Abstract
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.
1. Introduction
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 [1–7], stability and stabilization of nonlinear fractional-order systems [8–13], finite-time stability of fractional-order neural networks [14, 15], synchronization of fractional-order chaotic systems [16], consensus analysis of fractional-order multiagent systems [17–19], etc. The estimate on Mittag-Leffler function was first proposed in [20]. The definition of Mittag-Leffler function and the estimate of Mittag-Leffler function can be described by Definition 1 and Lemma 2, respectively, as follows.
Definition 1 (see [21]). The Mittag-Leffler function with one parameter is defined as where and .
The Mittag-Leffler function with two parameters is defined aswhere and . When , one has , and when and , one further has .
Lemma 2 (see [20]). For Mittag-Leffler function, the following properties hold.
(i) There exist constants such that, for any ,where denotes a matrix and denotes any vector or induced matrix norm.
(ii) If , then for If is a diagonal stability matrix, then there exists a constant such that for where is the largest eigenvalue of the diagonal matrix .
However, we have to point out that Lemma 2 is incorrect.
In [20], inequalities (3) and (4) are proved as follows:Actually, there are two problems in the above-mentioned proof. First,does not necessarily hold. In fact, (9) holds if all the elements of matrix are nonnegative, because matrix norms, such as -norm, -norm, and -norm, have the property of weak monotony. In other words, (9) may not hold when there exist negative elements in or . Second, does not exist when and . To confirm this point, let ; when and , the behavior of with and is shown in Figures 1(a) and 1(b), respectively. It can be obviously observed from Figure 1 that goes to infinity as goes to infinity, so has no supremum when or as goes to infinity for a fixed value of .
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Next, a counterexample is presented to show that does not satisfy inequality (3) or (4). Assume that ; by simple calculation, it has three different eigenvalues, i.e., , , and . Hence, a nonsingular matrix can be determined to make andFor each fixed value of , , can be calculated by means of the OPC algorithm [22]. Thus, can be calculated through (11). When , the behaviors of with and are displayed in Figures 2(a) and 2(b), respectively. It is obvious that goes to infinity as goes to infinity, so inequalities (3) and (4) are incorrect.
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The conclusion on inequality (6) is straightforward if inequalities (3) and (4) are correct. Because inequalities (3) and (4) are not correct, inequality (6) is not correct either. For example, let ; the behavior of for and is shown in Figures 3(a) and 3(b), respectively. It is clear that goes to infinity as goes to infinity for and , so inequality (6) does not hold.
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Next, we consider the case that . In [20], inequality (5) is proved as follows:
With the same argument as stated for (8), does not necessarily hold and it is true if all the elements of matrix are nonnegative. The other problem with (12) is that does not hold for and as . For example, when , . The behavior of Gamma function and its derivative is depicted in Figures 4(a) and 4(b), respectively. From Figure 4, it is clear that for . Hence, we can conclude that for when .
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From the above discussions, we can infer that inequality (5) holds only under some particular conditions; that is, we have to impose some restrictions on matrix and .
Conclusion 3. Suppose all the elements of matrix are nonnegative; if , then for , inequality (5) holds.
Conclusion 4. Suppose all the elements of matrix are nonnegative; if , then for , inequality (5) holds.
Now, a counterexample is presented to show that if the conditions in Conclusion 3 or Conclusion 4 are not satisfied, inequality (5) may not be correct. For instance, let ; then the behavior of for , , and is shown in Figures 5(a)–5(c), respectively. It is obvious that goes to infinity as goes to infinity, so inequality (5) does not hold.
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Similarly, (7) is incorrect because (5) is incorrect. The behavior of for , , and is shown in Figures 6(a)–6(c), respectively, which is in contradiction to inequality (7).
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2. Main Results
In this section, some sufficient conditions are derived to guarantee that can be bounded by for some , which can be formulated by the following two theorems.
Theorem 5. If matrix is diagonalizable, and the largest real part of eigenvalues of is positive, then for and anyone of the following two conditions:(i),(ii) and has no zero eigenvalue, and further there exists a positive constant such that for where denotes 1-norm, 2-norm, or -norm of a matrix.
To prove Theorem 5, another two lemmas are presented as follows, which will be used later.
Lemma 6 (see [21]). If , is an arbitrary real number, satisfies , and and are real constants, thenwhere
Lemma 7 (see [21]). If , is an arbitrary real number, satisfies , and is a real constant, thenwhere
Now, the proof of Theorem 5 can proceed.
Proof. Because is diagonalizable, there exists a nonsingular matrix such that . Then and . Since and are with the same characteristic polynomial and eigenvalues, so . Let ; then .
Let be the principal value of the argument of . According to the magnitude of the principal value of the argument of , the cases where and are considered, separately.
Case 1. If , it follows from Lemma 6 thatwhere . is the numerator of , where , .
Case 2. If , it follows from Lemma 7 thatThen, it follows from (19) and (20) thatwhere .
Thus, we haveWhen any one of the following two conditions is satisfied: (1),(2) and such that , we have When and , the following equality can be derived by L'Hospital's rule,As and it can be obtained from (24) that when and . To summarize, if conditions or in Theorem 5 are satisfied. It is obvious that . According to (22), we have . Due to the continuity of matrix norms, there exists a positive constant such that , which implies that inequality (15) holds. This completes the proof.
Now, an example is presented to verify the correctness of the newly established Theorem 5. Assume that ; it is diagonalizable and the eigenvalues are , respectively. Hence, the largest real part of the eigenvalues is ; then the behavior of for , , , , , and is shown in Figures 7(a)–7(f), respectively. It can be seen from Figure 7 that converges to zero, which indicates the validity of Theorem 5.
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Remark 8. Note that the condition that is needed in Theorem 5. If , then may go to infinity as goes to infinity, so the process of the proof cannot be carried out and the conclusion in Theorem 5 may not be obtained. To get the similar estimate of Mittag-Leffler function for , an extra restriction has to be imposed on the eigenvalues of matrix , which is given in the following Theorem 9.
Theorem 9. If matrix is diagonalizable and satisfies the following two conditions:(i)the largest real part of eigenvalues of is positive,(ii)the principal value of the argument of satisfies , , then for and ,and further there exists a positive constant such that for where denotes 1-norm, 2-norm, or -norm of a matrix.
Proof. According to condition in Theorem 9, (20) holds for each , . Thus we haveSimilarly, we can prove that there exists a positive constant such that , which implies that inequality (15) holds. This completes the proof.
Now, an example is presented to verify the correctness of Theorem 9. Assume that ; it is diagonalizable and the eigenvalues are , , , respectively, where . It is obvious that the largest real part of the eigenvalues is . Choose ; then , . Hence the two conditions in Theorem 9 are all satisfied, and Figures 8(a)–8(c) depict the behavior of for , , and , respectively, which shows that converges to zero and indicates the validity of Theorem 9.
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Remark 10. Note that the condition that matrix is diagonalizable is needed in Theorems 5 and 9; otherwise, there exists a nonsingular matrix such that , where andOne can obtain that and . In the proof of Theorems 5 and 9, the lower bound of and the upper bound of are needed. When A is not diagnosable, still holds but the upper bound of is difficult to obtain. The difficulties are described as follows.
For the Jordan block (31), we havewhere is a nonsingular matrix such that . As is composed of , it is difficult calculate via (32), so the upper bound of is difficult to obtain and is difficult to estimate.
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.
3. Conclusion
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 holds for some and numerical examples are given to verify the correctness of the newly developed results.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgments
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).