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.

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 [

Fractional calculus involves the operation of convolution with a power law function. Multiplying by an exponential factor results in tempered fractional derivatives and integrals [

As in classical calculus, stability analysis is still one of the most important tasks in fractional differential system [

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 [

This paper is organized as follows. In Section

Tempered fractional calculus plays an important role in different fields [

The tempered fractional integral of order

The tempered fractional Caputo derivative of tempered parameter

The tempered fractional Riemann–Liouville derivative of tempered parameter

In order to study the stability of tempered fractional systems, several lemmas are needed.

Let

Let

The Laplace transform of tempered fractional integral and Caputo derivative (

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

In this section, we establish tempered fractional comparison principles.

Assume that

Following from

By Lemma

According to

Taking the inverse Laplace transform on (

According to

Assume that

From Lemma

That is

In this section, we construct some inequalities for tempered fractional derivatives or systems.

From Lemma

The relationship between

If

We take

Multiplying both sides of equation (

Using Definition

Consider the following tempered fractional system

For the real-valued continuous function

It follows from (

If

By applying the tempered fractional integral operator

There exists a function

Combining with Lemma

In this section, some sufficient conditions are established for the tempered Mittag–Leffler stability of system (

If and only if

Assume

Tempered Mittag–Leffler stability is a generalization of Mittag–Leffler stability. When

Both Mittag–Leffler stability and tempered Mittag–Leffler stability imply asymptotic stability, that is,

Assume

It follows from equations (

There exists a function

Taking the Laplace transform to (

Applying the inverse Laplace transform to (

Because

Substituting (

Because

Assume all conditions in Theorem

It follows from Lemma

A similar proof method in Theorem

For the tempered fractional system (

From (

Using (

We can use Lemmas

In this section, we will give three examples to demonstrate theoretical analysis. The Adams–Bashforth–Moulton method [

Consider the tempered fractional Riemann–Liouville system:

By Theorem

Then,

Time evolution of system states

Consider the tempered fractional Caputo Hopfield neural networks:

Let

By inequalities (

To illustrate the effectiveness of Example

It is obvious that condition (

Time evolution of system states

Consider the following tempered fractional system:

Let the Lyapunov function

Then, the conditions of Theorem

Time evolution of system states

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.

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

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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).