Lyapunov Direct Method for Nonlinear Hadamard-Type Fractional Order Systems

: In this paper, a rigorous Lyapunov direct method (LDM) is proposed to analyze the stability of fractional non-linear systems involving Hadamard or Caputo–Hadamard derivatives. Based on the characteristics of Hadamard-type calculus, several new inequalities are derived for different deﬁnitions. By means of the developed inequalities and modiﬁed Laplace transform, the sufﬁcient conditions can be derived to guarantee the Hadamard–Mittag–Lefﬂer (HML) stability of the systems. Lastly, two illustrative examples are given to show the effectiveness of our proposed results.


Introduction
Over the last two decades, fractional calculus has been shown to be a powerful tool for modeling some non-classical phenomena in nature and society [1,2]. Fractional differential equations can describe materials and processes with memory, inheritance, and non-locality more compatible than the corresponding integer order models [3], such as viscoelastic systems, signal processing, electrochemistry, biology, biophysics, and so on [4][5][6][7][8][9]. In order to characterize the differences between these features, many different types fractional calculus have been proposed, such as Riemann-Liouville, Caputo, and Hadamard [10,11].
A large number of papers and books have studied the typical fractional derivatives (Riemann-Liouville and Caputo) [12][13][14][15][16][17][18], but Hadamard and Caputo-Hadamard derivatives are also worth further study. There are many differences between the Hadamard calculus and typical fractional ones: the kernel function of the former is logarithmic form (log t t 0 ) α−1 , while that of the latter is power form (t − s) α−1 ; the former can be viewed as a generalization of the form (t d dt ) n , and the latter can be thought of as a extension of classical derivatives ( d dt ) n . The solutions of Hadamard-type differential equations can own logarithmic decay (log t t 0 ) −α , but typical fractional differential equations have the power-law decay t −α [19]. In addition, the Hadamard-type calculus are widely applied to practical problems in mechanics and engineering, such as crack problems, fracture analysis [20], and igneous rock [21,22].
There is no doubt that stability analysis is a core problem for fractional systems. Many papers have focused on the stability of Riemann-Liouville and Caputo fractional systems, such as Caputo linear systems [23], Caputo non-linear systems [24][25][26], Caputo time-delay systems [27], Riemann-Liouville non-linear systems [28], Riemann-Liouville time-varying delays systems [29], and so on. However, there are few topics on the stability of Hadamard and Caputo-Hadamard fractional systems. It should be note that Li et al. [19] investigated the logarithmic decay of fractional Hadamard and Caputo-Hadamard systems. Ma et al. [30] discussed the finite-time stability of Hadamard-type systems.
It is well known that the LDM provides a handy tool for the stability analysis of fractional non-linear systems. There are two main aspects to illustrate its importance: on where 0 < t 0 < t and α > 0.

Hadamard-Type Fractional Inequalities
In this part, some Hadamard-type fractional inequalities are given, which are very important in the stability analysis.
Proof. (i) By applying Definition 3, let Taking integrate by parts on (13), it follows in which According to (14), we get y(t) ≤ 0. The proof of (8) is completed.
(ii) Similarly, by Definition 3, we derive where . By means of integrating by parts on (16), one deduces that where lim s→t y(s) By means of Young's inequality [36], one has Moreover, Therefore, from Formula (17), it holds that This concludes the proof of (9). (iii) Using Definition 3 concludes that Integrating by parts on (20), we see that in which lim s→t y(s) Employing Young's inequality [36] implies that Furthermore, there holds This concludes the proof of (10).
Theorem 3. Let x(t) ∈ R, α ∈ (0, 1), X(t) ∈ R n and M ∈ R n×n is a positive definite matrix. Then, the following inequalities hold: From (15), we gets The proof of (25) is complete.
(iii) With the help of Definition 2 and (1), we get By means of (22) and (23), we gets The proof of (27) is complete.

Remark 4.
The results of Theorem 3 also hold for the Riemann-Liouville fractional derivative, which have not been discussed until now.

Remark 5. The Theorems 2 and 3 bridge the gap from Hadamard and Caputo-Hadamard fractional derivatives of Lyapunov functions to non-linear systems.
Using the newly established inequalities, the stability problem of Hadamard-type system can be well solved by LDM. Moreover, two Theorems 2 and 3 almost have the same form, which illustrates the uniformity of the two definitions.
Proof. Let CH D α t 0 ,t x(t) − c 1 x(t) = g(t), using modified Laplace transform, one has Using Lemma 3 and (7) gives With the aid of the inequality g(t) ≤ c 2 (t), one obtains This ends the proof.

Theorem 5. Let x(t) be an absolutely continuous non-negative function and satisfy the inequality
where 0 < t 0 < t ≤ T, c 3 > 0, and c 4 (t) ≥ 0 is an integrable function. Then, Proof. Defining a function g(t) = H D α t 0 ,t x(t) − c 3 x(t), taking modified Laplace transform, it follows By applying Lemma 3 and Definition 7, the following equation follows: Using the inequality g(t) ≤ c 4 (t), one obtains These complete the proof.

Proof. By the linearity property of Caputo-Hadamard derivative, (31) becomes
By using CH D α t 0 ,t x * = 0 and multiplying x(t), we have Rewriting the inequality (32), we get Using Definition 3, inequality (33) can be read as Setting the auxiliary variable w(s) = x(s)−x(t) x(t) , we know that dw(s) ds . Inequality (34) is expressed as By means of integrate by parts, setting: In the first term of (36), there has an indetermination at s = t. The corresponding limit can be given as follows, By virtue of L'Hopital's rule, (37) yields that In view of w(s) > −1 and w(s) − ln (w(s) + 1) ≥ 0, (36) is read as It is obvious that the inequality (31) is true.

Stability of Hadamard-Type Systems
In this part, asymptotic stability theorems of Hadamard-type systems are obtained.
Consider the following two Hadamard-type systems: and where 0 < t 0 < t, α ∈ (0, 1), f (t, x(t)) : (t 0 , ∞) × Ω → R n is piecewise continuous in t and locally Lipschitz in x, and domain Ω ⊆ R n contains the origin x = 0. For convenience, we always suppose that the equilibrium x e is the origin [19], that is x e = 0.
Theorem 7. If x e = 0 is an equilibrium of system (38), f (t, x) is a Lipschitz function of x (with constant η > 0), then In particular, when α = 1, then x(t) ≤ x 0 Proof. By taking Hadamard fractional integral H D −α t 0 ,t for (38), together with (3), Theorem 1 and the Lipschitz condition, one has There exists a function G(t) ≥ 0 satisfying Taking the modified Laplace transform for (42) yields where (43) implies that Using the inverse modified Laplace transform to (44), one derives that where * denotes the modified convolution operator.
Theorem 9. If all the assumptions in Theorem 8 are hold except replacing Caputo-Hadamard derivative CH D t 0 ,t by Hadamard derivative H D t 0 ,t , then we can get lim t→∞ x(t) = 0.
Theorem 10. For the Caputo-Hadamard system (38), f (t, x(t)) is a Lipschitz function of x(t) (with constant η > 0). Suppose there exists a Lyapunov function W(t, x(t)) satisfies where β 1 , β 2 , β 3 , a ∈ R + . Then x(t) = E α A log lim inf Example 1. Consider the following Caputo-Hadamard system: where x(t) ∈ R. The Lyapunov function is chosen as W(t, x(t)) = 1 2 x 2 (t). Then, As can be seen from (77) and Theorem 8, x e = 0 of the system (76) is HML stable. The corresponding curves in Figure 1 illuminate the stability clearly.

Conclusions
A class of Lyapunov theorems has been developed for non-linear Hadamard-type fractional systems. Additionally, several useful Hadamard-type fractional inequalities are investigated. Based on these inequalities, it is very easy to design a appropriate Lyapunov function and calculate their Hadamard-type fractional derivative. According to the modified Laplace transform and the properties of Hadamard fractional calculus, the asymptotic stability theories of Hadamard-type systems are discussed, which enriched the knowledge of fractional calculus. Using these results, LDM can be applied to analyze the HML stability of Hadamard-type systems. At last, two examples are given to check the results of the systems by using the developed theory. In the future, we may focus on the following meaningful topics: • Extend the fractional order α of Hadamard-type system (38) and (39)