Novel stability criteria for neutral systems with multiple time delays

Chaos, Solitons and Fractals 32 (2007) 1735–1741 www.elsevier.com/locate/chaos

Novel stability criteria for neutral systems with multiple time delays Wenjun Xiong b

a,*

, Jinling Liang

b

a College of Science, Three Gorges University, Yichang 443002, China Department of Mathematics, Southeast University, Nanjing 210096, China

Accepted 1 December 2005

Abstract Based on the eigenvalues of characteristic equations, some new criteria are derived to ensure the asymptotic stability for a class of neutral differential equations with multiple time delays. Conditions obtained here are independent of the time delays and easy to be checked. When suitable fj(Æ) (j = 1, 2, . . . , m) are chosen, the model studied in this paper will reduce to a simple form. Moreover, our results can resolve some nonlinear neutral problems which are seldom discussed. Finally, an example with numerical simulation is given to show the effectiveness of our method.  2005 Elsevier Ltd. All rights reserved.

1. Introduction In recent years, dynamical behavior of delayed network systems has received increasing attention (see [4,16–21]). In particular, the interest in neutral differential equations has been growing rapidly due to their successful applications in practical fields such as circuit theory [23], bioengineering [26], population dynamics [8], automatic control [1,25] and so on. For instance, the delayed neutral differential system studied in [5] is as follows: x0 ðtÞ ¼ AxðtÞ þ Bxðt  sÞ þ Cx0 ðt  sÞ; where xðtÞ 2 Cn1 is the state vector; s > 0 denotes the delay; A, B and C are n · n matrices. And in [2,10,22], according to the eigenvalues of characteristic equations, the authors derived some sufficient conditions to ensure the asymptotical stability of the following neutral model with multiple time delays: x0 ðtÞ ¼ AxðtÞ þ

m X ½Bj xðt  sj Þ þ C j x0 ðt  sj Þ; j¼1

where sj P 0 (1 6 j 6 m) and s ¼ max fsj g. For the above two equations, similar results are also obtained with different 16j6m

methods. For example, based on the Lyapunov function and the matrix inequalities, some stability criteria have been derived in [6,9,12,13]; employing the linear matrix inequality (LMI) approach, sufficient conditions ensuring the neutral *

Corresponding author. E-mail addresses: [email protected], [email protected] (W. Xiong).

0960-0779/$ - see front matter  2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.12.020

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W. Xiong, J. Liang / Chaos, Solitons and Fractals 32 (2007) 1735–1741

system with multiple time delays to be stable have been obtained in [11,24]; in terms of Fourier series theory and techniques of real analysis inequality, stable periodic solutions for neutral models are studied in [14–17]. Obviously, the two equations mentioned above are linear delayed neutral systems. As we know, discussion on the stability of nonlinear neutral differential equations is of great more importance in practical applications such as pattern recognition, image processing and combinatorial optimization. Therefore, in this paper, our main purpose is to discuss the following nonlinear neutral system with multiple time delays: m X x0 ðtÞ ¼ AxðtÞ þ ½Bj fj ðxðtÞÞ þ C j fj ðxðt  sj ÞÞ þ Dj x0 ðt  sj Þ; ð1Þ j¼1 n1

where fj ðÞ 2 C is differentiable and fj(0) = 0. Based on the properties of eigenvalues of the characteristic equations, some delay independent conditions will be established to ensure the asymptotical stability of nonlinear model (1). Initial values associated with system (1) are assumed to be of the following form: xi ðtÞ ¼ /i ðtÞ;

t 2 ½s; 0; i ¼ 1; 2; . . . ; n;

where /i ðtÞ : ½s; 0 ! R is continuously differentiable. Throughout this paper, for matrices P = (pij)n·n and Q = (qij)n·n, P 6 Q means pij 6 qij for i, j = 1, 2, . . . , n. k(P) denotes the eigenvalue of matrix P, jPj = (jpijj)n·n. A1 represents the inverse of matrix A. The notation q(P) denotes the spectral norm of matrix P. And I denotes the unitary matrix with appropriate dimensions. The organization of this paper is as follows: In Section 2, the main results in this paper are given. In Section 3, an example with two numerical simulations will be provided to show the effectiveness of our results. Finally, conclusions are drawn in Section 4.

2. Main results Since fj( Æ ) is differentiable and fj(0) = 0, system (1) can be linearized as m X ½Bj fj0 ð0ÞxðtÞ þ C j fj0 ð0Þxðt  sj Þ þ Dj x0 ðt  sj Þ. x0 ðtÞ ¼ AxðtÞ þ

ð2Þ

j¼1

And the characteristic equations of neutral system (2) are as follows: " # m X 0 0 ðBj fj ð0Þ þ ðC j fj ð0Þ þ sDj Þ expðssj ÞÞ ¼ 0. Dðs; sj Þ, det sI  A 

ð3Þ

j¼1

To get the main results of this paper, we need the following lemmas. Lemma 1. Neutral differential equation (1) is asymptotically stable if all roots of Eq. (3) have rigorously negative real parts, i.e., b , sup{Re(s) : D(s, sj) = 0} < 0. Proof. From Ref. [3], we know that linear system (2) is asymptotically stable if all roots of Eq. (3) have rigorously negative real parts. Then, system (1) is also asymptotically stable. And the proof is completed. h Lemma 2 [7]. Suppose V 2 Cnn and q(V) < 1, then (I  V)1 exists and ðI  V Þ1 ¼ I þ V þ V 2 þ V 3 þ    . Lemma 3 [7]. For n · n complex matrices Q, P and V, if jQj 6 V, then (1) q(Q) 6 q(jQj) 6 q(V), (2) q(QP) 6 q(jQjjPj) 6 q(VjPj), (3) q(Q + P) 6 q(jQ + Pj) 6 q(jQj + jPj) 6 q(V + jPj).

Theorem 1. Neutral differential equation (1) is asymptotically stable if the following conditions hold: (1) There exists sj0 such that all eigenvalues of characteristic equation (3) have rigorously negative real parts when sj ¼ sj0 ; j ¼ 1; 2; . . . ; m.

W. Xiong, J. Liang / Chaos, Solitons and Fractals 32 (2007) 1735–1741

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(2) Dðin; s0j Þ 6¼ 0 for all n 2 R and s0j 2 ½sj0 ; sj  when sj0 < sj or s0j 2 ½sj ; sj0  when sj0 P sj ; j ¼ 1; 2; . . . ; m.

Proof. Obviously, x = 0 is an equilibrium point of neutral system (1) and D(s, sj) in Eq. (3) is continuous with respect to s and sj. Without loss of generality, suppose sj0 < sj . Condition (2) ensures that characteristic equation (3) has no root which is purely imaginary or zero for s0j 2 ½sj0 ; sj ; j ¼ 1; 2; . . . ; m. That is to say, neutral system (1) has no periodic solutions. From condition (1) and the continuity of D(s, sj), we conclude that Eq. (3) has no roots with positive real part. In fact, if this is not true, according to the mean value theorem, there must exist a real number n and sj 2 ðsj0 ; sj Þ such that Dðin; sj Þ ¼ 0, which contradicts to condition (2). From the analysis above, we conclude that all eigenvalues of characteristic equation (3) have rigorously negative real parts. And from Lemma 1, we know neutral differential equation (1) is asymptotically stable. The proof is completed. h As a special case, when sj0 ¼ 0, neutral differential equation (1) is degenerated as a neutral system without time delays. At this time, the term exp{ss} will not appear in the characteristic equation (3) so that the eigenvalues are easy to be obtained. Especially, the eigenvalues can be expressed in an analytic form so that the analysis for low-order systems is simplified. And it can be restated in the following form. Corollary 1. Neutral system (1) is asymptotically stable if the following conditions hold: (1) All eigenvalues of characteristic equation D(s, sj) = 0 have rigorously negative real parts when sj ¼ sj0 ¼ 0; j ¼ 1; 2; . . . ; m. (2) Dðin; s0j Þ 6¼ 0 for all n 2 R and s0j 2 ½0; sj . From Corollary 1 above, theoretically, it is convenient to obtain the asymptotic stability of neutral system (1) if one can prove conditions (1) and (2) hold with a specific sj0 ðj ¼ 1; 2; . . . ; mÞ. However, the conditions are so mathematically abstract that it is difficult to be checked in practice. So, here we present another result. Theorem 2. Neutral system (1) is asymptotically stable if A is Hurwitz and " # m X 1 0 0 sup q jðsI  AÞ jðjBj fj ð0Þj þ jC j fj ð0Þ þ sDj jÞ < 1. s2C;ReðsÞP0

ð4Þ

j¼1

Proof. We first show that all roots of characteristic equation (3) have negative real parts. Denote gj ðsÞ ¼ expðssj Þ;

j ¼ 1; 2; . . . ; m.

Since A is Hurwitz, all eigenvalues of A have negative real parts. And from Lemma 2, we know (sI  A)1 exists and jgj(s)j 6 1 for all Re(s) P 0. So " # m X Dðs; sj Þ ¼ det sI  A  ðBj fj0 ð0Þ þ ðC j fj0 ð0Þ þ sDj Þ expðssj ÞÞ j¼1

"

¼ detðsI  AÞ det I  ðsI  AÞ

1

# m X 0 0 ðBj fj ð0Þ þ ðC j fj ð0Þ þ sDj Þgj ðsÞÞ ¼ 6 0 j¼1

if Re(s) P 0 and sup s2C;ReðsÞP0

"

# m X 1 0 0 q ðsI  AÞ ðBj fj ð0Þ þ ðC j fj ð0Þ þ sDj Þgj ðsÞÞ < 1.

ð5Þ

j¼1

From Lemma 3, for all Re(s) P 0, we have "

# " # m m X X 1 1 0 0 0 0 ðsI  AÞ ðBj fj ð0Þ þ ðC j fj ð0Þ þ sDj Þgj ðsÞÞ 6 q jðsI  AÞ jðjBj fj ð0Þj þ jC j fj ð0Þ þ sDj jjgj ðsÞjÞ q j¼1

j¼1

6q

" m X j¼1

# 1

jðsI  AÞ

jðjBj fj0 ð0Þj

þ

jC j fj0 ð0Þ

þ sDj jÞ ;

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which implies that inequality (5) holds when condition (4) is true. As a result, the jth root sj of characteristic equation (3) has a negative real part, i.e., Re(sj) < 0, j = 1, 2, . . . , m. Next, we show that there is no root sequence of characteristic equation (3) whose real parts converge to zero. Assume this is not true, i.e., characteristic equation (3) has a sequence of roots {sn} such that Re(sn) ! 0 and Re(sn) < 0. Since matrix A is Hurwitz, we know that " # m X 1 0 0 ki ðsI  AÞ ðBj fj ð0Þ þ ðC j fj ð0Þ þ sDj Þgj ðsÞÞ ð6Þ j¼1

is an analytic function of s when Re(s) P 0. From (6) and condition (4), we have that there exist a constant e (0 < e < 1) such that " # m X sup q ðsI  AÞ1 ðBj fj0 ð0Þ þ ðC j fj0 ð0Þ þ sDj Þgj ðsÞÞ s2C;ReðsÞP0

¼

j¼1

sup s2C;ReðsÞP0

and

 " # m  X    max ki ðsI  AÞ1 ðBj fj0 ð0Þ þ ðC j fj0 ð0Þ þ sDj Þgj ðsÞÞ  ¼ 1  e; i  j¼1 

"

sup f2C;ReðfÞ¼0

# m X 1 0 0 q ðfI  AÞ ðBj fj ð0Þ þ ðC j fj ð0Þ þ fDj Þgj ðfÞÞ 6 1  e.

ð7Þ

j¼1

For a given constant e1 (0 < e1 < e), since Re(sn) ! 0, there must exist an integer n* such that jRe(sn)j is sufficiently small, Re(sn) < 0 and  "  # m  X      1 0 0 ðsn I  AÞ ðBj fj ð0Þ þ ðC j fj ð0Þ þ sn Dj Þgj ðsn ÞÞ  max ki   i  j¼1 " # m  X  ðfI  AÞ1 ðBj fj0 ð0Þ þ ðC j fj0 ð0Þ þ fDj Þgj ðfÞÞ  < e1 ; ð8Þ  sup q  f2C;ReðfÞ¼0 j¼1 for n > n*. Therefore, for n > n*  " # m  X    1 0 0 ðsn I  AÞ ðBj fj ð0Þ þ ðC j fj ð0Þ þ sn Dj Þgj ðsn ÞÞ  ki  j¼1  " # m X 1 0 0 ðfI  AÞ ðBj fj ð0Þ þ ðC j fj ð0Þ þ fDj Þgj ðfÞÞ þ e1 6 1  e þ e1 < 1. 6 sup q f2C;ReðfÞ¼0

ð9Þ

j¼1

Hence, for Re(sn) ! 0 and Re(sn) < 0, " # m X 0 0 Dðsn ; sj Þ ¼ det Isn  A  ðBj fj ð0Þ þ ðC j fj ð0Þ þ sn Dj Þ expðsn sj ÞÞ j¼1

"

¼ detðIsn  AÞ det I  ðIsn  AÞ

1

# m X 0 0 ðBj fj ð0Þ þ ðC j fj ð0Þ þ sn Dj Þgj ðsn ÞÞ ¼ 6 0; j¼1

which contradicts the assumption that {sn} are the roots of characteristic equation (3). From the discussions above and Lemma 1, we know neutral system (1) is asymptotically stable. And the proof is completed. h Especially, when matrix B = 0, neutral system (1) is transformed into the following delay neutral model: x0 ðtÞ ¼ AxðtÞ þ

m X 

 C j fj ðxðt  sj ÞÞ þ Dj x0 ðt  sj Þ .

j¼1

Using the similar method in Theorem 2, we can easily obtain the following corollary.

ð10Þ

W. Xiong, J. Liang / Chaos, Solitons and Fractals 32 (2007) 1735–1741

Corollary 2. Neutral differential equation (10) is asymptotically stable if A is Hurwitz and " # m X 1 0 jðsI  AÞ ðC j fj ð0Þ þ sDj Þj < 1. sup q s2C;ReðsÞP0

1739

ð11Þ

j¼1

According to Corollary 1 and Theorem 2, the following conclusion will be obtained. Corollary 3. The neutral system (1) is asymptotically stable if A is Hurwitz, and " # m X 1 0 0 (1) jðsI  AÞ ðBj fj ð0Þ þ C j fj ð0Þ þ sDj Þj < 1. sup q s2C;ReðsÞP0

j¼1

(2) Dðin; s0j Þ 6¼ 0, for all n 2 R and s0j 2 ½0; sj ; j ¼ 1; 2; . . . ; m.

Remark 1. In neutral system (1), when different fj(Æ) are taken, some different nonlinear problems will be resolved. Although condition (4) in Theorem 2 seems to be complicated, if suitable fj(Æ) is chosen, we may even gain some conditions which are more simpler and easier than previous ones, which will be exhibited in the following example. Remark 2. In comparison with Refs. [2,5], the neutral system in our paper is nonlinear, it is more general and practical in applications. And Theorem 2 in our paper includes the results of [2,5]. Moreover, our criteria are easily checked in practice.

3. An illustrative example In this section, an example will be given to demonstrate the effectiveness our results. Example 1. Consider the following neutral system: x0 ðtÞ ¼ AxðtÞ þ

2 X   Bj fj ðxðtÞÞ þ C j fj ðxðt  sj ÞÞ þ Dj x0 ðt  sj Þ ;

ð12Þ

j¼1

where f1(x) = x2, f2(x) = x3 and

xðtÞ ¼

x1 ðtÞ

!

0

0

1

A¼

; 1

!

1

; 2

3

2 1 B1 ¼

;

x2 ðtÞ C1 ¼

!

1

2 C2 ¼

3

1

4

1 B2 ¼

; 3 1

! ;

!

D1 ¼

1 3

2

1 8

0

0

1 16

! ;

D2 ¼

1! 4

1 8

1 8

1

3 16

;

! .

Since sup

q

" m X

s2C;ReðsÞP0

¼

# 1

jðsI  AÞ

jðjBj fj0 ð0Þj

þ

jC j fj0 ð0Þ

þ sDj jÞ

j¼1

sup s2C;ReðsÞP0

"   s   q  s þ 1

1 4

1 8

1

1 4

!# 6

sup s2C;ReðsÞP0

( pffiffiffi pffiffiffi) pffiffiffi    s  1 þ 2 1  2 1þ 2   max ; < 1; 6 s þ 1 4 4 4

from Theorem 2, we acquire that system (12) is asymptotically stable. The numerical simulations of Example 1 are shown in Figs. 1 and 2. With the two figures, one can easily find that our results are independent of the multiple time delays sj. However, using the criteria in [2,5], one cannot obtain the stability of Example 1. So our results are new and general.

1740

W. Xiong, J. Liang / Chaos, Solitons and Fractals 32 (2007) 1735–1741 0.5

0

–0.5

–0.1 –10

0

10

20

30

40

50

Fig. 1. Numeric simulation for the asymptotic stability of Eq. (12) with delay s1 = s2 = 1.

0.4 0.3 0.2 0.1 0 –0.1 –0.2 –0.3 –0.4 –0.5 –0.6 0

5

10

15

20

25

Fig. 2. Numeric simulation for asymptotic stability of Eq. (12) with delay s1 = s2 = 0.75.

4. Conclusions In this paper, several new sufficient conditions have been obtained to ensure the asymptotic stability for a neutral system with multiple time delays. Delay independent criteria have been derived based on the properties of characteristic equations. The system discussed in our paper not only includes those models in [2,5], but also can resolve problems of nonlinear systems. Moreover, compared with the results in the previous literature, our criteria are easily to be checked in practice.

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