Generalized Mittag–Leffler Stability of Hilfer Fractional Order Nonlinear Dynamic System

This article studies the generalized Mittag–Leffler stability of Hilfer fractional nonautonomous system by using the Lyapunov direct method. A new Hilfer type fractional comparison principle is also proved. The novelty of this article is the fractional Lyapunov direct method combined with the Hilfer type fractional comparison principle. Finally, our main results are explained by some examples.

Lately, fractional calculus has become more common in control problems. Different fractional order controllers are significant in almost every field of the control subject. Stability is one of the important properties of the control problem. Therefore several researchers have investigated the stability of fractional order systems. Up to now, it has made great strides [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. The Lyapunov direct method (LDM) is one of the more important methods to analyze stability of fractional order systems.
For nonlinear systems, the solutions of nonlinear differential equations are often difficult to express. LDM [33][34][35][36][37][38][39] offers an excellent method to analyze the property of the solution without solving this differential equation. Since LDM can be used in any order system, it shows that this method has great superiority. This method directly infers the stability of the system through a Lyapunov function for the system. LDM is a sufficient condition for judging system stability. In other words, even if the Lyapunov function candidate is not found, the system may also be stable.
In 2009, Li et al. [20] investigated the Mittag-Leffler stability of fractional order nonlinear dynamic systems: with initial condition x(t 0 ), where D denotes either the Caputo or Riemann-Liouville fractional operator, α ∈ (0, 1), f : [t 0 , ∞] × Ω → R n is piecewise continuous in t and locally Lipschitz in x on [t 0 , ∞] × Ω, and Ω ∈ R n is a domain that contains the origin x = 0.
In 2014, Aguila-Camacho et al. [25] considered the stability of fractional order nonlinear time-varying systems: where α ∈ (0, 1), c t 0 D α t denotes the Caputo fractional derivative and t represents the time. In 2017, Yang et al. [28] investigated the Mittag-Leffler stability of nonlinear fractional-order systems with impulses where D α 0,t denotes the Caputo fractional derivative of order α, (0 < α < 1), u(t) ∈ R n is the state vector, A ∈ R n×n is a constant matrix, g(t, u(t)) ∈ R n is the nonlinear term with g(t, 0) = 0, J k (·) standing for the jump operator of impulsive, and the impulsive moments satisfy 0 = t 0 < t 1 < t 2 < · · · < t k < t k+1 < · · · with lim k→+∞ t k = ∞.
To the best of our knowledge, while some research has been carried on the stability of the Riemann-Liouville or Caputo fractional order systems, no single study exists which has investigated the stability of the Hilfer fractional order system by using LDM. In this context, the dual index of the Hilfer fractional derivative is complex but fascinating. To overcome the difficulty of proving the stability of the given system, we developed a new Hilfer type fractional comparison principle, which plays a vital role in this article. This paper has three key aims. Firstly, the generalized Mittag-Leffler (G-M-L) stability is proposed. Then a new Hilfer type fractional comparison principle is proved. Finally, the analysis of fractional LDM is studied.

Preliminaries
In this section, we give some definitions and related lemmas.

Definition 1 ([9]
). The Hilfer fractional derivative of order α and type β for a function g is defined as t denotes Riemann-Liouville fractional integral. .
Consider the stability of the Hilfer fractional nonautonomous system with fractional integral type initial condition t 0 I Lipschitz on x ∈ R n with Lipschitz constant k 0 .
Now, we develop a new Hilfer type fractional comparison principle, which plays a vital role in the proof of our main theorems.
Taking the inverse Laplace transform of (5), we have Since

Main Theory
In this section, let us firstly give a simple introduction to the LDM. If one can seek out a Lyapunov function for the given system, then the system is stable. Note that LDM is a sufficient condition for judging system stability. In other words, when the Lyapunov function is not found, the system may also be stable, so we cannot conclude that the system is unstable. In this article, we get G-M-L stability of the Hilfer fractional nonautonomous system by using the LDM. What is more, we apply a new Hilfer type fractional comparison principle and class-K functions to investigate the fractional LDM, which is a completely new attempt of stability analysis of the Hilfer fractional dynamic system.

Theorem 1.
Let an equilibrium point of system (1) be x = 0 and U ⊂ R n be a domain containing the origin. Let W(t, x(t)) : [0, ∞) × U → R be a continuously differentiable function and locally Lipschitz with respect to x satisfying a 1 x m ≤ W(t, x(t)) ≤ a 2 x mc , where t ≥ 0, x ∈ U, α ∈ (0, 1), β ∈ [0, 1], a 1 , a 2 , a 3 , m and c are arbitrary positive constants. Then the equilibrium point x = 0 is G-M-L stable.
Proof. According to (16) and (17), we have Because V(t, x(t)) is bounded by the unique non-negative solution of the scalar differential equation It follows from Definition 3 that Then we can get the asymptotic stability of (19) by contradiction. Case 1: Assume that there is a constant t 1 ≥ 0 such that which means that for any t ≥ t 1 . According to Definition 3, the equilibrium point of Taking (23) in (19), we obtain By using Theorem 1, we obtain which contradicts the assumption f (t) ≥ . On the basis of the discussions in Case 1 and Case 2, we get lim t→∞ f (t) = 0. Then from (16) and V(t,x(t)) is bounded by f (t), we obtain lim t→∞ x(t) = 0. Remark 3. When β = 0 or β = 1, the stability of t 0 D α,β t x(t) = g(t, x) has been proved by Li, Chen and Podlubny [21]. Our main Theorems generalize and improve Theorems 5.1 and 6.2 of literature [21].

Remark 4.
When β = 1, the Hilfer fractional derivative becomes the Caputo fractional derivative. In this case, Example 2 is an extension of Example 14 [20].

Conclusions
In this paper, we studied the generalized Mittag-Leffler stability of Hilfer fractional nonautonomous system by using the Lyapunov direct method. The definition of the generalized Mittag-Leffler stability and a new Hilfer type fractional comparison principle were proposed, which enriches the knowledge of the system theory. Since the Hilfer fractional derivative includes many classical fractional derivatives, our conclusions can also be widely applied to many fractional order systems.
At present, research on the Caputo-Fabrizio fractional differential equations, which is a new research engine in the field of fractional calculus, is becoming more and more active. For its new development, see [42][43][44][45][46]. As an extension of our conclusion, we present an open question, namely how to develop the stability of the Caputo-Fabrizio fractional nonautonomous system by using the Lyapunov direct method. The biggest difficulty for this is to perfectly establish a new Caputo-Fabrizio type fractional comparison principle.