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Theory and Modern Applications

On stability of discrete-time systems under nonlinear time-varying perturbations

Abstract

We give some explicit stability bounds for discrete-time systems subjected to time-varying and nonlinear perturbations. The obtained results are extensions of some well-known results in (Hinrichsen and Son in Int. J. Robust Nonlinear ControlĀ 8:1169-1188, 1998; Shafai et al. in IEEE Trans. Autom. ControlĀ 42:265-270, 1997) to nonlinear time-varying perturbations. Two examples are given to illustrate the obtained results. Finally, we present an Aizerman-type conjecture for discrete-time systems and show that this conjecture is valid for positive systems.

MSC:39A30, 93D09.

1 Introduction and preliminaries

Discrete-time equations have numerous applications in science and engineering. They are used as models for a variety of phenomena in the life sciences, population biology, computing sciences, economics, etc.; see, e.g., [6, 10, 15].

Motivated by many applications in control engineering, problems of stability and robust stability of dynamical systems have attracted much attention from researchers for a long time, see, e.g., [2, 3, 5, 9ā€“11, 14ā€“21] and references therein. In this paper, we investigate exponential stability of discrete-time systems subjected to nonlinear time-varying perturbations. Some explicit stability bounds for discrete-time systems subjected to nonlinear time-varying perturbations are given. Furthermore, we present an Aizerman-type conjecture for discrete-time systems and show that it is valid for positive systems. Two examples are given to illustrate the obtained results.

Let R be the set of all real numbers and let N be the set of all natural numbers. Set Z + :=NāˆŖ{0}. For given NāˆˆN, let us denote N Ģ² :={1,2,ā€¦,N}. Let n, l, q be positive integers. Inequalities between real matrices or vectors will be understood componentwise, i.e., for two real lƗq-matrices A=( a i j ) and B=( b i j ), the inequality Aā‰„B means a i j ā‰„ b i j for i=1,ā€¦,l; j=1,ā€¦,q. In particular, if a i j > b i j for i=1,ā€¦,l; j=1,ā€¦,q, then we write Aā‰«B instead of Aā‰„B. The set of all nonnegative lƗq-matrices is denoted by R + l Ɨ q . If x=( x 1 , x 2 ,ā€¦, x n )āˆˆ R n and P=( p i j )āˆˆ R l Ɨ q we define |x|=(| x i |) and |P|=(| p i j |). It is easy to see that |CD|ā‰¤|C||D|. For any matrix Aāˆˆ R n Ɨ n the spectral radius of A is denoted by Ļ(A)=max{|Ī»|:Ī»āˆˆĻƒ(A)}, where Ļƒ(A):={zāˆˆC:det(z I n āˆ’A)=0} is the set of all eigenvalues of A. A norm āˆ„ā‹…āˆ„ on R n is said to be monotonic if |x|ā‰¤|y| implies āˆ„xāˆ„ā‰¤āˆ„yāˆ„ for all x,yāˆˆ R n . Every p-norm on R n ( āˆ„ x āˆ„ p = ( | x 1 | p + | x 2 | p + ā‹Æ + | x n | p ) 1 p , 1ā‰¤p<āˆž and āˆ„ x āˆ„ āˆž = max i = 1 , 2 , ā€¦ , n | x i |), is monotonic. Throughout this paper, the norm āˆ„Māˆ„ of a matrix Māˆˆ R l Ɨ q is always understood as the operator norm defined by āˆ„Māˆ„= max āˆ„ y āˆ„ = 1 āˆ„Myāˆ„ where R q and R l are provided with some monotonic vector norms. Then, the operator norm āˆ„ā‹…āˆ„ has the following monotonicity property (see, e.g., [11])

Pāˆˆ R l Ɨ q ,Qāˆˆ R + l Ɨ q ,|P|ā‰¤Qā‡’āˆ„Pāˆ„ā‰¤ āˆ„ | P | āˆ„ ā‰¤āˆ„Qāˆ„.
(1)

The next theorem summarizes some basic properties of nonnegative matrices which will be used in what follows.

Theorem 1.1 ([8, 12])

LetAāˆˆ R p Ɨ p be a nonnegative matrix. Then the following statements hold.

  1. (i)

    (Perron-Frobenius Theorem) Ļ(A) is an eigenvalue of A and there exists a nonnegative eigenvector xāˆˆ R p , xā‰ 0 such that Ax=Ļ(A)x.

  2. (ii)

    Given Ī±āˆˆ R + , there exists a nonzero vector xā‰„0 such that Axā‰„Ī±x if and only if Ļ(A)ā‰„Ī±.

  3. (iii)

    ( t I n āˆ’ A ) āˆ’ 1 exists and is nonnegative if and only if t>Ļ(A).

  4. (iv)

    Given Bāˆˆ R + p Ɨ p , Cāˆˆ R p Ɨ p . Then

    |C|ā‰¤BāŸ¹Ļ(A+C)ā‰¤Ļ(A+B).

2 Stability of discrete-time systems under nonlinear time-varying perturbations

Consider a nonlinear discrete-time system of the form

x(k+1)=f ( k , x ( k ) ) ,kā‰„ k 0 ,
(2)

where f: Z + Ɨ R n ā†’ R n is a given function such that f(k,0)=0, for all kāˆˆ Z + (i.e., Ī¾=0 is an equilibrium of the system (2)).

It is clear that for given k 0 āˆˆ Z + and x 0 āˆˆ R n , (2) has a unique solution, denoted by x(ā‹…, k 0 , x 0 ), satisfying the initial condition

x( k 0 )= x 0 .
(3)

Definition 2.1 The zero solution of (2) is said to be exponentially stable if there exist Mā‰„0 and Ī²āˆˆ[0,1) such that

āˆ€k, k 0 āˆˆ Z + ,kā‰„ k 0 ;āˆ€ x 0 āˆˆ R n : āˆ„ x ( k , k 0 , x 0 ) āˆ„ ā‰¤M Ī² k āˆ’ k 0 āˆ„ x 0 āˆ„.
(4)

We first give a simple sufficient condition for exponential stability of (2) which is used in what follows.

Proposition 2.2 Suppose there exists Aāˆˆ R + n Ɨ n such that

| f ( k , x ) | ā‰¤A|x|,āˆ€kāˆˆ Z + ,āˆ€xāˆˆ R n .
(5)

IfĻ(A)<1then the zero solution of (2) is exponentially stable.

Proof Let x(k):=x(ā‹…, k 0 , x 0 ), kā‰„ k 0 , be the solution of (2)-(3). It follows from (2) and (5) that

| x ( k + 1 ) | = | f ( k , x ( k ) ) | ā‰¤A | x ( k ) | ,āˆ€kā‰„ k 0 .

This gives

| x ( k + 1 ) | ā‰¤A | x ( k ) | ā‰¤ A 2 | x ( k āˆ’ 1 ) | ā‰¤ā‹Æā‰¤ A k āˆ’ k 0 + 1 | x ( k 0 ) | = A k āˆ’ k 0 + 1 | x 0 |,āˆ€kā‰„ k 0 .

Without loss of generality, let āˆ„ā‹…āˆ„= āˆ„ ā‹… āˆ„ p (1ā‰¤pā‰¤āˆž). Hence,

āˆ„ x ( k + 1 ) āˆ„ ā‰¤ āˆ„ A k āˆ’ k 0 + 1 āˆ„ āˆ„ x 0 āˆ„,āˆ€kā‰„ k 0 .
(6)

Since Ļ(A)<1, there exist Mā‰„1, Ī²āˆˆ[0,1) such that

āˆ„ A k āˆ„ ā‰¤M Ī² k ,āˆ€kāˆˆ Z + ,
(7)

see, e.g., [11]. By (6) and (7),

āˆ„ x ( k , k 0 , x 0 ) āˆ„ ā‰¤M Ī² k āˆ’ k 0 āˆ„ x 0 āˆ„,āˆ€kā‰„ k 0 .

This completes the proof.ā€ƒā–”

Remark 2.3 In particular, if for each kāˆˆ Z + , f(k,ā‹…) is continuously differentiable on R n and there exists Aāˆˆ R + n Ɨ n such that

| J ( k , x ) | ā‰¤A,āˆ€kāˆˆ Z + ,āˆ€xāˆˆ R n ,
(8)

then (5) holds. Here J(k,x):=( d f i d x j (k,x))āˆˆ R n Ɨ n , kāˆˆ Z + , xāˆˆ R n , denotes the Jacobian matrix of f(k,ā‹…) at x. Indeed, we have f(k,x)=f(k,x)āˆ’f(k,0)=( āˆ« 0 1 J(k,tx)dt)x, by the mean value theorem, see, e.g., [4]. Therefore, (8) yields,

| f ( k , x ) | = | ( āˆ« 0 1 J ( k , t x ) d t ) x | ā‰¤ ( āˆ« 0 1 | J ( k , t x ) | d t ) | x | ā‰¤ A | x | , āˆ€ k āˆˆ Z + , āˆ€ x āˆˆ R n .

Suppose all hypotheses of Proposition 2.2 hold. Thus, the zero solution of (2) is exponentially stable. Consider a perturbed system of the form

x(k+1)=f ( k , x ( k ) ) + āˆ‘ i = 1 N D i ( k , x ( k ) ) P i ( k , E i ( k , x ( k ) ) ) ,kāˆˆ Z + ,
(9)

where N is a given positive integer and D i : Z + Ɨ R n ā†’ R n Ɨ l i , E i : Z + Ɨ R n ā†’ R q i (iāˆˆ N Ģ² ) are given and P i : Z + Ɨ R q i ā†’ R l i (iāˆˆ N Ģ² ) are uncertainties. Furthermore, we assume that

( H 1 ) P i (k,0)=0, āˆ€kāˆˆ Z + and E i (k,0)=0, āˆ€kāˆˆ Z + for each iāˆˆ N Ģ² ;

( H 2 ) there exist D i āˆˆ R + n Ɨ l i , E i āˆˆ R + q i Ɨ n and P i āˆˆ R + l i Ɨ q i (iāˆˆ N Ģ² ) such that

| D i ( k , x ) | ā‰¤ D i ,āˆ€kāˆˆ Z + ,āˆ€xāˆˆ R n
(10)

and

(11)

The main problem here is to find a positive number Ī³ such that the zero solution of an arbitrary perturbed system of the form ( 9 ) remains exponentially stable whenever the size of perturbations is less than Ī³.

Remark 2.4 In particular, if

and

P i (k,y):= P i (k)y, P i (k)āˆˆ R l i Ɨ q i ,yāˆˆ R q i ,

then the perturbation āˆ‘ i = 1 N D i (k,x(k)) P i (k, E i (k,x(k))) becomes āˆ‘ i = 1 N D i (k) P i (k) E i (k)x(k). The problem of robust stability of linear infinite dimensional time-varying system

x(k+1)=A(k)x(k),kāˆˆ Z + ,
(12)

under the time-varying multi-perturbations

A(k)ā†ŖA(k)+ āˆ‘ i = 1 N D i (k) P i (k) E i (k),
(13)

has been analyzed in [21] and an abstract stability bound is given in terms of input-output operators.

We are now in the position to prove the main result of this paper.

Theorem 2.5 Assume that all hypotheses of Proposition 2.2 hold andAāˆˆ R + n Ɨ n satisfies (5). If ( H 1 )-( H 2 ) hold and

āˆ‘ i = 1 N āˆ„ P i āˆ„< 1 max i , j āˆˆ N Ģ² āˆ„ E i ( I n āˆ’ A ) āˆ’ 1 D j āˆ„ ,
(14)

then the zero solution of (9) remains exponentially stable.

Proof Since (5) and (10)-(11), it follows that

We show that Ļ(A+ āˆ‘ i = 1 N D i P i E i )<1 and then the zero solution of (9) is exponentially stable by Proposition 2.2.

Since A and D i , E i , P i (iāˆˆ N Ģ² ) are nonnegative, so is A+ āˆ‘ i = 1 N D i P i E i . Assume on the contrary that Ļ 0 :=Ļ(A+ āˆ‘ i = 1 N D i P i E i )ā‰„1. By the Perron-Frobenius Theorem (Theorem 1.1(i)), there exists xāˆˆ R + n , xā‰ 0 such that

( A + āˆ‘ i = 1 N D i P i E i ) x= Ļ 0 x.

Let Q(t):=t I n āˆ’A, tāˆˆR. Since Ļ(A)<1, Q( Ļ 0 ) is invertible. It follows that

Q ( Ļ 0 ) āˆ’ 1 āˆ‘ i = 1 N D i P i E i x=x.
(15)

Let i 0 be an index such that āˆ„ E i 0 xāˆ„= max i āˆˆ N Ģ² āˆ„ E i xāˆ„. Then (15) implies that āˆ„ E i 0 xāˆ„>0. Multiply both sides of (15) from the left by E i 0 , to get

āˆ‘ i = 1 N E i 0 Q ( Ļ 0 ) āˆ’ 1 D i P i E i x= E i 0 x.

Taking norms, we get

āˆ‘ i = 1 N āˆ„ E i 0 Q ( Ļ 0 ) āˆ’ 1 D i āˆ„ āˆ„ P i āˆ„āˆ„ E i xāˆ„ā‰„āˆ„ E i 0 xāˆ„.

This implies

max i , j āˆˆ N Ģ² āˆ„ E i Q ( Ļ 0 ) āˆ’ 1 D j āˆ„ ( āˆ‘ i = 1 N āˆ„ P i āˆ„ ) āˆ„ E i 0 xāˆ„ā‰„āˆ„ E i 0 xāˆ„,

or equivalently,

max i , j āˆˆ N Ģ² āˆ„ E i Q ( Ļ 0 ) āˆ’ 1 D j āˆ„ āˆ‘ i = 1 N āˆ„ P i āˆ„ā‰„1.
(16)

On the other hand, the resolvent identity gives

Q ( 1 ) āˆ’ 1 āˆ’Q ( Ļ 0 ) āˆ’ 1 =( Ļ 0 āˆ’1)Q ( 1 ) āˆ’ 1 Q ( Ļ 0 ) āˆ’ 1 .
(17)

Since Aāˆˆ R + n Ɨ n and Ļ(A)<1ā‰¤ Ļ 0 , Theorem 1.1(iii) yields Q ( 1 ) āˆ’ 1 ā‰„0 and Q ( Ļ 0 ) āˆ’ 1 ā‰„0. Then (17) implies Q ( 1 ) āˆ’ 1 ā‰„Q ( Ļ 0 ) āˆ’ 1 ā‰„0. Hence, E i Q ( 1 ) āˆ’ 1 D j ā‰„ E i Q ( Ļ 0 ) āˆ’ 1 D j ā‰„0, for any i,jāˆˆ N Ģ² . By (1), āˆ„ E i Q ( 1 ) āˆ’ 1 D j āˆ„ā‰„āˆ„ E i Q ( Ļ 0 ) āˆ’ 1 D j āˆ„, for any i,jāˆˆ N Ģ² . It follows from (16) that

āˆ‘ i = 1 N āˆ„ P i āˆ„ā‰„ 1 max i , j āˆˆ N Ģ² āˆ„ E i Q ( 1 ) āˆ’ 1 D j āˆ„ .

However, this conflicts with (14). This completes the proof.ā€ƒā–”

In particular, suppose (12) satisfies

| A ( k ) | ā‰¤A,āˆ€kāˆˆ Z + ,
(18)

for some Aāˆˆ R + n Ɨ n . Consider a perturbed system of the form

x(k+1)=A(k)x(k)+ āˆ‘ i = 1 N D i ( k , x ( k ) ) P i ( k , E i ( k , x ( k ) ) ) ,kāˆˆ Z + ,
(19)

where D i , P i and E i (iāˆˆ N Ģ² ) are as above.

The following is immediate from Theorem 2.5.

Corollary 2.6 Suppose (18) and ( H 1 )-( H 2 ) hold andĻ(A)<1. If (14) holds then the zero solution of (19) is exponentially stable.

Corollary 2.7 LetAāˆˆ R + n Ɨ n andĻ(A)<1. Suppose D i (ā‹…): Z + ā†’ R n Ɨ l i , E i (ā‹…): Z + ā†’ R q i Ɨ n (iāˆˆ N Ģ² ), are given and P i (ā‹…): Z + ā†’ R l i Ɨ q i (iāˆˆ N Ģ² ) are unknown. If there exist D i āˆˆ R n Ɨ l i , E i āˆˆ R q i Ɨ n and P i āˆˆ R l i Ɨ q i (iāˆˆ N Ģ² ) such that

| D i ( k ) | ā‰¤ D i ; | E i ( k ) | ā‰¤ E i ; | P i ( k ) | ā‰¤ P i ,āˆ€kāˆˆ Z + ,

and (14) holds then the zero solution of the perturbed system

x(k+1)= ( A + āˆ‘ i = 1 N D i ( k ) P i ( k ) E i ( k ) ) x(k),āˆ€kāˆˆ Z + ,
(20)

is exponentially stable.

Remark 2.8 If Aāˆˆ R + n Ɨ n , then the system

x(k+1)=Ax(k),kāˆˆ Z + ,
(21)

is positive. That is, for any initial state x 0 āˆˆ R + n , the corresponding trajectory of the system x(k, x 0 ), kāˆˆ Z + , remains in R + n for all kāˆˆ Z + . Positive dynamical systems play an important role in the modeling of dynamical phenomena whose variables are restricted to be nonnegative. They are often encountered in applications, for example, networks of reservoirs, industrial processes involving chemical reactors, heat exchangers, distillation columns, storage systems, hierarchical systems, compartmental systems used for modeling transport and accumulation phenomena of substances, see, e.g., [6, 10, 13].

In particular, the problem of robust stability of the positive linear discrete-time system (21) under the time-invariant structured perturbations

Aā†ŖA+DPE,

has been studied in [12, 19]. More precisely, it has been shown in [12, 19] that if (21) is exponentially stable and positive and D, E are given nonnegative matrices then a perturbed system of the form

x(k+1)=(A+DPE)x(k),kāˆˆ Z + ,

remains exponentially stable whenever

āˆ„Pāˆ„< 1 āˆ„ E ( I n āˆ’ A ) āˆ’ 1 D āˆ„ .

Furthermore, the problem of robust stability of the positive system (21) under the time-invariant multi-perturbations

Aā†ŖA+ āˆ‘ i = 1 N D i P i E i ,

has been analyzed in [12] by techniques of Ī¼-analysis.

Although there are many works devoted to the study of robust stability of discrete-time systems, to the best of our knowledge, the problem of robust stability of the positive system (21) under the time-varying multi-perturbations

Aā†ŖA+ āˆ‘ i = 1 N D i (k) P i (k) E i (k),

has not been studied yet, and a result like Corollary 2.7 cannot be found in the literature.

We illustrate the obtained results by a couple of examples.

Example 2.9 Consider the nonlinear time-varying equation

x(k+1)= 1 4 e āˆ’ k x(k)+sin ( k k 2 + 1 x ( k ) ) ,kāˆˆ Z + .
(22)

Clearly, (22) is of the form (2) with f(k,x):= 1 4 e āˆ’ k x+sin( k k 2 + 1 x). Since

| f ( k , x ) | = | 1 4 e āˆ’ k x + sin ( k k 2 + 1 x ) | ā‰¤ 1 4 |x|+ | k k 2 + 1 x | ā‰¤ 3 4 |x|,āˆ€kāˆˆ Z + ,āˆ€xāˆˆR,

the zero solution of (22) is exponentially stable, by Proposition 2.2.

Consider a perturbed equation given by

x(k+1)= ( 1 4 e āˆ’ k + a e āˆ’ k 2 āˆ’ 1 ) x(k)+sin ( k k 2 + 1 x ( k ) ) +arctan ( b x ( k ) ) ,kāˆˆ Z + ,
(23)

where a,bāˆˆR are parameters.

Note that |a e āˆ’ k 2 āˆ’ 1 x|ā‰¤ e āˆ’ 1 |a||x| and |arctan(bx)|ā‰¤|b||x|, for all kāˆˆ Z + , xāˆˆR. By Theorem 2.5, the zero solution of (23) is exponentially stable if e āˆ’ 1 |a|+|b|< 1 4 .

Example 2.10 Consider a linear discrete-time equation in R 2 defined by

x(k+1)=Ax(k),kāˆˆ Z + ,
(24)

where

A:=( 1 2 1 2 1 4 1 2 ).

Clearly, (24) is positive and exponentially stable. Consider a perturbed system given by

x(k+1)= ( A + D 1 ( k ) P 1 ( k ) E 1 ( k ) + D 2 ( k ) P 2 ( k ) E 2 ( k ) ) x(k),kāˆˆ Z + ,
(25)

where

and P 1 (k):=(a(k),b(k))āˆˆ R 1 Ɨ 2 ; P 2 (k):=(c(k),d(k))āˆˆ R 1 Ɨ 2 , kāˆˆ Z + are unknown perturbations.

Note that for any kāˆˆ Z + , we have

and

Let R 2 be endowed with 2-norm. By Corollary 2.7, (25) is exponentially stable provided

( sup k āˆˆ Z + | a ( k ) | ) 2 + ( sup k āˆˆ Z + | b ( k ) | ) 2 + ( sup k āˆˆ Z + | c ( k ) | ) 2 + ( sup k āˆˆ Z + | d ( k ) | ) 2 < 1 2 65 .

3 Aizerman-type problem

As an application, we now deal with an Aizerman-type problem for discrete-time systems.

Aizerman-type conjecture for discrete-time systems (ATC-DTS) LetAāˆˆ R n Ɨ n , Dāˆˆ R n Ɨ l , Eāˆˆ R q Ɨ n be given. For anyĪ³>0the linear systems

x(k+1)=(A+DPE)x(k),Pāˆˆ R l Ɨ q ,āˆ„Pāˆ„<Ī³,
(26)

are asymptotically stable if and only if the origin is globally asymptotically stable for all nonlinear systems

x(k+1)=Ax(k)+DN ( k , E x ( k ) ) ,
(27)

whereN: Z + Ɨ R q ā†’ R l , N(k,0)=0, āˆ€kāˆˆ Z + , satisfies

| N ( k , y ) | ā‰¤P|y|,āˆ€kāˆˆ Z + ,āˆ€yāˆˆ R q and Pāˆˆ R l Ɨ q ,āˆ„Pāˆ„<Ī³.
(28)

In particular, when N:Rā†’R, yā†¦N(y), is a scalar function and D, E T āˆˆ R n , the above conjecture is exactly a discrete-time version of the original Aizerman conjecture which was formulated first for ordinary differential systems, see [1]. It is well known that in general, the Aizerman classical conjecture does not hold, see, e.g., [7]. So a natural question arising here is that under what conditions of A, D, E and N does the ATC-DTS hold?

Theorem 3.1 IfAāˆˆ R + n Ɨ n andDāˆˆ R + n Ɨ l , Eāˆˆ R + q Ɨ n then the ATC-DTS holds.

In other words, the ATC-DTS holds for positive systems.

Proof Suppose (26) is asymptotically stable for any Pāˆˆ R l Ɨ q , āˆ„Pāˆ„<Ī³, for some Ī³>0. In particular, the unperturbed system (21) is asymptotically stable. It follows from Corollary 2.7 that (26) is asymptotically stable for any Pāˆˆ R l Ɨ q , āˆ„Pāˆ„< 1 āˆ„ E ( I n āˆ’ A ) āˆ’ 1 D āˆ„ (see also Remark 2.8). Furthermore, there exists P 0 āˆˆ R + l Ɨ q , āˆ„ P 0 āˆ„= 1 āˆ„ E ( I n āˆ’ A ) āˆ’ 1 D āˆ„ such that (26) is not asymptotically stable for P:= P 0 , see, e.g., [12, 19]. It remains to show that the zero solution of (27) is globally asymptotically stable for any nonlinearity N satisfying (28) with Ī³:= 1 āˆ„ E ( I n āˆ’ A ) āˆ’ 1 D āˆ„ . Let N satisfy (28) with Ī³:= 1 āˆ„ E ( I n āˆ’ A ) āˆ’ 1 D āˆ„ . Since Pāˆˆ R l Ɨ q , āˆ„Pāˆ„< 1 āˆ„ E ( I n āˆ’ A ) āˆ’ 1 D āˆ„ , the zero solution of (27) is globally asymptotically stable, by Corollary 2.6.

Conversely, assume that the zero solution of (27) is globally asymptotically stable for any nonlinearity N satisfying (28) for some Ī³>0. Then the unperturbed system (21) is asymptotically stable. As mentioned above, (26) is asymptotically stable for any Pāˆˆ R l Ɨ q , āˆ„Pāˆ„< 1 āˆ„ E ( I n āˆ’ A ) āˆ’ 1 D āˆ„ . So we assume that Ī³ā‰„ 1 āˆ„ E ( I n āˆ’ A ) āˆ’ 1 D āˆ„ . Note that (26) is not asymptotically stable for some P 0 āˆˆ R + l Ɨ q , āˆ„ P 0 āˆ„= 1 āˆ„ E ( I n āˆ’ A ) āˆ’ 1 D āˆ„ . This means that the zero solution of (27) is not globally asymptotically stable for N defined by N(k,y):= P 0 y, kāˆˆ Z + , yāˆˆ R q . This completes the proof.ā€ƒā–”

Remark 3.2 In general, the question ā€˜Under what conditions of A, D, E and N does the ATC-DTS hold?ā€™ is still open.

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Ngoc, P.H.A., Hieu, L.T. On stability of discrete-time systems under nonlinear time-varying perturbations. Adv Differ Equ 2012, 120 (2012). https://doi.org/10.1186/1687-1847-2012-120

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