Physica D 212 (2005) 126–136www.elsevier.com/locate/physd

LMI-based stability criteria for neural networks with multipletime-varying delays

Yong Hea,b, Qing-Guo Wanga,∗, Min Wub

a Department of Electrical and Computer Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260,Singapore

b School of Information Science and Engineering, Central South University, Changsha 410083, China

Received 3 January 2005; received in revised form 20 May 2005; accepted 27 September 2005Available online 2 November 2005

Communicated by H. Levine

Abstract

In this paper, the stability of neural networks with multiple time-varying delays is studied. A new class ofLyapunov–Krasovskii functionals is constructed and the S-procedure and free-weighting matrix method are employed to derivea delay-dependent stability criterion, from which a delay-independent criterion is obtained as a special case. Moreover, theresult is also extended to delay-dependent and rate-independent stability criteria for multiple unknown time-varying delays.Finally, numerical examples are given to illustrate the effectiveness of our methods and improvement over the existingones.c© 2005 Elsevier B.V. All rights reserved.

Keywords: Neural networks; Stability; Delay-dependent; Time-varying delay; Linear matrix inequality (LMI)

1. Introduction

Hopfield neural networks [1] have been extensively studied recently and involved in many different applicationareas such as pattern recognition, associative memory, and combinatorial optimization. Such applications heavilydepend on the dynamical behaviors. Though considerable efforts have been devoted to the analysis of the stabilityof neural networks without time delay, in reality, dynamics in a neural network often have time delays due, forexample, to the finite switching speed of amplifiers in electronic neural networks, or to the finite signal propagationtime in biological networks. As a result, the stability of different classes of neural networks with time delay has

∗ Corresponding author. Tel.: +65 6874 2282; fax: +65 6779 1103.E-mail address: elewqg@nus.edu.sg (Q.-G. Wang).

0167-2789/$ - see front matter c© 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2005.09.013

Y. He et al. / Physica D 212 (2005) 126–136 127

become an important topic and various stability conditions have been reported for such neural networks for singledelay [2–14] and multiple delays [15–28].

Recently, several LMI-based conditions, which can be easily checked numerically using interior-pointalgorithms, have been presented for the stability problems of neural networks [8,9,28,14]. However, there isroom for improvement of their results. Firstly, they handled the nonlinear parts by using the inequalities whilethe S-procedure [29] is known to be the most effective method to deal with the nonlinearities. Secondly,the Lyapunov–Krasovskii functionals employed in [8,9,28] can also be improved because the information onnonlinearities was not included in the Lyapunov–Krasovskii functional.

Note that the stability criteria for systems with delays can be classified into two categories, namely, delay-independent and delay-dependent. Since delay-independent criteria tend to be conservative, especially when thedelay is small, considerable attention has been paid to the delay-dependent type. Delay-dependent exponentialstability problems for neural networks with delay have been discussed [9,14]. However, they did not make useof the free-weighting matrix method [30–33], in which the free-weighting matrices are employed to express therelationship between the terms in the Leibniz–Newton formula. The free-weighting method has two advantages.First, it deals with the system model directly and does not employ any system transformation, thus avoidingthe conservatism that results from such a transformation. Second, it does not use any inequalities or theimproved inequalities to estimate the cross terms. In fact, it is the most efficient for systems with time-varyingdelay.

In this paper, a class of Lyapunov–Krasovskii functionals which contains the information on nonlinearitiesis constructed for neural networks with multiple time-varying delays and the S-procedure and free-weightingmatrix method are employed to derive a delay-dependent stability criterion. It is shown that the delay-dependentstability criterion includes the delay-independent one. The result is also extended to get a delay-dependent and rate-independent stability criterion for multiple unknown time-varying delays. Finally, numerical examples are given toillustrate effectiveness and an improvement over the existing ones.

2. Preliminaries

Consider the following neural networks with multiple time-varying delays:

u(t) = −Au(t) + W (0)g(u(t)) +

r∑k=1

W (k)g(u(t − τk(t))) + I, (1)

where u(t) = [u1(t), u2(t), . . . , un(t)]T is the neuron state vector, A = diag(a1, a2, . . . , an) is apositive diagonal matrix, W (k)

= (w(k)i j )n×n, k = 0, 1, 2, . . . , r , are the interconnection matrices, g(u) =

[g1(u1), g2(u2), . . . , gn(un)]T denotes the neuron activation with g(0) = 0, and I = [I1, I2, . . . , In]T is a constant

input vector. The time delays, τk(t), k = 1, 2, . . . , r , are time-varying differentiable functions and satisfy

0 ≤ τk(t) ≤ τk, (2)

and

τk(t) ≤ µk, (3)

where τk, µk, k = 1, 2, . . . , r are constants. In addition, it is assumed that each neuron activation function insystem (1), g j (·), j = 1, 2, . . . , n, satisfies the following condition:

0 ≤g j (x) − g j (y)

x − y≤ σ j , ∀x, y ∈ R, x 6= y, j = 1, 2, . . . , n, (4)

128 Y. He et al. / Physica D 212 (2005) 126–136

where σ j , j = 1, 2, . . . , n are positive constants. The existence and uniqueness of system (1) can be seenin [34,35].

In the following, we will shift the equilibrium point u∗= [u∗

1, u∗

2, . . . , u∗n]

T of system (1) to the origin by thetransformation x(·) = u(·) − u∗. Since u∗ is the equilibrium point of system (1), the following holds:

0 = −Au∗(t) + W (0)g(u∗(t)) +

r∑k=1

W (k)g(u∗(t − τk(t))) + I. (5)

(1) minus (5) leaves

x(t) = −Ax(t) + W (0) f (x(t)) +

r∑k=1

W (k) f (x(t − τk(t))), (6)

where x = [x1, x2, . . . , xn]T is the state vector of the transformed system, with f (x) =

[ f1(x1), f2(x2), . . . , fn(xn)]T and f j (x j ) = g j (x j + u∗

j ) − g j (u∗

j ), j = 1, 2, . . . , n. Note that functions f j (·)

here satisfy the following condition:

0 ≤f j (x j )

x j≤ σ j , ∀x j 6= 0, j = 1, 2, . . . , n, (7)

which is equivalent to the following one:

f j (x j )[

f j (x j ) − σ j x j]

≤ 0, j = 1, 2, . . . , n. (8)

The S-procedure, which is stated as follows, is employed to investigate the asymptotic stability for system (1)and (6).

Lemma 1 ([29] (S-procedure)). Let Ti ∈ Rn×n(i = 0, 1, . . . , p) be symmetric matrices. The conditions onTi (i = 0, 1, . . . , p),

ζTT0ζ > 0 for all ζ 6= 0 such that ζTTiζ ≥ 0 (i = 1, 2, . . . , p), (9)

hold if there exist τi ≥ 0 (i = 1, 2, . . . , p) such that

T0 −

p∑i=1

τi Ti > 0. (10)

3. Delay-dependent stability criteria

In this section, a new class of Lyapunov–Krasovskii functionals is constructed which contains the informationon nonlinearities. In addition, the S-procedure will be employed to deal with the nonlinearities and free-weightingmethod will be used to establish the following delay-dependent stability criterion.

Theorem 2. The origin of system (6) subject to conditions (7), (2) and (3) is asymptotically stable if there existP = PT > 0, Qk = QT

k ≥ 0, k = 1, 2, . . . , r , Rk = RTk ≥ 0, k = 1, 2, . . . , r, Zk = ZT

k > 0, k = 1, 2, . . . , r,Λ = diag(λ1, λ2, . . . , λn) ≥ 0, T = diag(t1, t2, . . . , tn) ≥ 0, Sk = diag(sk1, sk2, . . . , skn) ≥ 0, k = 1, 2, . . . , r ,and any appropriate dimensional matrices, Nk j , Mk j , k = 0, 1, 2, . . . , r; j = 1, 2, . . . , r , such that a feasible

Y. He et al. / Physica D 212 (2005) 126–136 129

solution exists for

Ξ =

Ξ11 Ξ12 Ξ13 + ΣT Ξ14 τ1 M01 · · · τr M0r

Ξ T12 Ξ22 Ξ23 Ξ24 + Σs τ1 M1 · · · τr Mr

Ξ T13 + TΣ Ξ T

23 Ξ33 − 2T Ξ34 τ1 N01 · · · τr N0r

Ξ T14 Ξ T

24 + Σs Ξ T34 Ξ44 − 2S τ1 N1 · · · τr Nr

τ1 MT01 τ1 MT

1 τ1 N T01 τ1 N T

1 −τ1 Z1 · · · 0...

......

......

. . ....

τr MT0r τr MT

r τr N T0r τr N T

r 0 · · · −τr Zr

< 0, (11)

where

Ξ11 = Φ11 + AT H A +

r∑k=1

(M0k + MT

0k

),

Φ11 = −P A − AP +

r∑k=1

Qk,

Ξ12 =

[r∑

k=1

MT1k − M01,

r∑k=1

MT2k − M02, . . . ,

r∑k=1

MTrk − M0r

],

Ξ13 = Φ13 − AT H W (0)+

r∑k=1

N T0k,

Φ13 = PW (0)− ATΛ,

Ξ14 = Φ14 − AT H W1r +

[r∑

k=1

N T1k,

r∑k=1

N T2k, . . . ,

r∑k=1

N Trk

],

Φ14 = PW1r ,

Ξ22 = Φ22 +

−M11 − MT

11 −M12 − MT21 · · · −M1r − MT

r1−MT

12 − M21 −M22 − MT22 · · · −M2r − MT

r2...

.... . .

...

−MT1r − Mr1 −MT

2r − Mr2 · · · −Mrr − MTrr

,

Φ22 = diag [−(1 − µ1)Q1, −(1 − µ2)Q2, . . . ,−(1 − µr )Qr ] ,

Ξ23 = [−N01, −N02, . . . ,−N0r ]T ,

Ξ24 =

−N T

11 −N T21 · · · −N T

r1−N T

12 −N T22 · · · −N T

r2...

.... . .

...

−N T1r −N T

2r · · · −N Trr

,

Σs = diag [Σ S1,Σ S2, . . . ,Σ Sr ] ,

Ξ33 = Φ33 + [W (0)]T H W (0),

Φ33 =

r∑k=1

Rk + ΛW (0)+ [W (0)

]TΛ,

Ξ34 = Φ34 + [W (0)]T H W1r ,

Φ34 = ΛW1r ,

130 Y. He et al. / Physica D 212 (2005) 126–136

Ξ44 = Φ44 + W T1r H W1r ,

Φ44 = diag [−(1 − µ1)R1, −(1 − µ2)R2, . . . ,−(1 − µr )Rr ] ,

S = diag [S1, S2, . . . , Sr ] ,

Σ = diag [σ1, σ2, . . . , σr ] ,

W1r = [W (1), W (2), . . . , W (r)],

H =

r∑k=1

τk Zk,

Mk = [MT1k, MT

2k, . . . , MTrk]

T, k = 1, 2, . . . , r,

Nk = [N T1k, N T

2k, . . . , N Trk]

T, k = 1, 2, . . . , r.

Proof. Construct the following Lyapunov–Krasovskii functional:

V (xt ) = xT(t)Px(t) + 2n∑

j=1

λ j

∫ x j

0f j (s)ds

+

r∑k=1

∫ t

t−τk (t)

[xT(s)Qk x(s) + f T(x(s))Rk f (x(s))

]ds,

+

r∑k=1

∫ 0

−τk

∫ t

t+θ

xT(s)Zk x(s) ds dθ, (12)

where P = PT > 0, Qk = QTk ≥ 0, k = 1, 2, . . . , r , Rk = RT

k ≥ 0, k = 1, 2, . . . , r , Zk = ZTk > 0, k =

1, 2, . . . , r , Λ = diag(λ1, λ2, . . . , λn) ≥ 0 are to be determined. Calculating the derivative of V (xt ) along thesolution of system (6) yields:

V (xt ) = 2xT(t)Px(t) + 2n∑

j=1

λ j f j (x j (t))x j (t)

+

r∑k=1

[xT(t)Qk x(t) − (1 − τk(t))xT(t − τk(t))Qk x(t − τk(t))

]+

r∑k=1

[f T(x(t))Rk f (x(t)) − (1 − τk(t)) f T(x(t − τk(t)))Rk f (x(t − τk(t)))

]+

r∑k=1

[τk xT(t)Zk x(t) −

∫ t

t−τk (t)xT(s)Zk x(s) ds

]≤ 2xT(t)Px(t) + 2 f T(x(t))Λx(t)

+

r∑k=1

[xT(t)Qk x(t) − (1 − µk)xT(t − τk(t))Qk x(t − τk(t))

]+

r∑k=1

[f T(x(t))Rk f (x(t)) − (1 − µk) f T(x(t − τk(t)))Rk f (x(t − τk(t)))

]+

r∑k=1

[τk xT(t)Zk x(t) −

∫ t

t−τk (t)xT(s)Zk x(s) ds

]. (13)

Y. He et al. / Physica D 212 (2005) 126–136 131

Using the Leibniz–Newton formula, for any appropriate dimensional matrices N jk, M jk, k = 1, 2, . . . , r, j =

0, 1, 2, . . . , r , the following equations hold for k = 1, 2, . . . , r :[xT(t)M0k +

r∑j=1

xT(t − τ j (t))M jk + f T(x(t))N0k +

r∑j=1

f T(x(t − τ j (t)))N jk

]

·

[x(t) − x(t − τk(t)) −

∫ t

t−τk (t)x(s) ds

]= 0. (14)

On the other hand, for k = 1, 2, . . . , r , for any appropriately dimensioned matrices

Xk =

X (k)

11 X (k)12 X (k)

13 X (k)14[

X (k)12

]TX (k)

22 X (k)23 X (k)

24[X (k)

13

]T [X (k)

23

]TX (k)

33 X (k)34[

X (k)14

]T [X (k)

24

]T [X (k)

34

]TX (k)

44

≥ 0,

the following holds:

τkξT(t)Xkξ(t) −

∫ t

t−τk (t)ξT(t)Xkξ(t) ds ≥ 0, (15)

where

ξ(t) = [xT(t), xT(t − τ1(t)), . . . , xT(t − τr (t)), f T(x(t)), f T(x(t − τ1(t))), . . . , f T(x(t − τr (t)))]T.

Then, adding the terms on the left of Eqs. (14) and (15) to V (xt ) in (13) allows us to express V (xt ) as:

Vd(xt ) ≤ ξT(t)

[Ξ +

r∑k=1

τk Xk

]ξ(t) −

r∑k=1

∫ t

t−τk (t)ζT(t, s)Ψkζ(t, s) ds, (16)

where

ζ(t, s) =

[ξT(t) xT(s)

]T,

Ξ =

Ξ11 Ξ12 Ξ13 Ξ14

Ξ T12 Ξ22 Ξ23 Ξ24

Ξ T13 Ξ T

23 Ξ33 Ξ34

Ξ T14 Ξ T

24 Ξ T34 Ξ44

,

Ψk =

X (k)11 X (k)

12 X (k)13 X (k)

14 M0k[X (k)

12

]TX (k)

22 X (k)23 X (k)

24 Mk[X (k)

13

]T [X (k)

23

]TX (k)

33 X (k)34 N0k[

X (k)14

]T [X (k)

24

]T [X (k)

34

]TX (k)

44 Nk

MT0k MT

k N T0k N T

k Zk

, k = 1, 2, . . . , r.

It is clear from (8) that there hold:

f j (x j (t))[

f j (x j (t)) − σ j x j (t)]

≤ 0, j = 1, 2, . . . , n, (17)

and

f j (x j (t − τk(t)))[

f j (x j (t − τk(t))) − σ j x j (t − τk(t))]

≤ 0, j = 1, 2, . . . , n; k = 1, 2, . . . , r. (18)

132 Y. He et al. / Physica D 212 (2005) 126–136

Then, by applying the S-procedure, system (6) is asymptotically stable if there exist T = diag(t1, t2, . . . , tn) ≥ 0and Sk = diag(sk1, sk2, . . . , skn) ≥ 0, k = 1, 2, . . . , r , such that

V (xt ) − 2n∑

j=1

t j f j (x j (t))[

f j (x j (t)) − σ j x j (t)]

−2r∑

k=1

n∑j=1

{sk j f j (x j (t − τk(t)))

[f j (x j (t − τk(t))) − σ j x j (t − τk(t))

]}≤ ξT(t)

[Ξ +

r∑k=1

τk Xk

]ξ(t) −

r∑k=1

∫ t

t−τk (t)ζT(t, s)Ψkζ(t, s) ds,

< 0, (19)

for ξ(t) 6= 0, where

Ξ =

Ξ11 Ξ12 Ξ13 + ΣT Ξ14

Ξ T12 Ξ22 Ξ23 Ξ24 + Σs

Ξ T13 + TΣ Ξ T

23 Ξ33 − 2T Ξ34

Ξ T14 Ξ T

24 + Σs Ξ T34 Ξ44 − 2S

.

It is clear that condition (19) is satisfied if Ξ +∑r

k=1 τk Xk < 0 and Ψk ≥ 0, k = 1, 2, . . . , r . Specifically, Xk canbe chosen to be

Xk =

M0kMkN0kNk

Z−1k

M0kMkN0kNk

T

, (20)

which ensures that Xk ≥ 0, k = 1, 2, . . . , r and Ψk ≥ 0, k = 1, 2, . . . , r . In this case, according to the Schurcomplement [29], Ξ +

∑rk=1 τk Xk < 0 is equivalent to Ξ < 0. This completes the proof. �

Remark 3. It is clear that the term, 2∑n

j=1 λ j∫ x j

0 f j (s) ds, in the Lyapunov–Krasovskii functional (12) includesthe information on nonlinearities. Furthermore, since the S-procedure is employed in our proof, all the Lyapunovmatrices are of full block, while some of them are confined to be diagonal in Theorem 1 of [28].

Remark 4. If the matrices Nk j , Mk j , k = 0, 1, 2, . . . , r, j = 1, 2, . . . , r in Eq. (11) are set to be zero, andZk = εk I (k = 1, 2, . . . , r), where εk, k = 1, 2, . . . , r are sufficiently small positive scalar and I is the identitymatrix, then Theorem 2 can be extended to the following delay-independent stability criterion.

Corollary 5. The origin of system (6) subject to conditions (7) and (3) is asymptotically stable if there existP = PT > 0, Qk = QT

k ≥ 0, k = 1, 2, . . . , r , Rk = RTk ≥ 0, k = 1, 2, . . . , r,, Λ = diag(λ1, λ2, . . . , λn) ≥ 0,

T = diag(t1, t2, . . . , tn) ≥ 0, Sk = diag(sk1, sk2, . . . , skn) ≥ 0, k = 1, 2, . . . , r , such that a feasible solution existsfor

Φ =

Φ11 0 Φ13 + ΣT Φ14

0 Φ22 0 Σs

ΦT13 + TΣ 0 Φ33 − 2T Φ34

ΦT14 Σs ΦT

34 Φ44 − 2S

< 0, (21)

where Φ11,Φ13,Φ14,Φ22,Φ33,Φ34,Φ44, Σs,Σ , S are defined in Theorem 2.

If Qk, k = 1, 2, . . . , r , and Rk, k = 1, 2, . . . , r , are all set to be zero, a delay-dependent and rate-independentcriterion can be derived, in which the derivative of delay may be unknown.

Y. He et al. / Physica D 212 (2005) 126–136 133

Corollary 6. The origin of system (6) subject to conditions (7) and (2) is asymptotically stable if there existP = PT > 0, Zk = ZT

k > 0, k = 1, 2, . . . , r , Λ = diag(λ1, λ2, . . . , λn) ≥ 0, T = diag(t1, t2, . . . , tn) ≥ 0,Sk = diag(sk1, sk2, . . . , skn) ≥ 0, k = 1, 2, . . . , r , and any appropriate dimensional matrices Nk j , Mk j , k =

0, 1, 2, . . . , r, j = 1, 2, . . . , r , such that a feasible solution exists for

Ξ11 Ξ12 Ξ13 + ΣT Ξ14 τ1 M01 · · · τr M0r

Ξ T12 Ξ22 Ξ23 Ξ24 + Σs τ1 M1 · · · τr Mr

Ξ T13 + TΣ Ξ T

23 Ξ33 − 2T Ξ34 τ1 N01 · · · τr N0r

Ξ T14 Ξ T

24 + Σs Ξ T34 Ξ44 − 2S τ1 N1 · · · τr Nr

τ1 MT01 τ1 MT

1 τ1 N T01 τ1 N T

1 −τ1 Z1 · · · 0...

......

......

. . ....

τr MT0r τr MT

r τr N T0r τr N T

r 0 · · · −τr Zr

< 0, (22)

where

Ξ11 = −P A − AP + AT H A +

r∑k=1

(M0k + MT

0k

),

Ξ22 =

−M11 − MT

11 −M12 − MT21 · · · −M1r − MT

r1−MT

12 − M21 −M22 − MT22 · · · −M2r − MT

r2...

.... . .

...

−MT1r − Mr1 −MT

2r − Mr2 · · · −Mrr − MTrr

,

Ξ33 = ΛW (0)+ [W (0)

]TΛ + [W (0)

]T H W (0),

Ξ44 = W T1r H W1r ,

and the other terms are defined in Theorem 2.

Remark 7. For the stability conditions of neural networks with time-varying delays, the previous results neededthe restriction that the derivative of delays are less than 1. Corollary 6 does not have this limitation since thefree-weighting matrices approach is employed. Then, the stability criterion for neural networks with unknowntime-varying delay can be derived.

4. Numerical examples

Example 8. Let system (6) with r = 2 have the following parameters:

A =

[1 00 1

], W (0)

=

[0.01 0.100.10 0.03

],

W (1)=

[0.05 −0.01

−0.10 0.35

], W (2)

=

[0.10 −0.03

0 −0.35

].

Suppose time-varying delays with µ1 = 0.2, µ2 = 0.1. Consider first that σ1 = 5.5 and σ2 = 1.19. Then,the LMI (21) in Corollary 5 is feasible, which implies that the system is delay-independent asymptotically stable,while LMI (4) in [28] is infeasible. In fact, LMI (4) in [28] is also infeasible for σ1 = 4.8, σ2 = 1.0, indicatingconservativeness of the method in [28]. Consider next that τ1 = 2 and τ2 = 1. Then, LMI (21) in Corollary 5 isinfeasible when one of σ1 exceeds 5.5 and σ2 is fixed to be 1.19, or σ1 is fixed to be 5.5 and σ2 exceeds 1.19, or σ1and σ2 exceed 5.5 and 1.19, respectively. But the LMI (11) in Theorem 2 is feasible for σ1 = 5.6 and σ2 = 1.39,showing improvement over the delay-independent criterion in Corollary 5. In fact, let g1(x) = 5.6 tanh(x),

134 Y. He et al. / Physica D 212 (2005) 126–136

Fig. 1. The time response curve of u1(t) for Example 8 with time-varying delays.

Fig. 2. The time response curve of u2(t) for Example 8 with time-varying delays.

g2(x) = 1.39 tanh(x), u1(θ) = 0.2, u2(θ) = −0.5, (θ ∈ [−2, 0]), I1 = I2 = 1, τ1(t) = 1.8 + 0.2 sin t ,τ2(t) = 0.9 + 0.1 sin t , the unique equilibrium u∗

= [1.933 1.520]T of system (1) is asymptotically stable, whose

convergence dynamics are shown in Figs. 1 and 2.Finally, suppose that no information on the derivatives of delays is available, that is, µ1 and µ2 could be any

values. If the upper bound of delays τ1 and τ2 are known to be 2 and 1 respectively, then LMI (22) in Corollary 6is feasible for σ1 = 4.2 and σ2 = 0.92.

Example 9 ([27]). Let system (6) with r = 2 have the following parameters:

A =

[1 00 1

], W (0)

=

[−1/2 −5

5 −2

], W (1)

=

[2/3 0

−2/3 0

], W (2)

=

[0 −2/30 2/3

].

In [27], the case for µ1 = µ2 = 0 and g1(x) = g2(x) = (| x + 1 | − | x − 1 |)/2, that is, σ1 = σ2 = 1,is discussed. It is seen that the above system is delay-independent asymptotically stable by using Corollary 5.In addition, the system is delay-independent asymptotically stable by using Corollary 5 for σ1 = σ2 = 4.5 andµ1 = µ2 = 0, which is significantly better than [27].

5. Conclusion

In this paper, a new class of Lyapunov–Krasovskii functionals which contains the information on nonlinearitiesis constructed for neural networks with multiple time-varying delays and S-procedure and free-weighting matrix

Y. He et al. / Physica D 212 (2005) 126–136 135

method are employed to derive a delay-dependent stability criterion. It can be seen that the delay-independentstability criterion is contained in the delay-dependent one. The result is also extended to get a delay-dependent andrate-independent stability criterion for multiple unknown time-varying delays.

References

[1] J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons, Proc. Natl. Acad. Sci.USA 81 (1984) 3088–3092.

[2] C.M. Marcus, R.M. Westervelt, Stability of analog neural network with delay, Phys. Rev. A 39 (1989) 347–359.[3] C.M. Marcus, F.R. Waugh, R.M. Westervelt, Nonlinear dynamics and stability of analog neural networks, Physica D 51 (1991) 234–247.[4] J. Belair, Stability in a model of a delayed neural network, J. Dynam. Differential Equations 5 (1993) 607–623.[5] J. Belair, S. Campbell, P. van den Driessche, Frustration, stability and delay-induced oscillations in a neural network model, SIAM J.

Appl. Math. 56 (1996) 245–255.[6] K. Gopalsamy, I. Leung, Delay induced periodicity in a neural network of excitation and inhibition, Physica D 89 (1996) 395–426.[7] H. Ye, A.N. Michel, K. Wang, Global stability and local stability of Hopfield neural networks with delays, Phys. Rev. E 50 (1994)

4206–4213.[8] X.F. Liao, G. Chen, E.N. Sanchez, LMI-based approach for asymptotically stability analysis of delayed neural networks, IEEE Trans.

Circuits Syst. 49 (7) (2002) 1033–1039.[9] X.F. Liao, G. Chen, E.N. Sanchez, Delay-dependent exponential stability analysis of delayed neural networks: an LMI approach, Neural

Netw. 15 (2002) 855–866.[10] J. Cao, J. Wang, Global asymptotic stability of a general class of recurrent neural networks with time-varying delays, IEEE Trans. Circuits

Syst. I 50 (2003) 34–44.[11] G.J. Yu, C.Y. Lu, J.S.H. Tsai, B.D. Liu, T.J. Su, Stability of cellular neural networks with time-varying delay, IEEE Trans. Circuits Syst.

I 50 (2003) 677–678.[12] S. Arik, An analysis of exponential stability of delayed neural networks with time varying delays, Neural Netw. 17 (2004) 1027–1031.[13] E. Yucel, S. Arik, New exponential stability results for delayed neural networks with time varying delays, Physica D 191 (2004) 314–322.[14] Q. Zhang, X. Wen, J. Xu, Delay-dependent exponential stability of cellular neural networks with time-varying delays, Chaos Solitons

Fractals 23 (2005) 1363–1369.[15] K. Gopalsamy, X.Z. He, Stability in asymmetric Hopfield networks with transmission delays, Physica D 76 (1994) 344–358.[16] H. Ye, A.N. Michel, K. Wang, Qualitative analysis of Cihen-Grossberg neural networks with multiple delays, Phys. Rev. E 51 (1995)

2611–2618.[17] P. Baldi, A.F. Atiya, How delays affect neural dynamics and learning, IEEE Trans. Neural Netw. 5 (1994) 612–621.[18] P. van den Driessche, X. Zou, Global attractivity in delayed Hopfield neural network models, SIAM J. Appl. Math. 58 (1998) 1878–1890.[19] S. Arik, Stability analysis of delayed neural networks, IEEE Trans. Circuits Syst. I. 47 (2000) 1089–1092.[20] T. Roska, C.W. Wu, M. Balsi, L.O. Chua, Stability and dynamics of delay-type general neural networks, IEEE Trans. Circuits Syst. 39

(1992) 487–490.[21] X.F. Liao, J. Yu, Robust stability for interval Hopfield neural networks with time delay, IEEE Trans. Neural Netw. 9 (1998) 1042–1046.[22] X.F. Liao, J. Yu, Qualitative analysis of bi-directional associative memory networks with time delays, Int. J. Circuit Theory Appl. 26

(1998) 219–229.[23] X.F. Liao, K.W. Wong, J. Yu, Novel stability conditions for cellular neural networks with time delay, Int. J. Bifurc. Chaos 11 (7) (2001)

1835–1864.[24] X.F. Liao, K.W. Wong, Z. Wu, G. Chen, Novel robust stability criteria for interval-delayed Hopfield neural networks, IEEE Trans. Circuits

Syst. 49 (11) (2001) 1355–1359.[25] X.F. Liao, Z. Wu, J. Yu, Stability analysis of cellular neural networks with continuous time delays, J. Comput. Appl. Math. 143 (2002)

29–47.[26] X.F. Liao, J. Yu, G. Chen, Novel stability criteria for bi-directional associative memory neural networks with time delays, Int. J. Circuit

Theory Appl. 30 (2002) 519–546.[27] X.X. Liao, J. Wang, Algebraic criteria for global exponential stability of cellular neural networks with multiple time delays, IEEE Trans.

Circuits Syst. 50 (2003) 268–275.[28] X.F. Liao, C. Li, An LMI approach to asymptotical stability of multi-delayed neural networks, Physica D 200 (2005) 139–155.[29] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, PA,

1994.[30] Y. He, M. Wu, J.H. She, G.P. Liu, Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays, Syst. Control

Lett. 51 (2004) 57–65.[31] Y. He, M. Wu, J.H. She, G.P. Liu, Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type

uncertainties, IEEE Trans. Automat. Control 49 (2004) 828–832.

136 Y. He et al. / Physica D 212 (2005) 126–136

[32] M. Wu, Y. He, J.H. She, G.P. Liu, Delay-dependent criteria for robust stability of time-varying delay systems, Automatica 40 (2004)1435–1439.

[33] M. Wu, Y. He, J.H. She, New delay-dependent stability criteria and stabilizing method for neutral systems, IEEE Trans. Automat. Control49 (2004) 2266–2271.

[34] J.Y. Zhang, Absolute stability of a class of neural networks with unbounded delay, Int. J. Circuit Theory Appl. 32 (2004) 11–21.[35] J.Y. Zhang, Absolute stability analysis in cellular neural networks with variable delays and unbounded delay, Comput. Math. Appl. 47

(2004) 183–194.