Discrete Dynamics in Nature and Society
Volume 2017 (2017), Article ID 9050289, 15 pages
https://doi.org/10.1155/2017/9050289

Research Article

Analysis of a Class of Fractional Nonlinear Multidelay Differential Systems

Zhuoyan Gao,¹ JinRong Wang,² and Yong Zhou3,4

¹College of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, Shanxi 030031, China
²Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China
³Department of Mathematics, Xiangtan University, Xiangtan, Hunan 411105, China
⁴Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Correspondence should be addressed to Yong Zhou; yzhou@xtu.edu.cn

Received 31 July 2017; Accepted 30 August 2017; Published 15 October 2017

Academic Editor: Antonio Iannizzotto

Copyright © 2017 Zhuoyan Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We address existence and Ulam-Hyers and Ulam-Hyers-Mittag-Leffler stability of fractional nonlinear multiple time-delays systems with respect to two parameters’ weighted norm, which provides a foundation to study iterative learning control problem for this system. Secondly, we design PID-type learning laws to generate sequences of output trajectories to tracking the desired trajectory. Two numerical examples are used to illustrate the theoretical results.

1. Introduction

Fractional differential equations have been used to deal with many problems from physics, engineering, and other fields. For some basic results in the theory of fractional differential equations, one can read the monographs [1–3] or the survey [4] and reference therein. Recently, considerable attention has been given to the control and stability of fractional differential equations; one can refer to [5–25] via Ulam’s type stability concepts and the references therein. We also note that there are some contributions on Mittag-Leffler stability of fractional order systems and stabilization [26–29]. We remark that there are some difference between the concept of Mittag-Leffler stability and Ulam-Hyers-Mittag-Leffler stability. The concept of Mittag-Leffler stability of solution follows the idea of stability of zero solution for the classical ODEs and gives an estimate inequality for the norm of solution via Mittag-Leffler function. The concept of Ulam-Hyers-Mittag-Leffler stability follows the idea of Ulam-Hyers stability of functional equations and gives an approximate relation via small parameter and Mittag-Leffler function between the solution of equations and the solution of inequalities, which is a special case of Ulam-Hyers-Rassias stability. That is, we try to find a solution of approximate inequalities close to the solution of the original equations in the sense of Ulam-Hyers-Mittag-Leffler stability. The main idea for this concept will provide an approach to find the explicit solution. However, there are only few works on existence and Ulam’s type stability for the nonlinear fractional time-delays differential equations.

Iterative learning control has become a popular strategy in the intelligent control community since it was proposed by Uchiyama [30] and developed by Arimoto et al. [31]. Recently, iterative learning control problems of P-type, D-type, I-type, or their combination schemes have been widely applied to various types of repetitive or batch dynamical systems (see, e.g., [32–38]). The problem on designing an ILC for uncertain plants with time-delays has not been fully investigated, and only a limited number of the results are available so far (see, e.g., [39–41]). However, most of the existing literatures focus on iterative learning control of the nonlinear fractional differential system without time-delays, especially multiple time-delays. Note that PID-type ILC learning algorithm is one of the popular updating laws. The advantage of PID-type ILC learning algorithm is simple and very easy to be realized in tracking problem. The disadvantage of PID-type ILC is that the error characterization for the signal is not the best and there is not a uniform method to design the weighting coefficients.

Delay systems are widely used to model dynamical systems in many scientific and engineering areas, for example, biology, climatology, and economy. Comparing with systems with single delay, systems with multidelay are more realistic models in the interacting complex systems. In fact, dynamics of multifeedback systems are representative examples of the multidelay systems.

Motivated by [15, 42], we firstly discuss existence, Ulam-Hyers stability, and Ulam-Hyers-Mittag-Leffler stability of solutions to fractional order nonlinear Cauchy problems with multiple time-delays of the form:

where is a positive constant; is the Caputo fractional derivative of order with the lower limit zero; ; are positive constant time-delays; ; is the initial continuous function of the system in ; ; and .

Secondly, we turn to study PID-type ILC learning algorithm of the following fractional order nonlinear system with output equation:

where denotes the th learning iteration; and are the states and control input and output of the system, respectively; is the initial continuous function of the system in ; , , and are given continuous functions in ; and is a continuous function.

The rest of the paper is organized as follows. Section 2 collects some notations and preparation results. Section 3 presents existence and uniqueness of solutions and shows Ulam-Hyers stability and Ulam-Hyers-Mittag-Leffler stability of solutions by using Picard operator method. Section 4 presents convergence result for PID-type ILC updating law. Section 5 gives two illustrative examples.

2. Preliminaries

Denote as the Banach space of continuous functions from endowed with the ()-norm (,  ).

Definition 1 (see [2]). The Riemann-Liouville fractional integrals are defined by

and the Riemann-Liouville fractional derivatives are defined by

where is Gamma function.

Definition 2 (see [2]). The Caputo derivative of order for a function can be written as

Definition 3 (see [43]). Let be a metric space. An is a Picard operator if there exists such that (i) , where is the fixed point set of ; (ii) the sequence converges to for all .

Lemma 4 (see [43]). Let be an ordered metric space and be an increasing Picard operator . Then, for implies .

The following Gronwall inequalities will be used in the sequel.

Lemma 5 (see [44, Lemma  3.1]). Let be a continuous function on and let be continuous and nonnegative on the triangle . Moreover, let be a positive continuous and nondecreasing function on . If

then

Lemma 6 (see [44, Lemma  7.1.1]). Let be continuous functions where . If is nondecreasing and there are constants and such that

then

Remark 7 (see [44]). Under the hypothesis of Lemma 6, let be a nondecreasing function on . Then we have .

By [45, Lemma 2.12], one can adopt the similar idea to prove the following result.

Lemma 8. Let and . Set

Then

where is a positive number and denotes the integer part of .

Proof. For completeness we supply the proofs. Denote

Case 1. If then . Obviously,

Case 2. If then . Obviously,

Hence, we get

Furthermore, we can get

The proof is finished.

3. Existence and Stability Results

We introduce the following assumptions:

Assume that , , and . In addition, set .

() For arbitrary , there exist positive constants such that

() Suppose the following inequalities hold:

where .

Theorem 9. Assume that , , and are satisfied. Then problem (1) has a unique solution in .

Proof. Define an operator () as follows:

Next, we show that defined in (19) is a contraction mapping on with respect to the previous ()-norm .
For all and , we have . This yields that .
For any and , according to , we have

Let

Therefore, (20) can be written as

Consider defined in (21); we have

Substituting (23) into (22), using Lemma 8, we can obtain

This implies that

where is defined in ().
Due to (), we can derive that is a contraction via the ()-norm on . The rest of the proof follows from the Banach contraction principle.

Let . Consider (1) and the following inequality:

Definition 10. Equation (1) is Ulam-Hyers stable if there exists such that for each and for each solution of the inequality (26) there exists a solution of (1) with

Remark 11. A function is a solution of inequality (26) if and only if there exists a function (which depend on ) such that

(i).(ii).

Theorem 12. Assume that , , and are satisfied; then (1) is Ulam-Hyers stable.

Proof. Let be a solution of inequality (26) and be a solution of

Then

Obviously,

which can be turned to

where (which depend on ).
According to Remark 11, one has

Therefore,

Now multiplying by the fact on both side of the above inequalities, one can derive that

So, we obtain

Furthermore, according to , combined with the fact of , we can get

where

Therefore, (1) is Ulam-Hyers stable.

Different from the above stability result in the Theorem 12, in the following part, we will discuss Ulam-Hyers-Mittag-Leffler stability of (1) on the time interval . So we first introduce the following Ulam-Hyers-Mittag-Leffler stability definition.

Consider problem (1) with

where is the Mittag-Leffler function [2] defined by

Definition 13 ([46]). Equation (1) is Ulam-Hyers-Mittag-Leffler stable with respect to if there exists such that for each and for each solution to (38), there exists a solution to (1) with

Remark 14 ([46]). A function is a solution of inequality (38) if and only if there exists a function (which depends on ) such that

(i) for all ;(ii).

Theorem 15. Assume that , , and are satisfied; then (1) is Ulam-Hyers-Mittag-Leffler stable.

Proof. Let be a solution to (38) and be the unique solution of the following problem:

Obviously,

On the other hand, from [47, Remark  2] via Remark 14, we can know that satisfied the following inequality:

Then, for , according to (), we have

Note the fact Consider the operator defined by

where .
Next, we verify that is a Picard operator. In fact, for all and arbitrary , it follows the proof in Theorem 9; one can show that is a contraction via the ()-norm on due to ().
Applying the Banach contraction principle to , we derive that is a Picard operator and . Then, we have , for and

We go on to verify that the solution is increasing. Now, denote and .
Then, for , we have

So, is increasing. Thus, due to and

Using Lemma 6, Remark 7, and the fact of , for , we obtain

where .
In particular, if , from (44), and applying the Lemma 4, we obtain , where is a Picard and an increasing operator. As a result, we know

Thus, (2) is Ulam-Hyers-Mittag-Leffler stable.

Remark 16. One can find that we use Gronwall’s inequality method to derive asymptotic stability of the corresponding systems instead of using the Lyapunov direct method in [28, 29]. We do not need to assume that Lyapunov function satisfies some certain condition, for example, [28, Theorem , -]. Next, note that if ; then the definition of Mittag-Leffler stability can be turned to stability of zero solutions. However, the concept of Ulam-Hyers-Mittag-Leffler is more general since is not necessarily less than 1 on the whole interval .

4. PID-Type ILC

In this section, we consider the open-loop and close-loop PID-type ILC updating laws of fractional order nonlinear system with multiple time-delays (2) via -weighted norm .

4.1. Open-Loop Case

For system (2), consider the open-loop PID-type ILC updating law with initial state learning:

where and are unknown parameters to be determined.

For the sake of brevity, the following notations will be used:

And we denote that , , and which is called tracking error at th repetition, where denotes desired output bounded trajectory satisfying

and is a desired control.

In this section, we imposed the following assumptions on the class of system described by (2).

The function is continuous and differentiable for all and with partial derivatives and . For the constant , set

The function is uniformly bounded on ; that is, there exists such that for any .

Theorem 17. Assume that – hold. If

then the nonlinear fractional multiple time-delays differential system (2) with the open-loop PID-type ILC updating law (51) guarantees that tends to as in the sense of -norm for all , where is the desired initial state on and is the desired output trajectory.

Proof. Note that

where lies in the segment with end point and for .
In what follows, we show that as for a.e. . By using mean value theorem we have

Substituting (51) into (58) and taking the absolute value, we have

It follows from (55) that

Now we turn to give an estimation for the upper bound of . One has

Repeating the same procedure in Theorem 9, we can get

where are defined in ().
Using Lemma 5, we get

where .
We know

Taking (51) into (64), one obtains

Combining with (64) and Lemma 8, we can get

where .
Multiplying on both sides of (66) and taking the maximum value on , we can get

Set

There exists a sufficiently large such that is very small and using (56) and (60) we can derive . The proof is completed.

4.2. Closed-Loop Case

For system (2), consider the close-loop PID-type ILC updating law with initial state learning:

where and are unknown parameters to be determined.

Theorem 18. Assume that – hold. If

then the nonlinear multiple time-delays system (2) with the close-loop PID-type ILC updating law (69) guarantees that tends to as in the sense of -norm for all , where is the desired initial state on and is the desired output trajectory.

Proof. Note that

where lies in the segment with end points and for .
In what follows, we show that as for a.e. . By using mean value theorem we have

Substituting (69) into (73) and taking the absolute value, we have

It follows from (70) that

Similar to the proof of Theorem 17, we can get

Then, one has

where .
Multiplying on both sides of (77) and taking -norm, we can derive

Set

There exists a sufficiently large such that is very small and using (71) and (75) we can derive . This completes the proof of Theorem 18.

5. Examples

In this section, we give two examples to illustrate our results above.

Example 1. Let , , , , , and . We consider the following nonlinear fractional multiple time-delays differential equations

and the inequalities

Define and and set . Thus ; then . Now all the assumptions in Theorems 9, 12, and 15 are satisfied, problem (80) has a unique solution, and the first equation in (80) is Ulam-Hyers stable with and Ulam-Hyers-Mittag-Leffler stable with

where .

Example 2. Let , , , , , and . We consider the following nonlinear fractional multiple time-delays differential equations

and the desired reference trajectory
Consider problem (83) and the open-loop PID-type ILC updating law with initial state learning:

Obviously, and . All the conditions of Theorem 17 are satisfied.

Next, consider problem (83) and the close-loop PID-type ILC updating law with initial state learning:

Thus and . All the conditions of Theorem 18 are satisfied.

Numerical simulation diagram for the open-loop control is shown in Figure 1(a).

(a)

(b)

Figure 1: The system output (blue line), the desired trajectory (red line), and the tracking error for (83).

Figure 1(b) shows the supremum norm of the tracking error in each iteration and the 100th error is 0.0979.

6. Conclusions

This paper is twofold: in the first part we show the existence and uniqueness result and present Ulam-Hyers and Ulam-Hyers-Mittag-Leffler stability results for fractional nonlinear multiple time-delays systems. In the second part we apply PID-type learning updating laws to obtain sequences of tracking trajectory to approximate a given trajectory. In the future, we will study the related topic of noninstantaneous impulsive systems [48].

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The first and second author’s work was partially supported by National Natural Science Foundation of China (11661016), Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), Unite Foundation of Guizhou Province ([2015]7640), and Graduate ZDKC([2015]003). The third author acknowledges the support by National Natural Science Foundation of China (11671339). The authors are grateful to Associate Professor Dr. Qian Chen for her discussion, careful reading of the manuscript, and valuable comments.

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