stability ft

7/31/2019 Stability FT

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IEEE TRANSACTIONS ON AUTOMATICAL CONTROL VOL. , NO. 2

This paper aims to study the finite-time stability of stochas-

tic nonlinear system in Lyapunov sense. First, a lemma that

ensures the existence and uniqueness of the global solution

of stochastic nonlinear system is proven under the local

Lipschitz and local boundnedness conditions on the integral of

diffusion operator with respect to the product measure ( Lemma

2.1). Then, the definition of finite-time stability in probability

for stochastic nonlinear system is presented ( Definition 3.1).

The contribution of the present paper are two folds: first, a

sufficient condition (Theorem 3.1) for the finite-time stability

in probability of stochastic nonlinear system is derived and

proved. A useful lemma ( Lemma 3.1), extended from Bihari’s

inequality is presented, which plays an important role in the

prove of our main theorem. Second, a continuous finite-time

control (Theorem 4.1) is designed to guarantee the finite-

time stability in probability for a class of stochastic nonlinear

systems. Based on Lemma 2.1, it is seen that the stochastic

differential equation satisfying the conditions in Theorem 3.1

admits a unique solution for any finite initial condition. For the

control design, since our proposed control falls in the family

of ρ-function, the stochastic closed-loop control system admitsa unique solution.

The rest of the paper is organized as follows: in Section

II, a class of stochastic nonlinear systems to be considered in

this paper, the notations, and some definitions are formulated.

Some preliminary results on the existence and uniqueness

are presented under the conditions that the coefficients of

a stochastic nonlinear equation satisfy the local Lipschitz

condition and the integral of diffusion operator for a radially

unbounded C 2 function is local bounded with respect to

product measure. In Section III, the finite-time stability in

probability for stochastic nonlinear systems is defined, and

the extension of Bihari’s inequality (see, [16]), is derived.

Then, the main result on the stochastic Lyapunov theorem:a sufficient condition on the finite-time stability in probability

of stochastic nonlinear systems is presented. An simulation

example is given to illustrate the theoretical analysis. In

section IV, we use the stochastic Lyapunov theorem to show

that a state feedback control can be designed to stabilize a

class of stochastic closed-loop systems in finite time. Finally,

concluding remarks are given in section V.

I I . PRELIMINARY RESULTS

In this paper, we will consider an n-dimensional stochastic

nonlinear system of the form

dx(t) = f (t, x(t))dt+g(t, x(t))dw(t), x(0) = x0 ∈ Rn, t ≥ 0,(2.1)

where x(·) ∈ Rn is the state process, w(·) is an m-dimensional

Brownian motion defined on a complete probability space

(Ω,F , P ) with the augmented filtration F tt≥0 generated

by w(·), and the functions f : R+ × Rn → Rn and g :R+ × Rn → Rn×m, also called coefficients of the equation,

are Borel measurable and satisfy f (t, 0) = 0, g(t, 0) = 0 for

all t ≥ 0.

For notional convenience, in above and in the sequel,

Rn always denotes the n-dimensional Euclidean space, R+

denotes all nonnegative real numbers and Rn×m denotes the

space of n×m matrices with real entries. For a vector x ∈ Rn,

|x| will denote the Euclidean norm |x| = (n

i=1 x2i )1/2. For

a matrix A, A will denote the Frobenius norm: A =(traceAT A)1/2, where the superscript “T” denotes the

transpose of a vector or a matrix. a ∧ b means the minimum

of a and b, while a ∨ b means the maximum.

A function V : Rn → R is said to be C k if it is k-times

continuously differentiable. Let ∂V/∂x denote the gradient of

a C 1 function V , we always write ∂V/∂x as a row vector.

For a C 2 function V , ∂ 2V /∂ 2x denotes the Hessian of V ,the n × n matrix of second-order partial derivatives of V .A function V : Rn → R is said to be positive-definite if

V (0) = 0 and V (x) > 0 for all x ∈ Rn \ 0.

Since we have assumed that f (t, 0) = 0 and g(t, 0) = 0for all t ≥ 0, this implies that Equation (2.1) admits a trivial

zero solution. It is well known that, in order for a stochastic

differential equation to have a unique global solution for any

give initial data, the coefficients of the equation are generally

needed to satisfy the linear growth condition and the local

Lipschitz condition. We will impose the following assumption.

Assumption 1: Both f (t, x) and g(t, x) satisfy the local

Lipschitz condition, that is, for any R > 0, there exists a

constant C R ≥ 0 such that

|f (t, x1) − f (t, x2)| ∨ |g(t, x1) − g(t, x2)| ≤ C R|x1 − x2|

for all t ∈ R+ and |x1| ∨ |x2| ≤ R.

If Assumption 1 holds, for any initial value x0 ∈ Rn, there

is a unique maximal local solution to (2.1) for all t ∈ [0, σe),

where σe is the explosion time [see, [15], pp.95]. In order to

show the solution is global, we only need to show that σe = ∞a.s.. For a C 2 function V , let LV denote the diffusion operator

of V with respect to the equation (2.1) defined by

LV (x) =∂V (x)

∂xf (t, x) +

1

2tracegT (t, x)

∂ 2V (x)

∂x2

g(t, x).

For the existence and uniqueness result of global solutions

to (2.1), the following lemma gives a positive answer under

some appropriate conditions. Even though some authors [9],

[23], have proved the existence and uniqueness result based

on a special C 2 function V , we here should point out that

our result has many refinements and improvements even

generalizes some of existing results.

Lemma 2.1: Let Assumption 1 hold. Suppose that there

exists a C 2 function V : Rn → R+, whose diffusion operator

LV with respect to (2.1) satisfies

E

σk∧T

0

LV (x(s))ds =

T

0

E

I s≤σkLV (x(s))

ds ≤ C T ,

(2.2)

for any stopping time σk = inf t; |x(t)| ≥ k, k ∈ N and

T > 0, where C T ≥ 0 is a constant depending on T only. If

the function V is radially unbounded, that is,

lim|x|→∞

V (x) = ∞ (2.3)

holds, then there exists a unique global solution to (2.1) for

any x0 ∈ Rn.

Proof: We will show that σe = ∞ a.s.. For each integer k ∈ N ,we define a stopping time as follows:

σk = inf

t : |x(t)| ≥ k




with the traditional setting inf ∅ = ∞, where ∅ denotes the

empty set. Clearly, σk is also increasing as k increases. If we

set σ∞ = limk→∞ σk, then σ∞ ≤ σe a.s. Furthermore, if we

can show that σ∞ = ∞ a.s., then σe = ∞ a.s., which implies

that (2.1) admits a unique global solution for all t ≥ 0. Let

T > 0 be arbitrary. Define σk ∧ T = σT k . For any t ≥ 0, by

Ito’s formula, we have

V (x(t ∧ σT k )) = V (x(0)) + t∧σT k

0

L(V (x(s)))ds

+

t∧σT k0

∂V (x(s))

∂xg(s, x(s))dw(s). (2.4)

Since ∂V/∂x and g(., x) are bounded whenever x is restricted

to a compact set, the local martingale term in (2.4) is a

martingale for t ∈ [0, σk ∧ T ]. Thus, by (2.2), we have

EV (x(σT k )) = V (x(0)) + E

σT k0

LV (x(s))ds

≤ V (x(0)) + C T < ∞. (2.5)

Note that for everyω ∈ σk ≤ T ,

|x(σ

T

k )| = |x(σk)| = k.

Hence it follows from (2.5) that

P σk ≤ T inf |x|=k

V (x) ≤ E

I σk≤T V (x(σk))

≤ V (x(0)) + C T < ∞. (2.6)

For any given k, the set x ∈ Rn : |x| = k is compact,

therefore the continuous function V attains a minimum on it.

By (2.3), it is not hard to get limk→∞ inf |x|=k V (x) = ∞.Letting k → ∞ on both sides of (2.6) gives limk→∞ P (σk ≤T ) = 0, and hence P (σ∞ ≤ T ) = 0. Since T > 0 is arbitrary,

we must have P (σ∞ < ∞) = 0, which implies that P (σ∞ =∞) = 1 and the required assertion follows.

Remark 2.1: The existence of a unique solution for stochas-

tic nonlinear system is very important in theory, which is also

the basis of dealing with practical control problems. However,

it is found that this aspect has not drawn enough attention in

the design of stochastic control systems.

Remark 2.2: Obviously, for a function V ∈ C 2(R2, R+),

if its diffusion operator LV ≤ 0, more generally, LV is

bounded above or, for example, is controlled by a nonnegative

continuous function η(·) such that ∞0 η(t)dt < ∞ , then

the condition (2.2) holds. An typical example of that (2.3)

holds is that there exists a class K∞ function µ such that

V (x) ≥ µ(|x|).

Definition 2.1: A function µ : R+ → R+ is said to be

a class K function if it is continuous, strictly increasing andµ(0) = 0. A class K function µ is said to belong to class K∞

if µ(r) → ∞ as r → ∞.

III. FINITE-TIME STABILITY THEOREM FOR STOCHASTIC

NONLINEAR SYSTEM

In this section, we shall propose and study the finite-time

stability theorem for the stochastic nonlinear system (2.1). The

technique used is based on the stochastic Lyapunov theory. It is

worth noting that although some existing results on stochastic

Lyapunov theory for stochastic nonlinear system can be found

in Has’minskii [9], Kusher [13], Mao [16], and Deng et al [4],

our stochastic Lyapunov theory is the extension to the finite-

time stability case. In fact, the following theorem can be seen

as the stochastic counterpart of the finite-time stability theory

for deterministic system in Bhat and Bernstein [2]. To our

best knowledge, it seems to be the first theorem on finite-time

stability of stochastic nonlinear system.

Let us first present a precise definition of finite-time stability

in probability for the trivial solution of (2.1). The following

definition is motivated by the definition of finite-time stablility

and stochastic Lyapunov stability in [2] and [16].

Definition 3.1: The trivial solution of equation (2.1) is said

to be finite-time stable in probability, if equation (2.1) admits

a unique solution for any initial condition x0 ∈ Rn, denoted

by x(t; x0), moreover, the following statements hold:

(i) Finite-time asymptotic stability in probability: given any

initial condition x0 ∈ Rn \ 0, the stochastic settling time,

τ = inf t; x(t; x0) = 0, is finite almost surely, that is, P

τ <∞

= 1;

(ii) Lyapuvov stability in probability: For every pair of ε ∈(0, 1) and r > 0, there exists a δ = δ(ε, r) > 0 such that

P |x(t; x0)| < r, for all t ≥ 0 ≥ 1 − ε, (3.1)

whenever |x0| < δ.

Remark 3.1: Note that, given an initial value x0 ∈ Rn,

if equation (2.1) admits a unique solution x(t, x0), then it

is clear that x(·, x0) is continuous, therefore, the finite-time

asymptotic stability in probability in Definition 3.1 implies

that P

x(τ ; x0) = 0

= 1. Besides, if x(τ ; x0) = 0 holds a.s.,

then x(t, x0) = 0 holds a.s. for all t ≥ τ from the uniqueness.

At the moment, the inequality (3.1) is equivalent to

P

sup

0≤t≤τ |x(t; x0)| < r

≥ 1 − ε. (3.2)

Now, we are in the position to present the main result of finite-

time stability of stochastic nonlinear system.

Theorem 3.1: Let Assumption 1 hold. If there exists a C 2

function V : Rn → R+, K∞ class functions µ1 and µ2,

positive real numbers C > 0 and 0 < γ < 1, such that for all

x ∈ Rn and t ≥ 0,

µ1(|x|) ≤ V (x) ≤ µ2(|x|), (3.3)

LV (x) ≤ −C

V (x)γ

, (3.4)

then the trivial solution of stochastic system (2.1) is finite-time

stable in probability.To proceed, we need the following lemmas.

Lemma 3.1: Let 0 < γ < 1 and λ > 0. Assume that there

exists a continuous function h : [0, ∞) → [0, ∞) with h(0) >0 such that, for any 0 ≤ u ≤ t,

h(t) − h(u) ≤ −λ

tu

(h(s))γds. (3.5)

Then there exists a real number T > 0 such that

h(t) ≤

h(0)1−γ − λ(1 − γ )t 11−γ

, ∀t ∈

0, T

. (3.6)




Proof: Let T 0 = inf t ≥ 0; h(t) = 0. Since h(0) > 0 and

h(t) ≥ 0 for all t > 0, it then follows that 0 < T 0 ≤ ∞. We

now define

l(t) =

h(0)1−γ − λ(1 − γ )t 11−γ

. (3.7)

It is not hard to verify that

l(t) − l(u) = −λ t

u (l(s))γ

ds (3.8)

for any 0 ≤ u ≤ t ∈

0, T 0 ∧ h(0)1−γ

λ(1−γ)

:= [0, T 0]. Note that

l(0) = h(0) and l(t) ≥ 0 as t ∈ [0, T 0]. Let S = t ∈[0, T 0]; h(t) > l(t). If S = ∅, then the required assertion

follows by taking T = T 0. Suppose that there exists a t ∈ S .This means that 0 < t ≤ T 0 and h(t) > l(t) ≥ 0. Let t =inf t < t; ∀s ∈ (t, t], h(t) > l(t). By the continuity of h(·)and l(·), we have that h(t) = l(t), and thus h(t) > l(t) > 0for any t ∈ (t, t). On the other hand, by (3.5), it is clear that

h(t) ≤ p(t) for all t ∈ [t, T 0], where

p(t) = h(t

) − λ t

t(h(s))

γ

ds.(3.9)

For any t ∈ [t, t), we have that

dp(t)

dt= −λ(h(t))γ , p(t) > 0,

anddl(t)

dt= −λ(l(t))γ , l(t) > 0.

Note that p(t)

l(t)

=

p(t)l(t) − l(t) p(t)

l(t)2

=−λ(h(t))γl(t) + λ(l(t))γ p(t)

l(t)2

≥λ(l(t))γh(t) − λ(h(t))γ l(t)

l(t)2

=λ(l(t))γ(h(t))γ

(h(t))1−γ − (l(t))1−γ

l(t)2

≥ 0.

Thus p(t)/l(t) is an increasing function as t ∈ [t, t). Since

p(t)/l(t) = h(t)/l(t) = 1, we immediately obtain that

p(t) ≥ l(t), t ∈ [t, t). By this, (3.8) and (3.9), we have tt

(h(s))γds ≤

tt

(l(s))γds, ∀t ∈ [t, t),

which implies that there exists a t ∈ (t, t) at least such thath(t) ≤ l(t), however, this is a contradiction. The proof is

complete.

Remark 3.2: Note that Lemma 3.1 is the extension of

Bihari’s inequality. In this lemma, T can be chosen as

T = T 0 ≤h(0)1−γ

λ(1 − γ ). (3.10)

Indeed, it is obvious that l(t) = 0 at t = h(0)1−γ

λ(1−γ) . If T 0 >h(0)1−γ

λ(1−γ) , then h(h(0)1−γ

λ(1−γ) ) > 0 from the definition of T 0, which

contradicts the fact of (3.6) because l(h(0)1−γ

λ(1−γ) ) = 0.

Lemma 3.2: Under the assumptions of Theorem 3.1, equa-

tion (2.1) admits a unique solution for any initial value

x0 ∈ Rn.

Proof: The conclusion is a direct consequence of Lemma

2.1 and Remark 2.2.

Proof of Theorem 3.1: By Lemma 3.2, equation (2.1) admits

a unique solution denoted by x(t; x0) for any initial value

x0 ∈ R

n. Let us first prove Lyaponov stability in probability

for the stochastic system (2.1). Let 0 < ε < 1 and r > 0be arbitrary. Define σr = t; |x(t; x0)| > r. Applying Ito’s

formula gives that

EV (x(t ∧ σr)) = V (x0) + E

t∧σr0

LV (x(s))ds ≤ V (x(0)),

(3.11)

where we have used (3.4) and the fact that t∧σr0

∂V ∂x g(s, x(s))dw(s) is a martingale. By (3.3), we

have

P (σr ≤ t)µ1(r) ≤ E

I σr≤tV (x(σr))

≤ E V (x(t ∧ σr))≤ V (x0) ≤ µ2(|x0|). (3.12)

Taking δ = µ−12

µ1(r)ε

, we obtain from (3.12) that P (σr ≤t) ≤ ε whenever |x0| ≤ δ. Letting t → ∞, we get P (σr <∞) ≤ ε, which implies that

P (supt≥0

|x(t; x0)| ≤ r) ≥ 1 − ε

as required.

We now turn our attention to prove finite-time stability in

probability. Obviously, the assertion holds for x0 = 0 since

x(t; x0) ≡ 0. We therefore only need to show the result forx0 ∈ Rn \ 0. Define

τ k = inf t ≥ 0; |x(t; x0)| ∈ (1

k, k),

where k ∈ N and satisfies 1k < |x0| < k. It is clear that τ k

is an increasing stopping time sequence. We now set τ ∞ =limk→∞ τ k. By Ito’s formula, for arbitrary 0 ≤ u ≤ t, we

have

EV (t ∧ τ k) = EV (u ∧ τ k) + E

t∧τ ku∧τ k

LV (x(s))ds

= EV (u ∧ τ k) + tu

E I s≤τ kLV (x(s))ds.

(3.13)

On one hand, it is easily seen that

EV (xt∧τ k) − EV (xu∧τ k)

= E

I t≤τ kV (x(t))

− E

I u≤τ kV (x(u))

+E

I t>τ k − I u>τ k

V (x(τ k))

≥ E

I t≤τ kV (x(t))

− E

I u≤τ kV (x(u))(3.14)




since V is positive definite. On the other hand, for any s ∈[0, ∞), by (3.3) and the definition of τ k, we can derive that

I s≤τ kµ1(1

k) ≤ I s≤τ kµ1(|x(s)|) ≤ I s≤τ kV (x(s))

≤ I s≤τ kµ2(|x(s)|) ≤ I s≤τ kµ2(k),

(3.15)

which, together with (3.4), gives

E

I s≤τ kLV (x(s))

≤ −CE

I s≤τ kV (x(s))γ

= −CE

I s≤τ kV (x(s))γ

≤ −C µ1( 1k )γ

µ2(k)γ

EI s≤τ kV (x(s))

γ.

(3.16)

Let C k = C µ1(

1k)γ

µ2(k)γ. It is obvious that C k is strictly decreasing

and converges to zero as k → ∞. By (3.13), (3.14) and (3.16),

we can deduce that

E

I t≤τ kV (x(t))

− E

I u≤τ kV (x(u))

≤ −C k tu

EI s≤τ kV (x(s))γds. (3.17)

Now define h(t) = E

I t≤τ kV (x(t))

, it then follows from

Lemma 3.1 and Remark 3.2 that there exists a sequence

0 < T k ≤V (x0)

1−γ

C k(1 − γ )

such that

h(T k) = E

I T k≤τ kV (x(T k))

= 0

which implies that P (T k ≤ τ k) = 0 from the definition of τ kand the fact that V is positive definite. Note that T k ↑ ∞ as

k → ∞. By the dominated convergence theorem, we have that

P (τ ∞ = ∞) = 0, which also implies that P (τ < ∞) = 1,

where

τ = inf

t; |x(t; x0)| = x(t; x0) = 0

,

and the required conclusion follows. The proof is thus com-

pleted.

Remark 3.3: We note that, in the proving process of Theo-

rem 3.1, assumptions (3.3) and (3.4) will guarantee finite-time

stability in probability for the trivial solution of (2.1), if system

(2.1) has a unique solution.

When the coefficients in equation (2.1) are continuous and

satisfy some ρ-conditions, we can establish the following

existence theorem of a unique solution, which follows from

Theorem 170 in Situ [21]. We state it as a lemma and willuse it in later analysis. Indeed, this lemma is a special case of

Theorem 170 in [21].

Lemma 3.3: Assume that f (t, x) and g(t, x) are continuous

in x. Assume also that there exist two non-negative functions

c1(t), c2(t), and a family of ρ-functions such that P - a.s., for

each 0 < T < ∞,

|f (t, x)| ≤ c1(t)(|1 + |x|), (3.18)

|g(t, x)|2 ≤ c1(t)(|1 + |x|2), (3.19)

2x1 − x2, f (t, x1) − f (t, x2) + |g(t, x1) − g(t, x2)|2

≤ c2(t)ρT (|x1 − x2|2), t ∈ [0, T ], (3.20)

0.0 0.5 1.0 1.5 2.0

− 2

− 1

0

1

2

Time (t)

S t a t e

( x )

Fig. 1. Simulation results for (3.21) with initial values x0 = 1 and x0 = −1

where T 0 ci(t)dt < ∞, i = 1, 2, and ρT (u) ≥ 0 as

u ≥ 0, is strictly increasing, continuous and concave such

that 0+ du/ρT (u) = ∞. Then for any given initial valuex0 ∈ Rn, (2.1) has a unique strong solution.

Example 3.1: Consider a one-dimensional stochastic au-

tonomous system in the form

dx(t) = f (x(t))dt + g(x(t))dw(t), x0 = 0, (3.21)

where

f (x) = c1x − c2xβ, c2 > 0, 1 > β > 0,

g(x) = c3x.

It is easy to check that f and g satisfy all assumptions in

Lemma 3.3 with c1(t) = [|c1| + c2] ∨ c23, c2(t) = [c21 + c23] andρT (u) = ρ(u) = u. In fact, it is obvious that c2(x1−x2)(xβ2 −xβ1 ) ≤ 0. So there is a unique strong solution to (3.21) from

Lemma 3.3. Consider Lyapunov function V (x) = |x|α, α ≥ 2.It is not hard to compute

LV (x) = αc1|x|α − αc2|x|α−2x1+β +1

2c23α(α − 1)|x|α.

Let ℵ denote the set of all odd numbers in N . Now set

β = pq , p, q ∈ ℵ, p < q,

c1 = −12c23(α − 1),

γ = α−1+β

α

,

we thus get

LV (x) = −αc2|x|αγ = −αc2|V (x)|γ,

and conclude that the trivial solution of (3.21) is finite-time

stable in probability.

Simulations have been carried out for system (3.21) with

c1 = −2, c2 = 1, c3 = 2, α = 2, β = 13 , in order to verify

Theorem 3.1. Figure 1 shows that the state of (3.21) converges

to zero in a finite time no matter the initial state is either

positive or negative.




IV. FINITE-TIME STABILIZATION OF A CLASS OF

STOCHASTIC NONLINEAR SYSTEMS

In this section, we will apply Theorem 3.1 and show that

we can find a state feedback control to guarantee finite-time

stability in probability of a class of stochastic closed-loop

control systems. To this end, we consider stochastic system

in the following form:

dx(t) = [f (t, x(t)) + u]dt + g(t, x(t))dw(t),

x(0) = x0 ∈ Rn \ 0, (4.1)

where x ∈ Rn is the system state, u ∈ Rn is the state

feedback control in the form u = u(x), and w(·) is a

one-dimensional Brownian motion. Assume that the coef-

ficients f (t, x) = (f 1(t, x), · · · , f n(t, x))T and g(t, x) =(g1(t, x), · · · , gn(t, x))T are continuous in x for all t ∈ R+

and satisfy f (t, 0) = 0 and g(t, 0) = 0. The problem is to find

a state feedback control u such that (4.1) is finite-time stable

in probability for any nonzero initial value.

We note that, when g(t, x) ≡ 0, system (4.1) reduces

to a deterministic closed-loop system. This system includes

the strict-feedback system and the MIMO system. Although

many stabilization control designs for stochastic nonlinear sys-

tems have been proposed based on Lyaponov-type functions,

including backstepping design [4], [20], [7], and adaptive

backstepping design [22]. However, it is still an open problem

whether these existing design methods can be used to make

system (4.1) to be finite-time stable in probability. For the full

state feedback control case, we find a stabilization design that

makes the system simultaneously stable instead of step by step

fashion.

Here, we first give an elementary inequality, which can befound in [11].

Lemma 4.1: For any xi ∈ R, i = 1, · · · , n, 0 ≤ b ≤ 1, the

following inequality holds:

|x1| + |x2| + · · · + |xn|

b≤ |x1|b+ |x2|b+ · · · + |xn|b. (4.2)

We now state our main result in this section.

Theorem 4.1: Assume that f (t, x) and g(t, x) are contin-

uous in x and satisfy f (t, 0) = 0 and g(t, 0) = 0 for each

t ≥ 0. If there exist constants C i ≥ 0, C i ≥ 0, i = 1, · · · , n,

and a ρ-function, such that P -a.s.

|f i(t, x)| ≤ C i|x|, (4.3)

|gi(t, x)|2 ≤ C i|x|2, (4.4)

2x1 − x2, f (t, x1) − f (t, x2) + |g(t, x1) − g(t, x2)|2

≤ ρ(|x1 − x2|2), ∀x1, x2 ∈ Rn, (4.5)

then system (4.1) can be finite-time stabilized in probability

by choosing a continuous state feedback control u.

Proof: Choose a Lyapunov function V (x) = |x|2, x ∈ Rn.

We now compute

LV (x) =∂V (x)

∂x[f (t, x) + u]

+1

2trace

gT (t, x)

∂ 2V (x)

∂x2g(t, x)

= 2xT · [f (t, x) + u] + |g(t, x)|2

= 2

n

i=1

xif i(t, x) + 2

n

i=1

xiui +

n

i=1

|gi(t, x)|2

≤ni=1

|xi|2 +

ni=1

|f i(t, x)|2

+2

ni=1

xiui +

ni=1

|gi(t, x)|2. (4.6)

By (4.3) and (4.4), we have

ni=1

|f i(t, x)|2 ≤ ni=1

C 2i

|x|2 (4.7)

andni=1

|gi(t, x)|2 ≤ ni=1

C i|x|2. (4.8)

We now set

ui = ui(x1, · · · , xn)

= −1

2xi

1 +

ni=1

[C 2i + C i]

−x2γ−1i , γ =p

q, p, q ∈ N,

1

2< γ < 1.(4.9)

Substituting (4.7)-(4.9) into (4.6) yields that

L

V (x) ≤ −2

n

i=1 x

2γ

i ≤ −2x

2

1 + x

2

2 + · · · + |xn|

2γ

= −2(V (x))γ (4.10)

by using Lemma (4.1). It is obvious that u is continuous. It

is suffices to show that system (4.1) admits a unique solution.

Let

u = (u1, · · · , un)T ,

ui = −1

2xi(1 +

ni=1

(C 2i + C i)) − x2γ−1i

In fact, the following inequality holds:

x − x, u − u =

n

i=1

(xi − xi) · (ui − ui)

= −1

2

ni=1

|xi − xi|2

1 +ni=1

(C 2i + C i))

−ni=1

(xi − xi) · (x2γ−1i − x2γ−1i )

≤1

2

ni=1

|xi − xi|2

1 +

ni=1

(C 2i + C i)

, (4.11)

which, together with (4.3)-(4.5), gives the required assertion

from Lemma 3.3, if we choose a new ρ-function ρ(u) =




ρ(u)+ 12(1+

ni=1(C 2i +C i))u. By Theorem 3.1, we therefore

conclude that the trivial solution of (4.1) is finite-time stable

in probability. The proof is complete.

Remark 4.1: Let us give two common examples for the ρ-

function:

ρ1(u) = K 1u, u ≥ 0, K 1 > 0; (4.12)

ρ2(u) = K 2 arctan(u), u ≥ 0, K 2 > 0. (4.13)

It can be shown that if ρ1 and ρ2 are ρ-functions, then ρ3 =ρ1 + ρ2 is also a ρ-function (see, [21]).

V. CONCLUSION

The main objective of this paper is to prove a stochastic

Lyapunov theorem on finite-time stability in probability for

stochastic nonlinear system. Although many theoretical criteria

for judging finite-time stability of deterministic nonlinear

system have been proposed over the last decade. Generally

speaking, these criteria can not be directly applied to stochastic

nonlinear system. The first obstacle is how to define finite-

time stability of stochastic nonlinear system, in other words,

what is the meaning of finite-time stability for a stochasticnonlinear system? For a stochastic nonlinear system, it is

impossible to find a fixed time at which the state of the system

is zero. Therefore, there is a key difference on finite-time

stability between deterministic system and stochastic system.

The other lies in that the Ito differentiation introduces not only

the gradient but also the Hessian of the Lyapunov function in

the Lyapunov analysis.

As we stated in Remark 2.1, the existence of a unique

solution for stochastic nonlinear system is a precondition of

various stochastic nonlinear control designs. However, This

has not drawn enough attention in stochastic control designs.

Every control design, in principle, should guarantee the ex-

istence of a unique solution for stochastic nonlinear system.

But this will bring some technical difficulties. Let us consider

stochastic system (4.1) again. Suppose there exist nonnegative

C 1 functions, ϕi(x1, · · · , xn), ψi(x1, · · · , xn), i = 1, · · · , nsatisfying

|f i(t, x)| ≤ (|x1| + · · · + |xn|)ϕi(x1, · · · , xn), (5.1)

|gi(t, x)| ≤ (|x1| + · · · + |xn|)ψi(x1, · · · , xn). (5.2)

Similar to Theorem 4.1, we can choose a continuous state

feedback control u ∈ Rn with the form

ui = ui(x1, · · · , xn)

= − 12

xi1 + n

ni=1

ϕ2i (x1, · · · , xn)

+nni=1

ψ2i (x1, · · · , xn)

−x2γ−1i , γ =p

q, p, q ∈ N,

1

2< γ < 1, (5.3)

such that

LV (x) ≤ −2ni=1

x2γi ≤ −2

x21 + x22 + · · · + |xn|2γ

= −2(V (x))γ . (5.4)

But it is clear that u is not local Lipschitz continuous, and such

a control design can not guarantee the existence of a unique

solution for (4.1). This is the reason why we emphasize that

a discussion on the existence and uniqueness of solutions for

stochastic nonlinear systems is both necessary and important.

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