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The estimate of Mittag-Leffler function has been widely applied in the dynamic analysis of fractional-order systems in some recently published papers. In this paper, we show that the estimate for Mittag-Leffler function is not correct. First, we point out the mistakes made in the estimation process of Mittag-Leffler function and provide a counterexample. Then, we propose some sufficient conditions to guarantee that part of the estimate for Mittag-Leffler function is correct. Meanwhile, numerical examples are given to illustrate the validity of the two newly established estimates.

Fractional calculus can date back to the seventeenth century, and now it has attracted considerable research interests due to its widespread applications in many fields. There are mainly two types of methods in the dynamic analysis of fractional-order nonlinear systems, that is, Lyapunov function based method and estimation based method. When estimation based method is employed, the solution of the fractional-order system being studied is usually expressed in terms of the Mittag-Leffler function. Obviously, the correctness of the estimate of Mittag-Leffler function is crucial to the whole estimation process and plays an important role if the estimation based method is adopted. Recently, estimation based method has been widely applied to the study of finite-time stability and synchronization of fractional-order memristor-based neural networks [

The Mittag-Leffler function with one parameter is defined as

The Mittag-Leffler function with two parameters is defined as

For Mittag-Leffler function, the following properties hold.

(i) There exist constants

(ii) If

However, we have to point out that Lemma

In [

The behavior of

Next, a counterexample is presented to show that

The behavior of

The conclusion on inequality (

The behavior of

Next, we consider the case that

With the same argument as stated for (

The behavior of (a)

From the above discussions, we can infer that inequality (

Suppose all the elements

Suppose all the elements

Now, a counterexample is presented to show that if the conditions in Conclusion

The behavior of

Similarly, (

The behavior of

In this section, some sufficient conditions are derived to guarantee that

If matrix

and further there exists a positive constant

To prove Theorem

If

If

Now, the proof of Theorem

Because

Let

Thus, we have

we have

Now, an example is presented to verify the correctness of the newly established Theorem

The behavior of

Note that the condition that

If matrix

the largest real part of eigenvalues

the principal value

then for

According to condition

Similarly, we can prove that there exists a positive constant

Now, an example is presented to verify the correctness of Theorem

The behavior of

Note that the condition that matrix

For the Jordan block (

To the best of our knowledge, the estimate of Mittag-Leffler function by the exponential function is still an open problem due to the complexity of Mittag-Leffler function and deserves further research.

In this paper, several counterexamples are presented to numerically show that the estimate for Mittag-Leffler function used in some recently published papers is not completely correct and the mistakes made in the estimation process are mainly due to the misuse of the properties of matrix norms. Besides, some sufficient conditions are developed to guarantee that the estimate

No data were used to support this study.

The authors declare that they have no conflicts of interest.

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

This work was supported by the National Natural Science Foundation of China (Nos. 61573008, 61473178, 61703233); Natural Science Foundation of Shandong Province (Nos. ZR2016FQ09, ZR2018MF005); the Postdoctoral Science Foundation of China (No. 2016M602112); the Open Fund of Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education (No. MCCSE2016A04); and SDUST Research Fund (No. 2018TDJH101).