# The Mean Stability Criteria in terms of Two Measures for Stochastic Differential Equations with Coefficient’s Uncertainty

Authors: Rui Zhang; Yinjing Guo; Xiangrong Wang; Xueqing Zhang
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

This paper extends the stochastic stability criteria of two measures to the mean stability and proves the stability criteria for a kind of stochastic Itô’s systems. Moreover, by applying optimal control approaches, the mean stability criteria in terms of two measures are also obtained for the stochastic systems with coefficient’s uncertainty.


---

## Body

## 1. Introduction

Lyapunov’s method, which makes an essential use of auxiliary functions (also called Lyapunov functions), is an important approach to study the stability of differential systems including ordinary differential equations (ODEs) and stochastic differential equations (SDEs). This method started in Lyapunov’s original work in 1892 [1] for demonstrating stability of ODEs. In the 1960s, Movchan [2] studied the stability with two measures; such works were also developed and can be seen in [3]. In the past decades, Lyapunov’s method is modified to the study of stability of Markovian processes [4], stochastic differential systems based on Brownian motions [5], semimartingales [6], or Lévy processes [7] and is also developed with the form of exponential stability [8] or LaSalle theorem [9], and so forth. Recently, the stability for systems with unknown parameters is also discussed [10], and the theorems of stability are widely applied in aerospace [11], state-feedback control [12], automatic control [13], neural networks [14], and other fields.In this paper, we will discuss the following stochastic Itô’s systems:(1)
d
x
=
f
t
,
x
d
t
+
σ
t
,
x
d
B
t
,
x
t
0
=
x
0
,
t
≥
t
0
∈
R
+
,where f
,
σ satisfy the usual Lipschitzian conditions and B is d-dimensional standard Brownian motion. It is well known that, for a stochastic process x
(
t
) and a given positive function h
(
t
,
x
), lim
t
→
∞
⁡
h
(
t
,
x
(
ω
,
t
)
)
=
0 almost surely discussed in [15] does not imply that lim
t
→
∞
⁡
E
h
(
t
,
x
(
t
)
)
→
0. So, we extend Lyapunov’s methods used by [3] for ODEs to the stochastic cases and study the mean stability criteria in terms of two measures for system (1).This paper is organized as follows: In Section2, we first introduce Lyapunov’s derivatives for (1) and deduce the basic comparison results in terms of Lyapunov’s function. In Section 3, we prove the stochastic two-measure stability criteria for Itô systems, which can be seen as the extension of that of the ODEs. As described in [16], stability, robustness, and optimality can be considered systematically and simultaneously. In Section 4, the optimal control approach is extended to the stochastic systems with coefficient’s uncertainty.


## 2. Basic Comparison Results for Stochastic Differential Equations

Let(
Ω
,
F
,
P
) be a given completed probability space, and {
B
t
}
t
≥
0 is a standard Brownian motion with filtration:(2)
F
t
=
σ
B
s
,
s
≤
t
.Let f
∈
C
(
R
+
×
R
n
,
R
n
), σ
∈
C
(
R
+
×
R
n
,
R
n
×
R
d
) be deterministic functions and satisfy the following Lipschitz condition and linear growth condtion: there exists L
>
0, for every x
,
y
∈
R
n, such that (3)
f
t
,
x
-
f
t
,
y
+
σ
t
,
x
-
σ
t
,
y
≤
L
x
-
y
,
f
t
,
x
≤
L
1
+
x
.For a given function V
∈
C
(
R
+
×
R
n
,
R
), (
t
,
x
)
∈
R
+
×
R
n, we denote (4)
D
+
V
t
,
x
1
=
lim
sup
h
→
0
+
⁡
1
h
E
V
t
+
h
,
x
+
h
f
t
,
x
+
h
σ
t
,
x
ξ
-
V
t
,
x
,where ξ represents a d-dimensional random variable with standard normal distribution; that is, ξ
~
N
(
0
,
I
) and I is a d-order identity matrix.Remark 1.
We use the notationD
+
V
(
t
,
x
)
(
1
) to emphasize the definition with respect to system (1). For convenience, we use the shortened form D
+
V to substitute D
+
V
(
t
,
x
)
(
1
).Remark 2.
IfV
∈
C
1,2
(
R
+
×
R
n
,
R
+
), D
+
V
(
t
,
x
) is Lyapunov’s operator L associated with (1); that is, (5)
D
+
V
1
=
L
V
≔
V
t
+
V
x
,
f
+
1
2
σ
σ
T
,
V
x
x
,where V
t is the partial derivative for t, V
x is the gradient of V for x, and V
x
x is the Hessian matrix of V for x.Since (4) is dependent on expectation calculating, it is not easy to check whether D
+
V exists or not. The following lemma gives a condition for the existence of D
+
V.Lemma 3.
LetV
∈
C
(
R
+
×
R
n
,
R
) be continuous one-order differentiable for t and also satisfy the following condition: (6)
V
t
,
x
+
y
-
V
t
,
x
-
α
t
,
x
,
y
≤
K
t
y
2
,
∀
x
,
y
∈
R
n
,where α
(
t
,
x
) is continuous on R
+
×
R
n and K
(
t
)
≥
0 is locally bounded for t. Then D
+
V
(
t
,
x
) exists at (
t
,
x
).Proof.
Denotey
=
h
f
(
t
,
x
)
+
h
σ
(
t
,
x
)
ξ; then, for fixed t
,
x, (7)
1
h
E
V
t
+
h
,
x
+
y
-
V
t
,
x
=
1
h
E
V
t
+
h
,
x
+
y
-
V
t
+
h
,
x
+
1
h
V
t
+
h
,
x
-
V
t
,
x
.By (6) and distribution of ξ, we have the first item of the right side of (7) that is bounded: (8)
α
t
,
x
,
f
t
,
x
-
K
t
,
x
,
y
E
h
f
t
,
x
+
σ
t
,
x
ξ
2
≤
1
h
E
V
t
+
h
,
x
+
y
-
V
t
+
h
,
x
≤
α
t
,
x
,
f
t
,
x
-
K
t
E
h
f
t
,
x
+
σ
t
,
x
ξ
2
.Since V is differentiable for t, so the last item of the right side of (7) is also bounded. Therefore, the supremum limit of (7) exists; that is, D
+
V
(
t
,
x
) exist exactly.The following lemmas will be used later.Lemma 4 (see Theorem7.1
.
2 in [6], or Theorems 4.3 and  4.4 in [17]).
Supposef
,
σ satisfy (3). Then, stochastic differential equation (1) admits a unique strong solution {
x
t
,
t
≥
0
} such that, for any h
>
0
(
[
t
,
t
+
h
]
⊂
[
0
,
T
]
,
l
≥
2
), there exists (K
T is a constant dependent only on T, l, and L): (9)
E
sup
t
≤
s
≤
t
+
h
⁡
x
s
l
≤
K
T
1
+
E
x
t
l
,
(10)
E
x
r
-
x
s
l
≤
K
T
1
+
E
x
s
l
r
-
s
l
/
2
.Lemma 5.
SupposeV satisfies (6), and x
(
t
) is the solution of (1); let m
(
t
)
=
E
V
(
t
,
x
(
t
)
); then (11)
D
+
m
t
≤
E
D
+
V
t
,
x
t
,where D
+
V
(
t
,
x
(
t
)
)
=
D
+
V
(
t
,
x
)
|
x
=
x
(
t
) and D
+
m
(
t
) is the usual right upper Dini derivative defined by (12)
D
+
m
t
=
lim
sup
h
→
0
+
⁡
m
t
+
h
-
m
t
h
.Proof.
For smallh
>
0, we have (13)
m
t
+
h
-
m
t
=
E
V
t
+
h
,
x
t
+
h
-
V
t
+
h
,
x
t
+
h
f
t
,
x
t
+
σ
t
,
x
t
B
t
+
h
-
B
t
+
V
t
+
h
,
x
t
+
h
f
t
,
x
t
+
σ
t
,
x
t
B
t
+
h
-
B
t
-
V
t
,
x
t
.We now prove that the first two items of the right side are the higher infinitesimal of h. By (1), we know that (14)
x
^
≔
x
t
+
h
-
x
t
+
h
f
t
,
x
t
+
σ
t
,
x
t
B
t
+
h
-
B
t
=
∫
t
t
+
h
f
s
,
x
s
-
f
t
,
x
t
d
s
+
∫
t
t
+
h
σ
s
,
x
s
-
σ
t
,
x
t
d
B
s
.For convenience, we denote f
^
s
≔
f
(
s
,
x
(
s
)
)
-
f
(
t
,
x
(
t
)
), similar meaning for σ
^
s. We have (15)
E
x
^
2
≤
2
E
∫
t
t
+
h
f
^
s
d
s
2
+
E
∫
t
t
+
h
σ
^
s
d
B
s
2
≤
2
h
∫
t
t
+
h
E
f
^
s
2
d
s
+
∫
t
t
+
h
E
σ
^
s
2
d
s
≤
2
L
h
∫
t
t
+
h
E
x
s
-
x
t
2
d
s
+
∫
t
t
+
h
E
x
s
-
x
t
2
d
s
.By Lemma 4 and inequality (10), we have (16)
E
x
^
2
≤
2
L
K
T
1
+
E
x
t
2
1
2
h
3
+
h
2
.Let x
~
t
≔
x
(
t
)
+
h
f
(
t
,
x
(
t
)
)
+
σ
(
t
,
x
(
t
)
)
(
B
t
+
h
-
B
t
); then (α
(
x
) replaces α
(
t
,
x
) for shortening)(17)
E
V
t
+
h
,
x
t
+
h
-
V
t
+
h
,
x
~
t
-
α
x
~
t
,
x
^
≤
C
h
h
2
.Now we estimate the order of (18)
E
α
x
~
t
,
x
^
=
E
α
x
t
,
x
^
+
E
α
x
~
t
-
α
x
t
,
x
^
.Since E
〈
α
(
x
(
t
)
)
,
x
^
〉
=
E
〈
α
(
x
(
t
)
)
,
E
x
^
∣
F
t
〉 and E
x
^
∣
F
t
=
E
∫
t
t
+
h
‍
f
^
s
d
s
∣
F
t, so (19)
E
α
x
t
,
x
^
=
E
α
x
t
,
∫
t
t
+
h
f
^
s
d
s
≤
C
1
h
h
3
/
2
,
E
α
x
~
t
-
α
x
t
,
x
^
≤
α
x
~
t
-
α
x
t
x
^
≤
C
2
h
h
3
/
2
,where C
1
(
h
)
,
C
2
(
h
) are continuous positive functions. By (19), we have (20)
E
α
x
~
t
,
x
^
≤
C
3
h
h
3
/
2
.Since (21)
E
V
t
+
h
,
x
t
+
h
-
V
t
+
h
,
x
~
t
≤
E
V
t
+
h
,
x
t
+
h
-
V
t
+
h
,
x
~
t
-
α
x
~
t
,
x
^
+
E
α
x
~
t
,
x
^
,by (17) and (20), we see that the first two items of right side are higher infinitesimal of t. So we have (22)
lim
h
→
0
+
⁡
1
h
E
V
t
+
h
,
x
t
+
h
-
V
t
+
h
,
x
~
t
=
0
.For the last two items of (13), since x
(
t
) is independent of B
t
+
h
-
B
t with normal distribution N
(
0
,
h
), so we have (23)
lim
sup
h
→
0
⁡
1
h
E
V
t
+
h
,
x
t
+
h
f
t
,
x
t
+
σ
t
,
x
t
B
t
+
h
-
B
t
-
V
t
,
x
t
≤
E
lim
sup
h
→
0
⁡
1
h
E
V
t
+
h
,
y
+
h
f
t
,
y
+
σ
t
,
y
B
t
+
h
-
B
t
-
V
t
,
y
y
=
x
t
=
E
D
+
V
t
,
y
y
=
x
t
.This proves (11).The following lemma will be used in the proof of Theorem7.Lemma 6 (see Theorem1.5
.
2 in [3]).
Letg
∈
C
(
R
+
×
R
+
,
R
) and r
(
t
) be the maximal solution of (24)
u
˙
=
g
t
,
u
,
u
t
0
=
u
0
,existing on [
t
0
,
∞
). Suppose m
∈
C
(
R
+
,
R
+
) and D
m
(
t
)
≤
g
(
t
,
m
(
t
)
), t
≥
t
0, where D is any fixed Dini derivative. Then m
(
t
0
)
≤
u
0 implies m
(
t
)
≤
r
(
t
), t
≥
t
0.Now we formulate the basic comparison results in terms of Lyapunov functionV.Theorem 7.
AssumeD
+
V
(
t
,
x
) satisfies (25)
D
+
V
t
,
x
≤
g
t
,
V
t
,
x
,
t
,
x
∈
R
+
×
R
n
,where g
∈
C
(
R
+
2
,
R
) is concave for u. Let r
(
t
)
=
r
(
t
,
t
0
,
u
0
) be the maximal solution of the differential equation(26)
u
˙
=
g
t
,
u
,
u
t
0
=
u
0
≥
0
.Then, for every solution of (1) x
(
t
)
=
x
(
t
,
t
0
,
x
0
), V
(
t
0
,
x
0
)
≤
u
0 implies (27)
E
V
t
,
x
t
≤
r
t
,
t
≥
t
0
.Proof.
Denotem
(
t
)
=
E
V
(
t
,
x
(
t
)
). By Lemma 5 and the concave of g we have(28)
D
+
m
t
≤
g
t
,
m
t
,
m
t
0
≤
u
0
.By Lemma 6, we can obtain the result (27).Remark 8.
IfV
∈
C
1,2
(
R
+
×
R
n
,
R
+
), the inequality (25) became (29)
L
V
t
,
x
≤
g
t
,
V
t
,
x
,
t
,
x
∈
R
+
×
R
n
.


## 3. Stability Criteria in terms of Two Measures

Now we discuss the two-measure stability criteria for the stochastic differential system (1). We assume f
(
t
,
0
)
=
σ
(
t
,
0
)
=
0 for all t
≥
t
0. Firstly, we give some definitions for stochastic stability.Definition 9.
The stochastic differential system (1) is said to be(
S
1
)
mean(
h
0
,
h
)-equistable, if for each ϵ
>
0 and t
0
∈
R
+, there exists a function δ
=
δ
(
t
0
,
ϵ
)
>
0 which is continuous in t
0 for each ϵ such that (30)
E
h
0
t
0
,
x
0
<
δ
implies
E
h
t
,
x
t
<
ϵ
,
t
≥
t
0
,
wherex
(
t
)
=
x
(
t
,
t
0
,
x
0
) is any solution of (1);
(
S
2
)
mean(
h
0
,
h
)-uniformly stable, if (
S
1
) holds with δ being independent of t
0;
(
S
3
)
mean(
h
0
,
h
)-quasiequiasymptotically stable, if for each ϵ
>
0 and t
0
∈
R
+, there exist positive number δ
0
=
δ
0
(
t
0
) and T
=
T
(
t
0
,
ϵ
) such that (31)
E
h
0
t
0
,
x
0
<
δ
0
implies
E
h
t
,
x
t
<
ϵ
,
t
≥
t
0
+
T
;
(
S
4
)
mean(
h
0
,
h
)-quasiuniform asymptotically stable if (
S
3
) holds with δ
0 and T being independent of t
0;
(
S
5
)
mean(
h
0
,
h
)-asymptotically stable if (
S
1
) holds and, given t
0
∈
R
+, there exists a δ
0
=
δ
0
(
t
0
)
>
0 such that (32)
E
h
0
t
0
,
x
0
<
δ
0
implies
l
i
m
t
→
∞
E
h
t
,
x
t
=
0
;
(
S
6
)
mean(
h
0
,
h
)-uniformly equiasymptotically stable, if (
S
1
) and (
S
3
) hold together;
(
S
7
)
mean(
h
0
,
h
)-uniformly asymptotically stable if (
S
2
) and (
S
4
) hold simultaneously;
(
S
8
)
mean(
h
0
,
h
)-unstable if (
S
1
) fails to hold.The following classes of functions will be used in this paper:K
=
{
a
∈
C
(
R
+
,
R
+
)
:
a
(
u
)
i
s
s
t
r
i
c
t
l
y
i
n
c
r
e
a
s
i
n
g
i
n
u
a
n
d
a
(
0
)
=
0
},
C
K
=
{
a
∈
C
(
R
+
2
,
R
+
)
:
a
(
t
,
s
)
∈
K
f
o
r
e
a
c
h
t
},
Γ
=
{
h
∈
C
(
R
+
×
R
n
,
R
+
)
:
inf
⁡
h
(
t
,
x
)
=
0
}
.Definition 10.
Leth
0
,
h
∈
Γ. Then, we say that h
0 is finer than h if there exists a function ϕ
∈
C
K such that h
(
t
,
x
)
≤
ϕ
(
t
,
h
0
(
t
,
x
)
). Furthermore, if ϕ is independent of t then we call h
0 uniformly finer than h.Definition 11.
LetV
∈
C
(
R
+
×
R
n
,
R
+
). If there exists a function convex b
∈
K such that b
(
h
(
t
,
x
)
)
≤
V
(
t
,
x
), then we call V
h-positive definite. If there exists a concave function a
∈
K such that V
(
t
,
x
)
≤
a
(
h
(
t
,
x
)
), then we call V
h-decrescent.Theorem 12.
Assume thatA
0
h
0
,
h
∈
Γ and h
0 is uniformly finer than h,
(
A
1
)
V
∈
C
(
R
+
×
R
n
,
R
+
), V
(
t
,
x
) satisfies (6), and V is h-positive definite and h
0-decrescent,
(
A
2
)
g
∈
C
(
R
+
2
,
R
) and g
(
t
,
0
)
≡
0,
(
A
3
)
D
+
V
(
t
,
x
)
≤
g
(
t
,
V
(
t
,
x
)
) for (
t
,
x
)
∈
R
+
×
R
n.
Then, the stability properties of the trivial solution of (26) imply the corresponding (
h
0
,
h
)-stability properties of (1).Proof.
SinceV is h-positive definite, so there exists a convex b
∈
K such that (33)
b
h
t
,
x
≤
V
t
,
x
,
t
,
x
∈
R
+
×
R
n
.Suppose the trivial solution of (26) is equistable and r
(
t
,
t
0
,
r
(
t
0
)
)  (
t
≥
t
0
) is its maximal solutions with initial time t
0 and initial value r
(
t
0
)
≥
0, then, for every ϵ
>
0, there exists δ
1
>
0, when r
(
t
0
)
<
δ
1, (34)
r
t
=
r
t
,
t
0
,
r
t
0
<
b
ϵ
,
t
≥
t
0
.Let r
(
t
0
)
=
E
V
(
t
0
,
x
(
t
0
)
); then, by Theorem 7, we have (35)
E
V
t
,
x
t
≤
r
t
,
t
≥
t
0
.Since V is h
0-decrescent, there exists a concave a
∈
K such that (36)
V
t
,
x
≤
a
h
0
t
,
x
,
t
,
x
∈
R
+
×
R
n
.So (37)
r
t
0
≤
E
a
h
0
t
0
,
x
t
0
≤
a
E
h
0
t
0
,
x
t
0
.Since a is continuous and strictly increasing, so let δ
=
a
-
1
(
δ
1
); then when E
h
0
(
t
0
,
x
(
t
0
)
)
<
δ, inequality (34) holds. Combining (33), (34), and (35) and using the strictly increase of b, we can gain (38)
E
h
t
,
t
0
,
x
t
0
<
ϵ
,
t
≥
t
0
,which implies (1) (
h
0
,
h
)-equistability.Remark 13.
IfV
∈
C
1,2
(
R
+
×
R
n
,
R
+
), then condition (
A
3
) can be replaced by (39)
L
V
t
,
x
≤
g
t
,
V
t
,
x
.Remark 14.
The stabilities of auxiliary ordinary differential equation (26) are defined by Definition  2.4
.
1 in [3].Example 15.
Consider the following 2-dimensional Itô’s system:(40)
d
x
1
t
=
x
2
t
d
t
,
d
x
2
t
=
-
b
x
1
t
+
a
x
2
t
d
t
+
c
x
2
t
+
e
x
1
t
d
B
t
.
Leth
(
t
,
x
)
=
h
0
(
t
,
x
)
=
e
γ
t
(
x
1
2
+
x
2
2
); suppose V
(
t
,
x
) has the form (41)
V
t
,
x
=
e
γ
t
α
x
1
2
+
2
β
x
1
x
2
+
x
2
2
.Then(42)
L
V
=
e
γ
s
γ
α
-
2
β
b
+
e
2
x
1
2
+
2
γ
β
+
2
α
-
2
β
a
-
2
b
+
2
c
e
x
1
s
x
2
+
γ
+
2
β
-
2
a
+
c
2
x
2
2
.
In order to makeV
(
t
,
x
)
h-positive, we let (43)
α
>
0
,
α
-
β
2
>
0
.Let g
(
t
,
u
)
=
-
θ
u, θ
>
0; in order to find θ
>
0 to satisfy(44)
L
V
≤
g
t
,
V
,we set (45)
2
β
γ
+
2
α
-
2
β
a
-
2
b
+
2
c
e
=
0
,
γ
α
-
2
β
b
+
e
2
<
0
,
γ
+
2
β
-
2
a
+
c
2
<
0
.When γ
<
a, combining (43) and (45), we have that, when (46)
max
⁡
1
2
a
-
γ
-
γ
-
a
2
+
4
b
-
c
e
,
γ
b
-
γ
c
e
+
e
2
γ
2
+
2
b
-
γ
a
<
β
<
min
⁡
b
-
c
e
γ
-
a
,
a
-
1
2
γ
+
c
2
,
1
2
a
-
γ
+
γ
-
a
2
+
4
b
-
c
e
,there exists θ
>
0 which satisfies (44). Moreover, the trivial solution of (47)
u
˙
=
-
θ
uis uniformly asymptotically stable; by Theorem 12, the stochastic differential is mean-(
h
0
,
h
)-asymptotically stable.However, in practice, the coefficients maybe have some uncertainty properties; that is, we only know the range of the parametersa
,
b. Then how to discuss the stability of such systems with uncertainty is still a very interesting problem. The following section will introduce an optimal control approach to discuss the stability of such systems with uncertainty.


## 4. The Stability for Systems with Uncertainty

Consider the following stochastic differential equations with uncertainty:(48)
d
x
=
f
x
+
g
x
k
x
d
t
+
σ
x
d
B
t
,where k
(
x
) is an uncertainty function, and f
(
0
)
=
k
(
0
)
=
σ
(
0
)
=
0, that is, the trivial solution, is 0 of (48). Furthermore, we also suppose there exists a function k
m
a
x
(
x
)
≥
0 such that (49)
k
x
≤
k
m
a
x
x
.Now we discuss how to determine the asymptotic stability of system (48) for all uncertainty function k. Similar to the methods applied by [18, 19], we can translate this stability problem into an optimal problem.For the nominal system(50)
d
x
=
f
x
+
g
x
u
d
t
+
σ
x
d
B
t
,suppose we can find a state-feedback control u
=
u
-
(
x
) that minimizes the cost functional(51)
J
x
·
,
u
=
E
y
∫
0
∞
f
m
a
x
2
x
+
ρ
h
0
x
+
3
u
T
u
d
t
,where h
0
∈
C
(
R
n
,
R
+
), y is the initial value of (50), ρ
>
0, and u is admissible on [
0
,
T
] for each T
<
∞, and satisfies (52)
J
x
·
,
u
<
∞
.Let the value function (53)
V
y
=
m
i
n
u
J
x
·
,
u
.Theorem 16.
Supposeu
-
(
x
) is an optimal control of problem (53), and there exists ρ
-
≤
ρ satisfying (54)
u
-
2
≤
ρ
-
2
h
0
x
,and the value function V
∈
C
2
(
R
n
,
R
+
), V is also h positive, and h
0 is decrescent, then system (48) is uniformly mean (
h
0
,
h
) uniformly asymptotically stable for all uncertainties k
(
x
).Proof.
The values functionV satisfies the Hamilton-Jacobi-Bellman equation: (55)
min
u
∈
R
p
⁡
f
m
a
x
2
+
x
T
x
+
u
T
u
+
V
x
,
f
+
g
u
+
1
2
σ
σ
T
,
V
x
x
=
0
.So, the optimal control u
- satisfies (56)
f
m
a
x
2
+
x
T
x
+
u
-
T
u
-
+
V
x
,
f
+
g
u
-
+
1
2
σ
σ
T
,
V
x
x
=
0
,
2
u
-
T
+
V
x
T
g
=
0
.Then, the Lyapunov generator of (48) for V is given as (57)
L
V
x
≔
V
x
,
f
+
g
k
+
1
2
V
x
x
,
σ
σ
T
=
V
x
,
g
k
-
V
x
,
g
u
-
-
h
0
x
-
k
m
a
x
2
-
u
-
T
u
-
≤
k
2
-
k
m
a
x
2
-
ρ
-
ρ
-
h
0
x
≤
-
ρ
-
ρ
-
h
0
x
.So, for the solutions x
(
t
) of (48) with uncertainty k
(
x
), applying Itô’s formula to V
(
x
(
t
)
), we have, when t
≥
t
0, (58)
E
V
x
t
-
E
V
x
t
0
=
E
∫
t
0
t
L
V
x
s
d
s
≤
-
ρ
-
ρ
-
E
∫
t
0
t
h
0
x
d
s
.Hence (59)
E
V
x
t
≤
E
V
x
t
0
-
E
∫
t
0
t
h
0
x
s
d
s
,
∀
t
≥
t
0
.So m
(
t
)
=
E
V
(
x
(
t
)
) is decreasing on [
0
,
∞
]. Now we show m
(
t
)
→
0 when t
→
∞. Suppose m
(
t
)
→
m
0
>
0. Since V is h
0 decrescent, so there exists c
>
0 and t
0
>
0, such that E
h
0
(
x
(
t
)
)
>
c
>
0 (t
≥
t
0) combining with (59); we have(60)
E
V
x
t
≤
E
V
x
t
0
-
c
t
-
t
0
⟶
-
∞
,
t
⟶
∞
.This contradicts the fact that V
(
x
)
≥
0. This implies that system (48) is uniformly mean (
h
0
,
h
) uniformly asymptotically stable.Corollary 17.
Specially, let(61)
f
x
=
A
x
,
g
x
=
G
,
σ
x
=
C
x
,
k
m
a
x
x
=
M
x
.The value function can be given by V
(
x
)
=
x
T
P
x and the optimal control u
-
(
x
)
=
-
G
T
P
x, where P
>
0 satisfies the following Riccati equation: (62)
M
2
+
1
I
+
P
A
+
A
T
P
+
C
T
P
C
-
P
G
G
T
P
=
0
.Let λ
m
a
x be the maximal eigenvalue of P
G
G
T
P. If λ
m
a
x
2
≤
ρ
/
2, then the corresponding system with uncertainty is uniformly mean (
h
0
,
h
)-equistable for all uncertainties k
(
x
) (which is also mean square asymptotically stable for all uncertainties k
(
x
)).Example 18.
Consider (41) with uncertainty coefficients. In (41), replace a
,
b by a
+
sin
⁡
x
1 and b
+
cos
⁡
x
2, respectively, and a takes values in [
1.4,1.6
], b in [
0.9,1.1
], c
=
-
1, and e
=
1. Then the system with uncertainty is obtained: (63)
d
x
1
t
=
x
2
t
d
t
,
d
x
2
t
=
-
b
+
0.1
cos
⁡
B
t
x
1
t
+
a
+
0.1
sin
⁡
B
t
x
2
t
d
t
+
c
x
2
t
+
e
x
1
t
d
B
t
.Let (64)
A
=
0
1
-
1
-
1.5
,
B
=
0
0
1.5
-
b
-
0.1
cos
⁡
B
t
1
-
a
-
0.1
sin
⁡
B
t
,
C
=
0
0
1
-
1
,And let G be the 2-order identical matrix. Let h
(
x
)
=
h
0
(
x
)
=
x
2; then (65)
f
x
=
A
x
,
g
x
=
G
,
k
x
=
B
x
,
σ
x
=
C
x
.We can take k
m
a
x
(
x
)
=
0.3
x
1
2
+
x
2
2 and the auxiliary optimal problem is (66)
d
x
=
A
x
+
G
u
d
t
+
C
x
d
B
t
,with the cost functional (67)
J
x
·
,
u
=
E
y
∫
0
∞
3
x
T
x
+
u
T
u
d
t
,
ρ
=
2.91, solving (62) with M
=
0.3; we have (68)
P
=
1
0
0
0.5
.By Corollary 17, we can determine that the stochastic system (63) is uniformly mean (
h
0
,
h
) uniformly stable for all uncertainties.


## 5. Conclusion

In this paper, we extend the stability criteria of two measures to the mean stability situations for the stochastic systems with uncertainty. For the usual SDE, we give the results of mean stability criteria which are the basic criteria for such systems. As far as the systems with uncertainty, in order to resolve the difficulties coming from the coefficient uncertainty, we use the optimal control results as an auxiliary method to determine the mean stability. Furthermore, the stability criteria in terms of two measures for other stochastic systems, such as systems with Markovian jumps or Poisson jumps, are worth further studying.



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*Source: 348235-2015-10-18.xml*