Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2010, Article ID 624619, 12 pages
doi:10.1155/2010/624619
Research Article
Existence and Global Exponential Stability
of Equilibrium Solution to Reaction-Diffusion
Recurrent Neural Networks on Time Scales
Kaihong Zhao1, 2 and Yongkun Li1
1
2
Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China
Department of Applied Mathematics, Kunming University of Science and Technology, Kunming,
Yunnan 650093, China
Correspondence should be addressed to Yongkun Li, yklie@ynu.edu.cn
Received 30 March 2010; Accepted 13 July 2010
Academic Editor: Francisco Solis
Copyright q 2010 K. Zhao and Y. Li. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The existence of equilibrium solutions to reaction-diffusion recurrent neural networks with
Dirichlet boundary conditions on time scales is proved by the topological degree theory and
M-matrix method. Under some sufficient conditions, we obtain the uniqueness and global
exponential stability of equilibrium solution to reaction-diffusion recurrent neural networks with
Dirichlet boundary conditions on time scales by constructing suitable Lyapunov functional and
inequality skills. One example is given to illustrate the effectiveness of our results.
1. Introduction
In the past few years, various neural network models have been extensively investigated
and successfully applied to signal processing, image processing, pattern classification,
quadratic optimization, associative memory, moving object speed detection, and so forth.
Such applications heavily depend on the dynamical behaviors of the neural networks.
Therefore, the analysis of the dynamical behaviors is a necessary step for practical design
of neural networks.
As is well known, both in biological and man-made neural networks, strictly speaking,
diffusion effects cannot be avoided in the neural network models when electrons are moving
in asymmetric electromagnetic fields, so we must consider that the activations vary in spaces
as well as in time. References 1–10 have considered the stability of neural networks with
diffusion terms, which are expressed by partial differential equations. It is also common to
consider the diffusion effects in biological systems such as immigration, see, e.g., 11–13.
2
Discrete Dynamics in Nature and Society
For more details of the literature related to models of reaction-diffusion neural networks and
their applications, the reader is referred to 14–21 and the references cited therein.
In fact, both continuous and discrete systems are very important in implementing
and applications. But it is troublesome to study the dynamics behavior for continuous and
discrete systems, respectively. Therefore, it is meaningful to study that on time scales which
can unify the continuous and discrete situations 22, 23.
To the best of our knowledge, few authors have considered global exponential stability
of reaction-diffusion recurrent neural networks with Dirichlet boundary conditions on time
scales, which is a very important in theories and applications and also is very challenging
problem. Motivated by the above discussion, in this paper, we will investigate the global
exponential stability of the following reaction-diffusion recurrent neural network with initial
value conditions and Dirichlet boundary conditions on time scales:
uΔ
i t, x m
n
∂
∂ui
aik
− bi ui t, x cij fj uj t, x Ii ,
∂xk
∂xk
j1
k1
ui 0, x φi x,
ui t, x 0,
t, x ∈ T × Ω,
x ∈ Ω,
1.1
1.2
t, x ∈ 0, ∞T × ∂Ω,
1.3
where i 1, 2, . . . , n, T ⊂ R is a time scale and T ∩ 0, ∞ 0, ∞T is unbounded, n is
the number of neurons in the networks, x x1 , x2 , . . . , xm T ∈ Ω ⊂ Rm , and Ω {x x1 , x2 , . . . , xm T : |xi | < li , i 1, 2, . . . , m} is a bounded compact set with smooth boundary
∂Ω in space Rm , ut, x u1 t, x, u2 t, x, . . . , un t, xT : T × Ω → Rn , and ui t, x is the
state of the ith neurons at time t and in space x. Smooth function aik > 0 corresponds to the
transmission diffusion operator along with the ith unit, bi > 0 represents the rate with which
the ith unit will reset its potential to the resting state in isolation when disconnected from
the network and external inputs, cij denotes the strength of the jth unit on the ith unit at
time t and in space x, dij is the synaptic connection strength of jth unit on ith unit at time t
and in space x, fj · denotes the activation function of the jth unit at time t and in space x,
φx φ1 x, φ2 x, . . . , φn xT ∈ CΩ, Rn , and I I1 , I2 , . . . , In T ∈ Rn is a constant input
vector.
2. Preliminaries
In this section, we first recall some basic definitions and lemmas on time scales which are
used in what follows.
Let T be a nonempty closed subset time scale of R. The forward and backward jump
operators σ, ρ : T → T and the graininess μ : T → R
are defined, respectively, by
σt inf{s ∈ T : s > t},
ρt sup{s ∈ T : s < t},
μt σt − t.
2.1
A point t ∈ T is called left-dense if t > inf T and ρt t, left-scattered if ρt < t,
right-dense if t < sup T and σt t, and right-scattered if σt > t. If T has a left-scattered
maximum m, then Tk T \ {m}; otherwise Tk T. If T has a right-scattered minimum m,
then Tk T \ {m}; otherwise Tk T.
Discrete Dynamics in Nature and Society
3
Definition 2.1 see 24. A function f : T → R is called regulated provided its right-side
limits exist finite at all right-side points in T and its left-side limits exist finite at all leftside points in T.
Definition 2.2 see 24. A function f : T → R is called rd continuous provided it is
continuous at right-dense point in T and its left-side limits exist finite at left-dense points in
T. The set of rd continuous functions f : T → R will be denoted by Crd Crd T Crd T, R.
Definition 2.3 see 24. Assume f : T → R and t ∈ Tk . Then we define f Δ t to be the
number if it exists with the property that given any ε > 0 there exists a neighborhood U of
t i.e., U t − δ, t δ ∩ T for some δ > 0 such that
fσt − fs − f Δ tσt − s < ε|σt − s|
2.2
for all s ∈ U. We call f Δ t the delta or Hilger derivative of f at t. The set of functions
f : T → R that are differentiable and whose derivative is rd continuous is denoted by
1
1
1
Crd
T Crd
T, R.
Crd
If f is continuous, then f is rd continuous. If f is rd continuous, the f is regulated. If
f is delta differentiable at t, then f is continuous at t.
Lemma 2.4 see 24. Let f be regulated, then there exists a function F which is delta differentiable
with region of differentiation D such that
F Δ t ft,
∀t ∈ D.
2.3
Definition 2.5 see 24. Assume f : T → R is a regulated function. Any function F as in
Lemma 2.4 is called a Δ-antiderivative of f. We define the indefinite integral of a regulated
function f by
ftΔt Ft C,
2.4
where C is an arbitrary constant and F is a Δ-antiderivative of f. We define the Cauchy
integral by
b
fsΔs Fb − Fa,
∀a, b ∈ T.
2.5
a
A function F : T → R is called an antiderivative of f : T → R provided that
F Δ t ft,
∀t ∈ Tk .
Lemma 2.6 see 24. If a, b ∈ T, α, β ∈ R, and f, g ∈ CT, R, then
b
b
b
i a αft βgtΔt α a ftΔt β a gtΔt;
2.6
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Discrete Dynamics in Nature and Society
ii if ft ≥ 0 for all a ≤ t < b, then
b
a
ftΔt ≥ 0;
iii if |ft| ≤ gt on a, b : {t ∈ T : a ≤ t < b}, then |
b
a
ftΔt| ≤
b
a
gtΔt.
A function p : T → R is called regressive if 1 μtpt / 0 for all t ∈ Tk . The set
of all regressive and rd-continuous functions f : T → R will be denoted by R RT RT, R. We define the set R
of all positively regressive elements of R by R
R
T, R {p ∈ R : 1 μtpt > 0 for all t ∈ T}. If p is a regressive function, then the generalized
t
exponential function ep is defined by ep t, s exp{ s ξμτ pτΔτ} for s, t ∈ T, with the
cylinder transformation
⎧
⎪
⎨ Log1 hz , if h / 0,
h
ξh z ⎪
⎩z,
if h 0.
2.7
Let p, q : T → R be two regressive functions, then we define
p ⊕ q p q μpq,
p −
p
,
1 μp
p q p ⊕ p q .
2.8
The generalized exponential function has the following properties.
Lemma 2.7 see 24. Assume that p, q : T → R are two regressive functions, then,
i ep σt, s 1 μtptep t, s;
ii 1/ep t, s ep t, s;
iii ep t, s 1/ep s, t ep s, t;
iv ep t, sep s, r ep t, r;
v ep t, sΔ ptep t, s;
vi ep c, ·Δ −pep c, ·σ for c ∈ T;
t
vii d/dzez t, s s 1/1 μτzΔτez t, s.
Lemma 2.8 see 24. Assume that f, g : T → R are delta differentiable at t ∈ Tk . Then,
fg
Δ
t f Δ tgt fσtg Δ t g Δ tft gσtf Δ t.
2.9
Next, let us introduce the Banach space which is suitable for 1.1–1.3 and some
assumed conditions which are needed in this paper.
Let Ω {x x1 , x2 , . . . , xm T : |xi | < li , i 1, 2, . . . , m} be an open bounded domain
m
in R with smooth boundary ∂Ω. Let Crd T × Ω, Rn be the set consisting of all the vector
functions ut, x which are rd-continuous with respect to t ∈ T and continuous with respect to
x ∈ Rm , respectively. For every t ∈ T and x ∈ Ω, we define the set Ct {ut, · : u ∈ CΩ, Rn }.
Then for every t ∈ T, Ct is a Banach space with the norm ut, · ni ui t, ·22 1/2 ,
where ut, x u1 t, x, u2 t, x, . . . , un t, xT , ui t, ·2 Ω |ui t, x|2 dx1/2 , i 1, 2, . . . , n.
Obviously, CΩ, Rn is a Banach space equipped with the norm ϕ0 ni ϕi 2τ 1/2 , where
ϕx ϕ1 x, ϕ2 x, . . . , ϕn xT ∈ CΩ, Rn , ϕi τ Ω |ϕi x|2 dx1/2 , i 1, 2, . . . , n.
Discrete Dynamics in Nature and Society
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Definition 2.9. A function u : T × Ω → Rn is called a solution of 1.1–1.3 if and only if u
satisfies 1.1, initial value conditions 1.2 and Dirichlet boundary conditions 1.3.
Definition 2.10. A constant vector u∗ u∗1 , u∗2 , . . . , u∗n T is said to be an equilibrium solution to
1.1–1.3, if it satisfies −bi u∗i nj1 cij fj u∗j Ii 0, i 1, 2, . . . , n.
Definition 2.11. The equilibrium solution u∗ u∗1 , u∗2 , . . . , u∗n T of recurrent neural network
1.1–1.3 is said to be globally exponentially stable if there exists a positive constant α ∈ R
and M ≥ 1 such that every solution ut, x u1 t, x, u2 t, x, . . . , un t, xT of 1.1–1.3
satisfies
ut, · − u∗ ≤ Meα t, 0,
t ∈ T
.
2.10
Definition 2.12 Lakshmikantham and Vatsala 25. For each t ∈ T, let N be a neighborhood
of t. Then, for V ∈ Crd T × Rn , R
, define D
V Δ t, xt to mean that, given ε > 0, there exists
a right neighborhood Nε ∩ N of t such that
1 V σt, xσt − V s, xσt − μs, tft, xt < D
V Δ t, xt ε
μt, s
2.11
for each s ∈ Nε , s > t, where μt, s ≡ σt − s. If t is rs and V t, xt is continuous at t, this
reduces to
D
V Δ t, xt V σt, xσt − V t, xσt
.
σt − t
2.12
3. Main Results
In this section, we will consider the existence, uniqueness, and global exponential stability of
equilibrium of 1.1–1.3. To proceed, we need the following lemma.
Lemma 3.1 see 14. Let Ω be a cube |xi | < li i 1, 2, . . . , m and let hx be a real-valued
function belonging to C1 Ω which vanish on the boundary ∂Ω of Ω, that is, hx|∂Ω 0. Then,
h xdx ≤
2
Ω
li2
∂h 2
dx.
Ω ∂xi
3.1
Throughout this paper, we always assume that
H1 fj · is Lipschitz continuous, that is, there exists constant Fj > 0 such that |fj ξ −
fj η| ≤ Fj |ξ − η|, for any ξ, η ∈ R, j 1, 2, . . . , n;
H2 W B − C
F is an M-matrix, where B diagb1 , b2 , . . . , bn , C
|cij |n×n , F diagF1 , F2 , . . . , Fn ,
n
2
H3 −2bi − m
j1 |cij |Fj |cji |Fi < 0, i 1, 2, . . . , n.
k1 2aik /lk Theorem 3.2. Assume that H1 and H2 hold, then 1.1–1.3 has at least one equilibrium
solution u∗ u∗1 , u∗2 , . . . , u∗n T .
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Discrete Dynamics in Nature and Society
Proof. By H1 , it follows that
fj s ≤ Fj |s| fj 0,
j 1, 2, . . . , n, ∀s ∈ R.
3.2
Let
hu, I Bu − Cfu − I 0,
3.3
where fu f1 u1 , f2 u2 , . . . , fn un T . It is obvious that solutions to 3.3 are equilibria of
1.1–1.3. Let us define homotopic mapping
Hu, λ λhu, I 1 − λu,
λ ∈ J 0, 1.
3.4
We have
⎡
⎞⎤
⎛
n
⎠
⎝
⎣
⎦
1 − λui cij fj uj Ii
|Hi u, λ| λ bi ui −
j1
n ≥ 1 λbi − 1|ui | − λ cij Fj uj 3.5
j1
⎤
n − λ⎣|Ii | cij fj 0⎦,
⎡
i 1, 2, . . . , n.
j1
That is,
H ≥ E λB − E, u
− λC
Fu
− λ I C
f 0
1 − λu
λB − C
Fu
− λ I C
f 0 ,
3.6
where H |H1 |, |H2 |, . . . , |Hn |T , u
|u1 |, |u2 |, . . . , |un |T , I |I1 |, |I2 |, . . . , |In |T , f 0 |f1 0|, |f2 0|, . . . , |fn 0|T , and E is an identity matrix.
Since W B − C
F is an M-matrix, we have W −1 B − C
F−1 ≥ 0 nonnegative
matrix and there exists a Q Q1 , Q2 , . . . , Qn T > 0 Qi > 0, i 1, 2, . . . , n such that B −
C
FQ > 0. Let
−1 UR0 u : u
≤ R0 Q B − C
F I C
f 0 .
3.7
Then, UR0 is not empty and it follows from 3.7 that for any u ∈ ∂UR0 boundary of
UR0 ,
−1 H ≥ 1 − λu
λB − C
F u
− B − C
F I C
f 0
1 − λu
λB − C
FQ > 0,
λ ∈ 0, 1,
3.8
Discrete Dynamics in Nature and Society
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which implies that Hu, λ /
0, for u ∈ ∂UR0 , λ ∈ 0, 1. So, from homotopy invariance
theorem, we have
degh, UR0 , 0 degHu, 1, UR0 , 0 degHu, 0, UR0 , 0 1,
3.9
where degh, UR0 , 0 denotes topological degree. By topological degree theory, we can
conclude that 3.3 has at least one solution in UR0 . That is, 1.1–1.3 has at least an
equilibrium solution u∗ u∗1 , u∗2 , . . . , u∗n T . This completes the proof.
Theorem 3.3. Assume that H1 –H3 hold, then the reaction-diffusion recurrent neural network
1.1–1.3 has a unique equilibrium solution u∗ u∗1 , u∗2 , . . . , u∗n T which is globally exponentially
stable.
Proof. The existence of equilibrium solutions for 1.1–1.3 follows from Theorem 3.2. Now
we only need to prove the uniqueness and global exponential stability of equilibrium
solutions for 1.1–1.3.
Suppose that ut, x and vt, x are two arbitrary solutions of 1.1–1.3 with
conditions φu x, φv x ∈ CΩ, Rn , and define zt, x ut, x−vt, x, φz x φu x−φv x,
then zt, x is governed by the following equations:
zΔ
i t, x m
n
∂
∂zi t, x
aik
− bi zi t, x cij fj uj t, x − fj vj t, x ,
∂xk
∂xk
j1
k1
3.10
where i 1, 2, . . . , n. Calculating the delta derivation of zi t, ·22 along the solution of 3.10,
we have, for i 1, 2, . . . , n,
zi t, ·22
Δ
zi t, x
Ω
Ω
2
Δ
dx zi t, x zi σt, xzi t, xΔ dx
2zi t, x μtzi t, xΔ zi t, xΔ dx
2
Ω
Δ
Ω
zi t, xzi t, x dx μt
Ω
zi t, xΔ
2
dx
m ∂
∂zi t, x
2
zi t, x
aik
dx − 2bi
zi t, x2 dx
∂x
∂x
k
k
Ω
k1 Ω
n
2 cij
zi t, x fj uj t, x − fj vj t, x dx
j1
Ω
2
μtzi t, ·Δ .
2
3.11
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Discrete Dynamics in Nature and Society
From Green formula 26, Dirichlet boundary condition, and Lemma 3.1, we have, for i 1, 2, . . . , n,
m k1
Ω
zi t, x
m ∂
∂zi t, x
∂zi t, x 2
aik
dx
aik
dx −
∂xk
∂xk
∂xk
k1 Ω
3.12
m aik
≤−
zi t, x2 dx.
2
l
k1 Ω k
By 3.11, 3.12, condition H1 , and Holder inequality, we get
zi t, ·22
Δ
≤−
m
2aik
k1
lk2
zi t, ·22 − 2bi zi t, ·22
n 2
2 cij Fj zi t, ·2 zj t, ·2 μtzi t, ·Δ 2
j1
−
m
2aik
k1
lk2
3.13
zi t, ·22
−
2bi zi t, ·22
n 2 cij Fj zi t, ·2 zj t, ·2 μtqtzi t, ·22 ,
j1
where zi t, ·Δ 22 qtzi t, ·22 , qt ≥ 0, i 1, 2, . . . , n.
If condition H3 holds, we can always choose a positive number σ > 0 may be very
small such that for i 1, 2, . . . , n,
−2bi −
m
2aik
k1
lk2
n cij Fj cji Fi σ < 0.
3.14
j1
Let us consider functions
m
n 2aik cij Fj cji Fi
qi yi yi ⊕ yi − 2bi −
2
j1
k1 lk
w yi μtqt max eyi ⊕yi σt, 0, ewyi −1μtqtzi t,·22 t, 0
eyi ⊕yi σt, 0
3.15
,
y
where wyi 0 i eyi −s /yi −s2 ds, i 1, 2, . . . , n. From 3.15, we obtain that qi 0 < −σ < 0
and qi yi is continuous for yi ∈ 0, ∞, furthermore, qi yi → ∞ as yi → ∞, thus there
Discrete Dynamics in Nature and Society
9
exist constant εi ∈ 0, ∞ such that qi εi∗ 0 and qi εi < 0, for εi ∈ 0, εi∗ . By choosing
ε min1≤i≤n {εi }, obviously ε > 0, we have, for i 1, 2, . . . , n,
qi ε ε ⊕ ε − 2bi −
m
2aik
k1
lk2
n cij Fj cji Fi
j1
wεμtqt max eε⊕ε σt, 0, ewε−1μtqtzi t,·22 t, 0
eε⊕ε σt, 0
3.16
≤ 0.
Now consider the Lyapunov functional
V t, zt n eε⊕ε t, 0zi t, ·22 ewε−1μtqtzi t,·22 t, 0 .
3.17
i1
Calculating the delta derivatives of V t, zt along the solution of 3.10 and noting
t
that d/dzez t, s s 1/1 μτzΔτez t, s > 0 if and only if z ∈ R
i.e., ez t, s is
increasing with respect to z if and only if z ∈ R
, we have
D
V Δ t, xt
n
Δ
ε ⊕ εeε⊕ε t, 0zi t, ·22 eε⊕ε σt, 0 zi t, ·22
i1
!
wε − 1μtqtzi t, ·22 ewε−1μtqtzi t,·22 t, 0
"
n
≤
ε ⊕ εeε⊕ε t, 0zi t, ·22 eε⊕ε σt, 0
i1
⎛
⎞
m
n 2a
ik
2
2
2
× ⎝−
zi t, ·2 −2bi zi t, ·2 2 cij Fj zi t, ·2 zj t, ·2 μtqtzi t, ·2⎠
2
j1
k1 lk
#
wε −
1μtqtzi t, ·22 ewε−1μtqtzi t,·22 t, 0
"
n
m
2aik
≤ eε⊕ε σt, 0
ε ⊕ εzi t, ·22 −
zi t, ·22 − 2bi zi t, ·22
2
l
i1
k1 k
n cij Fj zi t, ·2 zj t, ·2
2
j1
2
max eε⊕ε σt, 0, ewε−1μtqtzi t,·22 t, 0 wεμtqt
eε⊕ε σt, 0
zi t, ·22 ⎫
⎬
⎭
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Discrete Dynamics in Nature and Society
≤ eε⊕ε σt, 0
n
zi t, ·22
i1
⎧
⎨
ε ⊕ ε −
⎩
m
2aik
2
k1 lk
n cij Fj cji Fi
j1
⎫
max eε⊕ε σt, 0, ewε−1μtqtzi t,·22 t, 0 wεμtqt ⎬
⎭
eε⊕ε σt, 0
≤ 0.
3.18
From 3.17 and 3.18, we have, for t > 0,
eε⊕ε t, 0zt, ·2 eε⊕ε t, 0
n
zi t, ·22 ≤ V t, zt ≤ V 0, z0
i1
n zi 0, ·22
i1
n φz 2 1 φz 2 n,
1 i τ
0
3.19
i1
which implies that
zt, · ≤ Meε t, 0,
where M '
3.20
φz 20 n > 1.
∗∗
∗∗ T
Let u∗ u∗1 , u∗2 , . . . , u∗n T and u∗∗ u∗∗
1 , u2 , . . . , un be two arbitrary equilibrium
∗
∗∗
solutions of system
' 1.1–1.3. According to 3.20, we get u − u ≤ Meε t, 0 → 0 t →
∞, here M u∗ − u∗∗ 20 n > 1. It follows that u∗ u∗∗ , that is, the equilibrium solution
of 1.1–1.3 is unique.
Let ut, x u1 t, x, u2 t, x, . . . , un t, xT and u∗ u∗1 , u∗2 , . . . , u∗n T be arbitrary
solutions and an unique equilibrium solution of '
1.1–1.3, respectively. In the light of 3.20,
we obtain ut, x − u∗ ≤ Meε t, 0, here M φ − u∗ 20 n > 1. Thus, by Definition 2.11,
we obtain the global exponential stability of unique equilibrium solution of 1.1–1.3. The
proof is complete.
4. An Illustrative Example
Example 4.1. Consider the following reaction-diffusion recurrent neural network with
Dirichlet boundary conditions on time scales:
uΔ
i t, x
m
n
∂
∂ui
aik
− bi ui t, x cij fj uj t, x Ii ,
∂x
∂x
k
k
j1
k1
ui 0, x φi x,
ui t, x 0,
x ∈ Ω,
t, x ∈ T × Ω,
4.1
t, x ∈ 0, ∞T × ∂Ω,
where T ⊂ R is a time scale and T ∩ 0, ∞ 0, ∞T is unbounded, f1 v f2 v ev − e−v /ev e−v , Ω {x : |xi | < 1, i 1, 2}, and I I1 , I2 is the constant input vector.
Discrete Dynamics in Nature and Society
11
Obviously, fj v satisfies the Lipschitz condition with Fj 1. Let a11 0.5, a12 0.5, a21 0.3,
a22 0.7, b1 1.5, b2 0.4, c11 0.5, c12 0.4, c21 0.3, and c22 0.2. By simple calculation,
we have
(
B−C F −2b1 −
1.5 0
2
2a1k
lk2
2
2a2k
k1
lk2
(
−
0 0.4
k1
−2b2 −
)
0.5 0.4
0.3 0.2
)(
)
1 0
0 1
(
1
−0.4
−0.3 0.2
2 c1j Fj cj1 F1 −3.3 < 0,
)
,
4.2
j1
2 c2j Fj cj2 F2 −0.9 < 0,
j1
that is, H1 , H2 , and H3 hold. Hence, it follows from Theorem 3.3 that 4.1 has one
unique equilibrium solution which is globally exponentially stable.
Acknowledgment
This work is supported by the National Natural Sciences Foundation of People’s Republic of
China under Grant 10971183.
References
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