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Discrete Dynamics in Nature and Society
Volume 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

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; nc.ude.utx@uohzy

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 [13] 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 [525] 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 [2629]. 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., [3238]). 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., [3941]). 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 . SetThenwhere is a positive number and denotes the integer part of .

Proof. For completeness we supply the proofs. DenoteCase 1. If then . Obviously, Case 2. If then . Obviously, Hence, we getFurthermore, 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 haveLetTherefore, (20) can be written asConsider defined in (21); we haveSubstituting (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 ThenObviously, which can be turned to where (which depend on ).
According to Remark 11, one hasTherefore, 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 whereTherefore, (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) withwhere 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 haveNote the fact Consider the operator defined bywhere .
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