Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 494607, 14 pages
doi:10.1155/2010/494607
Research Article
Riccati Equations and Delay-Dependent BIBO
Stabilization of Stochastic Systems with Mixed
Delays and Nonlinear Perturbations
Xia Zhou and Shouming Zhong
School of Mathematical Sciences, University of Electronic Science and Technology of China,
Chengdu, Sichuan 611731, China
Correspondence should be addressed to Xia Zhou, zhouxia44185@sohu.com
Received 21 August 2010; Accepted 9 December 2010
Academic Editor: T. Bhaskar
Copyright q 2010 X. Zhou and S. Zhong. 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 mean square BIBO stability is investigated for stochastic control systems with mixed delays
and nonlinear perturbations. The system with mixed delays is transformed, then a class of suitable
Lyapunov functionals is selected, and some novel delay-dependent BIBO stabilization in mean
square criteria for stochastic control systems with mixed delays and nonlinear perturbations are
obtained by applying the technique of analyzing controller and the method of existing a positive
definite solution to an auxiliary algebraic Riccati matrix equation. A numerical example is given to
illustrate the validity of the main results.
1. Introduction
In recent years, Bounded-Input Bounded-Output BIBO stabilization has been investigated
by many researchers in order to track out the reference input signal in real world, see
1–6 and some references therein. On the other hand, because of the finite switching
speed, memory effects, and so on, time delays are unavoidable in technology and nature,
commonly exist in various mechanical, chemical engineering, physical, biological, and
economic systems. They can make the concerned control system be of poor performance
and instable, which cause the hardware implementation of the control system to become
difficult. It is necessary to introduce the distributed delay in control systems, which can
describe mathematical modeling of many biological phenomena, for instance, in preypredator systems, see 7–9. And so, BIBO stabilization analysis for mixed delays and
nonlinear systems is of great significance.
In 10, 11, the sufficient condition for BIBO stabilization of the control system with
no delays was proposed by the Bihari-type inequality. In 12, 13, employing the parameters
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technique and the Gronwall inequality investigated the BIBO stability of the system without
distributed time delays. In 14–16, based on Riccati-equations, by constructing appropriate
Lyapunov functions, some BIBO stabilization criteria for a class of delayed control systems
with nonlinear perturbations were established. In 17, the BIBO stabilization problem of a
class of piecewise switched linear system was further investigated. However, up to now, these
previous results have been assumed to be in deterministic systems, including continuous
time deterministic systems and discrete time deterministic systems, but seldom in stochastic
systems see 18, Fu and Liao get several mean square BIBO stabilization criteria in terms of
Razumikhin technique and comparison principle. In practice, stochastic control systems are
more applicable to problems that are environmentally noisly in nature or related to biological
realities. Thus, the BIBO stabilization analysis problems for stochastic case are necessary.
Up to now, to the best of authors knowledge, the method of existence of a positive
definite solution to an auxiliary algebraic Riccati matrix equation is only used to deal with
the BIBO stabilization for deterministic differential equations 14–17, not for stochastic
differential equations.
Motivated by the above discussions, the main aim of this paper is to study the
BIBO stabilization in mean square for the stochastic control system with mixed delays and
nonlinear perturbations. Based on the technique of analyzing controller and transforming
of the system, various suitable Lyapunov functionals are selected, different Riccati matrix
equations are established, and some sufficient conditions guaranteeing BIBO stabilization
in mean square are obtained. Finally, a numerical example is provided to demonstrate the
effectiveness of the derived results.
2. Problem Formulation and Preliminaries
Consider the stochastic control system described by the following equation:
dxt Axt B1 xt − h1 B2
t
xsds ft, xt Dut dt
t−h2
C1 xt − τ1 C2
t
xsds dwt,
t ≥ t0 ≥ 0,
t−τ2
2.1
yt Hxt,
xθ ϕθ ∈ CFb 0 t0 − τ, t0 ; Rn ,
θ ∈ t0 − τ, t0 ,
where xt, ut, yt are the state vector, control input, control output of the system,
respectively. τ1 > 0, h1 > 0 are discrete time delays, and h2 > 0, τ2 > 0 are distributed
time delays, τ max{τ1 , τ2 , h1 , h2 }. A, B1 , B2 , C1 , C2 , D, H are constant matrices with
appropriate dimensional, and C2 , D are nonsingular matrices. wt w1 t, w2 t, . . . , wn t
is an n-dimensional standard Brownian motion defined on a complete probability space
Ω, F, {Ft }t≥0 , P with a natural filtration {Ft }t≥0 . ft, xt is the nonlinear vector-valued
perturbation bounded in magnitude as
ft, xt ≤ αxt,
where α is known positive constant.
2.2
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To obtain the control law described by 2.1 of tracking out the reference input of the
system, we let the controller be in the form of
ut u1 t u2 t u3 t,
2.3
with
u1 t K1 txt rt,
u2 t K2 txt − h1 rt − h1 ,
2.4
u3 t K3 txt − h2 rt − h2 ,
where K1 t, K2 t, K3 t are the feedback gain matrices, rt, rt − h1 , rt − h2 are the
reference inputs.
To derive our main results, we need to introduce the following definitions and lemmas.
Definition 2.1 see 18. A vector function rt r1 t, r2 t, . . . , rn tT is said to be an element
of Ln∞ if r∞ supt∈t0 ,∞ rt < ∞, where · denotes the Euclid norm in Rn or the norm of a
matrix.
Definition 2.2 see 18. The nonlinear stochastic control system 2.2 is said to be BIBO stabilized
in mean square if one can construct a controller 2.5 such that the output yt satisfies
E yt2 ≤ N1 N2 r2∞ ,
2.5
where N1 , N2 are positive constants.
Definition 2.3 L-operator. Let Lyapunov functional V : C−τ, 0; Rn ×R → R; its infinitesimal
operator, L, acting on functional V is defined by
LV xt, t lim sup
Δ → 0
1
EV xt Δ, t Δ − V xt, t.
Δ
2.6
Lemma 2.4 see 19. For any constant symmetric matrix M ∈ Rn×n , M MT > 0, scalar r > 0,
vector function g : 0, r → Rn , such that the integrations in the following are well defined:
r
r
0
g sMgsds ≥
T
r
T r
gsds M
gsds .
0
2.7
0
Lemma 2.5 see 20. Let x, y ∈ Rn and any n × n positive-definite matrix Q > 0. Then, one has
2xT y ≤ xT Q−1 x yT Qy.
2.8
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3. BIBO Stabilization for Nonlinear Stochastic Systems
Transform the original system 2.1 to the following system:
d xt B1
t
xsds A B1 xt B2
t
t−h1
xsds ft, xt dt
t−h2
C1 xt − τ1 C2
t
xsds dwt Dutdt,
t ≥ t0 ≥ 0,
t−τ2
yt Hxt,
xθ ϕθ ∈ CFb 0 t0 − τ, t0 ; Rn ,
θ ∈ t0 − τ, t0 .
3.1
Theorem 3.1. The nonlinear stochastic control system 2.1 or 3.1 with the control law 2.3 is
BIBO stabilized in mean square if h1 B1 < 1 and there exist symmetric positive-definite matrices
Ri > 0, i 1, 2, . . . , 10, and Q1 > 0 such that
λmin Q1 − 2αP > 0
3.2
and P is the symmetric positive solution of the Riccati equation
P A B1 A B1 T P P Σ1 P Ξ1 Δ1 −Q1 ,
3.3
where
T
T
−1
−1
T
−1 T
T
Σ1 B2 R−1
1 B2 2DR10 D R5 R6 DR10 D P B1 R7 B1 P DR10 D
T
−1 T
B1 R−1
8 B1 B1 R9 B1 ,
Ξ1 h22 R1 R3 h21 R2 R7 R8 R9 R5 R6 τ22 R4 ,
T
T
T
−1 T
Δ1 A B1 T P B1 R−1
2 B1 P A B1 C1 P C1 C1 P C2 R4 C2 P C1
3.4
T
τ22 C2T P C2 h21 B1T P B2 R−1
3 B2 P B1 ,
K1 R10 DT P,
K2 K3 D−1 .
Proof. We define a Lyapunov functional V t, xt as
V t, xt V1 t, xt V2 t V3 t V4 t V5 t V6 t V7 t,
3.5
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where
V1 t, xt V2 t xt B1
t
t
t−h2
V4 t t
t−τ1
V5 t h2
xsds
P
xt B1
t
t−h1
t−h1
V3 t T t
xsds ,
t−h1
T
xT s R5 P B1 R−1
B
P
xsds,
1
8
T
xT s R6 P B1 R−1
B
P
xsds,
1
9
T
xsds,
xT s C1T P C1 C1T P C2 R−1
C
P
C
1
2
4
t
3.6
s − t h2 xT sR1 R3 xsds,
t−h2
V6 t τ2
t
t−τ2
V7 t h1
s − t τ2 xT s R4 C2T P C2 xsds,
t
t−h1
T
s − t h1 xT s R2 R7 R8 R9 B1T P B2 R−1
3 B2 P B1 xsds.
Taking the operator L of V1 t, xt along the trajectory of system 3.1, by Lemmas 2.4 and
2.5, we have
LV1 t, xt 2 xt B1
t
xsds
T
t−h1
× P A B1 xt B2
t
xsds Dut ft, xt
t−h2
t
t
1
trace C1 xt − τ1 C2
xsds 2P C1 xt − τ1 C2
xsds
2
t−τ2
t−τ2
T
≤ xT t P A B1 A B1 T P P B2 R−1
1 B2 P
T
xt 2xT P Dft, xt
A B1 T P B1 R−1
B
P
B
A
1
1
2
t
t
xT sR1 R3 xsds 2
xT sdsB1T P Dut
h2
t−h2
t−h1
T
xt − τ1 xT t − τ1 C1T P C1 C1T P C2 R−1
C
P
C
1
2
4
h1
2
t
t
t−h1
t−h1
T
xsds
xT s R2 B1T P B2 R−1
B
P
B
1
2
3
xT sdsB1T P ft, xt 2xT P Dut τ2
t
t−τ2
xT s R4 C2T P C2 ds.
3.7
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By the Lemma 2.5, 2.3 and 3.5, we conclude
T
−1
LV1 t, xt ≤ xT t P A B1 A B1 T P P B2 R−1
1 B2 P P R5 P
T
T
A B1 T P B1 R−1
2 B1 P A B1 2P DR10 D P
T
−1 T
T
P R−1
6 P P DR10 D P B1 R7 B1 P DR10 D P xt
T
xT t − τ1 C1T P C1 C1T P C2 R−1
4 C2 P C1 xt − τ1 T
xT t − h1 R5 P B1 R−1
8 B1 P xt − h1 3.8
T
x t − h2 R6 P B1 R−1
9 B1 P xt − h2 T
h1
t
t−h1
τ2
t
T
xsds
xT s R2 B1T P B2 R−1
B2
P
B
R
R
R
1
7
8
9
3
x s R4 T
t−τ2
C2T P C2
ds h2
t
xT sR1 R3 xsds
t−h2
2α P h1 B1T P xt2 6 P D h1 B1T P rt∞ xt.
Taking the operator L of Vi t, i 2, 3, . . . , 7 along the trajectory of system 3.1, we get
T
LV2 t xT t R5 P B1 R−1
8 B1 P xt
T
− xT t − h1 R5 P B1 R−1
8 B1 P xt − h1 ,
T
LV3 t xT t R6 P B1 R−1
9 B1 P xt
T
− xT t − h2 R6 P B1 R−1
9 B1 P xt − h2 ,
T
LV4 t xT t C1T P C1 C1T P C2 R−1
4 C2 P C1 xt
T
− xT t − τ1 C1T P C1 C1T P C2 R−1
4 C2 P C1 xt − τ1 ,
LV5 t h22 xT tR1 R3 xt − h2
t
xT sR1 R3 xsds,
t−h2
t
LV6 t τ22 xT t R4 C2T P C2 xt − τ2
LV7 t h21 xT t
− h1
t−τ2
R2 R7 R8 R9 t
t−h1
xT s R4 C2T P C2 xsds,
T
B1T P B2 R−1
3 B2 P B1
xt
T
xT s R2 R7 R8 R9 B1T P B2 R−1
3 B2 P B1 xsds.
3.9
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Combining 3.8 and 3.9, we have
LV t, xt ≤ xT t P A B1 A B1 T P P B2 R−1 B2T P P R−1
6 P
T
T
A B1 T P B1 R−1
2 B1 P A B1 2P DR10 D P R5
T
T
2
P DR10 DT P B1 R−1
7 B1 P DR10 D P h2 R1 R3 R6
T
−1 T
T
−1 T
P B1 R−1
8 B1 P P B1 R9 B1 P C1 P C2 R4 C2 P C1
T
T
h21 R2 B1T P B2 R−1
3 B2 P B1 R7 R8 R9 C1 P C1
2
T
τ22 R4 C2T P C2 P R−1
5 P xt 2αh1 B1 P xt
2αP xt2 6 P D h1 B1T P rt∞ xt
≤ − λmin Q1 − 2α P h1 B1T P xt2
6 P D h1 B1T P rt∞ xt.
3.10
Let ρ1 λmin Q1 − 2αP h1 B1T P , ρ2 6P D h1 B1T P rt∞ ; we have
LV t, xt ≤ −ρ1 xt2 ρ2 xt.
3.11
Set
β1 λmax P h1 λmax P B1 h1 λmax B1T P h21 λmax B1T P B1 ,
T
β2 h1 λmax R5 P B1 R−1
8 B1 P ,
T
β3 h2 λmax R6 P B1 R−1
9 B1 P ,
T
β4 τ1 λmax C1T P C1 C1T P C2 R−1
4 C2 P C1 ,
β5 h32 λmax R1 R3 ,
β6 τ23 λmax R4 C2T P C2 ,
3.12
T
β7 h31 λmax R2 B1T P B2 R−1
3 B2 P B1 R7 R8 R9 ,
under an assumption that V t, xt ≤ V t0 , xt0 for all t ≥ t0 , then
2
t
7
2
xsds ≤ V t, xt ≤ V t0 , xt0 ≤
βi Eϕθ ,
λmin P Ext B1
t−h1
i1
3.13
so
2 7
2
t
i1 βi E ϕθ
Ext B1
.
xsds ≤
λmin P t−h1
3.14
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Thus, according to 21, Theorem 1.3 page 331, we have
2
Ext ≤
1 h1 B1 1 − h1 B1 2 7
βi Eϕθ2
.
λmin P i1
3.15
If not, there exist t > t0 , such that V t, xt ≥ V s, xs for all s ∈ t0 , t, and one has
D EV t, xt ≥ 0.
3.16
D EV t, xt ELV t, xt.
3.17
In view of Ito’s formula, we obtain
By 3.16 and 3.17, it is easy to derive that
0 ≤ D EV t, xt ELV t, xt ≤ −ρ1 Ext2 ρ2 Ext,
3.18
so Ext ≤ ρ2 /ρ1 . By 3.18, we can conclude that
Ext2 ≤
ρ2
Ext ≤
ρ1
ρ2
ρ1
2
.
3.19
By 3.15 and 3.19, we get
Ext2 ≤
ρ22
ρ12
1 h1 B1 1 − h1 B1 2 7
βi Eϕθ2
,
λmin P i1
3.20
Thus
Eyt2 ≤ H2 Ext2
≤
H2 ρ22
ρ12
1 h1 B1 1 − h1 B1 2
H2
7
βi Eϕθ2
λmin P i1
3.21
≤ N1 N2 r2∞ ,
where
N1 1 h1 B1 1 − h1 B1 2 7
βi H2
Eϕθ2 ,
λmin P i1
36H2 P D h1 B1T P N2 .
λmin Q1 − 2α P − h1 B1T P 3.22
By Definition 2.2, the nonlinear stochastic control system 3.1 with the control law 2.3 is
said to be BIBO stabilized in mean square. This completes the proof.
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If we transform the original system 2.1 to the following system
d xt B2
t
s − t h2 xsds A h2 B2 xt B1 xt − h1 ft, xt dt Dutdt
t−h2
C1 xt − τ1 C2
t
xsds dwt,
t ≥ t0 ≥ 0
t−τ2
yt Hxt
xθ ϕθ ∈ CFb 0 t0 − τ, t0 ; Rn ,
θ ∈ t0 − τ, t0 ,
3.23
we can get the following result.
Theorem 3.2. The nonlinear stochastic control system 3.23 with the control law 2.3 is BIBO
stabilized in mean square if there exist symmetric positive-definite matrices Si > 0, i 1, 2, . . . , 10,
and Q2 > 0 such that
λmin Q2 − 2αP > 0
3.24
and P is the symmetric positive solution of the Riccati equation
P A h2 B2 A h2 B2 T P P Σ2 P Δ2 Ξ2 −Q2 ,
3.25
where
T
T
−1
−1
Σ2 B1 S−1
2 B1 2DS10 D S5 S6 ,
T T −1 2 T Δ2 A h2 B2 T P B2 S−1
1 B2 P A h2 B2 D P S7 P D τ2 C2 P C2
T T −1 T T −1 DT P S−1
8 P D C1 P C1 C1 P C2 S4 C2 P C1 D P S9 P D
T B1T P B2 S−1
3 B2 P B1 ,
3.26
1
Ξ2 S2 S5 S6 h42 S1 S3 S7 S8 S9 τ22 S4 ,
3
K1 S10 DT P ,
K2 K3 D−1 .
Proof. We define a Lyapunov functional V t, xt as
V t, xt V1 t, xt V2 t V3 t V4 t V5 t V6 t,
3.27
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where
V1 t, xt xt B2
T t
s − t h2 xsds P xt B2
t
t−h2
V2 t t
t−h1
V3 t t
t−h2
V4 t t
t−τ1
V5 t τ2
1
3
s − t h2 xsds ,
t−h2
T T −1 xT s S2 S5 B1T P B2 S−1
B
D
P
B
P
S
P
D
xsds,
1
2
3
8
xT s S6 DT P S−1
9 P D xsds,
T xT s C1T P C1 C1T P C2 S−1
4 C2 P C1 xsds,
t
t−τ2
V6 t t
s − t τ2 xT s S4 C2T P C2 xsds,
s − t h2 3 xT sS1 S3 S7 S8 S9 xsds.
t−h2
3.28
Let ρ1 λmin Q2 −2α1h22 P , ρ2 61h22 P Drt∞ . The rest of the proof is essentially
as that of Theorem 3.1, and hence is omitted. This completes the proof.
Remark 3.3. If we transform the original system 2.1 to the following system
d xt B1
t
xsds B2
t−h1
t
s − t h2 xsds
t−h2
Axt ft, xt Dut dt
t
C1 xt − τ1 C2
xsds dwt,
t ≥ t0 ≥ 0
3.29
t−τ2
yt Hxt
xθ ϕθ ∈ CFb 0 t0 − τ, t0 ; Rn ,
θ ∈ t0 − τ, t0 ,
using the same process of Theorem 3.1, we can get the corresponding BIBO stability in mean
square results. Here we omitted it.
Theorem 3.4. The nonlinear stochastic control system 2.1 with the control law 2.3 is BIBO
stabilized in mean square if there exist symmetric positive-definite matrices Ωi > 0, i 1, 2, . . . , 6,
and Q3 > 0 such that
λmin Q3 − 2αP > 0
3.30
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and P is the symmetric positive solution of the Riccati equation
P A AT P P Σ3 P Δ3 Ξ3 −Q3 ,
3.31
where
T
−1 T
T
−1
−1
Σ3 B1 Ω−1
1 B1 h2 B2 Ω2 B2 2DΩ6 D Ω4 Ω5 ,
T 2 T Δ3 C1T P C1 τ2 C2T P C1 Ω−1
3 C1 P C2 τ2 C2 P C2 ,
Ξ3 Ω1 h2 Ω2 τ2 Ω3 Ω4 Ω5 ,
K1 Ω6 DT P ,
3.32
K2 K3 D−1 .
Proof. We define a Lyapunov functional V t, xt as
V t, xt V1 t, xt V2 t V3 t V4 t V5 t,
3.33
where
V1 t, xt xtT P xt,
t
t
xT sΩ1 Ω4 xsds V2 t t−h1
V3 t t
xT sΩ5 xsds,
t−h2
s − t h2 xT sΩ2 xsds,
t−h2
V4 t t
t−τ1
V5 t t
t−τ2
3.34
T xsds,
xT s C1T P C1 τ2 C2T P C1 Ω−1
C
P
C
2
1
3
s − t τ2 xT s Ω3 τ2 C2T P C2 xsds.
Let ρ1 λmin Q3 − 2αP , ρ2 6P Drt∞ . The following proof runs as that of
Theorem 3.1., and hence is omitted. This completes the proof.
Remark 3.5. The systems 2.1, 3.1, and 3.23 are also asymptotically stable in mean square
when all the conditions in Theorems 3.1–3.4 are satisfied, if rt 0, rt − h1 0, rt − h2 0
in 2.3.
Remark 3.6. In 18, the authors studied the BIBO stabilization of stochastic delayed systems
with uncertainty in terms of Razumikhin technique and comparison principle. In the present
paper, we are the first to introduce a new way in the study of BIBO stabilization for stochastic
delayed systems by using algebraic Riccati matrix equation, which makes the stability
conditions be morefeasible.
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Remark 3.7. In 18, researchers investigate the BIBO stabilization of stochastic delayed
systems with uncertainty without distributed time delays and nonlinear perturbations. In
15, the authors discussed the BIBO stabilization of mixed time-delayed systems with
nonlinear perturbations, but the environmental noise is not taken into account in the models.
Therefore, compared with see 14–18, the systems reported in this paper are more general.
Remark 3.8. The criteria given in Theorem 3.1–3.4 are delay dependent with respect to delays.
Generally speaking, the delay-dependent stability criterion is less conservative than delayindependent stability when the time delay is small.
4. Example
In this section, a numerical example will be presented to show the effectiveness of the main
results derived in this paper.
Example 4.1. As a simple application of Theorem 3.4, consider the stochastic control system
2.1 with control law 2.3; the parameters are given by
A
−2 0
0 −1
,
−1 1
B1 1 −2
B2 C1 ,
1 0
,
0 1
D
−1 0
0 −1
,
4.1
C2 0, f 0.25 sint, 0.25 costT .
For
Ωi 1 0
0 1
i 1, 2, . . . , 6,
Q3 11 7
,
7 14
4.2
solving for P in the Riccati matrix 3.31 gives us
P
1 1
,
1 4
4.3
therefore, the stabilizing feedback gain matrix is given by
K1 −11 −7
−7 −14
,
K2 K3 −1 0
0 −1
.
4.4
Meanwhile, we obtain the maximum value τmax 0.6667.
5. Conclusions
The problem of delay-dependent BIBO stabilization in mean square for the stochastic control
systems with mixed delays and nonlinear perturbations was investigated. A suitable class of
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Lyapunov functional combined with the descriptor model transformation and decomposition
technique of controller were constructed to derive some novel mean square BIBO stability
criteria. This paper was the first to successfully introduce the method of Riccati matrix
equation to stochastic BIBO stabilization. A numerical example was given to illustrate the
validity of the main results.
Acknowledgments
The authors would like to thank the editor and the reviewers for their detailed comments and
valuable suggestions which have led to a much improved paper. This work was supported
by the National Natural Science Foundation of China Grant no. 60736029 and the National
Basic Research Program of China 2010CB732501.
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