Tempered Mittag–Leffler Stability of Tempered Fractional Dynamical Systems

Due to finite lifespan of the particles or boundedness of the physical space, tempered fractional calculus seems to be a more reasonable physical choice. Stability is a central issue for the tempered fractional system. )is paper focuses on the tempered Mittag–Leffler stability for tempered fractional systems, being much different from the ones for pure fractional case. Some new lemmas for tempered fractional Caputo or Riemann–Liouville derivatives are established. Besides, tempered fractional comparison principle and extended Lyapunov direct method are used to construct stability for tempered fractional system. Finally, two examples are presented to illustrate the effectiveness of theoretical results.


Introduction
Fractional derivatives were first proposed by Leibnitz soon after the more familiar classic integer order derivatives. In recent decades, the study of fractional differential systems has attracted wide attention. Compared with the classical calculus, fractional calculus can better characterize memory and hereditary properties of processes and materials. ey are now used to model the dynamical evolution in the fields of physics, chemistry, biology, and so on. Fractional calculus can be most easily understood in terms of probability. e relationships among random walks, Brownian motion, and diffusion processes were given in [1]. It is more reasonable to replace classic derivatives by fractional analogues in the diffusion equation [2].
Fractional calculus involves the operation of convolution with a power law function. Multiplying by an exponential factor results in tempered fractional derivatives and integrals [3], this exponential tempering has many merits both in mathematical and practical. A truncated Lévy flight was investigated to capture the natural cutoff in real physical systems [4]. Without a sharp cutoff, the tempered Lévy flight was studied as a smoother alternative [5]. Cartea and del-Castillo-Negrete [6] explored the tempered fractional diffusion equation by the tempered Lévy flight. In finance, the tempered stable process models describe price fluctuations with semiheavy tails [7][8][9][10]. Tempered fractional time derivatives can be also found in geophysics [11][12][13], Brownian motion [14], and so on.
As in classical calculus, stability analysis is still one of the most important tasks in fractional differential system [15][16][17][18][19][20]. It is a basic feature in fractional physical and biological systems, such as Duffing oscillator [21], neural networks [22][23][24], and predator-prey models [25]. At present, Lyapunov method has been applied to analyze Mittag-Leffler stability of different fractional systems [26][27][28][29][30]. Li et al. [26,27] obtained a series of conclusions on the Mittag-Leffler stable for nonlinear fractional equations. In [28], Mittag-Leffler stability of multiple equilibrium points of fractional recurrent neural networks was considered. In [29], a convex and positive definite function was used to analyze Mittag-Leffler stable for fractional system. In [30], the authors presented the Lyapunov stability analysis for fractional nonlinear systems with impulses.
As far as we know, no paper has discussed stability analysis for tempered fractional system. Motivated by this, we think it is very necessary and meaningful to study Mittag-Leffler stability of tempered fractional dynamical systems both in theoretical research and practical application. Because tempered fractional operators combine with nonlocal, weak singularity, and exponential factors [31][32][33], it has many differences to fractional case in stability analysis. In this paper, tempered Mittag-Leffler stability is first proposed. It is a more appropriate concept for tempered fractional system . Tempered comparison principle  and some inequalities are given for tempered fractional calculus  or systems.  en, sufficient conditions for tempered Mittag-Leffler stability are provided and verified by the Lyapunov  method. Finally, the theoretical results are applied to some  examples. is paper is organized as follows. In Section 2, some necessary definitions and lemmas are prepared. Section 3 mainly discusses the sufficient criterions ensuring Mittag-Leffler stability of the tempered fractional systems. In Section 4, two examples are presented to show the effectiveness of theoretical results. We conclude the paper with some discussions in Section 5.

Preliminaries
Tempered fractional calculus plays an important role in different fields [34,35]. In practical application, many different tempered fractional derivatives are proposed, such as Caputo, Riemmann-Liouville, and Riesz. Some definitions and lemmas are stated below, which will be used later.
Definition 1 (see [13]). e tempered fractional integral of order α > 0 and tempered parameter λ ≥ 0 is defined as where Γ presents the Euler gamma function.
Definition 2 (see [3]). e tempered fractional Caputo derivative of tempered parameter λ ≥ 0 is defined as where n − 1 ≤ α < n, n ∈ N, and C a D α t is the Caputo fractional derivative.
In order to study the stability of tempered fractional systems, several lemmas are needed.

Main Results
In this section, tempered fractional comparison principles, some inequalities, and tempered Mittag-Leffler stability are derived.

Tempered Fractional Comparison Principles.
In this section, we establish tempered fractional comparison principles.

Lemma 4. Assume that
By Lemma 3, equation (5) yields According to us, Taking the inverse Laplace transform on (8), solution of system (5) can be written as Mathematical Problems in Engineering According to m(t) ≥ 0, therefore we obtain In this section, we construct some inequalities for tempered fractional derivatives or systems.
From Lemma 1, we could easily obtain the following lemma.
is a continuously differentiable function, the following inequality holds: where for the Caputo fractional derivative. at is, Multiplying both sides of equation (14) by e − λt , it gives Using Definition 2, we obtain (12). Consider the following tempered fractional system □ Theorem 1. For the real-valued continuous function where α > 0, λ ≥ 0, and ‖·‖ denotes an arbitrary norm.
Proof. It follows from (1) that □ Theorem 2. If x � 0 is an equilibrium point of system (16) is Lipschitz on x with Lipschitz constant l and piecewise continuous with respect to t; then, we have Proof. By applying the tempered fractional integral operator 0 I α,λ t to system (16), it follows from Lemma 2 and Lipschitz condition in f(t, x) that ere exists a function M(t) ≥ 0 such that Combining with Lemma 3 and Laplace transform to (21), we obtain where ‖x(s)‖ � L ‖x(t)‖ { }. Taking the inverse Laplace transform to (22) gives where * denotes the convolution operator. Obviously, an equilibrium point of tempered fractional system (16).
Theorem 3. Assume x � 0 be an equilibrium point for (16) and domain D ⊂ R n contains the origin. Let V(t, x(t)) : [0, +∞) × D ⟶ R be a continuously differentiable function and locally Lipschitz with respect to x, such that where t ≥ 0, x ∈ D, β ∈ (0, 1), and α 1 , α 2 , α 3 , a, and b are given positive constants, then x � 0 is tempered Mittag-Leffler stable. If the assumptions hold globally on R n , then x � 0 is globally tempered Mittag-Leffler stable.

Applications
In this section, we will give three examples to demonstrate theoretical analysis.
e Adams-Bashforth-Moulton method [37] is employed for solving tempered fractional differential equations in the simulations. Example 1. Consider the tempered fractional Riemann-Liouville system: where x(0) > 0. e Lyapunov function candidate is chosen as V(t, x) � |x|. From Lemma 1, we obtain By eorem 3, we have en, x � 0 is tempered Mittag-Leffler stable. To verify the result, we choose parameters as α � 0.95 and x(0) � 4 and the tempered parameters as λ � 2, 4, 6, 8, respectively. e time evolution of the system states (42) is shown in Figure 1. It is presented that the solution of system (42) converges to the equilibrium point x � 0. e larger the tempered parameter λ is, the faster the convergence speed becomes.
x i (t) is the ith state, f j is the jth activation function, b ij is constant connection weight of the jth neuron on the ith neuron, a i > 0 denotes the resting rate when the ith neuron disconnected, and I i is the external inputs. Under the conditions system (45) is globally tempered Mittag-Leffler stable. Let x(t) � (x 1 , x 2 , . . . , x n ) T be any solution of system (45). We choose Lyapunov function as follows: By inequalities (46) and (47) and Lemma 7, tempered fractional Caputo derivative on V(t, x(t)) can be written as where c � min c 1 .c 2 , . . . , c n . From (49) and eorem 3, system (45) is globally tempered Mittag-Leffler stable.
To illustrate the effectiveness of Example 2, in system (45), we let It is obvious that condition (47) is satisfied. Let tempered parameters λ � 0, 2, 4, 6, respectively. As shown in Figure 2, the equilibrium point x � 0 is tempered Mittag-Leffler stable and the solution of system (45) converges to x � 0.

Conclusions
In this paper, we present some stability results for the tempered fractional systems. Based on the Laplace transform, we obtain the comparison principle for the tempered fractional systems. Some theorems about tempered Mittag-Leffler stability are derived, which enrich the knowledge of the system theory and the tempered fractional calculus and are helpful in characterizing the tempered fractional system models. Furthermore, we will study stability of tempered fractional systems with time-varying delays in future work.

Data Availability
e authors affirm that all data necessary for confirming the conclusions of the article are present in the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments is work was supported by the introduction of talent, the Northwest University for Nationalities, special Funds for Talents (nos. xbmuyjrc201916, xbmuyjrc201632), the