -L(E 1 ) = f (E, y).

(9) Let us consider the following Lyapunov function that has been proposed in Cruz-Zavala and Moreno ( 2018)

V (E) = n-1 i=1 β i g i (E i , E i+1 ) + β n p |E n | p , (10)
with

β 1 > 0, • • • , β n > 0 and g i (E i , E i+1 ) = ri p |E i | p r i - E i E i+1 p-r i r i+1 + p-ri p |E i+1 | p r i+1 , where for i = 1, • • • , n; we have r i = n + 1 -i and p ≥ r 1 + r 2 = 2n -1 ≥ r i + r i+1 ≥ 1 for n ≥ 2.
For simplicity, we introduce the variables

σ i = E i p-r i r i -E i+1 p-r i r i+1
,

ψ i = p -r i r i+1 |E i+1 | p-r i -r i+1 r i+1 E i+1 r i r i+1 -E i .
Note that g i , σ i and ψ i vanish on the same set. The partial derivatives of g i are

∂g i (E i , E i+1 ) ∂E i = σ i , ∂g i (E i , E i+1 ) ∂E i+1 = ψ i .
σ i and ψ i are continuous. Theorem 4.1. If we choose carefully the gains k i > 0, and

β i > 0 such that β n = 1, 0 < k n < 1 p-1 -2βn-2 βn-1(p-1) 1 βn-1 + 2(p-2) 2 p-1 , 0 < k n-1 < 1 p -1 - k n 1 βn-1 + 2(p-2) 2 p-1 - 2β n-2 β n-1 (p -1) 0 < k n-2 < 2 p -1 - 4β n-3 β n-2 (p -1) -2k n β n-1 (p -2) β n-2 (p -1) -(k n-1 + 1) β n-1 β n-2 + 2p -7 p -1 (11) for i = 3, • • • , n -2, 0 < k i-1 < r i-1 -1 p -1 - 2β i-2 r i β i-1 (p -1) -(k i+1 + 1) β i r i β i-1 (p -1) -(k i + 1) β i β i-1 + 2p -2r i -r i-1 p -1 - 2β i-2 r i β i-1 (p -1) (12) and          0 < k 1 < r 1 -1 p -1 -(k 3 + 1) β 2 r 2 β 1 (p -1) -(k 2 + 1) β 2 β 1 + 2p -2r 2 -r 1 p -1 , 1 + r 1 (1 + k 2 ) p -1 + n-1 i=2 2k i r i+1 p -1 (β i + β i-1 ) β 1 < k 1 , (13
) then ( 9) is uniformly GAS.

Proof.

As it has been shown in Cruz-Zavala and Moreno (2018), since the Lyapunov function V is the sum of non negative terms, then it is non negative and the fact that V (z) = 0 =⇒ z = 0, implies that V is positive definite. Due to homogeneity it is radially unbounded. Its derivative V is estimated as follows

V ≤ n i=1 δ i |E i | p-1 r i , where δ 1 = β 1 p -r 1 p -1 (1 -k 1 ) + β 1 r 1 (a 2 (y) + k 2 ) p -1 + n-1 i=2 2k i r i+1 p -1 (β i + β i-1 ) , δ 2 = β 1 p -r 1 p -1 k 1 -a 1 (y) p -1 p -r 1 -(a 1 (y)) p-r 1 r 2 + β 2 p -r 2 p -1 (a 2 (y) + k 2 ) +β 2 r 2 (a 3 (y) + k 3 ) p -1 + β 1 (a 2 (y) + k 2 ) 2(p -r 2 ) -r 1 p -1 . For all i = 3, • • • , n -2, δ i = β i-1 p -r i-1 p -1 k i-1 -a i-1 (y) p -1 p -r i-1 -(a i-1 (y)) p-r i-1 r i +β i p -r i p -1 (a i (y) + k i ) + β i r i (a i+1 (y) + k i+1 ) p -1 +β i-1 (a i (y) + k i ) 2(p -r i ) -r i-1 p -1 + β i-2 2r i a i-1 (y) p -1 , δ n-1 = β n-2 p -3 p -1 k n-2 -a n-2 (y) p -1 p -3 -(a n-2 (y)) p-3 2 +β n-1 p -2 p -1 (a n-1 (y) + k n-1 ) + β n-3 4a n-2 (y) p -1 +β n-2 (a n-1 (y) + k n-1 ) 2(p -2) -3 p -1 + 2k n β n-1 p -2 p -1 , δ n = β n-1 p -2 p -1 k n-1 -a n-1 (y) p -1 p -2 -(a n-1 (y)) p-2 + k n β n +β n-2 2a n-1 (y) p -1 + 2k n β n-1 (p -2) 2 p -1 .
We choose the gains k i > 0, and β i > 0 such that (11), ( 12) and ( 13) hold. Then, one has δ i < 0, ∀i ∈ {1, • • • , n}.

Which implies that V is negative definite. This implies that the system ( 9) is uniformly GAS.

Let F be a sub-homogeneous extension of f, which is given by {A(e r1s y)E -L(E 1 )} ,

F (E) = co
By construction, the set valued F (E) is nonempty (f (E, y) ∈ F (E)), compact, convex and upper semicontinuous (f is continuous, the proof is similar to the one of Lemma 3.1 in [START_REF] Braidiz | On fixed-time stability of a class of nonlinear time-varying systems[END_REF]) and d-subhomogeneous with degree of homogeneity ν = -1. Indeed, F (d(τ )E) ⊆ e -τ d(τ )F (E), ∀τ ≤ 0, and d(s) = diag{e ns , e (n-1)s , • • • , e s }. The system (7) is GAS, this implies that there exists a subset Ŝβ of GAS solutions of the DI ẋ(t) ∈ F (x(t)). The degree of homogeneity of F is ν = -1 < 0, then by using Theorem 3.3, ẋ(t) ∈ F (x(t)) is weakly globally FTS. Because the solutions of (7) are in Ŝβ , then it is globally FTS. For n = 2, Fig. 1 shows the FTS of solution of the system (9) for a : y ∈ R → (1 -1 2 cos(y)) ∈ R.

CONCLUSION

The notions of sub-homogeneity were introduced. As it was shown, this property allows us to analyze systems which do not admit homogeneous approximations. FTS of sub-homogenoeus GAS DIs is investigated using Lyapunov function. The notion of weak FTS for DI was introduced and it was investigated for sub-homogeneous DI, which is weakly GAS. Example from Section 4 shows that the notion of sub-homogeneity can be used to design a finitetime observer.

  )x = +∞ uniformly on the unit sphere. Property 2.1. The matrix G d ∈ R n×n defined as G d = lim s→0 d(s)-In s is known as the generator of the group d(s)

  homogeneous with degree of homogeneity k > max{-ν, 0} and positive definite; 2) W ∈ C ∞ (R n \ {0}, R + ), W is positive definite and d-homogeneous with degree of homogeneity k + ν; 3) max h∈F (x)

Fig. 1 .

 1 Fig. 1. The solutions of the system (9) with a(y) = 1 -1 100 cos(y), and the initial condition (x 0 , y 0 ) = (3, 40).

  and the function is strictly increasing. The function α : R + → R + belongs to the class K ∞ if α ∈ K and it is increasing to infinity. A continuous function β : R + ×R + → R + belongs to the class KL (resp., GKL) if β(•, t) ∈ K ∞ for each fixed t ∈ R + and lim

	t→+∞	β(s, t) = 0 (resp., if for each fixed
	t ∈ (0, T ), β is a strictly decreasing function of its
	second argument t ∈ R + for any fixed first argument
	s ∈ R + \ {0} and β(s, T ) = 0 for some 0 ≤ T < +∞)
	for each fixed s ∈ R + .