Stability of a Class of Fractional-Order Nonlinear Systems

Definition 1 (see [15].)The fractional integral $$ of function f(t) is defined as follows:

where fractional order α > 0, and $$ is the gamma function.

Definition 2 (see [15].)The Caputo derivative with order α of function f(t) is given as

where n − 1 < α < n, n ∈ Z⁺.

The formulas for Laplace transform of the Caputo fractional derivative $$ have the following form [16]:

where n − 1 ≤ α < n, and $$.

Definition 3 (see [17].)The Mittag-Leffler function is given as

where α > 0 and $$.

Remark 4. If β = 1, we have E_α,1(z) = E_α(z), especially, E_1,1(z) = E₁(z) = e^z.

Lemma 5 (see [15].)Considering the Laplace transform of Mittag-Leffler function with two parameters, we have

where t and s are, respectively, the variables in the time domain and Laplace domain, $$ stands for the real part of s, $$, and $$ denotes the Laplace transform.

Proof. The proof of this Lemma can be found in [15].

Lemma 6 (see [18], [19].)If 0 < α < 2, $$, and μ satisfies πα/2 < μ < min⁡{π, πα}, there exist C₁ > 0 and C₂ > 0 such that

where |arg(z)| < μ, |z| ≥ 0.

Lemma 7 (see [18], [19].)For the Mittage-Leffler function E_α,β(At^α), there exist finite real constants $$, $$, and $$ such that

where $$.

Proof. The proof of this Lemma can be found in [18].

Lemma 8 (Gronwall inequality [19, 20]). Let α > 0, u(t) is a nonnegative function locally integrable on [0, T) and a(t) is a nonnegative, nondecreasing continuous function defined on [0, T), a(t) < M (constant), and suppose z(t) is nonnegative and locally integrable on [0, T) with

on this interval. Then Moreover, if u(t) is a nondecreasing function on [0, T), we have

Theorem 9. The fractional order nonlinear system (12) is local asymptotically stable, if it satisfies the following conditions: (1)Re( eig (A)) < 0 and ω = − max Re( eig (A)) > Γ(α), where Γ(·) is the gamma function; (2)  g(x(t)) satisfies ‖g(x(t))‖ = o‖x(t)‖ as ‖x‖ → 0.

Proof. Applying the Laplace transform on (12), we have

that is, where X(S) is the Laplace transform of x(t), I is an n × n identity matrix, and $$ denotes the Laplace transform. By using the Laplace inverse transform, we obtain the solution of (16), It follows from Lemma 7 that there exist constants M₁ > 0 and M₂ > 0 such that Since matrix A is stable, there is a constant M₃ > 0 such that $$. Substituting it into (16), one has Based on the condition (2)∥g(x(t))∥ = o∥x(t)∥, that is, lim⁡_∥x∥→0(∥g(x(t))∥/∥x(t)∥) = 0, there is a constant δ > 0, such that where ‖x(t)‖ < δ. And (t − τ) ^α < t^α − τ^α when 0 < α < 1 and t > τ, then Multiplying the inequality by $$, we will get Applying Lemma 8 (Gronwall inequality) to (20), we have Then Therefore, when t → ∞, ‖x(t)‖ → 0 for ω > Γ(α), which implies that the system (12) is asymptotically stable.

Theorem 10. The fractional order nonlinear system (12) is globally asymptotically stable, if it satisfies the following conditions: (1)  g(x(t)) satisfies g(0) = 0 and the Lipschitz condition with respect to x, that is, ‖g(x₁) − g(x₂)‖ ≤ L‖x₁ − x₂‖; (2)Re( eig (A)) < 0 and ω = − max Re( eig (A)) > LM₃M₄Γ(α), where M₃ and M₄ satisfy ∥e^At∥≤M₃e^−ωt and $$.

Proof. Applying the Laplace transform and Laplace inverse transform on (12), we obtain the solution of (12),

It follows from Lemma 7 that there exist constants M₁ > 0, M₂ > 0, and M₃ > 0 such that Multiplying the inequality by $$, we will get Applying Lemma 8 (Gronwall inequality) to (25), we have Then Therefore, when t → ∞, ‖x(t)‖ → 0 for ω > LM₂M₃Γ(α), which implies that the system (12) is globally asymptotically stable.

Theorem 11. The controlled fractional order nonlinear system (28) is local asymptotically stable, if it satisfies the following conditions: (1)$$ and $$; (2)  g(x(t)) satisfies ‖g(x(t))‖ = o‖x(t)‖ as ‖x‖ → 0.

Proof. The proof is similar to that of Theorem 9.

Theorem 12. The controlled fractional order nonlinear system (28) is globally asymptotically stable, if it satisfies the following conditions: (1)  g(x(t)) satisfies g(0) = 0 and the Lipschitz condition with respect to x, that is, ‖g(x₁) − g(x₂)‖ ≤ L‖x₁ − x₂‖; (2)  $$ and $$, where M₃ and M₄ satisfy $$ and $$.

Proof. The proof is similar to that of Theorem 10.

Theorem 13. The fractional order nonlinear system (29) is local asymptotically stable, if it satisfies the following conditions: (1)Re(eig(A)) < 0 and ω = −max Re(eig(A)) > Γ(α) ^1/α; (2)  g(x(t)) satisfies ‖g(x(t))‖ = o‖x(t)‖ as ∥x∥→0.

Proof. Applying the Laplace transform on (29), we have

that is, where X(S) is the Laplace transform of x(t), and I is an n × n identity matrix. By using the Laplace inverse transform, we obtain the solution of (29), It follows from Lemma 7 that there exist constants M₁ > 0 and M₂ > 0 such that Since matrix A is stable, there is a constant M₄ > 0 such that $$. Substituting it into (33), one has Based on the condition (2)‖g(x(t))‖ = o‖x(t)‖, that is, lim⁡_∥x∥→0(‖g(x(t))‖/‖x(t)‖) = 0, there is a constant δ > 0, such that where ‖x(t)‖ < δ. And (t − τ) ^α > t − τ when 1 < α < 2, and then Multiplying the inequality by e^ωt, we will get Applying Lemma 8 (Gronwall inequality) and Lemma 6 to (37), we have Then Therefore, when t → ∞, ‖x(t)‖ → 0 for ω > Γ(α) ^1/α, which implies that the system (29) is asymptotically stable.

Theorem 14. The fractional order nonlinear system (29) is globally asymptotically stable, if it satisfies the following conditions: (1)  g(x(t)) satisfies g(0) = 0 and the Lipschitz condition with respect to x, that is, ‖g(x₁) − g(x₂)‖ ≤ L‖x₁ − x₂‖; (2)Re(eig(A)) < 0 and ω = −max Re(eig(A)) > LM₃M₄Γ(α), where M₃ and M₄ satisfy ‖e^At‖ ≤ M₃e^−ωt and $$.

Proof. Applying the Laplace transform and Laplace inverse transform on (29), we obtain the solution of (29),

It follows from Lemma 7 that there exist constants M₁ > 0, M₂ > 0, and M₃ > 0 such that Multiplying the inequality by e^ωt, we will get Applying Lemma 8 (Gronwall inequality) and Lemma 6 to (42), we have Then Therefore, when t → ∞, ‖x(t)‖ → 0 for ω > (LM₃M₄Γ(α)) ^1/α, which implies that the system (29) is globally asymptotically stable.

Theorem 15. The controlled fractional order nonlinear system (45) is local asymptotically stable, if it satisfies the following conditions: (1)$$ and $$; (2)  g(x(t)) satisfies ‖g(x(t))‖ = o‖x(t)‖ as ‖x‖ → 0.

Proof. The proof is similar to that of Theorem 13.

Theorem 16. The controlled fractional-order nonlinear system (45) is globally asymptotically stable, if it satisfies the following conditions: (1)  g(x(t)) satisfies g(0) = 0 and the Lipschitz condition with respect to x, that is, ‖g(x₁) − g(x₂)‖ ≤ L‖x₁ − x₂‖; (2)$$ and $$, where M₃ and M₄ satisfy $$ and $$.

Proof. The proof is similar to that of Theorem 14.

Remark 17. There are many fractional order chaotic (hyperchaotic) systems which satisfy ‖g(x(t))‖ = o‖x(t)‖ as ‖x‖ → 0 or the Lipschitz condition, such as fractional order Lorenz system, fractional order Chen system, fractional order Lü system, fractional order Liu system, and so forth [22]. Therefore, Theorems 9–16 can be used as the criteria to control chaos in a class of fractional-order systems. Compared with nonlinear control methods, the advantage of linear control lies in reducing control cost and is easy to implement.

Remark 18. The obtained sufficient conditions could be applied to a class of fractional order hyperchaotic systems [23–25]. On the one hand, complex multiscroll chaotic systems have garnered much attention in recent years. J. H. Lü has done a large amount of remarkable work. In fact, the sufficient conditions could be applied to a class of complex multiscroll chaotic systems, which could also generate a complex four-scroll chaotic attractor.