Abstract
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. This 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.
1. 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. They 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. The 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 [7–10]. Tempered fractional time derivatives can be also found in geophysics [11–13], Brownian motion [14], and so on.
As in classical calculus, stability analysis is still one of the most important tasks in fractional differential system [15–20]. It is a basic feature in fractional physical and biological systems, such as Duffing oscillator [21], neural networks [22–24], and predator-prey models [25]. At present, Lyapunov method has been applied to analyze Mittag–Leffler stability of different fractional systems [26–30]. 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 [31–33], 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. Then, sufficient conditions for tempered Mittag–Leffler stability are provided and verified by the Lyapunov method. Finally, the theoretical results are applied to some examples.
This 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.
2. 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]). The tempered fractional integral of order and tempered parameter is defined aswhere presents the Euler gamma function.
Definition 2 (see [3]). The tempered fractional Caputo derivative of tempered parameter is defined aswhere , and is the Caputo fractional derivative.
Definition 3 (see [9]). The tempered fractional Riemann–Liouville derivative of tempered parameter is defined aswhere , and is the Riemann–Liouville fractional derivative operator.
In order to study the stability of tempered fractional systems, several lemmas are needed.
Lemma 1 (see [36]). Let and , then
Lemma 2 (see [36]). Let , then(i)(ii) and
Lemma 3 (see [36]). The Laplace transform of tempered fractional integral and Caputo derivative (2) are given as(i)(ii), where denotes the Laplace transform of
3. Main Results
In this section, tempered fractional comparison principles, some inequalities, and tempered Mittag–Leffler stability are derived.
3.1. Tempered Fractional Comparison Principles
In this section, we establish tempered fractional comparison principles.
Lemma 4. Assume that , , and , then .
Proof. Following from , there exists a function such thatBy Lemma 3, equation (5) yieldsAccording to , we haveThus,Taking the inverse Laplace transform on (8), solution of system (5) can be written asAccording to , therefore we obtain .
Lemma 5. Assume that , , and , then .
Proof. From Lemma 1 and , we deriveThat is . From Lemma 4, we obtain .
3.2. Some Inequalities
In this section, we construct some inequalities for tempered fractional derivatives or systems.
From Lemma 1, we could easily obtain the following lemma.
Lemma 6. The relationship between and is as follows:where .
Lemma 7. If is a continuously differentiable function, the following inequality holds:where .
Proof. We take into Theorem 2 in [22]for the Caputo fractional derivative. That is,Multiplying both sides of equation (14) by , it givesUsing Definition 2, we obtain (12).
Consider the following tempered fractional systemsubjects to the proper initial conditions, where , , denotes either or , is piecewise continuous in and locally Lipschitz in , and is a domain contain the origin.
Theorem 1. For the real-valued continuous function in (16), we havewhere , and denotes an arbitrary norm.
Proof. It follows from (1) that
Theorem 2. If is an equilibrium point of system (16) with and is Lipschitz on with Lipschitz constant and piecewise continuous with respect to ; then, we have
Proof. By applying the tempered fractional integral operator to system (16), it follows from Lemma 2 and Lipschitz condition in thatThere exists a function such thatCombining with Lemma 3 and Laplace transform to (21), we obtainwhere . Taking the inverse Laplace transform to (22) giveswhere denotes the convolution operator. Obviously, , then inequality is obtained.
3.3. Tempered Mittag–Leffler Stability
In this section, some sufficient conditions are established for the tempered Mittag–Leffler stability of system (16).
Definition 4. If and only if , then is an equilibrium point of tempered fractional system (16).
Definition 5. Assume is an equilibrium point of system (16), the solution of (16) is said to be tempered Mittag–Leffler stable ifwhere , and satisfies locally Lipschitz condition.
Remark 1. Tempered Mittag–Leffler stability is a generalization of Mittag–Leffler stability. When , tempered Mittag–Leffler stability can be reduced to Mittag–Leffler stability [26, 27].
Remark 2. Both Mittag–Leffler stability and tempered Mittag–Leffler stability imply asymptotic stability, that is, as .
Theorem 3. Assume be an equilibrium point for (16) and domain contains the origin. Let be a continuously differentiable function and locally Lipschitz with respect to , such thatwhere , and are given positive constants, then is tempered Mittag–Leffler stable. If the assumptions hold globally on , then is globally tempered Mittag–Leffler stable.
Proof. It follows from equations (25) and (26) thatThere exists a function such thatTaking the Laplace transform to (28) giveswhere nonnegative constant and . We rewrite this in the formApplying the inverse Laplace transform to (30), unique solution of (28) isBecause and are nonnegative functions, we obtainSubstituting (32) into (25) satisfiesand if and only if .
Because is local Lipschitz condition with respect to and if and only if , then satisfies local Lipschitz condition about . So, system (16) is tempered Mittag–Leffler stable.
Theorem 4. Assume all conditions in Theorem 3 are satisfied except replacing by , then we have
Proof. It follows from Lemma 6 and thatA similar proof method in Theorem 3 shows result (35).
Theorem 5. For the tempered fractional system (10), where , is Lipschitz on with constant , and , if there exists a Lyapunov candidate yieldingwhere are given positive constants and , then
Proof. From (1) and (2), we find that tempered fractional derivative on can be represented in the formUsing (36) and (37) and Lipschitz condition on , we obtainWe can use Lemmas 7 and 2 to writewhere . By the same proof in Theorem 3, conclusion (38) holds.
4. Applications
In this section, we will give three examples to demonstrate theoretical analysis. The 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 . The Lyapunov function candidate is chosen as . From Lemma 1, we obtainBy Theorem 3, we haveThen, is tempered Mittag–Leffler stable. To verify the result, we choose parameters as and the tempered parameters as , respectively. The 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 . The larger the tempered parameter is, the faster the convergence speed becomes.
Example 2. Consider the tempered fractional Caputo Hopfield neural networks:where and is number of units. is the state, is the activation function, is constant connection weight of the neuron on the neuron, denotes the resting rate when the neuron disconnected, and is the external inputs. Under the conditionssystem (45) is globally tempered Mittag–Leffler stable.
Let 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 can be written aswhere . From (49) and Theorem 3, system (45) is globally tempered Mittag–Leffler stable.
To illustrate the effectiveness of Example 2, in system (45), we let , , and for andIt is obvious that condition (47) is satisfied. Let tempered parameters , respectively. As shown in Figure 2, the equilibrium point is tempered Mittag–Leffler stable and the solution of system (45) converges to .
(a)
(b)
(c)
Example 3. Consider the following tempered fractional system:where and .
Let the Lyapunov function . By Lemma 7, we obtainThen, the conditions of Theorem 3 are satisfied. Hence, is globally tempered Mittag–Leffler stable with respect to . Take . The numerical simulation is shown as Figure 3 with different tempered parameters . It is obvious and converge to 0. When tempered parameter of the system increase, the part solution of system converges faster.
(a)
(b)
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5. 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
The authors affirm that all data necessary for confirming the conclusions of the article are present in the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the introduction of talent, the Northwest University for Nationalities, special Funds for Talents (nos. xbmuyjrc201916, xbmuyjrc201632), the Humanities and Social Sciences Planning Projects of the Ministry of Education of P. R. China under Grant (no. 19YJAZH010), and Fundamental Research Funds for the Central Universities (no. 31920180119).