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Stability analysis of a class of generalized neural networks with delays
Physics Letters A 337 (2005) 203–215 www.elsevier.com/locate/pla
Stability analysis of a class of generalized neural networks with delays Lei Zhou a,b,∗ , Mingru Zhou b a Department of Mathematics, Nantong University, Nantong 226000, PR China b Department of Mathematics, Xuzhou Normal University, Xuzhou 221116, PR China
Received 25 March 2004; received in revised form 4 December 2004; accepted 25 January 2005
Communicated by A.P. Fordy
Abstract In this Letter, we discuss the global asymptotic stability and exponential stability of the equilibrium point of a class of generalized neural networks with delays. By introducing a new type of Lyapunov functionals and making use of matrix inequality technique, some sufficient conditions for existence and uniqueness of equilibrium and global exponential stability of the delayed Cohen–Grossberg neural networks are obtained. The conditions of the presented results are less restrictive than those of the earlier literature. 2005 Elsevier B.V. All rights reserved. Keywords: Neural networks; Delay; Global asymptotic stability; Global exponential stability; Matrix inequality; Lyapunov functional
1. Introduction Cohen and Grossberg [1] proposed a class of neural networks in 1983, which can be described by the following ordinary differential equations n dui (t) = −ai ui (t) bi ui (t) − Tij fj uj (t) , i = 1, 2, . . . , n, (1) dt j =1
where n denotes the number of neurons in the network, ui (t) denotes the state variable at time t associated with the ith neuron and the activation function fj (uj ) denotes the output of the j th neuron at time t, the n × n connection matrix T = (Tij ) represents the neuron interconnections. * Corresponding author.
E-mail addresses: [email protected] (L. Zhou), [email protected] (M. Zhou). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.01.067
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This neural network has been widely studied due to its promising potential for applications in associative memory, signal processing and solving some optimization problems. Obviously, this type of neural networks includes the popular Hopfield neural networks [2] ci
n ui (t) dui (t) =− + Tij fj uj (t) + Ii , dt Ri
i = 1, 2, . . . , n
(2)
j =1
and BSB models n dui (t) = −ui (t) + Tij fj uj (t) , dt
i = 1, 2, . . . , n.
(3)
j =1
In recent years, various generalizations and modifications of above models have also been proposed and studied. For example, considering models with time delay or reaction–diffusion terms. It is well known that if the output from neuron j excites or inhibits neuron i, then the Tij 0 or Tij 0, respectively. However, most of previous criteria for determining the stability of neural networks neglect the signs of entries of the connection matrices, thus the difference between the excitatory and inhibitory effects might be ignored and the true nature of neural networks cannot be described fully, therefore the obtained results turned out to be more restrictive and less applicable. Recently, some authors have noticed the disadvantage and some improvements have been made (see, e.g., [3–7, 9–12]). Lin Wang [6,7] considered the following delayed neural networks, respectively, m n dui (t) (k) = −ai ui (t) bi ui (t) − Tij fj uj (t − τk ) , i = 1, 2, . . . , n (4) dt k=0 j =1
and some sufficient conditions for the global asymptotic and exponential stability of this model are established. The purpose of this Letter is to study a class of generalized neural networks with multiple delays of the form m n dui (t) = −ai ui (t) bi ui (t) − Tij(k) fj uj t − τj(k) , i = 1, 2, . . . , n (5) dt k=0 j =1
under some weaker conditions. This Letter is organized as follows. In Section 2, we give some preliminaries. In Section 3, our main results on the global asymptotic stability and exponential stability of (5) will be presented. An example is given in Section 4. We present a summary in Section 5.
2. Preliminaries Let C denotes the set of all the continuous functions ϕ : [−τ, 0] → R n . In C, let ϕ = sup[−τ,0] |ϕ(θ )|, then C is a Banach space with the topology of uniform convergence. If δ > 0, then Bδ = {ϕ ∈ C | ϕ < δ} is a open ball in C with radius δ. For any map x(t) ∈ C([σ − τ, +∞), R n ), it is customary to introduce the following notation xt = xt (θ ) = x(t + θ ),
−τ θ 0, ∀t ∈ [σ, +∞].
Suppose f : Ω → R n is a continuous function, Ω is a open set in C. Consider the following autonomous functional differential equation with constant delays dx = f (xt ). dt
(6)
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Let x(t) ≡ x(t, σ, ϕ) denotes the solution of (6) satisfying the initial condition xσ = ϕ ∈ C. It is well known by the theory of functional differential equations that the stability of the equilibrium of (6) has no relation to the initial time, so we consider the case of σ = 0 only. Suppose W is a open set in C, V : C → R is continuous. We say V is a V functional on set W if V˙ 0 on the closure of W , where V (xh (ϕ)) − V (ϕ) . V˙ = lim sup h h→0+ Let S = {ϕ ∈ W¯ | V˙ = 0},
(7)
and M is the largest subset of S that is invariant with respect to (6), then we have the following lemma (see [8, Theorem 2.1]). Lemma 1. If V is a V functional on set W , xt (ϕ) is a bounded solution of (6) that remains in W , then xt (ϕ) tends to M as t → ∞, that is lim inf xt (ϕ) − ϕ : ϕ ∈ M = 0. t→∞
Suppose A is a n order square matrix, A > 0 means A is positive definite; A 0 means A is positive semidefinite. A < 0 if and only if −A > 0, A 0 if and only if −A 0. The following lemmas can be found in [13,14], respectively. Lemma 2. Given any real matrices A, B, C of appropriate dimensions and a scalar ε > 0, where C = C T > 0 (C T is the transpose of C). Then the following inequality holds: 1 AT B + B T A εAT CA + B T C −1 B. ε Lemma 3. Suppose Q(x) = QT (x), R(x) = R T (x), and S(x) depend affinely on x, then the following matrix inequality
Q(x) S(x) (8) >0 S T (x) R(x) is equivalent to R(x) > 0,
Q(x) − S(x)R −1 (x)S T (x) > 0.
3. Main results We consider the following Cohen–Grossberg neural network with time delays m n dui (t) (k) (k) = −ai ui (t) bi ui (t) − , i = 1, 2, . . . , n, Tij fj uj t − τj dt
(9)
k=0 j =1
(k)
where ai (ui (t)) > 0 is continuous and bounded, τj k = 1, 2, . . . , m.
(0)
are constants and τj
(k)
= 0, 0 < τj
τ , j = 1, 2, . . . , n,
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The initial conditions for system (9) are ui (t) = ϕi (t),
−τ t 0, i = 1, 2, . . . , n,
ϕ = (ϕ1 , ϕ2 , . . . , ϕn ) ∈ C,
ϕ =
T
sup
−τ t0
n 2 ϕi (t)
1/2 .
i=1
If we note T (k) (k) τk = τ1 , τ2 , . . . , τn(k) , T u(t) = u1 (t), u2 (t), . . . , un (t) , a u(t) = diag a1 u1 (t) , a2 u2 (t) , . . . , an un (t) , T b u(t) = b1 u1 (t) , b2 u2 (t) , . . . , bn un (t) , T f u(t) = f1 u1 (t) , f2 u2 (t) , . . . , fn un (t) , T (k) (k) f u(t − τk ) = f1 u1 t − τ1 , f2 u2 t − τ2 , . . . , fn un t − τn(k) , (k) Tk = Tij n×n , k = 0, 1, 2, . . . , m, then system (9) can be rewritten as m du(t) = −a u(t) b u(t) − Tk f u(t − τk ) . dt
(10)
k=0
Initial conditions became u(θ ) = ϕ(θ ),
−τ θ 0.
For notation convenience, we will note our following conditions as (H1 ): fj : R → R, there exists lj > 0, such that for any uj , vj ∈ R holds fj (uj ) − fj (vj ) lj |uj − vj |, j = 1, 2, . . . , n. (H∗1 ): f ∈ L, that is fi (ui ) is continuous, i = 1, 2, . . . , n, and there exists a L = diag(l1 , l2 , . . . , ln ), where li > 0, such that 0
fi (ξ ) − fi (η) li , ξ −η
∀ξ = η, i = 1, 2, . . . , n.
(H2 ): b ∈ D, that is bi (ui ) is continuous, i = 1, 2, . . . , n, and there exists a D = diag(d1 , d2 , . . . , dn ), where di > 0, such that bi (ξ ) − bi (η) di , ξ −η
∀ξ = η, i = 1, 2, . . . , n.
Theorem 1. Assume that system (9) satisfies conditions H∗1 , H2 . If there exist positive diagonal matrices P = diag(p1 , p2 , . . . , pn ), Q = diag(q1 , q2 , . . . , qn ), such that 2P DL
−1
− P T0 − T0T P
−
m
P Tk Q−1 TkT P − mQ > 0,
(11)
k=1
then system (10) has a unique equilibrium point, which is globally asymptotically stable, independent of the delays.
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Proof. Firstly, we prove the existence and the uniqueness of the equilibrium point. Constant vector u∗ = (u∗1 , u∗2 , . . . , u∗n ) is the equilibrium point of (10) if and only if u∗ is the solution of following equation b(u) −
m
Tk f (u) = 0.
(12)
k=0
Taking A = TkT P , B = I , C = Q−1 , ε = 1 (I is the unit matrix) in Lemma 2, then we have P Tk + TkT P P Tk Q−1 TkT P + Q,
k = 1, 2, . . . , m.
Thus 2P DL−1 − P T0 − T0T P −
m m P Tk + TkT P 2P DL−1 − P T0 − T0T P − P Tk Q−1 TkT P − mQ > 0. k=1
k=1
Therefore s m −1 Tk > 0, P DL − k=0
where [A]s = 12 (A + AT ). By the method proposed in [11], we can prove that there exist a unique equilibrium point, for more details refer to Theorem 1 in [11]. Secondly, we prove the asymptotic stability of the equilibrium point. Suppose that u∗ = (u∗1 , u∗2 , . . . , u∗n ) is the unique equilibrium point of (10). Let xi (t) = ui (t) − u∗i , gi (s) = fi s + u∗i − fi u∗i , T g x(t) = g1 x1 (t) , g2 x2 (t) , . . . , gn xn (t) , T (k) (k) , g2 x 2 t − τ 2 , . . . , gn xn t − τn(k) , g x(t − τk ) = g1 x1 t − τ1 then system (10) can be rewritten as m dx(t) = −a x(t) + u∗ b x(t) + u∗ − b u∗ − Tk g x(t − τk ) . dt
(13)
k=0
Correspondingly, the initial conditions become
x(θ ) = φ(θ ) = ϕ(θ ) − u∗ ,
−τ θ 0, i = 1, 2, . . . , n.
(14)
Obviously, u∗ is the globally asymptotically stable equilibrium of (10) if and only if the zero solution of (13) is globally asymptotically stable. To analyze the global stability of the zero solution of (13), consider the following Lyapunov functional n V x(t) = 2 pi i=1
xi (t)
0
gi (ρ) qi dρ + ∗ ai (ρ + ui ) m
n
k=1 i=1
t (k)
t−τi
gi2 xi (θ ) dθ.
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Calculating the derive of V along the trajectory of (13), we have n n m (k) (k) pi gi xi (t) bi xi (t) + u∗i − bi u∗i − Tij gj xj t − τj V˙ x(t) = −2 k=0 j =1
i=1
+m
n
qi gi2 xi (t) −
i=1
m
n
qi gi2 xi t − τi(k) .
k=1 i=1
Under the conditions of b ∈ D and f ∈ L, we have gi2 (xi ) li xi gi (xi ) li2 xi2 , i = 1, 2, . . . , n, gi (xi ) bi xi + u∗i − bi u∗i gi (xi )di xi di li−1 gi2 (xi ). If we assume that G1 is a n(m + 1) × 1 matrix, the transpose of G1 is GT1 = g T x(t) g T x(t − τ1 ) · · · g T x(t − τm ) , then V˙ x(t) −2g T x(t) P DL−1 g x(t) + 2g T x(t) P T0 g x(t) + mg T x(t) Qg x(t) +2
m
m g T x(t) P Tk g x(t − τk ) − g T x(t − τk ) Qg x(t − τk )
k=1
= −GT1
k=1
2P DL−1
− P T0 − T0T P −T1T P −T2T P
− mQ
··· −TmT P
By Lemma 3 and condition (11), we have 2P DL−1 − P T0 − T0T P − mQ −P T1 −T1T P Q T −T2 P 0 ··· ··· −TmT P 0
−P T2
−P T1
−P T2
Q
0
···
0 ··· 0
Q ··· 0
··· ··· ···
· · · −P Tm
0
···
Q ··· 0
··· ··· ···
· · · −P Tm
0 G1 . ··· Q 0
(15)
0 > 0. ··· Q 0
So, formula (15) implies V˙ (x(t)) 0. Further V˙ (x(t)) = 0 if and only if g(x(t)) = g(x(t − τ1 )) = · · · = g(x(t − τm )) = 0. Therefore V (x(t)) and all the solutions of (13) are bounded, and the set S in formula (7) is S = φ ∈ C | g φ(0) = g φ(−τ1 ) = g φ(−τ2 ) = · · · = g φ(−τm ) = 0 . By Lemma 1, all the solutions of (13) tend to M as t → ∞, where M is the largest invariant set with respect to (13). Clearly, M is generated by bounded solutions of the ordinary differential equation dx(t) = −a x(t) + u∗ b x(t) + u∗ − b u∗ . dt
(16)
The conditions of Theorem 1 imply that zero solution of system (16) is globally asymptotically stable. In fact, if we let V (x(t)) = ni=1 xi2 , then V is radially unbounded and along the trajectory of (16), we have
L. Zhou, M. Zhou / Physics Letters A 337 (2005) 203–215
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n xi (t)ai xi (t) + u∗i bi x(t) + u∗i − bi u∗i V˙ x(t) = −2
−2
i=1 n
di ai xi (t) + u∗i xi2 (t)
i=1
0 and V˙ = 0 only and only if x(t) = 0. In conclusion, the zero solution of (13) is globally asymptotically stable, so u∗ is the globally asymptotically stable equilibrium of (10). 2 Suppose m = 1, then the model (9) can be rewritten as n n dui (t) = −ai ui (t) bi ui (t) − aij fj uj (t) − bij fj uj (t − τj ) , dt j =1
i = 1, 2, . . . , n,
(17)
j =1
where ai (ui (t)) > 0 is continuous and bounded. Assume A = (aij )n×n and B = (bij )n×n , then we have the following result. Corollary 1. Suppose that system (17) satisfies conditions H∗1 , H2 . If there exist positive diagonal matrices P = diag(p1 , p2 , . . . , pn ), Q = diag(q1 , q2 , . . . , qn ), such that 2P DL−1 − P A − AT P − P BQ−1 B T P − Q > 0, then system (17) has a unique equilibrium point, which is globally asymptotically stable, independent of the delays. Remark 1. Assume ai (s) ≡ 1, bi (ui (t)) = di ui (t), m = 1, Q = I (I is the unit matrix). Then the main results of [8] can be obtained. Remark 2. The conditions in Corollary 1 are weaker than those of Theorem 3 in [11], in which the boundedness of fi (ui ) is required to prove the existence of the equilibrium point of system (17). In fact, by the proof of Theorem 1, this condition is redundant. Remark 3. In [12], with the assumption that B = 0 in neural networks (17), the following stability criterion (Theorem 2) was obtained DL−1 − A ∈ LDS, that is there exist positive diagonal matrices P and ε > 0, such that 2P DL−1 − P A − AT P εI > 0. If we let Q = 2ε I , then we have ε 2P DL−1 − P A − AT P − Q I > 0. 2 Thus the above criterion is a special case of Corollary 1. In Theorem 1, we suppose f ∈ L, that is fi (i = 1, 2, . . . , n) is monotonically nondecreasing on R and Lipschitz continuous. However, if we assume that system (10) has an equilibrium point u∗ = (u∗1 , u∗2 , . . . , u∗n ). In this case, similar to the proof of Theorem 1, we are also only required to prove the global asymptotic stability of zero solution of system (13) with the initial condition (14), then we can derive the following two theorems.
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i (η) Theorem 2. Suppose that system (10) exists an equilibrium point u∗ , and satisfies condition H2 , and fi (ξξ)−f 0, −η ∀ξ = η, i = 1, 2, . . . , n. If there exist positive diagonal matrices P = diag(p1 , p2 , . . . , pn ), Q = diag(q1 , q2 , . . . , qn ), R = diag(r1 , r2 , . . . , rn ), such that 2P D −P T0 + QD −P T1 −P T2 · · · −P Tm −T T P + QD −2QT − mR −QT −QT · · · −QT 0 1 2 m 0 −T1T P −T1T Q R 0 ··· 0 > 0, (18) −T2T P −T2T Q 0 R ··· 0 ··· ··· ··· ··· ··· ···
−TmT P
−TmT Q
···
0
···
R
then the equilibrium point u∗ is globally asymptotically stable, independent of the delays. Proof. Define the following Lyapunov functional xi (t)
n V x(t) = 2 pi i=1
ρ dρ + 2 qi ∗ ai (ρ + ui ) n
i=1
0
xi (t)
0
t
gi (ρ) dρ + ri ∗ ai (ρ + ui ) m
n
k=1 i=1
gi2 xi (θ ) dθ.
(k)
t−τi
Calculating the V˙ along the trajectory of (13), we have n m n ∗ (k) (k) ∗ ˙ V x(t) = −2 pi xi (t) bi xi (t) + ui − bi ui − Tij gj xj t − τj i=1 n
−2
k=0 j =1
m n ∗ (k) (k) ∗ qi gi xi (t) bi xi (t) + ui − bi ui − Tij gj xj t − τj
i=1 n
+m
k=0 j =1
ri gi2 xi (t) −
i=1
m n
(k) ri gi2 xi t − τi
k=1 i=1
m −2x T (t)P Dx(t) + 2x T (t)(P T0 − QD)g x(t) + 2 x T (t)P Tk g x(t − τk ) k=1
+2
m
g
T
m g T x(t − τk ) Rg x(t − τk ) x(t) QTk g x(t − τk ) −
k=0 T
+ mg Let
GT2 = x T (t)
we get
g T x(t)
k=1
x(t) Rg x(t) . g T x(t − τ1 ) . . .
2P D
−T T P + QD 0 −T1T P V˙ x(t) −GT2 −T2T P ··· −TmT P
g T x(t − τm )
−P T0 + QD
−P T1
−P T2
−2QT0 − mR
−QT1
−QT2
−T1T Q
R
0
−T2T Q ···
0 ···
−TmT Q
0
···
−P Tm
R ···
· · · −QTm ··· 0 G2 . ··· 0 ··· ···
···
···
R
L. Zhou, M. Zhou / Physics Letters A 337 (2005) 203–215
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By the condition (18), we have V˙ x(t) 0, and V˙ (x(t)) = 0 if and only if x(t) = g x(t) = g x(t − τ1 ) = · · · = g x(t − τm ) = 0. Therefore by Lemma 1, we have the zero solution of system (13) is globally asymptotically stable.
2
Theorem 3. Suppose that system (10) has an equilibrium point u∗ and condition H2 is satisfied. If there exist positive diagonal matrices P = diag(p1 , p2 , . . . , pn ), Q = diag(q1 , q2 , . . . , qn ), such that
−1 T 2P D − m −P T0 k=1 P Tk Q Tk P > 0, (19) −T0T P −mQ then the equilibrium point u∗ is globally asymptotically stable, independent of the delays. Proof. Define the following Lyapunov functional n V x(t) = 2 pi i=1
xi (t)
0
ρ dρ + qi ∗ ai (ρ + ui ) m
n
k=1 i=1
t
gi2 xi (θ ) dθ.
(k)
t−τi
Then along the trajectory of (13), we have n m n (k) (k) pi xi (t) bi xi (t) + u∗i − bi u∗i − Tij gj xj t − τj V˙ x(t) = −2 i=1 n
+m
k=0 j =1
qi gi2 xi (t) −
i=1
−2x T (t)P Dx(t) + 2
m
n
(k) qi gi2 xi t − τi
k=1 i=1 m T
x (t)P Tk g x(t − τk ) + mg T x(t) Qg x(t)
k=0
−
m
g T x(t − τk ) Qg x(t − τk ) .
k=1
By Lemma 2, we have 2x T (t)P Tk g x(t − τk ) = x T (t)P Tk g x(t − τk ) + g T x(t − τk ) TkT P x(t) g T x(t − τk ) Qg x(t − τk ) + x T (t)P Tk Q−1 TkT P x(t),
k = 1, 2, . . . , m.
Thus, we obtain m T −1 T ˙ V x(t) −x (t) 2P D − P Tk Q Tk P x(t) + 2x T (t)P T0 g x(t) + mg T x(t) Qg x(t) k=1
= − x T (t)
g (x(t)) T
2P D −
m
−1 T k=1 P Tk Q Tk P T −T0 P
−P T0 −mQ
x(t) g(x(t))
.
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By the condition (19) we have V˙ 0. Similar to the previous discussion, we can complete the proof of Theorem 3. 2 Corollary 2. Suppose that system (10) has an equilibrium point u∗ and conditions H1 and H2 are satisfied. If there exist positive diagonal matrices P = diag(p1 , p2 , . . . , pn ), Q = diag(q1 , q2 , . . . , qn ), such that 2P D −
m
P Tk Q−1 TkT P − (m + 1)QL2 > 0,
(20)
k=0
then equilibrium point u∗ is globally asymptotically stable, independent of the delays. Proof. Define the following Lyapunov functional n V x(t) = 2 pi i=1
xi (t)
0
t
ρ dρ + qi ∗ ai (ρ + ui ) m
n
k=1 i=1
gi2 xi (θ ) dθ.
(k)
t−τi
Similar to the proof of Theorem 3, we have m P Tk Q−1 TkT P x(t) + (m + 1)g T x(t) Qg x(t) . V˙ x(t) −x T (t) 2P D − k=0
Since f ∈ L, thus gi2 (xi ) li xi gi (xi ) li2 xi2 , i = 1, 2, . . . , n, m T −1 T 2 ˙ V x(t) −x (t) 2P D − P Tk Q Tk P − (m + 1)QL x(t). k=0
By the condition (20), we get V˙ 0. Similar to the proof of Theorem 2, we can complete the proof of this corollary. 2 Theorem 4. Suppose that system (10) has an equilibrium point u∗ , conditions H1 , H2 are satisfied and there exist constants a and a, such that 0 < a ai (s) a. If there exist positive diagonal matrices P = diag(p1 , p2 , . . . , pn ), Q = diag(q1 , q2 , . . . , qn ) and a constant σ > 0, such that
m −1 T −P T0 2P D − 2σ k=1 P Tk Qk Tk P a P − (21) > 0, −T0T P −mQ (k)
(k)
(k)
where Qk = diag(q1 e−2σ τ1 , q2 e−2σ τ2 , . . . , qn e−2σ τn ). Then exist a c > 0, for any solution u(t) of system (10), the following inequality holds u(t) − u∗ cφe−σ t , that is the equilibrium point u∗ is globally exponentially stable. Proof. Similar to the previous discussion we only need to prove the zero solution of (13) is globally exponentially stable. Define the following Lyapunov functional n V x(t) = 2e2σ t pi i=1
xi (t)
0
ρ dρ + qi ∗ ai (ρ + ui ) m
n
k=1 i=1
t (k)
t−τi
e2σ θ gi2 xi (θ ) dθ.
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Along the trajectory of (13), we have V˙ = 4σ e2σ t
xi (t)
n
pi
i=1
+ 2e2σ t
m
0 n
n ρ 2σ t dρ − 2e pi xi (t) bi xi (t) + u∗i − bi u∗i ai (ρ + u∗i ) i=1
n
m n (k) (k) + e2σ t Tij pi xi (t)gj xj t − τj qi gi2 xi (t)
k=0 i=1 j =1
−
m n
(k)
e2σ (t−τi
)
k=1 i=1
(k) qi gi2 xi t − τi
k=1 i=1 m 2σ 2σ t T 2σ t T 2σ t e x (t)P x(t) − 2e x (t)P Dx(t) + 2e x T (t)P Tk g x(t − τk ) a k=0
m g T x(t − τk ) Qk g x(t − τk ) , + me2σ t g T x(t) Qg x(t) − e2σ t k=1
and T 2x T (t)P Tk g x(t − τk ) g T x(t − τk ) Qk g x(t − τk ) + x T (t)P Tk Q−1 k Tk P x(t). So we have
m 2σ −1 T 2σ t T T T ˙ x (t) P − 2P D + P Tk Qk Tk P x(t) + 2x (t)P T0 g x(t) + mg x(t) Qg x(t) V e a k=1
−1 T 2P D − 2σ x(t) P− m −P T0 k=1 P Tk Qk Tk P 2σ t T T a x (t) g (x(t)) = −e . g(x(t)) −T0T P −mQ By the condition (21), we have V˙ 0. Therefore V x(t) V x(0) , ∀t 0. On the other hand n V x(0) = 2 pi i=1
2 pi a n
i=1
xi (0)
0
ρ qi dρ + ∗ ai (ρ + ui ) m
k=1 i=1
xi (0)
ρ dρ + 0
n
m n k=1 i=1
0
e2σ θ gi2 xi (θ ) dθ
(k)
−τi
0 e2σ θ li2 xi2 (θ ) dθ
qi (k)
−τi
n 0 1 T 2 x (0)P x(0) + mlM qM e2σ θ xi2 (θ ) dθ a i=1 −τ
1 2 pM + nmτ lM qM φ2 , a
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where pM = max{p1 , p2 , . . . , pn }, lM = max{l1 , l2 , . . . , ln }, qM = max{q1 , q2 , . . . , qn }. Otherwise n pi V x(t) 2e2σ t i=1
xi (t)
0
ρ pm x(t)2 , dρ ne2σ t ∗ ai (ρ + ui ) a
where pm = min{p1 , p2 , . . . , pn }. Letting c2 =
a 2 pM + nmτ alM qM , napm
then we obtain x(t) cφe−σ t . Thus complete the proof of Theorem 4.
2
Remark 4. The obtained Theorems 2–4 are based on the assumption that there exists an equilibrium of system (10), in fact this assumption can be satisfied by the well-known Brouwer fixed point theorem if we assume that the activation function fi (i = 1, 2, . . . , n) is bounded. Remark 5. In Theorem 3 the activation fj (uj ) (j = 1, 2, . . . , n) is neither monotonically nondecreasing nor Lipschitz continuous, thus the criterion turns out to be less restrictive than those of in [6,11].
4. Example Consider
u˙ 1 (t) 1 0 u1 (t) =− + u˙ 2 (t) u2 (t) 0 1 It is clear that
1 0 a(u) = , 0 1
b(u) =
u1 u2
1 2 1 2
− 12 1 2
(1)
u1 (t − τ1 ) (1)
u2 (t − τ2 )
,
f (u) =
u1 u2
.
,
T0 = 0,
T1 =
1 2 1 2
− 12 1 2
.
Since fi (ui ) (i = 1, 2) is not bounded, Theorem 3.3 in [6] and Theorem 3 in [11] are not applicable. But it is clear that f ∈ L and b ∈ D, if we let P = Q = I (unit matrix) in (11), we have
1 0 −1 T −1 T 2 2P DL − P T0 − T0 P − P T1 Q T1 P − Q = > 0. 0 12 Therefore, from Theorem 1, we know the above system has a unique equilibrium point and it is globally asymptotically stable.
5. Conclusions In this Letter, we construct a new type of Lyapunov functionals and take the signs of the entries into account, by use of matrix inequality technique, and several criterions for the uniqueness and global asymptotic stability and
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exponential stability of the equilibrium point of neural networks with delays are obtained. The proposed criterions consider the difference between the neuron excitatory and inhibitory effects, hence it is more applicable in the practical designing of neural networks. Moreover, this approach is computational effective since it can be easily verified and solved by use of the LMI control toolbox with MATLAB.
Acknowledgements The authors would like to thank referees for their helpful suggestion and comments. This work was supported by the NSF of China under Grant No. 60474076, NSF Grant BK2003034 from Jiangsu Province.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
M.A. Cohen, S. Grossberg, IEEE Trans. Systems Man Cybernet. 13 (1983) 815. J.J. Hopfield, Proc. Natl. Acad. Sci. 79 (1982) 2554. P. van den Driessche, X. Zou, SIAM J. Appl. Math. 58 (1998) 1878. M. Gopalsamy, X. He, Physica D 76 (1994) 344. T. Chen, W. Lu, S.-I. Amari, Neural Comput. 14 (2002) 2947. L. Wang, X. Zou, Physica D 170 (2002) 162. L. Wang, X. Zou, Neural Networks 15 (2002) 415. M. Joy, J. Math. Anal. Appl. 232 (1999) 61. M. Joy, Neural Networks 13 (2000) 613. X. Liao, G. Chen, Neural Networks 15 (2002) 855. T. Chen, L. Rong, Phys. Lett. A 317 (2003) 436. W. Lu, T. Chen, Neural Comput. 15 (2003) 1173. E.N. Sanchez, E.N. Perez, J.P. Perez, IEEE Trans. Circuits Systems I 46 (1999) 1395. P. Boyd, IEEE Trans. Neural Networks 5 (1994) 612.
Stability analysis of a class of generalized neural networks with delays Lei Zhou a,b,∗ , Mingru Zhou b a Department of Mathematics, Nantong University, Nantong 226000, PR China b Department of Mathematics, Xuzhou Normal University, Xuzhou 221116, PR China
Received 25 March 2004; received in revised form 4 December 2004; accepted 25 January 2005
Communicated by A.P. Fordy
Abstract In this Letter, we discuss the global asymptotic stability and exponential stability of the equilibrium point of a class of generalized neural networks with delays. By introducing a new type of Lyapunov functionals and making use of matrix inequality technique, some sufficient conditions for existence and uniqueness of equilibrium and global exponential stability of the delayed Cohen–Grossberg neural networks are obtained. The conditions of the presented results are less restrictive than those of the earlier literature. 2005 Elsevier B.V. All rights reserved. Keywords: Neural networks; Delay; Global asymptotic stability; Global exponential stability; Matrix inequality; Lyapunov functional
1. Introduction Cohen and Grossberg [1] proposed a class of neural networks in 1983, which can be described by the following ordinary differential equations n dui (t) = −ai ui (t) bi ui (t) − Tij fj uj (t) , i = 1, 2, . . . , n, (1) dt j =1
where n denotes the number of neurons in the network, ui (t) denotes the state variable at time t associated with the ith neuron and the activation function fj (uj ) denotes the output of the j th neuron at time t, the n × n connection matrix T = (Tij ) represents the neuron interconnections. * Corresponding author.
E-mail addresses: [email protected] (L. Zhou), [email protected] (M. Zhou). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.01.067
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This neural network has been widely studied due to its promising potential for applications in associative memory, signal processing and solving some optimization problems. Obviously, this type of neural networks includes the popular Hopfield neural networks [2] ci
n ui (t) dui (t) =− + Tij fj uj (t) + Ii , dt Ri
i = 1, 2, . . . , n
(2)
j =1
and BSB models n dui (t) = −ui (t) + Tij fj uj (t) , dt
i = 1, 2, . . . , n.
(3)
j =1
In recent years, various generalizations and modifications of above models have also been proposed and studied. For example, considering models with time delay or reaction–diffusion terms. It is well known that if the output from neuron j excites or inhibits neuron i, then the Tij 0 or Tij 0, respectively. However, most of previous criteria for determining the stability of neural networks neglect the signs of entries of the connection matrices, thus the difference between the excitatory and inhibitory effects might be ignored and the true nature of neural networks cannot be described fully, therefore the obtained results turned out to be more restrictive and less applicable. Recently, some authors have noticed the disadvantage and some improvements have been made (see, e.g., [3–7, 9–12]). Lin Wang [6,7] considered the following delayed neural networks, respectively, m n dui (t) (k) = −ai ui (t) bi ui (t) − Tij fj uj (t − τk ) , i = 1, 2, . . . , n (4) dt k=0 j =1
and some sufficient conditions for the global asymptotic and exponential stability of this model are established. The purpose of this Letter is to study a class of generalized neural networks with multiple delays of the form m n dui (t) = −ai ui (t) bi ui (t) − Tij(k) fj uj t − τj(k) , i = 1, 2, . . . , n (5) dt k=0 j =1
under some weaker conditions. This Letter is organized as follows. In Section 2, we give some preliminaries. In Section 3, our main results on the global asymptotic stability and exponential stability of (5) will be presented. An example is given in Section 4. We present a summary in Section 5.
2. Preliminaries Let C denotes the set of all the continuous functions ϕ : [−τ, 0] → R n . In C, let ϕ = sup[−τ,0] |ϕ(θ )|, then C is a Banach space with the topology of uniform convergence. If δ > 0, then Bδ = {ϕ ∈ C | ϕ < δ} is a open ball in C with radius δ. For any map x(t) ∈ C([σ − τ, +∞), R n ), it is customary to introduce the following notation xt = xt (θ ) = x(t + θ ),
−τ θ 0, ∀t ∈ [σ, +∞].
Suppose f : Ω → R n is a continuous function, Ω is a open set in C. Consider the following autonomous functional differential equation with constant delays dx = f (xt ). dt
(6)
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Let x(t) ≡ x(t, σ, ϕ) denotes the solution of (6) satisfying the initial condition xσ = ϕ ∈ C. It is well known by the theory of functional differential equations that the stability of the equilibrium of (6) has no relation to the initial time, so we consider the case of σ = 0 only. Suppose W is a open set in C, V : C → R is continuous. We say V is a V functional on set W if V˙ 0 on the closure of W , where V (xh (ϕ)) − V (ϕ) . V˙ = lim sup h h→0+ Let S = {ϕ ∈ W¯ | V˙ = 0},
(7)
and M is the largest subset of S that is invariant with respect to (6), then we have the following lemma (see [8, Theorem 2.1]). Lemma 1. If V is a V functional on set W , xt (ϕ) is a bounded solution of (6) that remains in W , then xt (ϕ) tends to M as t → ∞, that is lim inf xt (ϕ) − ϕ : ϕ ∈ M = 0. t→∞
Suppose A is a n order square matrix, A > 0 means A is positive definite; A 0 means A is positive semidefinite. A < 0 if and only if −A > 0, A 0 if and only if −A 0. The following lemmas can be found in [13,14], respectively. Lemma 2. Given any real matrices A, B, C of appropriate dimensions and a scalar ε > 0, where C = C T > 0 (C T is the transpose of C). Then the following inequality holds: 1 AT B + B T A εAT CA + B T C −1 B. ε Lemma 3. Suppose Q(x) = QT (x), R(x) = R T (x), and S(x) depend affinely on x, then the following matrix inequality
Q(x) S(x) (8) >0 S T (x) R(x) is equivalent to R(x) > 0,
Q(x) − S(x)R −1 (x)S T (x) > 0.
3. Main results We consider the following Cohen–Grossberg neural network with time delays m n dui (t) (k) (k) = −ai ui (t) bi ui (t) − , i = 1, 2, . . . , n, Tij fj uj t − τj dt
(9)
k=0 j =1
(k)
where ai (ui (t)) > 0 is continuous and bounded, τj k = 1, 2, . . . , m.
(0)
are constants and τj
(k)
= 0, 0 < τj
τ , j = 1, 2, . . . , n,
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The initial conditions for system (9) are ui (t) = ϕi (t),
−τ t 0, i = 1, 2, . . . , n,
ϕ = (ϕ1 , ϕ2 , . . . , ϕn ) ∈ C,
ϕ =
T
sup
−τ t0
n 2 ϕi (t)
1/2 .
i=1
If we note T (k) (k) τk = τ1 , τ2 , . . . , τn(k) , T u(t) = u1 (t), u2 (t), . . . , un (t) , a u(t) = diag a1 u1 (t) , a2 u2 (t) , . . . , an un (t) , T b u(t) = b1 u1 (t) , b2 u2 (t) , . . . , bn un (t) , T f u(t) = f1 u1 (t) , f2 u2 (t) , . . . , fn un (t) , T (k) (k) f u(t − τk ) = f1 u1 t − τ1 , f2 u2 t − τ2 , . . . , fn un t − τn(k) , (k) Tk = Tij n×n , k = 0, 1, 2, . . . , m, then system (9) can be rewritten as m du(t) = −a u(t) b u(t) − Tk f u(t − τk ) . dt
(10)
k=0
Initial conditions became u(θ ) = ϕ(θ ),
−τ θ 0.
For notation convenience, we will note our following conditions as (H1 ): fj : R → R, there exists lj > 0, such that for any uj , vj ∈ R holds fj (uj ) − fj (vj ) lj |uj − vj |, j = 1, 2, . . . , n. (H∗1 ): f ∈ L, that is fi (ui ) is continuous, i = 1, 2, . . . , n, and there exists a L = diag(l1 , l2 , . . . , ln ), where li > 0, such that 0
fi (ξ ) − fi (η) li , ξ −η
∀ξ = η, i = 1, 2, . . . , n.
(H2 ): b ∈ D, that is bi (ui ) is continuous, i = 1, 2, . . . , n, and there exists a D = diag(d1 , d2 , . . . , dn ), where di > 0, such that bi (ξ ) − bi (η) di , ξ −η
∀ξ = η, i = 1, 2, . . . , n.
Theorem 1. Assume that system (9) satisfies conditions H∗1 , H2 . If there exist positive diagonal matrices P = diag(p1 , p2 , . . . , pn ), Q = diag(q1 , q2 , . . . , qn ), such that 2P DL
−1
− P T0 − T0T P
−
m
P Tk Q−1 TkT P − mQ > 0,
(11)
k=1
then system (10) has a unique equilibrium point, which is globally asymptotically stable, independent of the delays.
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Proof. Firstly, we prove the existence and the uniqueness of the equilibrium point. Constant vector u∗ = (u∗1 , u∗2 , . . . , u∗n ) is the equilibrium point of (10) if and only if u∗ is the solution of following equation b(u) −
m
Tk f (u) = 0.
(12)
k=0
Taking A = TkT P , B = I , C = Q−1 , ε = 1 (I is the unit matrix) in Lemma 2, then we have P Tk + TkT P P Tk Q−1 TkT P + Q,
k = 1, 2, . . . , m.
Thus 2P DL−1 − P T0 − T0T P −
m m P Tk + TkT P 2P DL−1 − P T0 − T0T P − P Tk Q−1 TkT P − mQ > 0. k=1
k=1
Therefore s m −1 Tk > 0, P DL − k=0
where [A]s = 12 (A + AT ). By the method proposed in [11], we can prove that there exist a unique equilibrium point, for more details refer to Theorem 1 in [11]. Secondly, we prove the asymptotic stability of the equilibrium point. Suppose that u∗ = (u∗1 , u∗2 , . . . , u∗n ) is the unique equilibrium point of (10). Let xi (t) = ui (t) − u∗i , gi (s) = fi s + u∗i − fi u∗i , T g x(t) = g1 x1 (t) , g2 x2 (t) , . . . , gn xn (t) , T (k) (k) , g2 x 2 t − τ 2 , . . . , gn xn t − τn(k) , g x(t − τk ) = g1 x1 t − τ1 then system (10) can be rewritten as m dx(t) = −a x(t) + u∗ b x(t) + u∗ − b u∗ − Tk g x(t − τk ) . dt
(13)
k=0
Correspondingly, the initial conditions become
x(θ ) = φ(θ ) = ϕ(θ ) − u∗ ,
−τ θ 0, i = 1, 2, . . . , n.
(14)
Obviously, u∗ is the globally asymptotically stable equilibrium of (10) if and only if the zero solution of (13) is globally asymptotically stable. To analyze the global stability of the zero solution of (13), consider the following Lyapunov functional n V x(t) = 2 pi i=1
xi (t)
0
gi (ρ) qi dρ + ∗ ai (ρ + ui ) m
n
k=1 i=1
t (k)
t−τi
gi2 xi (θ ) dθ.
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Calculating the derive of V along the trajectory of (13), we have n n m (k) (k) pi gi xi (t) bi xi (t) + u∗i − bi u∗i − Tij gj xj t − τj V˙ x(t) = −2 k=0 j =1
i=1
+m
n
qi gi2 xi (t) −
i=1
m
n
qi gi2 xi t − τi(k) .
k=1 i=1
Under the conditions of b ∈ D and f ∈ L, we have gi2 (xi ) li xi gi (xi ) li2 xi2 , i = 1, 2, . . . , n, gi (xi ) bi xi + u∗i − bi u∗i gi (xi )di xi di li−1 gi2 (xi ). If we assume that G1 is a n(m + 1) × 1 matrix, the transpose of G1 is GT1 = g T x(t) g T x(t − τ1 ) · · · g T x(t − τm ) , then V˙ x(t) −2g T x(t) P DL−1 g x(t) + 2g T x(t) P T0 g x(t) + mg T x(t) Qg x(t) +2
m
m g T x(t) P Tk g x(t − τk ) − g T x(t − τk ) Qg x(t − τk )
k=1
= −GT1
k=1
2P DL−1
− P T0 − T0T P −T1T P −T2T P
− mQ
··· −TmT P
By Lemma 3 and condition (11), we have 2P DL−1 − P T0 − T0T P − mQ −P T1 −T1T P Q T −T2 P 0 ··· ··· −TmT P 0
−P T2
−P T1
−P T2
Q
0
···
0 ··· 0
Q ··· 0
··· ··· ···
· · · −P Tm
0
···
Q ··· 0
··· ··· ···
· · · −P Tm
0 G1 . ··· Q 0
(15)
0 > 0. ··· Q 0
So, formula (15) implies V˙ (x(t)) 0. Further V˙ (x(t)) = 0 if and only if g(x(t)) = g(x(t − τ1 )) = · · · = g(x(t − τm )) = 0. Therefore V (x(t)) and all the solutions of (13) are bounded, and the set S in formula (7) is S = φ ∈ C | g φ(0) = g φ(−τ1 ) = g φ(−τ2 ) = · · · = g φ(−τm ) = 0 . By Lemma 1, all the solutions of (13) tend to M as t → ∞, where M is the largest invariant set with respect to (13). Clearly, M is generated by bounded solutions of the ordinary differential equation dx(t) = −a x(t) + u∗ b x(t) + u∗ − b u∗ . dt
(16)
The conditions of Theorem 1 imply that zero solution of system (16) is globally asymptotically stable. In fact, if we let V (x(t)) = ni=1 xi2 , then V is radially unbounded and along the trajectory of (16), we have
L. Zhou, M. Zhou / Physics Letters A 337 (2005) 203–215
209
n xi (t)ai xi (t) + u∗i bi x(t) + u∗i − bi u∗i V˙ x(t) = −2
−2
i=1 n
di ai xi (t) + u∗i xi2 (t)
i=1
0 and V˙ = 0 only and only if x(t) = 0. In conclusion, the zero solution of (13) is globally asymptotically stable, so u∗ is the globally asymptotically stable equilibrium of (10). 2 Suppose m = 1, then the model (9) can be rewritten as n n dui (t) = −ai ui (t) bi ui (t) − aij fj uj (t) − bij fj uj (t − τj ) , dt j =1
i = 1, 2, . . . , n,
(17)
j =1
where ai (ui (t)) > 0 is continuous and bounded. Assume A = (aij )n×n and B = (bij )n×n , then we have the following result. Corollary 1. Suppose that system (17) satisfies conditions H∗1 , H2 . If there exist positive diagonal matrices P = diag(p1 , p2 , . . . , pn ), Q = diag(q1 , q2 , . . . , qn ), such that 2P DL−1 − P A − AT P − P BQ−1 B T P − Q > 0, then system (17) has a unique equilibrium point, which is globally asymptotically stable, independent of the delays. Remark 1. Assume ai (s) ≡ 1, bi (ui (t)) = di ui (t), m = 1, Q = I (I is the unit matrix). Then the main results of [8] can be obtained. Remark 2. The conditions in Corollary 1 are weaker than those of Theorem 3 in [11], in which the boundedness of fi (ui ) is required to prove the existence of the equilibrium point of system (17). In fact, by the proof of Theorem 1, this condition is redundant. Remark 3. In [12], with the assumption that B = 0 in neural networks (17), the following stability criterion (Theorem 2) was obtained DL−1 − A ∈ LDS, that is there exist positive diagonal matrices P and ε > 0, such that 2P DL−1 − P A − AT P εI > 0. If we let Q = 2ε I , then we have ε 2P DL−1 − P A − AT P − Q I > 0. 2 Thus the above criterion is a special case of Corollary 1. In Theorem 1, we suppose f ∈ L, that is fi (i = 1, 2, . . . , n) is monotonically nondecreasing on R and Lipschitz continuous. However, if we assume that system (10) has an equilibrium point u∗ = (u∗1 , u∗2 , . . . , u∗n ). In this case, similar to the proof of Theorem 1, we are also only required to prove the global asymptotic stability of zero solution of system (13) with the initial condition (14), then we can derive the following two theorems.
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i (η) Theorem 2. Suppose that system (10) exists an equilibrium point u∗ , and satisfies condition H2 , and fi (ξξ)−f 0, −η ∀ξ = η, i = 1, 2, . . . , n. If there exist positive diagonal matrices P = diag(p1 , p2 , . . . , pn ), Q = diag(q1 , q2 , . . . , qn ), R = diag(r1 , r2 , . . . , rn ), such that 2P D −P T0 + QD −P T1 −P T2 · · · −P Tm −T T P + QD −2QT − mR −QT −QT · · · −QT 0 1 2 m 0 −T1T P −T1T Q R 0 ··· 0 > 0, (18) −T2T P −T2T Q 0 R ··· 0 ··· ··· ··· ··· ··· ···
−TmT P
−TmT Q
···
0
···
R
then the equilibrium point u∗ is globally asymptotically stable, independent of the delays. Proof. Define the following Lyapunov functional xi (t)
n V x(t) = 2 pi i=1
ρ dρ + 2 qi ∗ ai (ρ + ui ) n
i=1
0
xi (t)
0
t
gi (ρ) dρ + ri ∗ ai (ρ + ui ) m
n
k=1 i=1
gi2 xi (θ ) dθ.
(k)
t−τi
Calculating the V˙ along the trajectory of (13), we have n m n ∗ (k) (k) ∗ ˙ V x(t) = −2 pi xi (t) bi xi (t) + ui − bi ui − Tij gj xj t − τj i=1 n
−2
k=0 j =1
m n ∗ (k) (k) ∗ qi gi xi (t) bi xi (t) + ui − bi ui − Tij gj xj t − τj
i=1 n
+m
k=0 j =1
ri gi2 xi (t) −
i=1
m n
(k) ri gi2 xi t − τi
k=1 i=1
m −2x T (t)P Dx(t) + 2x T (t)(P T0 − QD)g x(t) + 2 x T (t)P Tk g x(t − τk ) k=1
+2
m
g
T
m g T x(t − τk ) Rg x(t − τk ) x(t) QTk g x(t − τk ) −
k=0 T
+ mg Let
GT2 = x T (t)
we get
g T x(t)
k=1
x(t) Rg x(t) . g T x(t − τ1 ) . . .
2P D
−T T P + QD 0 −T1T P V˙ x(t) −GT2 −T2T P ··· −TmT P
g T x(t − τm )
−P T0 + QD
−P T1
−P T2
−2QT0 − mR
−QT1
−QT2
−T1T Q
R
0
−T2T Q ···
0 ···
−TmT Q
0
···
−P Tm
R ···
· · · −QTm ··· 0 G2 . ··· 0 ··· ···
···
···
R
L. Zhou, M. Zhou / Physics Letters A 337 (2005) 203–215
211
By the condition (18), we have V˙ x(t) 0, and V˙ (x(t)) = 0 if and only if x(t) = g x(t) = g x(t − τ1 ) = · · · = g x(t − τm ) = 0. Therefore by Lemma 1, we have the zero solution of system (13) is globally asymptotically stable.
2
Theorem 3. Suppose that system (10) has an equilibrium point u∗ and condition H2 is satisfied. If there exist positive diagonal matrices P = diag(p1 , p2 , . . . , pn ), Q = diag(q1 , q2 , . . . , qn ), such that
−1 T 2P D − m −P T0 k=1 P Tk Q Tk P > 0, (19) −T0T P −mQ then the equilibrium point u∗ is globally asymptotically stable, independent of the delays. Proof. Define the following Lyapunov functional n V x(t) = 2 pi i=1
xi (t)
0
ρ dρ + qi ∗ ai (ρ + ui ) m
n
k=1 i=1
t
gi2 xi (θ ) dθ.
(k)
t−τi
Then along the trajectory of (13), we have n m n (k) (k) pi xi (t) bi xi (t) + u∗i − bi u∗i − Tij gj xj t − τj V˙ x(t) = −2 i=1 n
+m
k=0 j =1
qi gi2 xi (t) −
i=1
−2x T (t)P Dx(t) + 2
m
n
(k) qi gi2 xi t − τi
k=1 i=1 m T
x (t)P Tk g x(t − τk ) + mg T x(t) Qg x(t)
k=0
−
m
g T x(t − τk ) Qg x(t − τk ) .
k=1
By Lemma 2, we have 2x T (t)P Tk g x(t − τk ) = x T (t)P Tk g x(t − τk ) + g T x(t − τk ) TkT P x(t) g T x(t − τk ) Qg x(t − τk ) + x T (t)P Tk Q−1 TkT P x(t),
k = 1, 2, . . . , m.
Thus, we obtain m T −1 T ˙ V x(t) −x (t) 2P D − P Tk Q Tk P x(t) + 2x T (t)P T0 g x(t) + mg T x(t) Qg x(t) k=1
= − x T (t)
g (x(t)) T
2P D −
m
−1 T k=1 P Tk Q Tk P T −T0 P
−P T0 −mQ
x(t) g(x(t))
.
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By the condition (19) we have V˙ 0. Similar to the previous discussion, we can complete the proof of Theorem 3. 2 Corollary 2. Suppose that system (10) has an equilibrium point u∗ and conditions H1 and H2 are satisfied. If there exist positive diagonal matrices P = diag(p1 , p2 , . . . , pn ), Q = diag(q1 , q2 , . . . , qn ), such that 2P D −
m
P Tk Q−1 TkT P − (m + 1)QL2 > 0,
(20)
k=0
then equilibrium point u∗ is globally asymptotically stable, independent of the delays. Proof. Define the following Lyapunov functional n V x(t) = 2 pi i=1
xi (t)
0
t
ρ dρ + qi ∗ ai (ρ + ui ) m
n
k=1 i=1
gi2 xi (θ ) dθ.
(k)
t−τi
Similar to the proof of Theorem 3, we have m P Tk Q−1 TkT P x(t) + (m + 1)g T x(t) Qg x(t) . V˙ x(t) −x T (t) 2P D − k=0
Since f ∈ L, thus gi2 (xi ) li xi gi (xi ) li2 xi2 , i = 1, 2, . . . , n, m T −1 T 2 ˙ V x(t) −x (t) 2P D − P Tk Q Tk P − (m + 1)QL x(t). k=0
By the condition (20), we get V˙ 0. Similar to the proof of Theorem 2, we can complete the proof of this corollary. 2 Theorem 4. Suppose that system (10) has an equilibrium point u∗ , conditions H1 , H2 are satisfied and there exist constants a and a, such that 0 < a ai (s) a. If there exist positive diagonal matrices P = diag(p1 , p2 , . . . , pn ), Q = diag(q1 , q2 , . . . , qn ) and a constant σ > 0, such that
m −1 T −P T0 2P D − 2σ k=1 P Tk Qk Tk P a P − (21) > 0, −T0T P −mQ (k)
(k)
(k)
where Qk = diag(q1 e−2σ τ1 , q2 e−2σ τ2 , . . . , qn e−2σ τn ). Then exist a c > 0, for any solution u(t) of system (10), the following inequality holds u(t) − u∗ cφe−σ t , that is the equilibrium point u∗ is globally exponentially stable. Proof. Similar to the previous discussion we only need to prove the zero solution of (13) is globally exponentially stable. Define the following Lyapunov functional n V x(t) = 2e2σ t pi i=1
xi (t)
0
ρ dρ + qi ∗ ai (ρ + ui ) m
n
k=1 i=1
t (k)
t−τi
e2σ θ gi2 xi (θ ) dθ.
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Along the trajectory of (13), we have V˙ = 4σ e2σ t
xi (t)
n
pi
i=1
+ 2e2σ t
m
0 n
n ρ 2σ t dρ − 2e pi xi (t) bi xi (t) + u∗i − bi u∗i ai (ρ + u∗i ) i=1
n
m n (k) (k) + e2σ t Tij pi xi (t)gj xj t − τj qi gi2 xi (t)
k=0 i=1 j =1
−
m n
(k)
e2σ (t−τi
)
k=1 i=1
(k) qi gi2 xi t − τi
k=1 i=1 m 2σ 2σ t T 2σ t T 2σ t e x (t)P x(t) − 2e x (t)P Dx(t) + 2e x T (t)P Tk g x(t − τk ) a k=0
m g T x(t − τk ) Qk g x(t − τk ) , + me2σ t g T x(t) Qg x(t) − e2σ t k=1
and T 2x T (t)P Tk g x(t − τk ) g T x(t − τk ) Qk g x(t − τk ) + x T (t)P Tk Q−1 k Tk P x(t). So we have
m 2σ −1 T 2σ t T T T ˙ x (t) P − 2P D + P Tk Qk Tk P x(t) + 2x (t)P T0 g x(t) + mg x(t) Qg x(t) V e a k=1
−1 T 2P D − 2σ x(t) P− m −P T0 k=1 P Tk Qk Tk P 2σ t T T a x (t) g (x(t)) = −e . g(x(t)) −T0T P −mQ By the condition (21), we have V˙ 0. Therefore V x(t) V x(0) , ∀t 0. On the other hand n V x(0) = 2 pi i=1
2 pi a n
i=1
xi (0)
0
ρ qi dρ + ∗ ai (ρ + ui ) m
k=1 i=1
xi (0)
ρ dρ + 0
n
m n k=1 i=1
0
e2σ θ gi2 xi (θ ) dθ
(k)
−τi
0 e2σ θ li2 xi2 (θ ) dθ
qi (k)
−τi
n 0 1 T 2 x (0)P x(0) + mlM qM e2σ θ xi2 (θ ) dθ a i=1 −τ
1 2 pM + nmτ lM qM φ2 , a
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L. Zhou, M. Zhou / Physics Letters A 337 (2005) 203–215
where pM = max{p1 , p2 , . . . , pn }, lM = max{l1 , l2 , . . . , ln }, qM = max{q1 , q2 , . . . , qn }. Otherwise n pi V x(t) 2e2σ t i=1
xi (t)
0
ρ pm x(t)2 , dρ ne2σ t ∗ ai (ρ + ui ) a
where pm = min{p1 , p2 , . . . , pn }. Letting c2 =
a 2 pM + nmτ alM qM , napm
then we obtain x(t) cφe−σ t . Thus complete the proof of Theorem 4.
2
Remark 4. The obtained Theorems 2–4 are based on the assumption that there exists an equilibrium of system (10), in fact this assumption can be satisfied by the well-known Brouwer fixed point theorem if we assume that the activation function fi (i = 1, 2, . . . , n) is bounded. Remark 5. In Theorem 3 the activation fj (uj ) (j = 1, 2, . . . , n) is neither monotonically nondecreasing nor Lipschitz continuous, thus the criterion turns out to be less restrictive than those of in [6,11].
4. Example Consider
u˙ 1 (t) 1 0 u1 (t) =− + u˙ 2 (t) u2 (t) 0 1 It is clear that
1 0 a(u) = , 0 1
b(u) =
u1 u2
1 2 1 2
− 12 1 2
(1)
u1 (t − τ1 ) (1)
u2 (t − τ2 )
,
f (u) =
u1 u2
.
,
T0 = 0,
T1 =
1 2 1 2
− 12 1 2
.
Since fi (ui ) (i = 1, 2) is not bounded, Theorem 3.3 in [6] and Theorem 3 in [11] are not applicable. But it is clear that f ∈ L and b ∈ D, if we let P = Q = I (unit matrix) in (11), we have
1 0 −1 T −1 T 2 2P DL − P T0 − T0 P − P T1 Q T1 P − Q = > 0. 0 12 Therefore, from Theorem 1, we know the above system has a unique equilibrium point and it is globally asymptotically stable.
5. Conclusions In this Letter, we construct a new type of Lyapunov functionals and take the signs of the entries into account, by use of matrix inequality technique, and several criterions for the uniqueness and global asymptotic stability and
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exponential stability of the equilibrium point of neural networks with delays are obtained. The proposed criterions consider the difference between the neuron excitatory and inhibitory effects, hence it is more applicable in the practical designing of neural networks. Moreover, this approach is computational effective since it can be easily verified and solved by use of the LMI control toolbox with MATLAB.
Acknowledgements The authors would like to thank referees for their helpful suggestion and comments. This work was supported by the NSF of China under Grant No. 60474076, NSF Grant BK2003034 from Jiangsu Province.
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