Let α > 0 andThen the Riemann–Liouville fractional integral of order . with respect to . is defined as

where Γ(·) is the gamma function, and ∗ denotes the convolution operator.

The Riemann–Liouville fractional integral operator have the following properties [8].

Let α, β > 0 and Then

(i) is nondecreasing with respect to f. In particular,;

(ii) The operatoris compact, andwhere σ(·) denotes the spectral set of;

(iii);

(iv) For the real-valued continuous function f, it has where ǁ·ǁ denotes an arbitrary norm.

For the fractional derivatives, there are two types that are commonly used: the Riemann– Liouville fractional derivative and the Caputo derivative.

Let The Riemann–Liouville fractional derivative and Caputo fractional derivative of order . with respect to . are defined, respectively, as

Further, ifthen Caputo fractional derivative can also be defined as

.which is known as a smooth fractional derivative.

Note that if, then coincides with

The Caputo fractional derivative shares many similar properties with the ordinary derivative, and it is suitable for initial value problems, and so it can be applied into a lot of engineering and physical problems in real world.

LetThe following formulas hold:

The generalized Mittag-Leffler function is defined by

where.

In particular, when p = 1, it becomes the two-parameter Mittag-Leffler function, i.e. when it becomes the one-parameter Mittag-Leffler function, i.e.,.

[20] The Wright function is defined by

Note that the case is trivial since . In particular, for the case where Mv(z) is the Mainardi’s function defined as

The Laplace transform of the Mainardi’s function is

On the other hand, Mv(z) satisfies the following two equalities:

where

The special functions listed as above play an important role in the investigation of fractional differential equations. In the following, we give the solutions of some kinds of fractional differential equations with help of the special functions.

(See [12].) Let The solution to the initial value problem

has the form

(See [12].) Let. The solution to the initial value problem

has the form

where E_ν,m,l(z) is defined by the following series:

with

(See [10].) Let 0 < α ≤ 1, and let a(t)be a bounded and continuous function on Ω. The solution to the initial value problem

has the form, is a bounded linear operator defined on

R0 is an identity operator, and .k denotes the k-times composition operator of ...

(See [9].) Letwith µ. unique solution of the following initial value problem

has the form

where.

Next, we list some properties about the special functions and two integral inequalities,

which will be used in the latter discussion.

(See [11].) Let ρ, µ, γ, ν, σ > 0, and let t > 0, then

In particular

(See [31].) Let 0 < α ≤ 1, a(t) and l(t)be continuous, nonnegative functions on Ω, and u(t)be a continuous, nonnegative function on Ω with

Then it has

If a(t)is nondecreasing on Ω, then the inequality is reduced to

If a(t) ≡ 0 on Ω, then u(t) ≡ 0, where 0 < β < α ≤ 1, and

(See [8].) Suppose β > 0, a(t) and u(t) are nonnegative, locally integrable functions on Ω, g(t) is a nonnegative, nondecreasing continuous function on Ω, and they satisfy the following relationship:

Then, for any we have

Finally, we introduce some concepts of stability in the case of a fractional-order sys- tem with randomly time-varying parameters. Stability studies about differential systems are essentially problems of convergence. For the system with randomly time-varying parameters, it is only possible to investigate the convergence in some stochastic sense such as convergence in mean sense, convergence in probability, or convergence with probability one. In this paper, we will mainly consider the stability of system in mean sense.

A constant vector is an equilibrium solution of system (1) if and only if .

For convenience, we state all definitions and theorems for the case when the equilibrium solution of system (1) is the origin of There is no loss of generality in doing so because any equilibrium solution can be changed to the origin via a change of variables. Suppose the equilibrium solution for system (1) is x* = 0 and consider the change of variable y = x - x*. Then system (1) with respect to the new variable y is

where g(t, ω(t), 0) = 0, and the new system (4) has equilibrium solution at the origin.

The equilibrium solution x(t) = 0 of system (1) is said to be stable in mean sense if for any ϵ > 0, there exists a such that for any initial condition satisfying, the expected value of the norm of the solution x(t) satisfiesfor any t ≥ 0.

The equilibrium solution x(t) = 0 of system (1) is said to be asymptotically stable in mean sense if, in addition, to being stable in mean sense, it is true that for each x₀, there exists a δ > 0 such that whenever.

The equilibrium solution x(t) = 0 of system (1) is said to be Mittag-Leffler stable in mean sense if for each x₀, the expected value of the norm of the solution satisfies

where 0 < α ≤ 1, and λ > 0.

If the equilibrium solution x(t) = 0 of system (1) is said to be stable (asymptotically stable, Mittag-Leffler stable) in mean sense, then we also call system (1) stable (asymptotically stable, Mittag-Leffler stable) in mean sense.

In this subsection, we will use the generalized Lyapunov functional method to analyze the stability of system (1). The method is to construct a scalar functional V (t, x), which is continuous in both t and x, has first partial derivatives in these variables and equals zero only at the equilibrium solution x(t) = 0. Such a functional is called a Lyapunov functional. The basic idea of the Lyapunov method is that if we can construct a Lyapunov functional, which represents some tubes surrounding the equilibrium solution x(t) = 0 such that all solutions cross through the tubes towards x(t) = 0.

For any solution x, the a-order Caputo derivative of the scaler functional . is calcu- lated as

where

Let 0 < α ≤ 1, and letbe a continuously differential function and locally Lipschitz continuous with respect to x and satisfy the following conditions:

(i) V (t, 0) = 0;

(ii) V (t, x) ≥ aǁxǁ;

(iii)

Then the equilibrium solution x(.) = 0 of system (1)is stable in mean sense.

Proof. Note that V (t, x(t)) can be written as

where V₀ = V (0, x₀) only depends on the initial state ...

Taking the expected value on the both sides of equality (5) leads to

Furthermore, combining condition (iii) and Property 1, we have

On the other hand, from condition (ii) we get

Since V (t, x) is locally Lipschitz continuous with respect to ., there exists a constant . such that for some γ > 0 and 0 < ǁxǁ ≤ y, it has V (t, x) ≤ Lǁxǁ. Then, for any ϵ > 0, if is chosen as then, for it has

This implies that the equilibrium solution x(t) = 0 of system (1) is stable in mean sense. The proof is completed.

According to Theorem 1 and the definition of Caputo fractional derivative, we can immediately obtain the following corollary.

Letbe a continuously differential function and locally Lipschitz continuous with respect to x and satisfy the conditions

(i) V (t, 0) = 0;

(ii) V (t, x) ≥ aǁxǁ;

(iii) E(dV (t, x(t))/dt) ≤ 0.

Then the equilibrium solution x(.) = 0 of system (1)is stable in mean sense.

Next, we will discuss the asymptotically stability of system (1). To do this, we state a lemma.

(See [1].) If g(r) is an always increasing function defined for all r ≥ 0 and

g(0) = 0, then if E{r} ≥ M > 0, there exists an L(M) > 0 such that E{g(r)} ≥ L.

Under the hypothesis of Theorem ., further assume that V (t, x) has the property thatis negative definite, i.e.,

where h(0) = 0, and h(||x||) is an increasing function. Then the equilibrium solution x(t) = 0 of system (1)is asymptotically stable in mean sense.

Proof. We will prove it by contradiction. Assume that does not tend to zero as Then it is possible to find a δ > 0 such that > δ for any t ≥ t₀. It follows from Lemma 8 that there exists a .(.) such that

On the other hand, from inequalities (6), (7) and Property 1 we get

This implies that if we choose t such that then it haswhich leads to a contradiction with condition (ii). It follows that tends to zero as t → ∞. The proof is completed.

Under the hypothesis of Theorem ., further assume that V (t, x) has the property that E(dV (t, x(t))/dt)is negative definite, i.e.

where h(0) = 0, and h(||x||) is an increasing function. Then the equilibrium solution x(t) = 0 of system (1)is asymptotically stable in mean sense.

Finally, we will consider Mittag-Leffler stability of system (1) in mean sense.

Under the hypothesis of Theorem ., further assume that there exist positive constants a., p, q (i = 1, 2, 3)and β (0 ≤ β < 1)such that V (t, x) has the properties:

Then the equilibrium solution x(t) = 0of system (1)is Mittag-Leffler stable in mean sense.

Proof. From the second inequality in condition (iv) we have

Also, from condition (v) we get

On the other hand, from the first inequality in condition (iv) we have

Then, combining inequalities (8) and (9), we can obtain

According to the well-known comparison principle in [17] and Lemma 4, we have

The proof is completed.

For the case Theorems 1 and 2 become Theorems 3.1 and 3.2 in [1], respectively. Theorem 3 improves and extends Theorems 5.1 and 5.4 in [19].

Example 1. Consider the stability of the following linear fractional-order system:

Construct a Lyapunov functional as

where is a constant positive definite matrix, and .. denotes the transpose of x.

According to Corollary 1, a sufficient condition for stability of system (10) is

That is to say, a sufficient condition for stability of system (10) is that

be negative definite for all t ≥ 0.

Now we consider a particular case when A(t) is a diagonal matrix, i.e.,

where a.(t) (i = 1, 2, . . . , n) are bounded continuous functions on or have the form as t. with γ > 0. Then, according to Lemmas 2 and 3, system (10) has a unique solution, and the unique solution can be expressed in a closed form. For convenience, we denote the unique solution as

Substituting (12) into (11), we obtain a sufficient condition for the stability in mean sense is that

be negative definite for all t ≥ 0. Let . be an identity matrix. Then it is stable in mean sense if ai(t) < 0 (i = 1, 2, . . . , n) for all t ≥ 0.

In the following, we demonstrate numerical simulation. For example, we take a = 0.5, and

Then, according to Theorem 1, the equilibrium solution x(t) = 0 is stable. Figure 1 is the numerical result. From Fig. 1 one sees that the numerical results agree with the theory analysis.

We obtained the stability criteria by the Lyapunov functional approach in this subsection. However, by Example 1 one can find that it is only in some special cases that a Lyapunouv function can be constructed. Zhou et al. in [30] studied the exponential stability for delayed neutral networks driven by fractional Brownian noise using some mathematical techniques and Gronwall inequality. It motivates us investigate the stability of system (1) by mathematical techniques and generalized Gronwall integral inequalities in the next subsection.

In this subsection, we firstly prove the boundness of the solution of system (1) by using some generalized Gronwall inequalities and then provide another alternative approach to verify the stability of system (1) in mean sense.

Let the function f (t, ω(t), x) in system (1)satisfies the relation.

and let L verifies the condition

whereis a nonnegative continuous function. Then we have the estimate

where0 < β < α < 1, and

Proof. Let x(b) be the solution of system (1). Then we have

Let us define the mapping. as

Then it has .(.) = .. + .(.). Noting the fact . (t, ω(t), 0) ≡ 0, we get

On the other hand, from (14) we have Then by (13) and (14) we have

According to Lemma 4, we can obtain estimate (15). The proof is completed.

By using Lemma 9 we can formulate the following lemma, which tells us that the solution of system (1) is bounded under suitable conditions.

If the function f satisfies the condition in Lemma . and also there exist two positive constants δ. and M such that

for all, then the equilibrium solution x(t) = 0of system (1)is stable in mean sense.

Proof. Let ε > 0 and x(t) be the solution of system (1). We choose . such that

where M is the constant boundedness in Lemma 10. Then, for ǁx₀ǁ < δ(ε), we have

This implies that the equilibrium solution x(t) = 0 of system (1) is stable in mean sense. The proof is completed.

Finally, we will discuss the Mittag-Leffler stability of system (1) using integral in- equality.

Assume that system (1) can be decomposed into

Also, assume that

(i) There exist two constants λ, K > 0 such that

(ii) For any t ≥ 0, it has g(t, ω(t), 0) = 0, and there exists a constant L > 0 with L < λ such that for, it has

Then the equilibrium solution x(t) = 0 of system (1) is Mittag-Leffler stable in mean sense.

Proof. Combining the fact g(t, ω(t), 0) ≡ 0 and inequality (17), we have

Let x(t) be the solution of system (16). Then by Lemma 1 we have

According to equality (2), we have the estimate

Using the similar arguments to (19), we can deduce that

Then, taking the expectation on the both sides of inequality (19), we get

Applying Lemma 6 to inequality (20), we have

In fact, it has

It follows that Eǁx(t)ǁ ≤ K2Eα((L − λ)tα)ǁx0ǁ. The proof is completed.

Example 2. We consider the stability of system (10) in Example 1. Obviously, it has L(t, u) = ǁA(t)ǁu. If ǁA(t)ǁ satisfies the condition

where M is a positive constant, then by Theorem 4 system (10) is stable in mean sense.

For the particular case when ǁA(t)ǁ ≤ e−λt (λ > 0), it has

Then by Theorem 5 system (10) is Mittag-Leffler stable in mean sense.

For example, we take α = 0.5, and

Then, according to Theorem 5, the equilibrium solution x(t) = 0 is stable. Figure 2 is the numerical result. From Fig. 2 one sees that the numerical results agree with the theory analysis.

Example 3. Consider the following fractional stochastic differential equation:

where B(t) is the standard Brownian motion.

Take α = 0.5, A(t) = 5t, and x0 = 1. Then it has L = 0 and λ = 1. Therefore, according to Theorem 5, the equilibrium solution x(t) = 0 is Mittag-Leffler stable in mean sense. Figure 3 is the numerical result, and from it one sees that the numerical result agrees with the theory analysis.