Dissipativity Analysis of Neural Networks with Time-varying Delays by warrent

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									International Journal of Automation and Computing                                                                         05(3), July 2008, 290-295
                                                                                                                    DOI: 10.1007/s11633-008-0290-x




             Dissipativity Analysis of Neural Networks with
                          Time-varying Delays
                                                  Yan Sun            Bao-Tong Cui∗
                   School of Communication and Control Engineering, Jiangnan University, Wuxi, Jiangsu 214122, PRC




Abstract: A new definition of dissipativity for neural networks is presented in this paper. By constructing proper Lyapunov func-
tionals and using some analytic techniques, sufficient conditions are given to ensure the dissipativity of neural networks with or without
time-varying parametric uncertainties and the integro-differential neural networks in terms of linear matrix inequalities. Numerical
examples are given to illustrate the effectiveness of the obtained results.

Keywords:     Neural network, dissipativity, Lyapunov functional.



1    Introduction                                                      preliminaries are presented. The dissipativity conditions of
                                                                       delayed neural networks are derived in Section 3 and the
   Neural networks have extensive applications in control,             result is extended to the case with time-varying parametric
signal processing, pattern recognition, image processing,              uncertainties in Section 4. In Section 5, the dissipative
and association in the recent years[1−10] . Many essential             conditions of the integro-differential neural networks with
features of these networks, such as qualitative properties             time-varying delay are derived. In Section 6, a numerical
of stability, oscillation, and convergence issues have been            example is given to illustrate the effectiveness of the
investigated by many authors[1−6,10−13] . In many engineer-            results. Conclusions are drawn in Section 7.
ing problems, stability issues are often linked to the theory
of dissipative systems which postulates that the energy dis-           Notations
sipated inside a dynamic system is less than the energy
                                                                        W = (wij ) ∈ Cm×n .
supplied from an external source. In the literature of non-
                                                                        W T : The transpose of square matrix W .
linear control, dissipativeness was initially introduced by
                                                                        W −1 : The inverse of square matrix W .
Willems[14] in terms of an inequality involving the storage
                                                                        λ(W ): The eigenvalues of the square matrix W .
function and supply rate. It is well known that the dissi-
                                                                        W > 0 (W < 0): A positive- (negative-) definite matrix
pativity theory plays an important role in both electrical
                                                                       W.
network and nonlinear control systems and provides a nice                               n
tool to analyze the stability of systems[15,16] .                         W ∞ = max        |wij |.
                                                                                            i   j=1
   The dissipative performance includes passivity as a spe-                                     m
cial case, so the study of passivity properties is very im-                  W    1=   max            |wij |.
                                                                                        j       j=1
portant. The passivity properties of static and dynamical
neural networks were studied in [7, 8, 13, 17, 18]. Recently,          2      Model description and preliminaries
in [19, 20], the researchers studied the passivity of linear de-
lay systems. Li and Liao[9] derived the passivity conditions               We consider the following delayed neural network model:
for delayed neural networks using linear matrix inequalities
                                                                           dx(t)
(LIMs). A new criterion for the passivity of neural net-                         = −Dx(t) + Af (x(t)) + Bf (x(t − τ (t))) + u(t) (1)
works was derived in [21]. Lou and Cui[12] established the                  dt
passivity of integro-differential neural networks with time-            where x(t) = [x1 (t), x2 (t), · · · , xn (t)]T ∈ Rn is the neuron
varying delays based on the Lyapunov method and LMI                    state vector, D = diag(d1 , d2 , · · · , dn ) is a positive diag-
framework. To the best of our knowledge, the dissipavity               onal matrix, An×n and B n×n are interconnection weight
of delayed neural networks and integro-differential neural              matrices, u(t) = [u1 (t), u2 (t), · · · , un (t)]T is the input vec-
networks has not yet been studied in the sense of our defi-             tor; f (x) = [f1 (x), f2 (x), · · · , fn (x)]T denotes the neuron
nition.                                                                activation function. Let y(t) = f (x(t)) be the output of
   This paper deals with the dissipativity analysis of the             the neural networks. The delay τ (t) is a differential and
delayed neural networks and integro-differential neural net-            bounded function with 0               τ (t)    τ0 , τ ˙   d < 1 for
works with time-varying delays. It is organized as follows.            i = 1, 2, · · · , n.
In Section 2, the problem to be studied is stated and some                There are some different definitions of dissipativity. A
                                                                       less restrictive definition of dissipativity is given in this pa-
 Manuscript received March 27, 2007; revised Januany 17, 2008          per.
 This work was supported by National Natural Science Foundation           The quadratic energy supply function E associated with
of China (No. 60674026), Key Project of Chinese Ministry of Edu-
cation (No. 107058), Jiangsu Provincial Natural Science Foundation     system (1) is defined by
of China (No. BK2007016).
 *Corresponding author. E-mail address: btcui@vip.sohu.com                       E(u, y, T ) = y, Qy            T   + 2 y, Su   T   + u, Ru   T   (2)
 Y. Sun and B. T. Cui / Dissipativity Analysis of Neural Networks with Time-varying Delays                                                                                  291

where                                                                                            f T (x(t))(P A + AT P + Pm − Q)f (x(t)) +
                                    T                                                            f T (x(t))P Bf (x(t−τ (t)))+f T(x(t))(P −S T)u(t)+
               z, v   T   =             z T vdt,          T      0.
                                0                                                                f T (x(t − τ (t)))B TP f (x(t)) −
Let L2 [0, ∞] be the space of square integrable functions on                                     f T (x(t−τ (t)))(1−d)Pmf (x(t−τ (t)))+uT(t)P x(t)+
[0, ∞]. Q, S, and R are real matrices of appropriate dimen-
                                                                                                  uT (t)(P −S)f (x(t))+uT(t)(αI −R)u(t).                                    (8)
sions with Q and R symmetric. Sometimes, the arguments
of a function will be omitted so that no confusion can arise.                                    Then, we obtain
   In order to present a precise formulation of our results,
                                                                                                       V (t) + αuT (t)u(t) − y T (t)Qy(t) − 2y T (t)Su(t)−
                                                                                                       ˙
we introduce the following definition.
   Definition 1. System (1) is strictly (Q, S, R)–dissipative                                                       uT (t)Ru(t)           δ(t)T Φ1 δ(t)                      (9)
for any T      0 and some scalar α > 0. Under zero initial                                       where
state, the following condition is satisfied.                                                                                                                                 T
                                                                                                     δ(t) =     xT (t)      f T (x(t))       f T (x(t − τ (t)))   uT (t)        .
                               E(u, y, T )             α u, u        T.                    (4)
                                                                                                 Since Φ1 < 0, it is easy to get
3        Dissipativity of delayed neural net-                                                            y T (t)Qy(t) + 2y T (t)Su(t) + uT (t)Ru(t) >
         works
                                                                                                              V (t) + αuT (t)u(t).
                                                                                                              ˙                                                            (10)
  In this section, we analyze the dissipativity of delayed                                       Integrating (10) from 0 to T , under zero initial conditions
neural network (1) and present the following results.                                            we obtain
  Theorem 1. If there exists a scalar α > 0 and a sym-
                                                                                                              E(y, u, T )       α u, u   T   + V (T ) − V (0)
metric diagonal matrix P = diag(p1 , p2 , · · · , pn ) > 0 and
Pm > 0, such that the following LMI holds,                                                                                  α u, u   T                                     (11)
         ⎡                                                                             ⎤         for all T  0. Therefore, when condition (4) is satisfied, the
               (1, 1)1                 ∗               ∗                         ∗               neural network (1) is strictly (Q, S, R)–dissipative accord-
         ⎢   AT P − P D                                                       P − ST   ⎥
         ⎢                          (2, 2)1           PB                               ⎥         ing to Definition 1.
Φ1 = ⎢                                                                                 ⎥ <0
         ⎣      BT P                 BT P         −(1 − d)Pm                     0     ⎦
                 P                  P −S               0                      αI − R
                                                                                                 4    Dissipativity of delayed neural net-
                                                                                           (5)        works with parametric uncertainties
where
                                                                                                   In this section, we consider the neural network model
(1, 1)1 = −(DT P + P D),  (2, 2)1 = P A + AT P + Pm − Q                                          with time-varying parametric uncertainties.
                                                       (6)                                       dx(t)
then the delayed neural network (1) is strictly (Q, S, R)–                                             = −(D + ΔD)x(t) + (A + ΔA)f (x(t)) +
                                                                                                  dt
dissipative.
                                                                                                               (B + ΔB)f (x(t − τ (t))) + u(t)                             (12)
    Proof. Choose a Lyapunov-Krasovskii functional as
                                                      n              xi (t)
                                                                                                     ΔD(t)     ΔA(t)        ΔB(t)        = M1 Δ(k)     N3  N1     N2
             V (t) = xT (t)P x(t) + 2                       pi                fi (s)ds+                                                                   (13)
                                                      i=1        0
                                                                                                 where ΔD, ΔA, and ΔB are unknown matrices represent-
                      t                                                                          ing time-varying parameter uncertainties, M1 , N1 , N2 , and
                               f T (x(ν))Pm f (x(ν))dν.                                    (7)   N3 are known real constant matrices, and Δ(t) satisfies
                     t−τ (t)
                                                                                                                        Δ(t) = J1 (t)[I − JJ1 (t)]−1                       (14)
Then, we have
                                                                                                                                         T
                                                                                                                                I − JJ        > 0.                         (15)
V (t) + αuT (t)u(t) −
˙
                                                                                                 The uncertain matrix J1 (k) satisfies
(y T (t)Qy(t) + 2y T (t)Su(t) + uT (t)Ru(t))
                                                                                                                               J1 (t)T J1 (t)     I.                       (16)
xT (t)P x(t) + xT(t)P x(t) + 2f T (x(t))P x(t) +
˙                     ˙                   ˙
                                                                                                                 [15]
                                                                                                    Lemma 1 . Given constant matrices Ξ1 , Ξ2 , and Ξ3
f T (x(t))Pm f (x(t)) − y T (t)Qy(t) −                                                           of appropriate dimensions with Ξ1 = ΞT , then
                                                                                                                                      1
(1 − d)f T (x(t − τ (t)))Pm f (x(t − τ (t))) +
                                                                                                                 Ξ1 + Ξ2 Δ(t)Ξ3 + ΞT Δ(t)T ΞT < 0
                                                                                                                                   3        2
     T                    T                   T
αu (t)u(t)−2y (t)Su(t)−u (t)Ru(t)
                                                                                                 where Δ(t) = J1 (t)[I − JJ1 (t)]−1 , J1 (t)T J1 (t)                   I, ∀t, if
     T         T                              T                      T
−x (t)(D P +P D)x(t)+x (t)(P A−D P )f (x(t))+                                                    and only if for some scalar ε > 0
xT (t)P Bf (x(t−τ (t)))+xT(t)P u(t)+                                                                                       T                         −1
                                                                                                              ε−1 ΞT
                                                                                                                   3               I         −J           ε−1 ΞT
                                                                                                                                                               3
 T               T                                T                           T                      Ξ1 +                                                               < 0.
f (x(t))(A P −P D)x(t)+f (x(t−τ (t)))B P x(t)+                                                                 εΞ2                −J T        I            εΞ2
292                                                                             International Journal of Automation and Computing 05(3), July 2008

                                                                                       ⎡                                   ⎤T
   Theorem 2.               If there exist a scalar η > 0, a                               −ε−1 N1 T
                                                                                                            εP M1
                                                                                       ⎢     −1 T                          ⎥
scalar α > 0, and a symmetric diagonal matrix P =                                      ⎢    ε N2            εP M1          ⎥
diag(p1 , p2 , · · · , pn ) > 0 and Pm > 0, such that the fol-                         ⎢                                   ⎥ < 0.                               (22)
                                                                                       ⎣         T
                                                                                            ε−1 N3            0            ⎦
lowing LMI holds,                                                                             0               0
    ⎡                                                                       ⎤         Placing the Schur complement in (22), it becomes
           (1, 1)1          ∗         ∗           ∗       ∗       ∗
    ⎢ AT P − P D         (2, 2)1      ∗           ∗       ∗       ∗         ⎥          ⎡                                               ⎤
    ⎢                                                                       ⎥
    ⎢    BT P             BT P                    ∗       ∗       ∗         ⎥                  (1, 1)1              ∗            ∗         ∗       ∗   ∗
    ⎢                              (3, 3)1                                  ⎥
    ⎢
    ⎢                    P −S                  αI − R     ∗       ∗
                                                                            ⎥
                                                                            ⎥          ⎢ AT P − P D              (2, 2)1         ∗         ∗       ∗   ∗    ⎥
    ⎢
          P                           0
                                                                            ⎥          ⎢                                                                    ⎥
    ⎣    −N1                                             −ηI      ∗         ⎦          ⎢    BT P                  BT P        (3, 3)1      ∗       ∗   ∗    ⎥
                           N2        N3           0                                    ⎢                         P −S                   αI − R     ∗   ∗
                                                                                                                                                            ⎥ < 0.
            P M1          P M1        0           0      JT     −η−1 I                 ⎢     P                                   0                          ⎥
                                                                                       ⎣ −ε−1 N1                 ε−1 N2       ε−1 N3       0      −I   ∗    ⎦
                                                                                               εP M1             εP M1           0         0      JT   −I
 <0                                                                        (17)
                                                                                                                                                  (23)
where                                                                                 Pre-multiplying (23) by diag(I, I, I, I, εI, ε−1 I) and post-
                       (1, 1)1 = −(DT P + P D)                                        multiplying the result by diag(I, I, I, I, εI, ε−1 I), and let-
                       (2, 2)1 = P A + AT P + Pm − Q                                  ting η = ε2 , it is easy to see that (23) is equivalent to (17).
                       (3, 3)1 = −(1 − d)Pm                                (18)
then the delayed neural                      network    (12)   is       strictly         Remark 1. The above (Q, S, R)−dissipative perfor-
(Q, S, R)−dissipative.                                                                mance includes H∞ and passivity[9,20] as special cases, and
                                                                                      strict (Q, S, R)−dissipativeness contains both the phase and
   Proof. By Theorem 1, and letting DΔ = D+ΔD, AΔ =                                   gain information concerning the system.
A + ΔA, and BΔ = B + ΔB replace D, A, and B, it is easy
to get
⎡                                                                   ⎤
                                                                                      5        Dissipativity of integro-differential
        −(DΔ P + P DΔ )
            T
                                ∗           ∗              ∗                                   neural networks with time-varying
⎢        A T P − P DΔ                       ∗              ∗        ⎥
⎢          Δ                 (2, 2)2                                ⎥
⎢                                                                   ⎥ <0                       delays
⎣              T
              BΔ P             T
                              BΔ P     −(1 − d)Pm          ∗        ⎦
               P             P −S           0           αI − R
                                                                                        In this part, we consider the integro-differential neural
                                                                           (19)       network model with both discrete delays and distributed
                                                                                      delays:
where
                     (2, 2)2 = P AΔ + AT P + Pm − Q.
                                       Δ                                   (20)       dx(t)
                                                                                            = −Dx(t) + Af (x(t)) + Bf (x(t − τ (t))) +
Substituting the uncertainty structure (13) into (20) and                              dt
rearranging, we get                                                                                   n             t
⎡                                                       ⎤                                                  cij          Kij (t − s)f (x(s))ds + u(t)            (24)
   −(DT P + P D) P A − DT P           PB           P                                                               −∞
⎢ AT P − P D                                                                                         j=1
⎢                     (2, 2)1         PB        P − ST ⎥⎥
⎢         T              T                              ⎥                             for i = 1, 2, · · · , n, where n denotes the number of the neu-
⎣       B P            B P       −(1 − d)Pm        0    ⎦
          P           P −S             0        αI − R                                rons in the neural networks; C = (cij )n×n is the connection
                                                                                      matrix. In this part, the following assumptions are made.
  ⎡         ⎤
     P M1                                                                                      ∞
  ⎢ PM ⎥                                                                                  1)        Kij (s)ds = 1;
  ⎢      1 ⎥                                                                                   0
+⎢          ⎥ Δ(t) −N1 N2 N3 0 +
  ⎣ 0 ⎦
                                                                                          2) There exists a positive number μ such that
       0                                                                               ∞
                                                                                       0
                                                                                           seμs Kij (s)ds < ∞.
                                  ⎡        ⎤T
                                     P M1
                          T       ⎢ PM ⎥                                                 Theorem 3. If there exist a scalar α and a symmetric
                                  ⎢      1 ⎥
      −N1 N2 N3 0           Δ(t)T ⎢        ⎥ < 0. (21)                                diagonal matrix P = diag(p1 , p2 , · · · , pn ) > 0 and Pm > 0,
                                  ⎣ 0 ⎦                                               such that the following LMI holds,
                                       0
                                                                                                ⎡                                                           ⎤
     By Lemma 1, if and only if for some ε > 0, we have                                               (1, 1)3                 ∗           ∗           ∗
⎡                                                                               ⎤               ⎢   AT P − P D                            ∗           ∗     ⎥
                                                                                                ⎢                          (2, 2)3                          ⎥
  −(DT P + P D) P A − DT P        PB                              P                     Φ3 = ⎢
                                                                                                ⎣      BT P                 BT P     −(1 − d)Pm       ∗
                                                                                                                                                            ⎥ <0
                                                                                                                                                            ⎦
⎢ AT P − P D                                                    P − ST          ⎥
⎢                   (2, 2)1       PB                                            ⎥                       P                  P −S           0        αI − R
⎢                                                                               ⎥
⎣     BTP            BTP      −(1 − d)Pm                           0            ⎦
        P           P −S           0                            αI − R                                                                                          (25)
  ⎡                 ⎤                                                                 where
            T
    −ε−1 N1 εP M1
  ⎢ ε−1 N T                          −1
  ⎢       2   εP M1 ⎥
                    ⎥       I −J                                                       (1, 1)3 = −(DT P + P D) + ||P C||∞ I
+ ⎢ −1 T            ⎥                   ×
  ⎣ ε N3        0   ⎦ −J T     I
       0        0                                                                      (2, 2)3 = P A+ATP +Pm −Q+2||P C||1 I +||P C||∞ I (26)
    Y. Sun and B. T. Cui / Dissipativity Analysis of Neural Networks with Time-varying Delays                                                                                           293

                                                                                                     n       n
then the integro-differential neural network (1) is strictly
                                                                                                                 pi |cij |x2 (t) +
                                                                                                                           i
(Q, S, R)−dissipative.                                                                              i=1 j=1

                                                                                                         n   n                       ∞
   Proof. Choose a radically unbounded and positive defi-                                                                                          2
                                                                                                                     pi |cij |            Kij (s)fj (xj (t − s))ds                      (29)
nite Lyapunov function as                                                                                                        0
                                                                                                        i=1 j=1
                                                     n             xi (t)                                n       n                   ∞
V (t) = xT (t)P x(t) + 2                                  pi                gi (s)ds +              2                pi cij              Kij (s)fi (xi (t))fj (xj (t−s))ds
                                                    i=1        0
                                                                                                        i=1 j=1                  0

            t                                                                                        n       n
                        f T (x(λ))Pm f (x(λ))dλ +                                                                pi |cij |fi2 (x2 (t)) +
                                                                                                                                i
        t−τ (t)
                                                                                                    i=1 j=1
                n       n                       ∞                   t
                                                                          2                              n   n                       ∞
        2                    pi |cij |              Kij (s)(             fj (xj (λ))dλ)ds.   (27)                                                 2
                                            0                      t−s                                               pi |cij |            Kij (s)fj (xj (t − s))ds.                     (30)
            i=1 j=1
                                                                                                        i=1 j=1                  0

Taking the derivative of the Lyapunov function (27) along
                                                                                                    We can easily obtain that
the solution of form (24) yields:
                                                                                                    V (t) + αuT (t)u(t) −
                                                                                                    ˙
V (t) + αuT (t)u(t) −
˙
                                                                                                        (y T (t)Qy(t) + 2y T (t)Su(t) + uT (t)Ru(t))
    (y T (t)Qy(t) + 2y T (t)Su(t) + uT (t)Ru(t))
                                                                                                    −xT (t)(DT P +P D)x(t)+xT(t)(P A−DT P )f (x(t))+
xT (t)P x(t)+xT(t)P x(t)+2f T (x(t))P x−uT (t)Ru(t)+
˙                   ˙                 ˙
                                                                                                    xT (t)P Bf (x(t−τ (t)))+xT(t)P u(t)+
f T (x(t))Pm f (x(t))−y T (t)Qy(t)−2y T(t)Su(t)−
                                                                                                    ||P C||∞ xT (t)x(t)+2||P C||1 f T (x(t))f (x(t))+
                    T                                                               T
(1−d)f (x(t−τ (t)))Pmf (x(t−τ (t)))+αu (t)u(t)+
                                                                                                    ||P C||∞ f T (x(t))f (x(t))+f T(x(t−τ (t)))B TP x(t)+
     n          n                           ∞
2                       pi |cij |                        2          2
                                                Kij (s)(fj (xi (t)−fj (xi (t−s)))ds                 f T (x(t))(AT P −P D)x(t)+f T (x(t−τ (t)))B TP f (x(t))+
    i=1 j=1                             0
                                                                                                    f T (x(t))(P A+ATP +Pm −Q)f (x(t))+
        T                T                                T                     T
−x (t)(D P +P D)x(t)+x (t)(P A−D P )f (x(t))+                                                       f T (x(t))P Bf (x(t−τ (t)))+f T(x(t))(P −S T)u(t)−
    T                                                     T
x (t)P Bf (x(t − τ (t))) + x (t)P u(t) +                                                            f T (x(t−τ (t)))(1−d)Pmf (x(t−τ (t)))+
     n          n                       ∞
2                       pi cij              Kij (s)xi (t)fj (xj (t−s))ds+                           uT (t)P x(t)+uT(t)(P −S)f (x(t))+uT (t)(αI−R)u(t) =
    i=1 j=1                         0
                                                                                                    −xT (t)(DT P + P D − ||P C||∞ I)x(t) +
f T (x(t))(ATP −P D)x(t)+                                                                           xT (t)(P A−DT P )f (x(t))+xT (t)P Bf (x(t−τ (t)))+
f T (x(t))(P A+ATP +Pm −Q)f (x(t))+                                                                 f T (x(t))(P A+ATP +Pm −Q+2||P C||1 I+||P C||∞ I)f (x(t))+
f T (x(t))P Bf (x(t−τ (t)))+f T(x(t))(P −S T )u(t)+                                                 f T (x(t))(AT P −P D)x(t)+f T (x(t))P Bf (x(t−τ (t)))+

f T (x(t−τ (t)))B TP x(t)+                                                                          f T (x(t))(P −S T )u(t))+uT (t)(αI−R)u(t)+
     n          n                       ∞                                                           f T (x(t−τ (t)))B TP f (x(t))+xT(t)P u(t)−
2                       pi cij              Kij (s)fi (xi (t))fj (xj (t−s))ds+
    i=1 j=1                         0                                                               f T (x(t−τ (t)))(1−d)Pmf (x(t−τ (t)))+
                                                                                                    f T (x(t−τ (t)))B TP x(t)+uT (t)P x(t)+
f T (x(t − τ (t)))B TP f (x(t)) −
                                                                                                    uT (t)(P −S)f (x(t)).
f T (x(t−τ (t)))(1−d)Pmf (x(t−τ (t)))+
                                                                                                                                                                                        (31)
     n          n
                                  2
                                                                                                         Then, we have
2                       pi |cij |fj (xj (t)) −
    i=1 j=1                                                                                                  V (t) + αuT (t)u(t) − y T (t)Qy(t) − 2y T (t)Su(t)−
                                                                                                             ˙
     n          n                           ∞
                                                                                                                                 uT (t)Ru(t)            δ(t)T Φ3 δ(t)                   (32)
2                       pi |cij |               Kij (s)fj (xj (t−s))ds+uT (t)P x(t)+
    i=1 j=1                             0                                                           where
                                                                                                                                                                                         T
 uT (t)(P − S)f (x(t)) + uT (t)(αI − R)u(t).                                                 (28)        δ(t) =           xT (k)           f T (x(t))     f T (x(t − τ (t)))   uT (t)        .
                                                                   2        2
By using the inequality 2ab                                    a + b for any a, b ∈ R, it is
easy to see that                                                                                    Since Φ3 < 0, it is easy to get

     n          n                       ∞                                                                                 V (t) + αuT (t)u(t) <
                                                                                                                          ˙
2                       pi cij              Kij (s)xi (t)fj (xj (t − s))ds
    i=1 j=1                         0                                                                                 y T (t)Qy(t) + 2y T (t)Su(t) + uT (t)Ru(t).                       (33)
294                                                                                          International Journal of Automation and Computing 05(3), July 2008


Integrating (33) from 0 to T , under zero initial condition                                          Remark 4. For numerical calculation of the paper, we
we obtain                                                                                          choose the diagonal matrix P = diag(1, 1) > 0, and

    E(u, y, T )     α u, u         T       + V (T ) − V (0)          α u, u       T       (34)
                                                                                                                                0.1     0
                                                                                                                        C=                    .
for all T    0. Therefore, by using Definition 1, when con-                                                                      0.1   0.25
dition (4) is satisfied, the neural network (24) is strictly
(Q, S, R)–dissipative.                                                                                It is easy to see that ||P C||∞ I and ||P C||1 I can be cal-
   By comparing our results with other previous work, we                                           culated, and (25) will be reduced to an LMI, which can
derive the following remarks.                                                                      be solved by the Matlab LMI control toolbox. In order to
   Remark 2. In [8, 9], the activation function of the model                                       determine the positive-definite solutions of LMIs (17) and
needed to be bounded and differentiable. However, our                                               (25), we apply the following computational procedure.
studies do not need the above conditions. Thus, it is easy                                            Step 1. Given the matrices A, B, C, D, M1 , N1 , N2 ,
to see that our results extend the earlier works.                                                  and N3 .
   Remark 3. When Q = 0, S = I, and work αI − R =                                                     Step 2. Choose the matrix P , which must be positive
−γI, our result in Theorem 3 corresponds to a passive                                              and diagonal. Without loss of generality, we make the iden-
problem[22] . Thus, the investigation in this paper improves                                       tity matrix as P . The choice of P should make it easy to
the existing literature.                                                                           calculate ||P C||∞ I and ||P C||1 I.
                                                                                                      Step 3. If (17) and (25) have feasible solutions, then
6     Numerical example                                                                            stop.        Otherwise, increase (or decrease) pi for i =
                                                                                                   1, 2, · · · , n, and repeat Step 3.
  In this section, we demonstrate the theory developed in
this paper by means of a simple example. Consider the                                              7    Conclusions
uncertain delayed neural networks (12) and (13) with the
parameters as follows:                                                                                In this paper, dissipativity is investigated for delayed
                                                                                                   neural networks with or without time-varying parametric
           0.1      0                                  −0.1        0                               uncertainteis and the integro-differential neural networks
    A=                         ,             B=                               ,
           0.1     0.1                                  0.1       −0.2                             with time-varying delays. Three theorems are presented to
           0.5      0                                    0.1         −0.1                          ensure dissipativity based on proper Lyapunov functionals
    D=                         ,             N1 =                                 ,                and some analytic techniques are used. The results are pre-
            0      0.8                                   −0.1        −0.1
                                                                                                   sented in terms of LMIs, which can be solved easily by using
            −0.1          0                            −0.1          −0.1                          the Matlab tools.
    N2 =                               ,     N3 =                                 ,
             0           0.1                            0            −0.1                             This paper is only a first step towards dissipative anal-
                                                                                                   ysis of the neural networks. In future, we will study the
             0.1    0.1
    M1 =                           .                                                               dissipativity of singular neural networks, and investigate
             0.1     0                                                                             of dissipativity properties of Cohen-Grossberg neural net-
  Then, consider the integro-differential neural networks                                           works with time delay and the dissipativity results to the
(24) with                                                                                          stability analysis of delayed neural networks.

                                            0.1     0                                              References
                          C=                                .
                                            0.1   0.25
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     World Congress on Intelligent Control and Automation,                                 Jiaotong University, PRC, from July 2003
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                                                                                           at Department of Electrical and Computer
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     Solitons & Fractals, vol. 25, no. 2, pp. 393–401, 2005.         chronization.

								
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