Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 748279, 23 pages
doi:10.1155/2012/748279
Research Article
Delay-Dependent Exponential Stability for
Uncertain Neutral Stochastic Systems with Mixed
Delays and Markovian Jumping Parameters
Huabin Chen
Department of Mathematics, Nanchang University, Nanchang 330031, China
Correspondence should be addressed to Huabin Chen, chb 00721@126.com
Received 9 December 2011; Accepted 5 March 2012
Academic Editor: Vimal Singh
Copyright q 2012 Huabin Chen. 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.
This paper is mainly concerned with the globally exponential stability in mean square of uncertain
neutral stochastic systems with mixed delays and Markovian jumping parameters. The mixed
delays are comprised of the discrete interval time-varying delays and the distributed time delays.
Taking the stochastic perturbation and Markovian jumping parameters into account, some delaydependent sufficient conditions for the globally exponential stability in mean square of such
systems can be obtained by constructing an appropriate Lyapunov-Krasovskii functional, which
are given in the form of linear matrix inequalities LMIs. The derived criteria are dependent on
the upper bound and the lower bound of the time-varying delay and the distributed delay and are
therefore less conservative. Two numerical examples are given to illustrate the effectiveness and
applicability of our obtained results.
1. Introduction
It is well known that many dynamical systems not only depend on the present and past
states but also involve the derivative with delays as well as the functional of the past history.
Neutral delay differential equations are often used to describe the following systems 1:
dxt − Dxt − τ
ft, xt, xt − τ.
dt
1.1
Many authors have considered the dynamical analysis of the neutral delay differential
equations see 2–7 and references therein. For example, Chen et al. in 3 and Wu et al.
in 4, 5 have given some LMI-based conditions ensuring the stability analysis and the
2
Discrete Dynamics in Nature and Society
stabilization of neutral delay systems. Taking the environmental disturbances into account,
the neutral stochastic delay differential equations can be given as follows:
dxt − Dxt − τ ft, xt, xt − τdt
gt, xt, xt − τdBt.
1.2
Some fundamental theories of neutral stochastic delay differential equations are introduced
in 1, 8. Since they can be extensively applied into many branches for the control field,
the problem about the exponential stability and the asymptotical stability of the neutral
stochastic delay systems has attracted many authors’ attention over the past few years,
and many less conservative results of delay-dependent conditions ensuring the stabilization
analysis and H∞ filtering design for such systems have been reported in many works, see,
for example, 9–14 and references therein. The methods used include the Razumikhin-type
theorems 10, the Lyapunov functional 13, the fixed point theorem 14, and the linear
matrix inequality 9, 11, 12. For example, Huang and Mao in 9 and Chen et al. in 11 have
given the exponential stability criteria of neutral stochastic delay systems. Some LMI-based
sufficient conditions for the mean-square exponential stability analysis of stochastic systems
of neutral type have been obtained by introducing an auxiliary vector in 12. In practice,
the parameter uncertainty is considered as one of the main sources leading to undesirable
behavior e.g., instability of dynamical systems, especially when implementing neural
networks in applications. The stability analysis of the uncertain neutral stochastic system
has received considerable research attention, see, for example, 15–18, and the problem of
the H∞ filter design of the uncertain neutral stochastic delay systems has been discussed in
19, 20.
On the other hand, Markovian jump systems introduced by 21 are the hybrid
systems with two components in the state. The first one refers to the mode that is
described by a continuous-time finite-state Markovian process, and the second one refers
to the state that is represented by a system of differential equations. The jump systems
have the advantage of modeling the dynamic systems subject to abrupt variation in
their structures, such as component failures or repairs, sudden environmental disturbance,
changing subsystem interconnections, and operating in different points of a nonlinear plant
22. The stability analysis and H∞ filter design of stochastic delay systems with Markovian
jumping parameters and delay systems with Markovian jumping parameters have been
widely studied, see, for example, 23–38. For example, in 24, Liu et al. have discussed
the exponential stability of delayed recurrent neural networks with Markovian jumping
parameters; Liu et al. in 24 and Wang et al. in 25 have considered some sufficient
conditions for the exponential stability of stochastic neural networks with mixed time delays
and Markovian switching; Mao in 13 has also given some sufficient conditions for the
exponential stability of stochastic delay interval systems with Markovian switching. More
recently, He and Liu in 39 and Balasubramaniam et al. in 40 have presented some LMIbased sufficient conditions for the exponential stability of uncertain neutral systems with
Markovian jumping parameters. Although Kolmanovskii et al. in 17, and Mao et al. in
18 have derived the exponential stability of the neutral stochastic delay systems with
Markovian jumping parameters, some sufficient conditions obtained by using the estimate
method are not easily checked. Thus, the problem of the stability analysis of the uncertain
neutral stochastic systems with mixed delays and Markovian jumping parameters has not
been fully investigated and there is still much room left for further consideration, which
constitutes the motivation for the present research.
Discrete Dynamics in Nature and Society
3
In this paper, the global exponential stability of a class of the uncertain neutral
stochastic systems with mixed delays and Markovian jumping parameters is discussed.
The delays include the discrete and distributed time delays, and the jumping parameters
are generated from a finite state Markov chain. By constructing an appropriate Lyapunov
functional, some LMIs-based sufficient conditions ensuring the exponential stability in mean
square of the uncertain neutral stochastic systems with mixed delays and Markovian jumping
parameters are obtained by using the stochastic analysis and some bounding technique. It is
worth pointing out that compared with the earlier works in 17, 18, the obtained results
given in the form of the linear matrix inequalities LMIs can be easily be solved by using
the standard software packages. Two illustrative examples are exploited to demonstrate the
effectiveness and applicability of the obtained results.
The content of the paper is arranged as follows. In Section 2, some necessary notations,
definitions, and lemmas will be introduced. In Section 3, we mainly study the exponential
stability in mean square of the uncertain neutral stochastic systems with mixed delays and
Markovian jumping parameters. Two illustrative numerical examples are given to show the
power of our obtained results in Section 4.
Notations. Unless otherwise specified, for a real square matrix A, the matrix A > 0 A ≥ 0,
A < 0, A ≤ 0 means that A is a positive definite positive semidefinite, negative definite,
and negative semidefinite, resp.; λmax A and λmin A denote the maximum and minimum
eigenvalues of the square matrix A, respectively. Let Ω, , {t }t≥0 , P be a probability space
with a natural {t }t≥0 and let E{·} stand for the mathematical expectation operator with
respect to
this probability measure. If A is a vector or matrix, its transpose is denoted by
AT . |B| traceBT B denotes the Euclidean norm of a vector B and its induced norm of a
matrix B. Unless explicitly stated, matrices are assumed to have real entries and compatible
dimensions. Let τ > 0 and C−τ, 0; Rn be the family of all continuous Rn -valued functions
φ on the interval −τ, 0 with the norm φ sup{|φθ| : −τ ≤ θ ≤ 0}. Denote by
L20 −τ, 0; Rn the family of all 0 -measurable C−τ, 0; Rn -valued random variables ξ {ξθ : −τ ≤ θ ≤ 0} such that supθ∈−τ,0 E|ξθ|2 < ∞. Bt B1 t, B2 t, . . . , Bn tT t ≥ 0
is an n-dimensional standard Brownian motion defined on the completed probability space
Ω, , P .
2. Problem Formulation
Let rt t ≥ 0 be a right-continuous Markov chain on the probability space taking values in
a finite state space S {1, 2, . . . , N} with generator Γ {γij }N×N given by
P rt
Δ j | rt i ⎧
⎨γij Δ
⎩
1
oΔ,
γii Δ
oΔ,
if i /
j,
if i j,
2.1
where Δ > 0. Here, γij ≥ 0 is the transition rate from i to j if i /
j while
γii −
γij .
j/
i
2.2
4
Discrete Dynamics in Nature and Society
Now, we assume that the Markov chain r· is independent of the Brownian motion B·. It is
well known that almost every sample path of rt is a right-continuous step function with a
finite number of simple jumps in any finite subinterval of R .
In this paper, we will consider the following uncertain neutral stochastic systems with
mixed delays and Markovian switching:
dxt − Drtxt − dt A1 rt, txt
A4 rt, t
A2 rt, tfxt
A3 rt, tfxt − dt
t
fxsds dt
t−σt
σ t, xt, xt − dt,
t
xsds, rt dBt,
t ≥ 0,
t−σt
2.3
with the initial value x0 ϕ ∈ L20 −τ, 0, Rn τ max{d2 , σ}, where xt ∈ Rn is the system
state vector associated with the neurons and dt and σt are the time-varying delays. Here,
we assume that the Markov chain r· is independent of the Brownian motion Bt t ≥ 0.
f : Rn → Rn is neuron activation function, and the noise perturbation σ : R × Rn × Rn × Rn ×
S → Rn×n is the noise intensity matrix. When rt i i ∈ S, Drt, A1 rt, t, A2 rt, t,
A3 rt, t, and A4 rt, t are, respectively, denoted as Di , A1i t, A2i t, A3i t, and A4i t,
and Di i ∈ S are known matrices with |Di | < 1 i ∈ S. A1i t, A2i t, A3i t, and A4i t are
matrix functions with time-varying uncertainties, that is,
A1i t A1i
ΔA1i t,
A2i t A2i
ΔA2i t,
A3i t A3i
ΔA3i t,
A4i t A4i
ΔA4i t,
2.4
where A1i , A2i , A3i , and A4i i ∈ S are known real constant matrices and ΔA1i t, ΔA2i t,
ΔA3i t, and ΔA4i t i ∈ S are unknown matrices representing time-varying parameter
uncertainties in system model. We assume that the uncertainties are norm-bounded and can
be described as
ΔA1i t ΔA2i t ΔA3i t ΔA4i t Mi Fi tN1i N2i N3i N4i ,
FiT tFi t ≤ I,
i ∈ S,
2.5
where Mi , N1i , N2i , N3i , and N4i i ∈ S are known real matrices and Fi t i ∈ S is unknown
real and possibly time-varying matrix for any given t. It is assumed that the elements of
Discrete Dynamics in Nature and Society
5
Fi t i ∈ S are Lebesgue measurable. When Fi t 0 i ∈ S, systems 2.3 have the following nominal case:
dxt − Di xt − dt A1i xt
A2i fxt
A3i fxt − dt
t
A4i
fxsds dt
t−σt
σ t, xt, xt − dt,
t
t ≥ 0.
xsds, i dBt,
t−σt
2.6
In order to obtain our results, we need some assumptions as follows.
Assumption 2.1. The neuron activation functions f· in 2.3 or 2.6 are bounded and satisfy the following Lipschitz condition:
fx − f y ≤ L x − y ,
∀x, y ∈ Rn ,
2.7
where L ∈ Rn×n is known constant matrix and f0 0.
Assumption 2.2. The noise perturbation σ satisfies the following condition:
trace σ T
t, xt, xt − dt,
t
xsds, i σ t, xt, xt − dt,
t−σt
≤x
T
tRT1i R1i xt
x t −
T
t
xsds, i
t−σt
dtRT2i R2i xt
− dt
T
t
xsds
t−σt
RT3i R3i
t
xsds ,
t−σt
2.8
where R1i , R2i , and R3i are known constant matrices with appropriate dimensions and σt, 0,
0, 0, i 0 i ∈ S.
Remark 2.3. Under Assumptions 2.1 and 2.2, it is easily shown that the system 2.3 with
uncertainties 2.4 admits a unique trivial solution when the initial data ξ 0. The readers
can refer to 41.
Assumption 2.4. dt and σt are two time-varying continuous functions that satisfy
0 ≤ d1 ≤ dt ≤ d2 ,
ḋt ≤ μ < 1,
0 ≤ σt ≤ σ,
σ̇t ≤ ν < 1,
where d1 and d2 are the lower and upper bounds of the time delay dt, respectively.
We present the definitions and three useful lemmas as follows.
2.9
6
Discrete Dynamics in Nature and Society
Definition 2.5. The neutral stochastic systems with mixed delays and Markovian jump
parameters 2.6 is said to be exponentially stable in mean square if there exist a pair of
positive scalar α > 0 and l > 0 such that every solution xt, ξ, i of systems 2.6 satisfies
2
E|xt, ξ, i|2 ≤ l sup Eϕs e−αt ,
∀t ≥ 0,
s∈−τ,0
2.10
for any ϕ ∈ L20 −τ, 0, Rn .
Definition 2.6. The uncertain neutral stochastic systems with mixed delays and Markovian
jump parameters 2.3 are said to be exponentially stable in mean square if 2.10 holds for
all admissible uncertainties 2.5.
Lemma 2.7 see 8. For any vectors a, b ∈ Rn , the inequality
±2aT b ≤ aT Xa
bT X −1 b
2.11
holds, in which X is any n × n matrix with X > 0.
Lemma 2.8 see 8. Let X ∈ Rn×n ; then
λmin XaT b ≤ aT Xb ≤ λmax XaT b
2.12
for any a ∈ Rn if X is a symmetric matrix.
Lemma 2.9 see 42 Schur complement. For a given matrix
S
S11 S12
2.13
ST12 S22
with S11 ST11 , S22 ST22 , the following conditions are equivalent:
1 S < 0,
T
2 S22 < 0, S11 − S12 S−1
22 S12 < 0,
3 S11 < 0, S22 − ST12 S−1
11 S12 < 0.
Lemma 2.10 see 42. Let U, V , W, and M be real matrices of appropriate dimensions with M
satisfying M MT ; then
M
UV W
W T V T UT < 0,
∀V T V ≤ I,
2.14
if and only if there exist a scalar ε > 0 such that
M
ε−1 UUT
εW T W < 0.
2.15
Discrete Dynamics in Nature and Society
7
Lemma 2.11 see 43. For any positive symmetric constant matrix M ∈ Rn×n and a scalar γ > 0,
a vector function ω : 0, γ → Rn such that the integrations concerned are well defined, and then the
following inequality holds:
γ
T ωsds M
0
γ
ωsds ≤ γ
γ
0
ωT sMωsds.
2.16
0
3. Main Results
Theorem 3.1. Suppose that Assumptions 2.1–2.4 hold and for any given positive scalar κ ∈ 0, 1,
the neutral stochastic systems with mixed delays and Markovian switching 2.6 are exponentially
stable in mean square if there exist λi > 0 (i ∈ S) and some positive definite matrices Pi > 0 (i ∈ S)
and Ql > 0 (l 1, 2, . . . , 10) such that the following linear matrix inequalities (LMIs) are satisfied: for
i ∈ S,
Pi ≤ λi I,
3.1
⎤
⎡
Ωi11 Ωi12 0 Pi A2i Pi A3i Pi A4i 0
0
0
⎥
⎢
⎢ ∗ Ωi22 0
0
0
0
Ωi27 Ωi28 Ωi29 ⎥
⎥
⎢
⎥
⎢
⎢ ∗
∗ Ωi33
0
0
0
0
0
0 ⎥
⎥
⎢
⎥
⎢
⎥
⎢ ∗
∗
∗
−Q
0
0
0
0
0
5
⎥
⎢
⎥
⎢
⎥ < 0,
∗
∗
∗
∗
−Q
Ωi ⎢
0
0
0
0
6
⎥
⎢
⎥
⎢
⎢ ∗
∗
∗
∗
∗
−Q7
0
0
0 ⎥
⎥
⎢
⎥
⎢
⎢ ∗
∗
∗
∗
∗
∗
−Q8 0
0 ⎥
⎥
⎢
⎥
⎢
⎥
⎢ ∗
∗
∗
∗
∗
∗
∗
−Q
0
9
⎦
⎣
∗
∗
∗
∗
∗
∗
∗
∗ −Q10
3.2
where ∗ denotes the entries that are readily inferred by symmetry of a symmetric matrix and
Ωi11 Pi A1i
AT1i Pi
N
γij Pj
λi RT1i R1i
LT Q5
Q8 L
Q1
d2 Q2
j1
Ωi12 AT1i Pi Di −
N
1 2
d2 − d12 Q3
2
σ 2 Q4 ,
γij Pj Di ,
j1
Ωi22 LT Q6
Q9 L
N
γij DiT Pj Di
λi RT2i R2i − 1 − uQ1 ,
j1
Ωi33 LT Q7
Ωi27 DiT Pi A2i ,
Q10 L
λi RT3i R3i − κ1 − vQ5 ,
Ωi28 DiT Pi A3i ,
Ωi29 DiT Pi A4i .
3.3
8
Discrete Dynamics in Nature and Society
Proof. Denote by C2,1 R × Rn × S; Rn the family of all nonnegative functions V t, x, i on R ×
Rn × S that are once differentiable with respect to the first variable t and twice differentiable
with respect to the second variable x. To obtain the stability conditions, we consider the
following Lyapunov functional:
V t, xt, i V1 t, xt, i
V2 t, xt, i
3.4
V3 t, xt, i,
where
V1 t, xt, i xt − Di xt − dtT Pi xt − Di xt − dt,
V2 t, xt, i t
xT sQ1 xsds
t−dt
V3 t, xt, i −d1
−d2
t
t s
t
t
t
t−dt
s
xT uQ3 xudu dθ ds
σ
θ
xT θQ2 xθdθ ds,
0
t
−σt
t s
3.5
xT θQ4 xθdθ ds.
The weak infinitesimal operator LV 17 along 2.6 from R × Rn × S to R is given by
LV t, xt, i LV1 t, xt, i
LV2 t, xt, i
LV3 t, xt, i,
3.6
where
LV1 t, xt, i
⎡
xT t⎣Pi A1i
AT1i Pi
⎤
N
γij Pj ⎦xt
2xT tPi A2i fxt
j1
2xT tPi A3i fxt − dt
2xT tPi A4i
t
fxsds
t−σt
2xT t − dtDiT Pi A1i xt − 2xT t − dtDiT Pi A2i fxt
− 2xT t − dtDiT Pi A3i fxt − dt − 2xT t − dtDiT Pi A4i
⎡
⎤
N
xT t⎣− γij Pj Di ⎦xt − dt
⎡
t
fxsds
t−σt
⎤
N
xT t − dt⎣− γij DiT Pi ⎦xt
j1
j1
⎡
⎤
N
xT t − dt⎣ DiT Pj Di ⎦xt − dt
j1
trace σ T t, xt, xt − dt,
t
t−σt
xsds V1xx σ t, xt, xt − dt,
t
xsds
.
t−σt
3.7
Discrete Dynamics in Nature and Society
9
By Lemma 2.7, we have
xT tPi A2i fxt, i ≤ xT tPi A2i Q5−1 AT2i Pi xt
f T xtQ5 fxt
≤ xT tPi A2i Q5−1 AT2i Pi xt
3.8
xT tLT Q5 Lxt,
2xT tPi A3i fxt − dt, i ≤ xT tPi A3i Q6−1 AT3i Pi xt
f T xt − dt, iQ6 fxt − dt, i
3.9
≤ xT tPi A3i Q6−1 AT3i Pi xt
xT t − dtLT Q6 Lxt − dt,
−2xT t − dtDiT Pi A2i fxt, i ≤ xT t − dtDiT Pi A2i Q8−1 AT2i Pi Di xt − dt
f T xtQ8 fxt
≤ xT t − dtDiT Pi A2i Q8−1 AT2i Pi Di xt − dt
3.10
xT tLT Q8 Lxt,
−2xT t − dtDiT Pi A3i fxt − dt, i ≤ xT t − dtDiT Pi A3i Q9−1 AT3i Pi Di xt − dt
f T xt − dtQ9 fxt − dt
≤ xT t − dtDiT Pi A3i Q9−1 AT3i Pi Di xt − dt
3.11
xT t − dtLT Q9 Lxt − dt.
From Lemmas 2.7 and 2.11, it implies that
2xT tPi A4i
t
t−σt
fxsds ≤ xT tPi A4i Q7−1 AT4i Pi xt
T
t
fxsds
t
Q7
t−σt
fxsds
t−σt
≤ xT tPi A4i Q7−1 AT4i Pi xt
T
t
LT Q7 L
xsds
t−σt
t
xsds ,
t−σt
3.12
−2xT t − dtDiT Pi A4i
t
t−σt
−1 T
fxsds ≤ xT t − dtDiT Pi A4i Q10
A4i Pi Di xt − dt
T
t
fxsds
t
Q10
t−σt
fxsds
t−σt
−1 T
≤ xT t − dtPi A4i Q10
A4i Pi xt − dt
T
t
xsds
t−σt
LT Q10 L
t
xsds .
t−σt
3.13
10
Discrete Dynamics in Nature and Society
Substituting 3.8–3.13 into 3.7 and from Assumption 2.2, it follows that
LV1 t, xt, i
⎡
≤ xT t⎣Pi A1i
N
γij Pj
AT1i Pi
λi RT1i R1i
LT Q5
Q8 L
j1
Pi A2i Q5−1 AT2i Pi
⎡
xT t⎣AT1i Pi Di −
N
⎤
Pi A4i Q7−1 AT4i Pi ⎦xt
Pi A3i tQ6−1 A−T
3i Pi
⎤
γij Pj Di ⎦xt − dt
xT t − dt
j1
⎤
N
× ⎣DiT Pi A1i − γij DiT Pj ⎦xt
⎡
xt − dt
j1
⎡
× ⎣DiT Pi A2i Q8−1 AT2i Pi Di
DiT Pi A3i Q9−1 AT3i Pi Di
−1 T
DiT Pi A4i Q10
A4i Pi Di
L Q6
N
γij DiT Pj Di
Q9 L
T
⎤
λi RT2i R2i ⎦xt
− dt
j1
T
t
xsds
LT Q7
Q10 L
t−σt
λi RT3i R3i
t
xsds .
t−σt
3.14
On the other hand, we can obtain
LV2 t, xt, i ≤ xT Q1
d2 Q2 xt − 1 − μ xT t − dtQ1 xt − dt
− 1−μ
t
t−dt
1 2
d2 − d12 Q3
LV3 t, xt, i ≤ xT t
2
− σ1 − ν
3.15
xT sQ2 xsds,
t
σ 2 Q4 xt −
−d1
t
−d2
t s
−d1
t
−d2
t s
xT θQ3 xθdθ ds
xT sQ5 xsds
t−σt
1 2
d2 − d12 Q3
≤ xT t
2
− σ1 − κ1 − ν
σ 2 Q4 xt −
t
xT θQ3 xθdθ ds
xT sQ5 xsds
t−σt
− κ1 − ν
T
t
xsds
t−σt
t
Q5
xsds .
t−σt
3.16
Discrete Dynamics in Nature and Society
11
Substituting 3.14–3.16 into 3.6, we have
LV t, xt, i
⎡
≤ xT t⎣Pi A1i
AT1i Pi
N
γij Pj
LT Q5
λi RT1i R1i
Q8 L
Pi A2i Q5−1 AT2i Pi
d2 Q2
1 2
d2 − d12 Q3
2
j1
Pi A3i tQ6−1 AT3i Pi
Pi A4i Q7−1 AT4i Pi
⎡
⎤
N
xT t⎣AT1i Pi Di − γij Pj Di ⎦xt − dt
Q1
⎤
σ 2 Q4 ⎦xt
⎡
⎤
N
xt − dt⎣DiT Pi AT1i − γij DiT Pj ⎦xt
j1
j1
⎡
xT t − dt⎣LT Q6
DiT Pi A2i Q8−1 AT2i Pi Di
Q9 L
−1 T
DiT Pi A4i Q10
A4i Pi Di
λi RT2i R2i
DiT Pi A3i Q9−1 AT3i Pi Di
⎤
N
γij DiT Pj Di − 1 − μ Q1 ⎦xt − dt
j1
T
t
xsds
LT Q7
Q10 L
t−σt
− 1−μ
t
xT sQ2 xsds −
t−dt
− σ1 − κ1 − ν
t
λi RT3i R3i − κ1 − νQ4
−d1
t
−d2
t s
t
xsds
t−σt
xT θQ3 xθdθ ds
xT sQ5 xsds
t−σt
≤ ξT tΠi ξt − 1 − μ
t
xT sQ2 xsds −
t−dt
− σ1 − κ1 − ν
t
−d1
t
−d2
t s
xT θQ3 xθdθ ds
xT sQ4 xsds,
t−σt
3.17
where ξt xT t xT t − dt
t
T
t−σt
xT sds and, for i ∈ S,
⎡
⎤
Π11 Π12 0
⎢
⎥
⎥
Πi ⎢
⎣ ∗ Π22 0 ⎦,
∗
∗ Π33
Π11 Pi A1i
AT1i Pi
N
γij Pj
λi RT1i R1i
LT Q5
Q8 L
j1
Pi A3i tQ6−1 AT3i Pi
Pi A4i Q7−1 AT4i Pi
Q1
d2 Q2
Pi A2i Q5−1 AT2i Pi
1 2
d2 − d12 Q3
2
σ 2 Q4 ,
12
Discrete Dynamics in Nature and Society
Π12 AT1i Pi Di −
N
Pj Di ,
j1
Π22 LT Q6
DiT Pi A2i Q8−1 AT2i Pi Di
Q9 L
−1 T
DiT Pi A4i Q10
A4i Pi Di
DiT Pi A3i Q9−1 AT3i Pi Di
N
λi RT2i R2i
γij DiT Pj Di − 1 − μ Q1 ,
j1
Π33 L Q7
Q10 L
T
λi RT3i R3i
− κ1 − νQ4 .
3.18
In view of 3.2, we have Πi < 0, for i ∈ S. By the Lyapunov functional V t, xt, i,
t
V t, xt, i xt − Di xt − dtT Pi xt − Di xt − dt
xT sQ1 xsds
t−dt
σ
t
t
t−dt
s
xT θQ2 xθdθ ds
0
t
−σt
t s
t
−d2
t s
t
t
−d2
t s
θ
xT uQ3 xudu dθ ds
xT θQ4 xθdθ ds
t
≤ ξT tΠi ξt
−d1
−d1
xT sQ1
d2 Q2 xsds
σ2
t−dt
t
xT sQ4 xsds
t−σt
xT θd2 Q3 xθdθ ds,
3.19
where
⎡
⎤
0
⎥
⎢ T
T
⎥.
−D
Πi ⎢
P
D
P
D
0
i
i
i
i
i
⎣
⎦
0
0
0
−Pi Di
Pi
3.20
Letting α > 0, for system 2.6, we can define another operator Leαt V t, xt, i : R ×
R × S → R as follows:
n
L eαt V t, xt, i eαt αV t, xt, i
≤ eαt ξT t αΠi
−d1
t
−d2
t s
t
t−σt
LV t, xt, i
Πi ξt
t
xT s αQ1
d2 Q2 − 1 − μ Q2 xsds
t−dt
xT θαd2 − 1Q3 xθdθ ds
xT s ασ 2 − σ1 − κ1 − ν Q4 ds.
3.21
Discrete Dynamics in Nature and Society
13
Now, we can choose α > 0 sufficiently small such that
αλmax Πi
αQ1
λmax Πi < 0,
d2 Q2 − 1 − μ Q2 < 0,
3.22
αd2 − 1 < 0,
ασ 2 − σ1 − κ1 − ν < 0,
for i ∈ S.
By the weak infinitesimal operator along 2.6, it is obtained from 3.21 and 3.22
that
t
eαt EV t, xt, i ≤ EV 0, x0, i
ELeαs V s, xs, ids
3.23
0
≤ EV 0, x0, i.
Using the definition of the Lyapunov functional 3.4 again, we have
EV 0, x0, i Ex0 − Di x−d0T Pi x0 − Di x−d0
0
0
−d0
σ
ExT θQ2 xθdθ ds
s
0
0
−σ0
s
≤ 2λmax Πi −d1
0
0
−d2
s
θ
0
−d0
ExT sQ1 xsds
ExT uQ3 xudu dθ ds
xT θQ4 xθdθ ds
d2 λmax Q1 τ 2 λmax Q2 d2 − d1 τ 2 λmax Q3 σ 3 λmax Q4 2
× sup Eϕθ
θ∈−τ,0
Mi ,
3.24
for i ∈ S.
Thus, from 3.23 and 3.24, it follows that
E|xt − Di xt − dt|2 ≤
where Θ mini∈S {λmin Pi } and M maxi∈S {Mi }.
1
Me−αt ,
Θ
3.25
14
Discrete Dynamics in Nature and Society
From |Di | < 1 i ∈ S, we obtain that there exist a positive scalar l > 0 such that
l maxi∈S |Di | < 1. So, for all ε ∈ 0, min{α, −2/τ log |l|} and any θ > 0, by using the
elementary inequality, we derive
Di xt − dt|2
eεt E|xt|2 eεt E|xt − Di xt − dt
≤ 1
≤
1
≤
1
θe E|xt − Di xt − dt|
εt
θ
Θ
θ
Θ
Me
−α−εt
!
1 εt
e E|Di xt − dt|2
θ
1
!
1 εt
e E|Di xt − dt|2
θ
1
l2 1
M
2
3.26
!
1 ετ εt−τt
E|xt − dt|2 .
e e
θ
From for all ε ∈ 0, min{α, −2/τ log |l|}, we have l2 eετ < 1. Thus, we can choose θ sufficiently large such that
Δl
2
!
1 ετ
e < 1.
θ
1
3.27
So,
eεt E|xt|2 ≤
1
θ
Θ
M
Δeεt−dt E|xt − dt|2 .
3.28
For all T > 0, from 3.28, it follows that
1
sup eεt E|xt|2 ≤
0≤t≤T
θ
Θ
M
2
ΔEϕ
Δ sup eεt E|xt|2 ,
0≤t≤T
3.29
that is,
1
sup eεt E|xt|2 ≤
0≤t≤T
When T →
2
θ/ΘM ΔEϕ
.
1−Δ
3.30
∞, it follows from 3.30 that
1
sup eεt E|xt|2 ≤
0≤t<∞
2
θ/ΘM ΔEϕ
.
1−Δ
3.31
So, we can obtain
2
E|xt| ≤
1
The proof of this theorem is completed.
2
θ/ΘM ΔEϕ −εt
e .
1−Δ
3.32
Discrete Dynamics in Nature and Society
15
Theorem 3.2. Suppose that Assumptions 2.1–2.4 hold and for any given positive scalar κ ∈ 0, 1, the
uncertain neutral stochastic systems with mixed delays and Markovian switching 2.3 are robustly
exponentially stable in mean square if there exist λi > 0 (i ∈ S), ε1i > 0, ε2i > 0 (i ∈ S), and some
positive definite matrices Pi > 0 (i ∈ S) and Ql > 0 (l 1, 2, . . . , 10) such that the following linear
matrix inequalities (LMIs) are satisfied:
i ∈ S,
Pi < λi I,
⎡
Ω11 Ω12 0 Ω14 Ω15
⎢
⎢ ∗ Ω22 0
0
0
⎢
⎢
⎢ ∗
∗ Ω33 0
0
⎢
⎢
⎢ ∗
∗
∗ Ω44 Ω45
⎢
⎢
⎢ ∗
∗
∗
∗ Ω55
⎢
⎢
⎢
Ωi ⎢ ∗
∗
∗
∗
∗
⎢
⎢ ∗
∗
∗
∗
∗
⎢
⎢
⎢ ∗
∗
∗
∗
∗
⎢
⎢
⎢ ∗
∗
∗
∗
∗
⎢
⎢
⎢ ∗
∗
∗
∗
∗
⎣
∗
∗
∗
∗
∗
Ω16 Ω17 Ω18 Ω19 Ω110
Ω27 Ω28 Ω29
0
0
0
0
0
Ω46
0
0
0
Ω56
0
0
0
Ω66
0
0
0
∗
Ω77 Ω78 Ω79
∗
∗
Ω88 Ω89
∗
∗
∗
Ω99
∗
∗
∗
∗
∗
∗
∗
∗
0
⎤
⎥
Ω211 ⎥
⎥
⎥
0
0 ⎥
⎥
⎥
0
0 ⎥
⎥
⎥
0
0 ⎥
⎥
⎥
0
0 ⎥
⎥ < 0,
⎥
0
0 ⎥
⎥
⎥
0
0 ⎥
⎥
⎥
0
0 ⎥
⎥
⎥
−ε1i I 0 ⎥
⎦
0
∗
i ∈ S,
3.33
−ε2i I
where ∗ denotes the entries that are readily inferred by symmetry of a symmetric matrix and
Ω11 Pi A1i
N
AT1i Pi
γij Pj
λi RT1i R1i
LT Q5
Q8 L
1 2
d2 − d12 Q3
2
d2 Q2
Q1
j1
ε1i
σ 2 Q4
Ω12 AT1i Pi Di −
T
ε2i N1i
N1i ,
N
γij Pj Di ,
T
Ω14 Pi A2i ε1i N1i
N2i ,
Ω15 Pi A3i
T
ε1i N1i
N3i , Ω110 Pi Mi ,
j1
Ω16 Pi A4i
Ω22 LT Q6
T
ε1i N1i
N4i ,
Q9 L
N
T
Ω17 ε2i N1i
N2i ,
γij DiT Pj Di
T
Ω18 ε2i N1i
N3i ,
λi RT2i R2i − 1 − μ Q1 ,
T
Ω19 ε2i N1i
N4i ,
Ω27 DiT Pi A2i ,
j1
Ω28 DiT Pi A3i ,
Ω33 LT Q7
Ω29 DiT Pi A4i ,
Q10 L
T
Ω46 ε1i N2i
N4i ,
Ω211 DiT Pi Mi ,
λi RT3i R3i − κ1 − υQ4 ,
Ω55 −Q6
Ω44 −Q5
T
ε1i N2i
N2i ,
T
ε1i N3i
N3i ,
T
Ω56 ε1i N3i
N4i ,
Ω77 −Q8
T
ε2i N2i
N2i ,
T
Ω78 ε2i N2i
N3i ,
T
Ω79 ε2i N2i
N4i ,
Ω88 −Q9
T
ε2i N3i
N3i ,
T
Ω89 ε2i N3i
N4i ,
Ω99 −Q10
T
Ω45 ε1i N2i
N3i ,
Ω66 −Q7
T
ε2i N4i
N4i .
T
ε1i N4i
N4i ,
3.34
16
Discrete Dynamics in Nature and Society
Proof. Replacing A1i , A2i , A3i , and A4i in 3.2 with A1i ΔA1i t, A2i ΔA2i t, A3i ΔA3i t,
and A4i ΔA4i t, ΔA1i t, ΔA2i t, ΔA3i t, ΔA4i t are described in 2.4 and 2.5, in view
of Lemma 2.9 and Lemma 2.10, we obtain
Πi t Πi
Πi
ΞTi Fi tE1i
−1
ε1i
Ξi ΞTi
T T
E1i
Fi tΞi
T
ε1i E1i
E1i
ΨTi Fi tE2i
−1
ε2i
Ψi ΨTi
T T
E2i
Fi tΨi
3.35
T
ε2i E2i
E2i
Ωi < 0,
where
T
Ξi MiT Pi 0 0 0 0 0 0 0 0 ,
T
T
T
T
T
0 0 N2i
N3i
N4i
0 0 0 ,
E1i N1i
T
Ψi 0 MiT Pi Di 0 0 0 0 0 0 0 ,
T
3.36
T
T
T
T
E2i N1i
0 0 0 0 0 N2i
N3i
N4i
,
where ε1i > 0 and ε2i > 0, for any i ∈ S. The proof of the remainder can be easily finished by
following a similar line as in the proof of Theorem 3.1. The proof is completed.
Remark 3.3. The delay-dependent sufficient conditions ensuring the robustly exponential
stability in mean square of the uncertain neutral stochastic systems with mixed delays and
Markovian switching 2.3 are provided in Theorem 3.2. It should be pointed out that such
conditions are given in the form of LMIs, which could be easily solved by using the standard
software packages. Besides, the criteria derived are dependent upon both the upper and
lower bound of the time-varying delay and the distributed delay, which are less conservative.
Remark 3.4. Besides, by the Borel-Cantelli Lemma, we can also obtain the almost surely
exponential stability of systems 2.3. Here, for the sake of brevity, we omit it and the readers
can refer to 17. Besides, we can easily come to a conclusion that the uncertain neutral
stochastic delay systems with mixed delays and Markovian jumping parameters 2.3 are
asymptotically stable in mean square from the conditions |Di | < 1 i ∈ S. Thus, the results
given in 15 are generalized.
Case 1. Consider the problem of delay-dependent robust exponential stability for a special
case of the uncertain neutral stochastic systems with mixed delays and Markovian switching
in 2.6, that is,
dxt − Di xt − dt
A1i t xt
A2i tfxt A3i tfxt − dt
A4i t
t
fxsds dt
t−σt
A5i txt
A6i txt − dt
A7i t
t
xsds dBt,
t ≥ 0,
t−σt
3.37
Discrete Dynamics in Nature and Society
17
where A1i t, A2i t, A3i t, and A4i t are given in 2.3, while A5i t, A6i t, and A7i t are of
the following form:
A5i t A6i t A7i t A5i A6i A7i 3.38
Mi Fi tN5i N6i N7i ,
where A5i , A6i , A7i , Mi , N5i , N6i , and N7i i ∈ S are known constant matrices. For systems
3.37, we can also obtain the robust exponential stability. The proof can be easily established
by following a similar line as in the proof of Theorem 3.1 and then is omitted here.
Theorem 3.5. Suppose that Assumptions 2.1 and 2.2 hold and for any given positive scalar κ ∈
0, 1, the uncertain neutral stochastic systems with mixed delays and Markovian switching 3.37
are robustly exponentially stable in mean square if there exist λi > 0, ε1i > 0, ε2i > 0, ε3i > 0 (i ∈ S)
and some positive definite matrices Pi > 0 (i ∈ S) and Qi > 0 (i 1, 2, . . . , 10) such that the following
linear matrix inequalities (LMIs) are satisfied:
Pi ≤ λi I,
⎡
Γ11
⎢
⎢∗
⎢
⎢
⎢∗
⎢
⎢
⎢∗
⎢
⎢
⎢∗
⎢
⎢
⎢∗
⎢
⎢
⎢
⎢∗
⎢
⎢
⎢∗
⎢
⎢
⎢∗
⎢
⎢
⎢∗
⎢
⎢
⎢∗
⎢
⎢
⎢∗
⎣
∗
Γ12 Γ13 Γ14 Γ15 Γ16
Γ22 Γ23
0
0
0
0
0
0
0
0
i ∈ S,
0
AT5i Pi Pi Mi
0
Γ27 Γ28 Γ29 AT6i Pi
0
DiT Pi Mi
∗
Γ33
∗
∗
∗
∗
∗
∗
∗
∗
∗
Γ66
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
0
0
0
AT7i Pi
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Γ99
0
0
0
∗
∗
−Pi
0
0
∗
∗
∗
∗
−ε1i I
0
∗
∗
∗
∗
∗
∗
−ε2i I
∗
∗
∗
∗
∗
∗
∗
Γ44 Γ45 Γ46
Γ55 Γ56
Γ77 Γ78 Γ79
Γ88 Γ89
⎤
0
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥
⎥
0 ⎥ < 0,
⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥
Pi Mi ⎥
⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎦
−ε3i I
where ∗ denotes the entries that are readily inferred by symmetry of a symmetric matrix and
Γ11 Pi A1i
AT1i Pi
N
LT Q5
γij Pj
Q8 L
Q1
d2 Q2
j1
T
ε1i N1i
N1i
Γ12 AT1i Pi Di −
N
σ 2 Q5
T
ε3i N5i
N5i ,
γij Pj DiT
T
ε3i N5i
N6i ,
T
Γ13 ε3i N5i
N7i ,
j1
Γ15 Pi A3i
1 2
d2 − d12 Q3
2
T
ε1i N1i
N3i ,
Γ16 Pi A4i
T
ε1i N1i
N4i ,
Γ14 Pi AT2i
T
ε1i N1i
N2i ,
3.39
18
Discrete Dynamics in Nature and Society
Γ22 LT Q6
Q9 L
N
γij DiT Pj Di − 1 − μ Q1
T
Γ23 ε3i N6i
N7i ,
T
ε3i N6i
N6i ,
j1
Γ27 DiT Pi A2i ,
Γ33 LT Q7
Γ28 DiT Pi A3i ,
Q10 L − κ1 − νQ5
T
N4i ,
Γ46 ε1i N2i
Γ77 −Q8
Γ55 −Q6
T
ε2i N2i
N2i ,
T
N4i ,
Γ89 ε2i N3i
Γ29 DiT Pi A3i ,
Γ44 −Q5
ε3i N7iT N7i ,
T
ε1i N2i
N2i ,
T
Γ45 ε1i N2i
N3i ,
T
ε1i N3i
N3i ,
T
Γ56 ε1i N3i
N4i ,
Γ66 −Q7
T
ε1i N4i
N4i ,
T
Γ78 ε2i N2i
N3i ,
T
Γ79 ε2i N2i
N4i ,
Γ88 −Q9
T
ε2i N3i
N3i ,
T
Γ99 ε2i N4i
N4i .
3.40
Case 2. When σ 0, the uncertain neutral stochastic systems 2.3 are described as
dxt − Di xt − dt A1i txt
A3i tfxt − dt
A2i tfxt
3.41
t
A4i t
fxsds dt,
t ≥ 0.
t−σt
Theorem 3.6. Suppose that Assumptions 2.1 and 2.2 hold and for any given positive scalar κ ∈
0, 1, the uncertain neutral stochastic systems with mixed delays and Markovian switching 3.41
are robustly exponentially stable in mean square if there exist λi > 0 (i ∈ S), εi > 0 (i ∈ S) and some
positive definite matrices Pi > 0 (i ∈ S) and Qi > 0 (i 1, 2, . . . , 10) such that the following linear
matrix inequalities (LMIs) are satisfied: for i ∈ S,
⎡
Ω11 Ω12 0 Ω14 Ω15
⎢
⎢ ∗ Ω22 0
0
0
⎢
⎢
⎢ ∗
∗ Ω33 0
0
⎢
⎢
⎢
∗
∗ Ω44 Ω45
⎢ ∗
⎢
⎢
⎢ ∗
∗
∗
∗ Ω55
⎢
⎢
∗
∗
∗
∗
Ωi ⎢
⎢ ∗
⎢
⎢ ∗
∗
∗
∗
∗
⎢
⎢
⎢
∗
∗
∗
∗
⎢ ∗
⎢
⎢
⎢ ∗
∗
∗
∗
∗
⎢
⎢
⎢ ∗
∗
∗
∗
∗
⎣
∗
∗
∗
∗
∗
Ω16 Ω17 Ω18 Ω19 Ω110
0
Ω27 Ω28 Ω29
0
0
0
0
Ω46
0
0
0
Ω56
0
0
0
Ω66
0
0
0
∗
Ω77 Ω78 Ω79
∗
∗
Ω88 Ω89
∗
∗
∗
Ω99
∗
∗
∗
∗
∗
∗
∗
∗
0
⎤
⎥
Ω211 ⎥
⎥
⎥
0
0 ⎥
⎥
⎥
⎥
0
0 ⎥
⎥
⎥
0
0 ⎥
⎥
⎥
0
0 ⎥
⎥ < 0,
⎥
0
0 ⎥
⎥
⎥
⎥
0
0 ⎥
⎥
⎥
0
0 ⎥
⎥
⎥
−ε1i I 0 ⎥
⎦
∗
−ε2i I
0
i ∈ S,
3.42
Discrete Dynamics in Nature and Society
19
where ∗ denotes the entries that are readily inferred by symmetry of a symmetric matrix and
Ω11 Pi A1i
N
γij Pj
AT1i Pi
LT Q5
Q8 L
Q1
1 2
d2 − d12 Q3
2
d2 Q2
j1
ε1i
σ 2 Q4
T
ε2i N1i
N1i ,
Ω22 LT Q6
N
Q9 L
3.43
γij DiT Pj Di − 1 − μ Q1 ,
Ω27 DiT Pi A2i ,
j1
Ω33 LT Q7
Q10 L − κ1 − υQ4 ,
and Ω12 , Ω14 , Ω15 , Ω16 , Ω17 , Ω18 , Ω19 , Ω27 , Ω28 , Ω29 , Ω33 , Ω44 , Ω45 , Ω46 , Ω55 , Ω56 , Ω66 , Ω77 , Ω78 ,
Ω79 , Ω88 , Ω89 , and Ω99 are given in Theorem 3.2.
Remark 3.7. The sufficient conditions ensuring the exponential stability in mean square of
the uncertain neutral delay systems with Markovian jumping parameters in 39, 40 are
dependent upon the upper bound of the exponential stability rate. But, Theorem 3.6 can
remove this restrictive conditions. So, we can improve and generalize the results in 39, 40.
4. Numerical Examples
In this section, two examples are provided to illustrate the feasibility and applicability of our
obtained results.
Example 4.1. Consider the case of 2D Brownian motion, and rt
3is right-continuous
Markovian chain taking values in S {1, 2} with its generator Γ −3
4 −4 . And the parameters in systems 2.3 are given as follows:
A11 −1.5
0
0
−1.8
A31 −0.7 −0.5
0.4
D1 A12 ,
0.6
1.0 0.1
0
0
−1.8
A32 ,
−0.2 0.0
−1.8
,
D2 0.7 0.5
,
−0.3 0.3
0.1 0.0
0.7 0.2
A21 ,
,
A41 M1 0.3
0.2
,
−0.6 −0.8
0.2 0.3
−0.1 0.2
0.1 0.2
0.0 0.1
,
,
0.6 −0.5
A22 A42 M2 −0.4 −0.3
0.3 0.2
−0.2 0.4
0.2 0.1
0.1 0.2
,
,
,
N11 N12 N21 N22 N31 N32 N41 N42 0.1 0.1 .
4.1
The delays dt 0.45 sint 0.3, σt 0.4 cost 0.4, and it is easily obtained that d1 0.15,
d2 0.75, μ 0.45, ν 0.4, and σ 0.8. Assume that the activation function f satisfies
20
Discrete Dynamics in Nature and Society
0.0 Assumption 2.1 with the Lipschitz coefficient matrix L 0.2
0.0 0.2 and the function σ satisfies
Assumption 2.2 with the matrices Rij i 1, 2; j 1, 2, 3 given by
R11 0.15 0.0
0.20 0.35
,
R12 R31 0.5
0.1
0.28 0.20
0.1 0.2
0.1 0.2
,
,
0.3 0.0
R21 R32 0.2 0.1
0.1 0.1
,
R22 0.1 0.0
0.2 0.1
,
4.2
.
0.2 0.1
With the parameters above, when κ 0.9999, by using Matlab LMI Toolbox, according to
Theorem 3.2, we solve LMIs 3.33 and obtain there feasible solutions as follows:
P1 Q2 70.0930 12.2285
12.2285 21.8248
,
P2 ,
Q3 1.8693 −0.0065
−0.0065 0.6213
117.1703 −17.8696
Q5 −17.8696 119.1560
Q8 57.0181 12.7280
12.7280 58.0073
λ1 75.3750,
λ2 80.6197,
Q9 Q1 ,
15.4873 20.7284
4.8828 −0.0153
114.7061 67.0438
67.0438 91.0274
41.6921 14.6919
14.6919 40.8478
ε11 66.1002,
134.4662 25.4223
25.4223 20.5296
Q4 ,
−0.0153 1.6598
Q6 ,
,
74.1549 15.4873
Q10 ,
16.7672 −0.0153
−0.0153 1.6598
ε12 97.9532,
,
58.3284 23.0811
Q7 ,
,
,
23.0811 70.1310
40.5029 −0.2295
−0.2295 33.2920
ε21 47.7433,
,
ε22 56.9671.
4.3
Example 4.2. Let
rt
be right-continuous Markovian chain taking values in S {1, 2} with its
2 . Consider the following uncertain neutral systems with mixed delays
generator Γ −2
1 −1
and Markovian switching: for any i ∈ S,
d
xt − Di xt − dt A1i txt
dt
A2i txt − dt,
4.4
t ≥ 0,
where
A11 A12 D2 N11
−2 0.0
0.0 −3
0.1 0.0
,
,
A21 A22 M1 0.1 0.1
−1 0
−1 −1
,
,
M2 D1 0.3 0
0 0.3
0.2 −0.1
,
,
0.0 0.1
0.2 0.1
0.1 0.1
0.1 0.0 , N12 0.0 0.1 , N21 −0.0 0.1 , N22 0.1 0.1 .
4.5
Discrete Dynamics in Nature and Society
21
Applying Matlab toolbox, by Theorem 3.6, for μ 0, it can be obtained that the uncertain neutral systems with delays and Markovian jumping parameters 4.4 are robustly exponentially
stable in mean square with the delay dt satisfying 0 ≤ dt ≤ 6.0 × 1017 .
5. Conclusions
In this paper, the exponential stability in mean square of the uncertain neutral stochastic
systems with mixed delays and Markovian jumping parameters has been considered.
The mixed delays consist of the discrete delay and the distributed delay. The LMI-based
conditions ensuring the exponential stability in mean square of such systems are obtained
by constructing an appropriate Lyapunov functional, which are dependent upon the upper
bound and lower bound of the discrete time delays and distributed delays. It is worth
pointing out that, compared with the previous works 17, 18, the stability criteria in this
paper can be easily checked by using some standard numerical packages. Two illustrative
examples are provided to show the effectiveness and applicability of the proposed results.
Acknowledgments
The author would like to thank the anonymous referees for their very helpful comments
and suggestions that greatly improved this paper and the editors for their careful reading
of this paper. This work is supported by the National Natural Science Foundation of China
under Grant no. 11126278 and Grant no. 11101202, the Natural Science Foundation of Jiangxi
Province under Grant no. 20114BAB211001 and Grant no. 2009GQS0018, and the Youth
Foundation of Jiangxi Provincial Educations of China under Grant no. GJJ11045 and Grant
no. GJJ10051.
References
1 S. E. A. Mohammed, Stochastic Functional Differential Equations, vol. 99 of Research Notes in Mathematics,
Pitman, Boston, Mass, USA, 1984.
2 J. Zhu, Q. Zhang, and C. Yang, “Delay-dependent robust stability for Hopfield neural networks of
neutral-type,” Neurocomputing, vol. 72, no. 10–12, pp. 2609–2617, 2009.
3 W.-H. Chen and W. X. Zheng, “Delay-dependent robust stabilization for uncertain neutral systems
with distributed delays,” Automatica, vol. 43, no. 1, pp. 95–104, 2007.
4 M. Wu, Y. He, and J.-H. She, “New delay-dependent stability criteria andstabilizing method for
neutral systems,” IEEE Transactions on Automatic Control, vol. 49, no. 12, pp. 2266–2271, 2004.
5 Y. He, M. Wu, J.-H. She, and G.-P. Liu, “Delay-dependent robust stability criteria for uncertain neutral
systems with mixed delays,” Systems & Control Letters, vol. 51, no. 1, pp. 57–65, 2004.
6 P. Balasubramaniam and R. Rakkiyappan, “Global exponential stability for neutral-type BAM neural
networks with time-varying delays,” International Journal of Computer Mathematics, vol. 87, no. 9, pp.
2064–2075, 2010.
7 R. Rakkiyappan, P. Balasubramaniam, and J. Cao, “Global exponential stability results for neutraltype impulsive neural networks,” Nonlinear Analysis: Real World Applications, vol. 11, no. 1, pp. 122–
130, 2010.
8 X. Mao, Stochastic Differential Equations and Applications, HorwoodPublication, Chichester, UK, 1997.
9 L. Huang and X. Mao, “Delay-dependent exponential stability of neutral stochastic delay systems,”
IEEE Transactions on Automatic Control, vol. 54, no. 1, pp. 147–152, 2009.
10 X. Mao, “Razumikhin-type theorems on exponential stability of neutral stochastic functionaldifferential equations,” SIAM Journal on Mathematical Analysis, vol. 28, no. 2, pp. 389–401, 1997.
11 H. Chen, Y. Zhang, and P. Hu, “Novel delay-dependent robust stability criteria for neutral stochastic
delayed neural networks,” Neurocomputing, vol. 73, no. 13–15, pp. 2554–2561, 2010.
22
Discrete Dynamics in Nature and Society
12 J. Wang, Y. Chen, and W. X. Zheng, “Mean-square exponential stability analysis for stochastic systems
of neutral-type,” in Proceedings of the 8th World Congress on Intelligent Control and Automation (WCICA
’10), pp. 3542–3547, 2010.
13 X. Mao, “Exponential stability of stochastic delay interval systems with Markovian switching,” IEEE
Transactions on Automatic Control, vol. 47, no. 10, pp. 1604–1612, 2002.
14 J. Luo, “Fixed points and stability of neutral stochastic delay differential equations,” Journal of
Mathematical Analysis and Applications, vol. 334, no. 1, pp. 431–440, 2007.
15 W. Su and Y. Chen, “Global asymptotic stability analysis for neutral stochastic neural networks with
time-varying delays,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 4, pp.
1576–1581, 2009.
16 L. Zhou and G. Hu, “Almost sure exponential stability of neutral stochastic delayed cellular neural
networks,” Journal of Control Theory and Applications, vol. 6, no. 2, pp. 195–200, 2008.
17 V. Kolmanovskii, N. Koroleva, T. Maizenberg, X. Mao, and A. Matasov, “Neutral stochastic differential
delay equations with Markovian switching,” Stochastic Analysis and Applications, vol. 21, no. 4, pp.
819–847, 2003.
18 X. Mao, Y. Shen, and C. Yuan, “Almost surely asymptotic stability of neutral stochastic differential
delay equations with Markovian switching,” Stochastic Processes and their Applications, vol. 118, no. 8,
pp. 1385–1406, 2008.
19 G. Chen and Y. Shen, “Robust H∞ filter design for neutral stochastic uncertain systems with timevarying delay,” Journal of Mathematical Analysis and Applications, vol. 353, no. 1, pp. 196–204, 2009.
20 S. Xu, P. Shi, Y. Chu, and Y. Zou, “Robust stochastic stabilization and H∞ control of uncertain neutral
stochastic time-delay systems,” Journal of Mathematical Analysis and Applications, vol. 314, no. 1, pp.
1–16, 2006.
21 N. N. Krasovskiı̆ and È. A. Lidskiı̆, “Analytic design of controllers in systems with random attributes.
II. Equations of optimum solutions. Approximate solutions,” Automation and Remote Control, vol. 22,
no. 10, pp. 1141–1146, 1961.
22 M. Mariton, Jump Linear Control Systems, Marcel Dekker, New York, NY, USA, 1990.
23 Z. Wang, Y. Liu, L. Yu, and X. Liu, “Exponential stability of delayed recurrent neural networks with
Markovian jumping parameters,” Physics Letters A, vol. 356, pp. 346–352, 2006.
24 Y. Liu, Z. Wang, and X. Liu, “On delay-dependent robust exponential stability of stochastic neural
networks with mixed time delays and Markovian switching,” Nonlinear Dynamics, vol. 54, no. 3, pp.
199–212, 2008.
25 G. Wang, J. Cao, and J. Liang, “Exponential stability in the mean square for stochastic neural networks
with mixed time-delays and Markovian jumping parameters,” Nonlinear Dynamics, vol. 57, no. 1-2, pp.
209–218, 2009.
26 Q. Zhu and J. Cao, “Exponential stability of stochastic neural networks with both markovian jump
parameters and mixed time delays,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 99, pp.
1–13, 2010.
27 Q. Zhu and J. Cao, “Robust exponential stability of Markovian jump impulsive stochastic CohenGrossberg neural networks with mixed time delays,” IEEE Transactions on Neural Networks, vol. 8, pp.
1314–1325, 2010.
28 D. Yue and Q.-L. Han, “Delay-dependent exponential stability of stochastic systems with timevarying delay, nonlinearity, and Markovian switching,” IEEE Transactions on Automatic Control, vol.
50, no. 2, pp. 217–222, 2005.
29 X. Lou and B. Cui, “Stochastic exponential stability for Markovian jumping BAM neural networks
with time-varying delays,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 3, pp. 713–719,
2007.
30 H. Zhang and Y. Wang, “Stability analysis of Markovian jumping stochastic Cohen-Grossberg neural
networks with mixed time delays,” IEEE Transactions on Neural Networks, vol. 2, pp. 366–370, 2008.
31 Y. Shen and J. Wang, “Almost surely exponential stability of recurrent neural networks with
Markovian switching,” IEEE Transactions on Neural Networks, vol. 5, pp. 840–855, 2009.
32 Y. Liu, Z. Wang, J. Liang, and X. Liu, “Stability and synchronization of discrete-time Markovian
jumping neural networks with mixed mode-dependent time delays,” IEEE Transactions on Neural
Networks, vol. 7, pp. 1102–1116, 2009.
33 S. Ma and E. K. Boukas, “Stability and robust stabilisation for uncertain discrete stochastic hybrid
singular systems with time delay,” IET Control Theory & Applications, vol. 3, no. 9, pp. 1217–1225,
2009.
Discrete Dynamics in Nature and Society
23
34 S. Xu, J. Lam, and X. Mao, “Delay-dependent H∞ control and filtering for uncertain Markovian jump
systems with time-varying delays,” IEEE Transactions on Circuits and Systems, vol. 54, no. 9, pp. 2070–
2077, 2007.
35 L. Huang and X. Mao, “On almost sure stability of hybrid stochastic systems with mode-dependent
interval delays,” IEEE Transactions on Automatic Control, vol. 55, no. 8, pp. 1946–1952, 2010.
36 L. Xie, “Stochastic robust stability analysis for Markovian jumping neural networks with time
delays,” in Proceedings of the IEEE Networking, Sensing and Control (ICNSC ’05), pp. 923–928, 2005.
37 S. Cong, H.-T. Zhang, and Y. Zou, “A new exponential stability condition for delayed systems with
Markovian switching,” Acta Automatica Sinica, vol. 36, no. 7, pp. 1025–1028, 2010.
38 C. Yuan and X. Mao, “Robust stability and controllability of stochastic differential delay equations
with Markovian switching,” Automatica, vol. 40, no. 3, pp. 343–354, 2004.
39 S. He and F. Liu, “Exponential stability for uncertain neutral systems with Markov jumps,” Journal of
Control Theory and Applications, vol. 7, no. 1, pp. 35–40, 2009.
40 P. Balasubramaniam, A. Manivannan, and R. Rakkiyappan, “Exponential stability results for
uncertain neutral systems with interval time-varying delays and Markovian jumping parameters,”
Applied Mathematics and Computation, vol. 216, no. 11, pp. 3396–3407, 2010.
41 X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press,
London, UK, 2006.
42 S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control
Theory, vol. 15 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics
SIAM, Philadelphia, Pa, USA, 1994.
43 K. Gu, “An integral inequality in the stability problem of time-delay systems,” in Proceedings of the 39th
IEEE Confernce on Decision and Control, Sysdney, NSW, vol. 3, pp. 2805–2810, Sydney NSW, Australia,
2000.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014