On improved delay-dependent stability criterion of certain neutral differential equations

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Applied Mathematics and Computation 199 (2008) 385–391 www.elsevier.com/locate/amc

On improved delay-dependent stability criterion of certain neutral differential equations O.M. Kwon

a,*

, Ju H. Park

b

a

b

School of Electrical and Computer Engineering, Chungbuk National University, 410 SungBong–Ro Heungduk-gu, Cheongju 361-763, Republic of Korea Department of Electrical Engineering, Yeungnam University, 214-1 Dae-Dong, Kyongsan 712-749, Republic of Korea

Abstract In this letter, the problem for an asymptotic stability of the following neutral differential equation: d ½xðtÞ þ pxðt  sÞ ¼ axðtÞ þ b tanh xðt  rÞ dt is considered. Based on the Lyapunov method, a stability criterion, which is delay-dependent on both s and r, is derived in terms of linear matrix inequality (LMI). Two numerical examples are included to show the effectiveness of the proposed method. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Neutral equation; Asymptotic stability; Lyapunov method; LMI

1. Introduction Consider the neural differential equations of the form d ½xðtÞ þ pxðt  sÞ ¼ axðtÞ þ b tanh xðt  rÞ; dt

t P 0;

ð1Þ

where a, s and r are positive real scalars, b and p are real numbers, and jpj < 1. It is assumed that the delays s . With each solution of Eq. (1), the initial condition is defined and r are bounded as 0 6 s 6 s and 0 6 r 6 r as: xðsÞ ¼ /ðsÞ;

*

s 2 ½d; 0;

g; where d ¼ maxfs; r

/ðsÞ 2 Cð½d; 0; RÞ:

Corresponding author. E-mail addresses: [email protected] (O.M. Kwon), [email protected] (J.H. Park).

0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.09.031

386

O.M. Kwon, J.H. Park / Applied Mathematics and Computation 199 (2008) 385–391

Recently, for the study of the dynamic characteristics of neutral networks of Hopfield type, various types including the delay differential equations Eq. (1) have attracted the attention of many researchers (for example, see [1–3] and references cited therein). More recently, delay-dependent asymptotic stability for Eq. (1), which gives a less conservative result than delay-independent one when the size of delay is small, have been studied in [4,5]. In this letter, we propose a new stability criterion for asymptotic stability of Eq. (1). The proposed criterion is delay-dependent on both s and r. In order to derive a less conservative results, a new integral inequality is introduced and two zero equations are utilized in deriving the stability criterion for Eq. (1). Unlike the methods in [4,5], we do not use the model transformation technique, which leads an additional dynamics. The proposed stability criterion is represented in terms of an LMI which can be solved efficiently by various convex optimization algorithms. Two numerical examples are given to show the effectiveness of the proposed method. Throughout the letter, w represents the elements below the main diagonal of a symmetric matrix. The notation X > Y, where X and Y are matrices of same dimensions, means that the matrix X  Y is positive definite. Cð½0 1Þ; Rn Þ denotes the Banach space of continuous vector functions from [0 1) to Rn . 2. Main results Let us consider the following zero equation with a scalar b: Z t bxðtÞ  bxðt  sÞ  b x_ ðsÞds ¼ 0:

ð2Þ

ts

With the above equation, we can represent the system (1) as, Z t x_ ðsÞds þ b tanh xðt  rÞ  px_ ðt  sÞ: x_ ðtÞ ¼ ða þ bÞxðtÞ  bxðt  sÞ  b

ð3Þ

ts

Note that b will be chosen to guarantee the asymptotic stability of the system (3). Before deriving the main result, we need the following facts and lemma. Fact 1 (Schur Complement). Given constant symmetric matrices R1, R2, R3 where R1 ¼ RT1 and 0 < R2 ¼ RT2 , then R1 þ RT3 R1 2 R3 < 0 if and only if "

#

R1

RT3

R3

R2

 < 0;

or

R2

R3

RT3

R1

 < 0:

Fact 2. For any real vectors a, b, and any matrix Q > 0 with appropriate dimension, the following inequality 2aT b 6 aT Qa þ bT Q1 b is always satisfied. To derive a less conservative stability criterion, let us introduce a new integral inequality which will be used to derive the upper bound of Lyapunov function. Lemma 1. For any scalars l > 0, and fi (i = 1, . . . ,8), the following inequality holds:

l

Z

t

x_ 2 ðsÞds 6 fT ðtÞ Fe fðtÞ þ l1sfT ðtÞF T F fðtÞ;

ð4Þ

ts

where

" T

T

T

f ðtÞ ¼ x ðtÞx ðt  sÞ

Z

t

ts

T

T

T

T

x_ ðsÞds x_ ðtÞ x_ ðt  sÞðtanh xðtÞÞ ðtanh xðt  rÞÞ

T

Z

t

tr

T # tanh xðsÞds ;

ð5Þ

O.M. Kwon, J.H. Park / Applied Mathematics and Computation 199 (2008) 385–391

F ¼ ½ f1 f2 2 0 0 6H 0 6 6 6H H 6 6H H 6 Fe ¼ 6 6H H 6 6H H 6 6 4H H H

f3

f4

f5

f6

ð6Þ

f8 ;

f7

3

f1 f2

0 0

0 0

0 0

0 0

2f 3

f4

f5

f6

f7

H H

0 0 H 0

0 0

0 0

H H

H H H H

0 H

0 0

0 0 7 7 7 f8 7 7 0 7 7 7: 0 7 7 0 7 7 7 0 5

H H

H H

H

H

0

Proof. By utilizing Fact 2, we have Z t Z t Z t 2 1 l fT ðtÞF T F fðtÞds x_ ðsÞds 6 2 x_ ðsÞF fðtÞds þ l ts ts ts 2 3 0 607 6 7 6 7 617 6 7 607 6 7 T 6 2f ðtÞ6 7F fðtÞ þ l1sfT ðtÞF T F fðtÞ ¼ fT ðtÞ Fe fðtÞ þ l1sfT ðtÞF T F fðtÞ: 607 6 7 607 6 7 6 7 405 0 This completes the proof.

h

Here, we define the following matrices for simplicity in matrix expression: R ¼ ðRij Þ; R11 ¼ 2ac þ q1 þ 2a  2am1 þ 2n1 þ h; R12 ¼ a  am2  n1 þ n2 ; R13 ¼ a þ f1  am3  n1 þ n3 ; R14 ¼ m1  am4 þ n4 ; R15 ¼ pc  pm1  am5 þ n5 ; R16 ¼ am6 þ n6 ; R17 ¼ bc þ bm1  am7 þ n7 ; R18 ¼ h1  am8 þ n8 ; R22 ¼ q1  2n2 ; R23 ¼ f2  n2  n3 ; R24 ¼ m2  n4 ; R25 ¼ pm2  n5 ; R26 ¼ n6 ; R27 ¼ bm2  n7 ; R28 ¼ h2  n8 ; R33 ¼ 2f 3  2n3 ; R34 ¼ f4  m3  n4 ; R35 ¼ f5  pm3  n5 ; R36 ¼ f6  n6 ; R37 ¼ f7 þ bm3  n7 ; R38 ¼ f8 þ h3  n8 ; R44 ¼ q2 þ ls  2m4 ; R45 ¼ pm4  m5 ; R46 ¼ m6 ; R47 ¼ bm5  pm7 ; R48 ¼ h4  pm8 ; R55 ¼ q2  2pm5 ; R56 ¼ pm6 ; R57 ¼ bm5  pm7 ; R58 ¼ h5  pm8 ; R66 ¼ g þ rm  h; R67 ¼ bm6 ; R68 ¼ h6 ; R77 ¼ g þ 2bm7 ; R78 ¼ h7 þ bm8 ; R88 ¼ 2h8 ; 3 2 0 0 0 0 0 0 0 h1 6 H 0 0 0 0 0 0 h2 7 7 6 6H H 0 0 0 0 0 h 7 6 3 7 7 6 7 6 e ¼ 6 H H H 0 0 0 0 h4 7; H 6H H H H 0 0 0 h 7 5 7 6 7 6 6 H H H H H 0 0 h6 7 7 6 4 H H H H H H 0 h7 5 H

H

H H

H

H

H

387

2h8

ð7Þ

ð8Þ

388

O.M. Kwon, J.H. Park / Applied Mathematics and Computation 199 (2008) 385–391

H ¼ ½ h1 h2 h3 h4 h5 h6 h7 h8 ; M ¼ ½ m1 m2 m3 m4 m5 m6 m7 m8 ; N ¼ ½ n1 n2 n3 n4 n5 n6 2 2am1 am2 am3 6H 0 0 6 6 6H H 0 6 6H H H 6 N1 ¼ 6 6H H H 6 6H H H 6 6 4H H H H 2 2n1 6H 6 6 6H 6 6H 6 N2 ¼ 6 6H 6 6H 6 6 4H H

n7 n8 ; m1  am4

pm1  am5

am6

bm1  am7

am8

m2

pm2

0

bm2

0

m3 2m4

pm3 pm4  m5

0 m6

bm3 bm4  m7

H H

2pm5 H

pm6 0

bm5  pm7 bm6

H

2bm7

H

H

H H H n1 þ n2 n1 þ n3 2n2 n2  n3

H

H H

2n3 H

n4 0

n5 0

n6 0

H

H

H

0

0

H H

H H

H H

H H

0 H

H H 3 n7 n8 n7 n8 7 7 7 n7 n8 7 7 0 0 7 7 7: 0 0 7 7 0 0 7 7 7 0 0 5

H

H

H

H

H

H

n4 n4

n5 n5

n6 n6

3

7 7 7 7 0 7 m8 7 7 7; pm8 7 7 7 0 7 7 bm8 5 0

ð9Þ

0

Then, we have the following theorem for asymptotic stability of Eq. (1).  > 0, every solution x(t) of Eq. (1) satisfies x(t) ! 0 as t ! 1, if there exist Theorem 1. For given s > 0, and r positive scalars c, qi(i = 1, 2), g, l, m, h and constant scalars a, fi, hi, mi, ni(i = 1, . . . ,8) such that the following LMI holds 2

R 6 4H H

sF T sl H

3 H T r 7 0 5 < 0;

ð10Þ

 rm

Proof. For positive scalars c, q1, q2, g, l, and m, let us consider the following Lyapunov function defined by V ¼V1þV2þV3

ð11Þ

where V 1 ¼ cx2 ðtÞ; Z t Z t Z t V 2 ¼ q1 x2 ðsÞds þ q2 tanh2 xðsÞds; x_ 2 ðsÞds þ g ts ts tr Z t Z t Z t Z t 2 V3 ¼l tanh2 xðuÞduds: x_ ðuÞduds þ m ts

s

tr

s

First, the time derivative of V1 along the solution of Eq. (3) is give by   Z t V_ 1 ¼ 2cxðtÞ ða þ bÞxðtÞ  bxðt  sÞ  b x_ ðsÞds þ b tanh xðt  rÞ  px_ ðt  sÞ :

ð12Þ

ts

Second, differentiating V2 leads to V_ 2 ¼ q1 x2 ðtÞ  q1 x2 ðt  sÞ þ q2 x_ 2 ðtÞ  q2 x_ 2 ðt  sÞ þ gtanh2 xðtÞ  gtanh2 xðt  rÞ:

ð13Þ

O.M. Kwon, J.H. Park / Applied Mathematics and Computation 199 (2008) 385–391

Last, V_ 3 can be obtained as Z t Z 2 2 _V 3 ¼ ls_x2 ðtÞ  l x_ ðsÞds þ mr tanh xðtÞ  m 6 lsx_ 2 ðtÞ  l

ts Z t

r tanh2 xðtÞ  m x_ 2 ðsÞds þ m

ts

389

t

tanh2 xðsÞds

tr Z t

tanh2 xðsÞds:

ð14Þ

tr

By utilizing Lemma 1, we have a new upper bound of V_ 3 as e þ m1 r H T H ÞfðtÞ; V_ 3 6 lsx_ 2 ðtÞ þ m r tanh2 xðtÞ þ fT ðtÞð Fe þ l1sF T F þ H

ð15Þ

e , and H are in (9). As a mathematical tool in reducing a where f(t), Fe , and F are defined in Lemma 1 and H conservatism of stability criterion, we consider the following two zero equations, 2fT ðtÞM ½_xðtÞ  axðtÞ þ b tanh xðt  rÞ  px_ ðt  sÞ ¼ 0;

ð16Þ

and 

T

2f ðtÞN xðtÞ  xðt  sÞ 

Z



t

x_ ðsÞds ¼ 0;

ð17Þ

ts

 and N will be chosen later. Eqs. (16) and (17) can be represented as in obtaining an upper bound of V_ where M fT ðtÞðN1 þ N2 ÞfðtÞ ¼ 0;

ð18Þ

where N1 are N2 are in (9). By utilizing the relation tanh2x (t) 6 x2(t), we have 2

hx2 ðtÞ 6 hðtanh xðtÞÞ ;

ð19Þ

where h is a positive constant to be chosen later. From Eqs. (12)–(19), we have the following upper bound of V_ as   Z t _V 6 2cxðtÞ ða þ bÞxðtÞ  bxðt  sÞ  b x_ ðsÞds þ b tanh xðt  rÞ  px_ ðt  sÞ þ q1 x2 ðtÞ ts

 q1 x ðt  sÞ þ q2 x_ ðtÞ  q2 x_ ðt  sÞ þ gtanh2 xðtÞ  gtanh2 xðt  rÞ þ lsx_ 2 ðtÞ þ m rtanh2 xðtÞ 2

2

2

e þ m1 r H T H ÞfðtÞ þ fT ðtÞðN1 þ N2 ÞfðtÞ þ hx2 ðtÞ  hðtanh xðtÞÞ þ fT ðtÞð Fe þ l1sF T F þ H

2

H T H ÞfðtÞ; ¼ fT ðtÞðX þ l1sF T F þ m1 r

ð20Þ

where 2

ð1; 1Þ ð1; 2Þ

6H 6 6 6H 6 6H 6 X¼6 6H 6 6H 6 6 4H H

3

ð1; 3Þ R14

R15

R16

R17

R18

R22 H

R23 R33

R24 R34

R25 R35

R26 R36

R27 R37

H H

H H

R44 H

R45 R55

R46 R56

R47 R57

H

H

H

H

R66

R67

H H

H H

H H

H H

H H

R77 H

R28 7 7 7 R38 7 7 R48 7 7 7; R58 7 7 R68 7 7 7 R78 5

ð1; 1Þ ¼ 2ac þ q1 þ 2cb  2am1 þ 2n1 þ h; ð1; 2Þ ¼ cb  am2  n1 þ n2 ; ð1; 3Þ ¼ cb þ f1  am3  n1 þ n3 ; and Rij is defined in (9).

R88

390

O.M. Kwon, J.H. Park / Applied Mathematics and Computation 199 (2008) 385–391

H T H < 0 is equivalent to the LMI (10). Therefore, if LMI (10) Let a = cb. By Fact 1, X þ l1sF T F þ m1 r _ _ holds, V is negative. This means that V 6 jkxðtÞk2 for sufficiently small j > 0. According to Theorem 9.8.1 in [6], we conclude that if jpj < 1 and the inequality (10) holds, then, system (1) is asymptotically stable. This completes the proof. h Remark 1. The problem for solving LMI (10) in Theorem 1 is to determine whether the problem is feasible or not. It is called the feasibility problem. The solutions of the LMI (10) can be found by solving eigenvalue problem with respect to c, qi, g, l, m, h, a, fi, hi, mi, and ni which is a convex optimization problem [7]. Various efficient convex optimization algorithms can be used to check the feasibility of the LMI in Theorem 1. In this letter, in order to solve the LMI of Theorem 1, we utilize Matlab’s LMI Control Toolbox [8], which implements state-of-the-art interior-point algorithms, which is significantly faster than classical convex optimization algorithm [7]. Therefore, all solutions c, qi, g, l, m, h, a, fi, hi, mi, and ni can be obtained simultaneously. Remark 2. The criterion of Theorem 1 is delay-dependent on both s and r. To demonstrate the effectiveness of the proposed criterion, we give the following two examples. Example 1. Consider the neutral differential equation in [5]: d ½xðtÞ þ 0:2xðt  sÞ ¼ 0:6xðtÞ þ 0:3 tanh xðt  rÞ: dt

ð21Þ

By applying Theorem to the system (21), we can see that the system (21) is asymptotic stable for any s P 0 and r P 0. However, the maximum allowable delay bound for stability with s = 0.1 in Park and Kwon [5] was  ¼ s ¼ 107 can be obtained as 1.902. For reference, the LMI solution of Theorem 1 with r c ¼ 5:6469  105 ;

q1 ¼ 6:3818  104 ; 5

4

a ¼ 1:5191  10 ;

m ¼ 0:0080; h ¼ 2:5171  10 ; f1 ¼ 1:8091  10

10

f5 ¼ 1:0918  1010 ; h1 ¼ 0;

h2 ¼ 0;

;

f 2 ¼ 2:3519  10

m5 ¼ 4:1288  104 ; 4

10

;

h4 ¼ 0;

h5 ¼ 0;

m2 ¼ 5:6137  104 ; m6 ¼ 0;

n1 ¼ 1:5191  10 ;

n2 ¼ 6:4240  10 ;

n5 ¼ 2:4321  104 ;

n6 ¼ 0;

f 4 ¼ 3:6358  1011 ;

f 8 ¼ 0;

h6 ¼ 0;

h7 ¼ 0;

m3 ¼ 4:8812  104 ;

h8 ¼ 8:0196  1010 ; m4 ¼ 3:9770  105 ;

m8 ¼ 0;

n3 ¼ 2:9162  104 ;

n7 ¼ 2:6114  104 ;

l ¼ 0:0509;

l ¼ 0:9950;

f 3 ¼ 4:1609  1010 ;

m7 ¼ 1:1377  105 ; 3

g ¼ 1:3809  105 ;

g ¼ 1:7010;

f 7 ¼ 7:3520  1011 ;

f 6 ¼ 0;

h3 ¼ 0;

m1 ¼ 1:8096  105 ;

q2 ¼ 1:6174  105 ;

n4 ¼ 5:4201  104 ;

n8 ¼ 0:

Thus, Theorem 1 in this letter provides a much less conservative result than the one in Park and Kwon [5]. Example 2. Consider the equation in [5]: d ½xðtÞ þ 0:35xðt  0:5Þ ¼ 1:5xðtÞ þ b tanh xðt  0:5Þ: dt

ð22Þ

Table 1 Upper bounds of b for guaranteeing stability in Example 2 Upper bounds of b Agrawal and Grace [3] El-Morshedy and Gopalsamy [1] Park [4] Park and Kwon [5] Theorem 1

0.318 0.423 0.423 0.669 1.49

O.M. Kwon, J.H. Park / Applied Mathematics and Computation 199 (2008) 385–391

391

For this system, the maximum allowable bound of b is given in Table 1. From Table 1, one can see our result gives a larger upper bound of b for guaranteeing stability than those in other literature. References [1] K. Gopalsamy, I. Leung, P. Liu, Global Hopf-bifuration in a neural netlet, Applied Mathematics and Computation 94 (1998) 171–192. [2] H.A. El-Morshedy, K. Gopalsamy, Nonoscillation, oscillation and convergence of a class of neutral equations, Nonlinear Analysis 40 (2000) 173–183. [3] R.P. Agarwal, S.R. Grace, Asymptotic stability of certain neutral differential equations, Mathematical and Computer Modelling 31 (2000) 9–15. [4] J.H. Park, Delay-dependent criterion for asymptotic stability of a class of neutral equations, Applied Mathematics Letters 17 (2004) 1203–1206. [5] J.H. Park, O.M. Kwon, Stability analysis of certain nonlinear differential equation, Chaos Solitons & Fractals, doi:10.1016/ j.chaos.2006.09.015. [6] J. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. [7] S. Boyd, L.E. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, Studies in Applied Mathematics, SIAM, Philadelphia, 1994. [8] P. Gahinet, A. Nemirovski, A. Laub, M. Chilali, LMI Control Toolbox User’s Guide, The Mathworks, Natick, Massachusetts, 1995.