# Analysis of a Class of Fractional Nonlinear Multidelay Differential Systems

Authors: Zhuoyan Gao; JinRong Wang; Yong Zhou
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## 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.


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## Body

## 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:(1)D0+αcxt=ft,xt,xt-τ1,…,xt-τm+It,xt,xt-ξ1,…,xt-ξnut,t∈0,T,0<α<1,xt=ψt,t∈-a,0,a=maxτ1,…,τm,ξ1,…,ξn,where T is a positive constant; Dc0+α is the Caputo fractional derivative of order α with the lower limit zero; u∈C([0,T],R); τ1,…,τm,ξ1,…,ξn are positive constant time-delays; x(t)∈R; ψ is the initial continuous function of the system in t∈[-a,0]; f∈C([0,T]×R×⋯R︸n,R); and I∈C([0,T]×R×⋯R︸n,R).Secondly, we turn to study PID-type ILC learning algorithm of the following fractional order nonlinear system with output equation:(2)Dc0+αxkt=ft,xkt,xkt-τ1,…,xkt-τm+It,xkt,xkt-ξ1,…,xkt-ξnukt,t∈0,T,0<α<1,xkt=ψkt,t∈-a,0,a=max⁡τ1,…,τm,ξ1,…,ξn,ykt=gt,xkt+d∫0tuksds,t∈0,T,where k denotes the kth learning iteration; uk(t)∈R and yk(t)∈R are the states and control input and output of the system, respectively; ψk is the initial continuous function of the system in t∈[-a,0]; f, I, and uk are given continuous functions in [0,T]; and g:[0,T]×R→R is a continuous function.The rest of the paper is organized as follows. Section2 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

DenoteX≔C([-a,T],R) as the Banach space of continuous functions from [-a,T]→R endowed with the (λ,α)-norm xλ,α=maxt∈[-a,T]e-λtαxt(x∈C([-a,T],R),  λ>0,0<α<1).Definition 1 (see [2]).
The Riemann-Liouville fractional integralsIa+αf are defined by (3)Ia+αfx≔1Γα∫axftx-t1-αdt,x>a;α>0,and the Riemann-Liouville fractional derivatives Da+αf are defined by (4)Da+αfLx≔ddxnIa+n-αfx=1Γn-αddxn∫axftx-tα-n+1dt,x>a;α>0;n=α+1,where Γ(·) is Gamma function.Definition 2 (see [2]).
The Caputo derivative of orderγ for a function f:[a,∞)→R can be written as(5)Dca+γft=DLa+γft-∑k=an-1tkk!fka,t>0,n-1<γ<n.Definition 3 (see [43]).
Let(X,d) be a metric space. An A:X→X is a Picard operator if there exists x∗∈X such that (i) FA=x∗, where FA={x∈X:A(x)=x} is the fixed point set of A; (ii) the sequence (An(x0))n∈N converges to x∗ for all x0∈X.Lemma 4 (see [43]).
Let(X,d,≤) be an ordered metric space and A:X→X be an increasing Picard operator (FA=x∗). Then, for x∈X,x≤A(x) implies x≤xA∗.The following Gronwall inequalities will be used in the sequel.Lemma 5 (see [44, Lemma  3.1]).
Letu(t) be a continuous function on t∈[0,T] and let v(t-τ) be continuous and nonnegative on the triangle 0≤τ≤t. Moreover, let w(t) be a positive continuous and nondecreasing function on t∈[0,T]. If (6)ut≤wt+∫0tvt-τuτdτ,t∈0,T,then(7)ut≤wte∫0tvt-τdτ,t∈0,T.Lemma 6 (see [44, Lemma  7.1.1]).
Letz,ω:[0,T)→[0,∞) be continuous functions where T≤∞. If ω is nondecreasing and there are constants κ≥0 and 0<α<1 such that (8)zt≤ωt+κ∫0tt-sα-1zsds,t∈0,T,then(9)zt≤ωt+∫0t∑n=1∞κΓαnΓnαt-snα-1ωsds,t∈0,T.Remark 7 (see [44]).
Under the hypothesis of Lemma6, let ω(t) be a nondecreasing function on [0,T). Then we have z(t)≤ω(t)Eα(κΓ(α)tα).By [45, Lemma 2.12], one can adopt the similar idea to prove the following result.Lemma 8.
Let0<α<1 and λ>0. Set(10)Z≔∫0tt-sα-1eλsαds,t∈0,T,T>0.Then(11)Z≤tαeλtααλ+1α,ifλ=0,Z≤eλtα1+tααλα,ifλ≥1,where T is a positive number and [λ] denotes the integer part of λ.Proof.
For completeness we supply the proofs. Denote(12)Z1≔∑k=1λ∫k-1t/λ+1kt/λ+1t-sα-1eλsαds,Z2≔∫λt/λ+1tt-sα-1eλsαds.
Case 1. If [λ]=0 then Z=Z2. Obviously, (13)Z2=∫λt/λ+1tt-sα-1eλsαds≤eλtα∫λt/λ+1tt-sα-1ds=tαeλtααλ+1α.
Case 2. If [λ]≥1 then Z=Z1+Z2. Obviously, (14)∫k-1t/λ+1kt/λ+1t-sα-1eλsαds≤∫k-1t/λ+1kt/λ+1sλ+1k-sα-1eλsαdss≤ktλ+1⟹t≥sλ+1k≤λ+1k-1α-1∫k-1t/λ+1kt/λ+1sα-1eλsαds≤1λαλ+1k-1α-1eλkt/λ+1α-eλk-1t/λ+1α.Hence, we get(15)Z1≔∑k=1λ∫k-1t/λ+1kt/λ+1t-sα-1eλsαds≤∑k=1λ1λαλ+1k-1α-1eλkt/λ+1α-eλk-1t/λ+1α≤1λαλα-1eλt/λ+1α-1+λ+12-1α-1eλ2t/λ+1α-eλt/λ+1α+⋯+λ+1λ-1α-1eλλt/λ+1α-eλλ-1t/λ+1α≤1λαλ+11-1α-1eλt/λ+1α-1+λ+12-1α-1eλ2t/λ+1α-1+⋯+λ+1λ-1α-1eλλt/λ+1α-1≤1λαλλ1-αeλλt/λ+1α-1≤1αλα-1eλλt/λ+1α-1≤1αλα-1eλλt/λ+1α≤eλtααλα.
Furthermore, we can get(16)Z=Z1+Z2≤eλtααλα+tαeλtααλ+1α≤eλtα1+tααλα.The proof is finished.


## 3. Existence and Stability Results

We introduce the following assumptions:(
H
1
) Assume that f∈C([0,T]×R×⋯R︸n,R), I∈C([0,T]×R×⋯R︸n,R), and u∈C([0,T],R). In addition, set ub=maxt∈[0,T]ut.(H2) For arbitrary x1,x2∈C([-a,T],R), there exist positive constants Lf,LI such that (17)ft,x2t,x2t-τ1,…,x2t-τm-ft,x1t,x1t-τ1,…,x1t-τm≤Lfx2t-x1t+∑l=1mx2t-τl-x1t-τl,It,x2t,x2t-ξ1,…,x2t-ξm-It,x1t,x1t-ξ1,…,x1t-ξm≤LIx2t-x1t+∑l=1nx2t-ξl-x1t-ξl.(H3) Suppose the following inequalities hold:(18)cTαΓα+1<1,λ=0,c1+TαΓα+1λα<1,λ≥1,where c=Lf(m+1)+LIbu(n+1).Theorem 9.
Assume that(H1), (H2), and (H3) are satisfied. Then problem (1) has a unique solution in C([-a,T],R).Proof.
Define an operatorA:X→X (X:=C([-a,T],R)) as follows:(19)Axt=ψt,t∈-a,0,ψ0+1Γα∫0tt-sα-1fs,xs,xs-τ1,…,xs-τm+Is,xs,xs-ξ1,…,xs-ξnusds,t∈0,T.
Next, we show thatA defined in (19) is a contraction mapping on X with respect to the previous (λ,α)-norm ·λ,α.
For allt∈[-a,0] and x(t),z(t)∈X, we have A(x)(t)-A(z)(t)=0. This yields that A(x)-A(z)λ,α=0.
For anyt∈[0,T] and x,z∈X, according to (H2), we have(20)Axt-Azt≤1Γα∫0tt-sα-1fs,xs,xs-τ1,…,xs-τm-fs,zs,zs-τ1,…,zs-τm+Is,xs,xs-ξ1,…,xs-ξn-Is,zs,zs-ξ1,…,zs-ξnusds≤LfΓα∫0tt-sα-1xs-zs+∑l=1mxs-τl-zs-τlds+LIbuΓα∫0tt-sα-1xs-zs+∑l=1nxs-ξl-zs-ξlds.
Let(21)Gθ=xt-θ-zt-θ,θ∈τ1,…,τm,ξ1,…,ξn.
Therefore, (20) can be written as(22)Axt-Azt≤LfΓα∫0tt-sα-1xs-zs+∑l=1mGτlds+LIbuΓα∫0tt-sα-1xs-zs+∑l=1nGξlds≤LfΓα∫0tt-sα-1eλsαe-λsαxs-zs+∑l=1mGτlds+LIbuΓα∫0tt-sα-1eλsαe-λsαxs-zs+∑l=1nGξlds.
ConsiderG(·) defined in (21); we have(23)∫0tt-sαeλsαe-λsαGθds=∫0tt-sαeλsαe-λsαxs-θ-zs-θds≤∫0tt-sαeλsαe-λsαψs-ψsds=0,t∈0,θ,∫0tt-sαeλsαx-zλ,αds,t∈θ,T,≤∫0tt-sαeλsαdsx-zλ,α.Substituting (23) into (22), using Lemma 8, we can obtain (24)Axt-Azt≤LfΓα∫0tt-sα-1eλsαe-λsαxs-zs+∑l=1mGτlds+LIbuΓα∫0tt-sα-1eλsαe-λsαxs-zs+∑l=1nGξlds≤Lfm+1+LIbun+1Γα∫0tt-sα-1eλsαdsx-zλ,α≤cTαeλtαΓα+11+λαx-zλ,α,λ=0,c1+TαeλtαΓα+1λαx-zλ,α,λ≥1.This implies that (25)Ax-Azλ,α≤cTαΓα+11+λαx-zλ,α,λ=0,c1+TαΓα+1λαx-zλ,α,λ≥1,where c is defined in (H3).
Due to (H3), we can derive that A is a contraction via the (λ,α)-norm ·λ,α on X. The rest of the proof follows from the Banach contraction principle.Letϵ>0. Consider (1) and the following inequality:(26)Dc0,tαzt-ft,zt,zt-τ1,…,zt-τm-It,zt,zt-ξ1,…,zt-ξnut≤ϵ,t∈-a,T,0<α<1,z∈X.Definition 10.
Equation (1) is Ulam-Hyers stable if there exists c^>0 such that for each ϵ>0 and for each solution z¯∈X of the inequality (26) there exists a solution x∈X of (1) with(27)z¯t-xt≤c^ϵ,t∈-a,T.Remark 11.
A functionz¯∈X is a solution of inequality (26) if and only if there exists a function w¯∈X (which depend on z¯) such that(i)
w¯t≤ϵ,t∈[-a,T].(ii)
Dc0,tαz¯(t)=f(t,z¯(t),z¯t-τ1,…,z¯(t-τm))+I(t,z¯(t),z¯(t-ξ1),…,z¯(t-ξn))u(t)+w¯(t),t∈[0,T],0<α<1.Theorem 12.
Assume that(H1), (H2), and (H3) are satisfied; then (1) is Ulam-Hyers stable.Proof.
Letz¯∈C([-a,T],R) be a solution of inequality (26) and x(t) be a solution of (28)Dc0+αxt=ft,xt,xt-τ1,…,xt-τm+It,xt,xt-ξ1,…,xt-ξnut,t∈0,T,0<α<1,xt=z¯t,t∈-a,0,a=max⁡τ1,…,τm,ξ1,…,ξn.Then(29)xt=z¯t,t∈-a,0,z¯0+1Γα∫0tt-sα-1fs,xs,xs-τ1,…,xs-τm+Is,xs,xs-ξ1,…,xs-ξnusds,t∈0,T.Obviously, (30)Dc0,tαz¯t=ft,z¯t,z¯t-τ1,…,z¯t-τm+It,z¯t,z¯t-ξ1,…,z¯t-ξnut+w¯t,t∈0,T,which can be turned to (31)z¯t=z¯0+1Γα∫0tt-sα-1fs,z¯s,z¯s-τ1,…,z¯s-τm+Is,z¯s,z¯s-ξ1,…,z¯s-ξnus+w¯sds,t∈0,T,where w¯∈X (which depend on z¯).
According to Remark11, one has(32)z¯t-z¯0-1Γα∫0tt-sα-1fs,z¯s,z¯s-τ1,…,z¯s-τm-Is,z¯s,z¯s-ξ1,…,z¯s-ξnusds≤1Γα∫0tt-sα-1w¯sds≤ϵΓα∫0tt-sα-1ds≤c1ϵ,c1=TαΓα+1.
Therefore,(33)z¯t-xt≤z¯t-z¯0-1Γα∫0tt-sα-1fs,z¯s,z¯s-τ1,…,z¯s-τm+Is,z¯s,z¯s-ξ1,…,z¯s-ξnus+w¯sds+1Γα∫0tt-sα-1×fs,xs,xs-τ1,…,xs-τm+Is,xs,xs-ξ1,…,xs-ξnus-fs,z¯s,z¯s-τ1,…,z¯s-τm+Is,z¯s,z¯s-ξ1,…,z¯s-ξnusds≤c1ϵ+∫0tt-sα-1LfΓαz¯s-xs+∑l=1mz¯s-τl-xs-τl+LIbuΓαz¯s-xs+∑l=1nz¯s-ξl-xs-ξlds≤c1ϵ+cTαΓα+11+λαz¯-xλ,α,λ=0,t∈0,T,c1ϵ+c1+TαΓα+1λαz¯-xλ,α,λ≥1,t∈0,T.Now multiplying by the fact e-λtα on both side of the above inequalities, one can derive that (34)1-cTαΓα+11+λαz¯-xλ,α≤c1ϵ,λ=0,t∈0,T,1-c1+TαΓα+1λαz¯-xλ,α≤c1ϵ,λ≥1,t∈0,T.So, we obtain (35)z¯t-xt≤c1ϵeλTα1-cTα/Γα+11+λα=c1ϵΓα+11+λαeλTαΓα+11+λα-cTα,λ=0,t∈0,T,z¯t-xt≤c1ϵeλTα1-c1+Tα/Γα+1λα=c1ϵΓα+1λαeλTαΓα+1λα-c1+Tα,λ≥1,t∈0,T.Furthermore, according to (H3), combined with the fact of z¯t-xt=0,t∈[-a,0], we can get (36)z¯t-xt≤c^ϵ,t∈-a,T,where(37)c^=max⁡c1Γα+11+λαeλTαΓα+11+λα-cTα,c1Γα+1λαeλTαΓα+1λα-c1+Tα.Therefore, (1) is Ulam-Hyers stable.Different from the above stability result in the Theorem12, in the following part, we will discuss Ulam-Hyers-Mittag-Leffler stability of (1) on the time interval [-a,T]. So we first introduce the following Ulam-Hyers-Mittag-Leffler stability definition.Consider problem (1) with(38)Dc0,tαzt-ft,zt,zt-τ1,…,zt-τm-It,zt,zt-ξ1,…,zt-ξnut≤ϵEαtα,t∈0,T,0<α<1,where Eα is the Mittag-Leffler function [2] defined by (39)Eαz≔∑k=0∞zkΓkα+1,z∈C,Re⁡α>0.Definition 13 ([46]).
Equation (1) is Ulam-Hyers-Mittag-Leffler stable with respect to Eα(tα) if there exists cEα>0 such that for each ϵ>0 and for each solution z∈X to (38), there exists a solution x∈X to (1) with(40)zt-xt≤cEαϵEαtα,t∈-a,T.Remark 14 ([46]).
A functionz∈X is a solution of inequality (38) if and only if there exists a function w∈X (which depends on z) such that(i)
wt≤ϵEα(tα) for all t∈[0,T];(ii)
Dc0,tαz(t)=f(t,z(t),z(t-τ1),…,z(t-τm))+I(t,z(t),z(t-ξ1),…,z(t-ξn))u(t)+w(t),t∈[0,T],0<α<1.Theorem 15.
Assume that(H1), (H2), and (H3) are satisfied; then (1) is Ulam-Hyers-Mittag-Leffler stable.Proof.
Letz∈X be a solution to (38) and x∈X be the unique solution of the following problem:(41)Dc0,tαxt=ft,xt,xt-τ1,…,xt-τm+It,xt,xt-ξ1,…,xt-ξnut,t∈0,T,xt=zt,t∈-a,0.Obviously, (42)xt=zt,t∈-a,0,z0+1Γα∫0tt-sα-1fs,xs,xs-τ1,…,xs-τm+Is,xs,xs-ξ1,…,xs-ξnusds,t∈0,T.
On the other hand, from [47, Remark  2] via Remark 14, we can know that z satisfied the following inequality:(43)zt-z0-1Γα∫0tt-sα-1fs,zs,zs-τ1,…,zs-τm+Is,zs,zs-ξ1,…,zs-ξnusds≤1Γα∫0tt-sα-1wsds≤ϵΓα∫0tt-sα-1Eαsαds≤ϵEαtα,t∈0,T.
Then, fort∈[0,T], according to (H2), we have(44)zt-xt≤zt-z0-1Γα∫0tt-sα-1fs,zs,zs-τ1,…,zs-τm+Is,zs,zs-ξ1,…,zs-ξnusds+1Γα∫0tt-sα-1fs,zs,zs-τ1,…,zs-τm+Is,zs,zs-ξ1,…,zs-ξnusds-∫0tt-sα-1fs,xs,xs-τ1,…,xs-τm+Is,xs,xs-ξ1,…,xs-ξnusds≤ϵEαtα+1Γα∫0tt-sα-1fs,zs,zs-τ1,…,zs-τm-fs,xs,xs-τ1,…,xs-τm+Is,zs,zs-ξ1,…,zs-ξn-Is,xs,xs-ξ1,…,xs-ξnusds≤ϵEαtα+LfΓα∫0tt-sα-1xs-zs+∑l=1mxs-τl-zs-τlds+LIbuΓα∫0tt-sα-1xs-zs+∑l=1nxs-ξl-zs-ξlds.
Note the factzt-xt=0,t∈[-a,0]. Consider the operator F:C([-a,T],R+)→C([-a,T],R+) defined by(45)Fx¯t=0,t∈-a,0,ϵEαtα+LfΓα∫0tt-sα-1x¯sds+∫0tt-sα-1∑l=1mx¯s-τlds+LIbuΓα∫0tt-sα-1x¯sds+∫0tt-sα-1∑l=1nx¯s-ξlds,t∈0,T,where x¯∈C([-a,T],R+).
Next, we verify thatF is a Picard operator. In fact, for all t∈[0,T] and arbitrary x¯,z¯∈C([-a,T],R+), it follows the proof in Theorem 9; one can show that F is a contraction via the (λ,α)-norm on C([-a,T],R+) due to (H3).
Applying the Banach contraction principle toF, we derive that F is a Picard operator and GF={x∗}. Then, we have x∗=0, for t∈[-a,0] and (46)x∗t=ϵEαtα+LfΓα∫0tt-sα-1x∗sds+∫0tt-sα-1∑l=1mx∗s-τlds+LIbuΓα∫0tt-sα-1x∗sds+∫0tt-sα-1∑l=1nx∗s-ξlds,t∈0,T.
We go on to verify that the solutionx∗ is increasing. Now, denote m1≔mins∈[0,T][x∗(s)+∑l=1mx∗(s-τl)]∈R+ and m2≔mins∈[0,T][x∗(s)+∑l=1nx∗(s-ξl)]∈R+.
Then, for0≤t1≤t2≤T, we have (47)x∗t2-x∗t1=ϵEαt2α-Eαt1α+LfΓα∫0t1t2-sα-1-t1-sα-1x∗s+∑l=1mx∗s-τlds+LIbuΓα∫0t1t2-sα-1-t1-sα-1x∗s+∑l=1nx∗s-ξlds+LfΓα∫t1t2t2-sα-1x∗s+∑l=1mx∗s-τlds+LIbuΓα∫t1t2t2-sα-1x∗s+∑l=1nx∗s-ξlds≥ϵEαt2α-Eαt1α+Lfm1Γα+LIbum2Γα∫0t1t2-sα-1-t1-sα-1ds+Lfm1Γα+LIbum2Γα∫t1t2t2-sα-1ds=ϵEαt2α-Eαt1α+Lfm1Γα+1+LIbum2Γα+1t2α-t1α>0.
So,x∗ is increasing. Thus, x∗(t-θ)≤x∗(t) due to t-θ≤t(θ∈{τ1,…,τm,ξ1,…,ξn}) and (48)x∗t≤ϵEαtα+Lf1+mΓα∫0tt-sα-1x∗sds+LIbun+1Γα∫0tt-sα-1x∗sds≤ϵEαtα+Lfm+1+LIbun+1Γα∫0tt-sα-1x∗sds.
Using Lemma6, Remark 7, and the fact of x∗=0, for t∈[-a,0], we obtain (49)x∗t≤ϵEαtαEαLfm+1+LIbun+1Tα≤cEαϵEαtα,t∈-a,T,where cEα=Eα((Lf(m+1)+LIbu(n+1))Tα).
In particular, ifx¯(t)=zt-xt, from (44), x¯≤Fx¯ and applying the Lemma 4, we obtain x¯≤xF∗, where F is a Picard and an increasing operator. As a result, we know (50)zt-xt≤cEαϵEαtα,t∈-a,T.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 5, (12)-(13)]. Next, note that Eα(z)≤1 if z≤0; 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 Eα(tα) is not necessarily less than 1 on the whole interval [-a,T].


## 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 (λ,α)-weighted norm ·λ,α.


### 4.1. Open-Loop Case

For system (2), consider the open-loop PID-type ILC updating law with initial state learning:(51)xk+10=xk0+ϱek0,uk+1t=ukt+η1ekt+η2e˙kt+η3∫0teksds,where ϱ and ηi(i=1,2,3) are unknown parameters to be determined.For the sake of brevity, the following notations will be used:(52)fkt=ft,xkt,xkt-τ1,…,xkt-τm,Ikt=It,xkt,xkt-ξ1,…,xkt-ξn,gkt=gt,xkt.And we denote that Δuk(t)=uk+1(t)-uk(t), Δxk(t)=xk+1(t)-xk(t), and ek(t)=yd(t)-yk(t) which is called tracking error at kth repetition, where yd(t) denotes desired output bounded trajectory satisfying (53)Dc0+αxdt=ft,xdt,xdt-τ1,…,xdt-τm+It,xdt,xdt-ξ1,…,xdt-ξnudt,t∈0,T,0<α<1,xdt=ψdt,t∈-a,0,a=max⁡τ1,…,τm,ξ1,…,ξn,ydt=gt,xdt+d∫0tudsds,t∈0,T,and ud is a desired control.In this section, we imposed the following assumptions on the class of system described by (2).(
H
4
) The function g is continuous and differentiable for all x and t with partial derivatives gx and gt. For the constant βj,j=1,2, set(54)0<β1≤gx·,x≔∂g·,x∂x≤β2.(
H
5
) The function I(t,x(t),x(t-ξ1),…,x(t-ξm)) is uniformly bounded on [0,T]; that is, there exists bI>0 such that It,xt,xt-ξ1,…,xt-ξm≤bI for any t∈[0,T].Theorem 17.
Assume that(H1)–(H5) hold. If(55)max⁡1-β1ϱ,1-β2ϱ≤r0<1,(56)r1≔1-dγ2+dγ1T+dγ3T22<1,then the nonlinear fractional multiple time-delays differential system (2) with the open-loop PID-type ILC updating law (51) guarantees that yk tends to yd as k→∞ in the sense of (λ,α)-norm for all t∈[0,T], where ψd(t) is the desired initial state on [-a,0] and yd(t) is the desired output trajectory.Proof.
Note that(57)ek+1t=ekt+gkt-gk+1t=ekt-gxμkt,tΔxkt-d∫0tΔuksds,where μk(t) lies in the segment with end point xk(t) and xk+1(t) for t∈[0,T].
In what follows, we show thatekλ,α→0 as k→∞ for a.e. t∈[0,T]. By using mean value theorem we have(58)ek+10=ek0+yk0-yk+10=ek0-gxμk0,0Δxk0.
Substituting (51) into (58) and taking the absolute value, we have (59)ek+10=ek0+yk0-yk+10=ek0-gxμk0,0Δxk0=ek0-gxμk0,0ϱek0≤1-ϱgxμk0,0ek0≤r0ek0.
It follows from (55) that(60)limk→∞⁡ek0λ,α=0.
Now we turn to give an estimation for the upper bound ofΔxk. One has(61)Δxkt=xk+1t-xkt=Δxk0+1Γα∫0tt-sα-1fk+1s+Ik+1suk+1s-fks+Iksuksds=Δxk0+1Γα∫0tt-sα-1fk+1s-fks+Ik+1s-Iksuk+1s+Iksuk+1s-uksds.
Repeating the same procedure in Theorem9, we can get (62)Δxkt=xk+1t-xkt≤Δxk0+1Γα∫0tt-sα-1fk+1s+Ik+1suk+1s-fks-Iksuksds≤Δxk0+1Γα∫0tt-sα-1fk+1s-fks+Ik+1s-Iksuk+1s+Iksuk+1s-uksds≤Δxk0+LfΓα∫0tt-sα-1xk+1s-xks+∑l=1mxk+1s-τl-xks-τlds+LIbuΓα∫0tt-sα-1xk+1s-xks+∑l=1nxk+1s-ξl-xks-ξlds+bIΓα∫0tt-sα-1Δuksds≤Δxk0+LfΓα∫0tt-sα-1Lfm+1+LIbun+1Δxksds+bIΓα∫0tt-sα-1Δuksds,where bI,bu are defined in (H6).
Using Lemma5, we get (63)Δxkt≤Δxk0+bIΓα∫0tt-sα-1Δuksdsec/Γα∫0tt-sα-1ds≤Δxk0+bIΓα∫0tt-sα-1ΔuksdsecTα/Γα+1,where c=Lf(m+1)+LIbu(n+1).
We know(64)ek+1t=ekt-yk+1t-ykt=ekt-gxμkt,tΔxkt-d∫0tΔuksds.
Taking (51) into (64), one obtains (65)ek+1t=ekt-yk+1t-ykt=ekt-gxμkt,tΔxkt-d∫0tΔuksds=ekt-gxμkt,tΔxkt-dη1∫0teksds+η2∫0te˙ksds+η3∫0t∫0sekτdτds=ekt-gxμkt,tΔxkt-dη1∫0teksds+η2ekt-ek0+η3∫0t∫τtekτdsdτ=1-dη2ekt-gxμkt,tΔxkt-dη1∫0teksds-η2ek0+η3∫0tt-seksds=1-dη2ekt+dη2ek0-gxμkt,tΔxkt-dη1∫0teksds-dη3∫0tt-seksds.
Combining with (64) and Lemma 8, we can get(66)ek+1t≤1-dη2ekt+dη2ek0+ξΔxkt+dη1∫0teksds+dη3∫0tt-seksds≤1-dη2ekt+dη2ek0+ξΔxk0+bIΓα∫0tt-sα-1ΔuksdsecTα/Γα+1+dη1∫0teksds+dη3∫0tt-seksds≤1-dη2ekt+dη2+ξϱecTα/Γα+1ek0+bIξecTα/Γα+1Γα·eλtαtααλ+1αΔukλ,α+dη1teλtαekλ,α+dη3·t22eλtαekλ,α,λ=0,1-dη2ekt+dη2+ξϱecTα/Γα+1ek0+bIξecTα/Γα+1Γα·eλtα1+tααλαΔukλ,α+dη1teλtαekλ,α+dη3·t22eλtαekλ,α,λ≥1,where ξ=max⁡{β1,β2}.
Multiplyinge-λtα on both sides of (66) and taking the maximum value on [0,T], we can get (67)ek+1λ,α≤1-dη2+dη1T+dη3·T22ekλ,α+dη2+ξϱecTα/Γα+1ek0λ,α+bIξecTα/Γα+1Γα·Tαα1+λαΔukλ,α,λ=0,1-dη2+dη1T+dη3·T22ekλ,α+dη2+ξϱecTα/Γα+1ek0λ,α+bIξecTα/Γα+1Γα·1+TααλαΔukλ,α,λ≥1.
Set(68)μ=max⁡bIξecTα/Γα+1Γα·Tαα1+λα,bIξecTα/Γα+1Γα·1+Tααλα.
There exists a sufficiently largeλ such that μ is very small and using (56) and (60) we can derive limk→∞ekλ,α=0. 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:(69)xk+10=xk0+ζek+10,uk+1t=ukt+κ1ek+1t+κ2e˙k+1t+κ3∫0tek+1sds,where ϱ and κi(i=1,2,3) are unknown parameters to be determined.Theorem 18.
Assume that(H1)–(H5) hold. If(70)min⁡1+β1ζ,1+β2ζ≥r¯0>1,(71)r2≔1+dκ2-dκ1-dκ3T22>1,then the nonlinear multiple time-delays system (2) with the close-loop PID-type ILC updating law (69) guarantees that yk tends to yd as k→∞ in the sense of λ,α-norm for all t∈[0,T], where ψd(t) is the desired initial state on [-a,0] and yd(t) is the desired output trajectory.Proof.
Note that(72)ek+1t=ekt+gkt-gk+1t=ekt-gxμkt,tΔxkt-d∫0tΔuksds,where μk(t) lies in the segment with end points xk(t) and xk+1(t) for t∈[0,T].
In what follows, we show thatekλ,α→0 as k→∞ for a.e. t∈[0,T]. By using mean value theorem we have(73)ek+10=ek0+yk0-yk+10=ek0-gxμk0,0Δxk0.
Substituting (69) into (73) and taking the absolute value, we have (74)1+gxμk0,0ζek+10=ek0.It follows from (70) that(75)limk→∞⁡ek0λ,α=0.
Similar to the proof of Theorem17, we can get(76)ek+1t=ekt-yk+1t-ykt=ekt-gxμkt,tΔxkt-d∫0tΔuksds=ekt-gxμkt,tΔxkt-dκ1∫0tek+1sds+κ2∫0te˙k+1sds+κ3∫0t∫0sek+1τdτds=ekt-gxμkt,tΔxkt-dκ1∫0tek+1sds+κ2ek+1t-ek+10+κ3∫0t∫τtekτdsdτ=ekt-dκ2ek+1t+dκ2ek+10-gxμkt,tΔxkt-dκ1∫0tek+1sds-dκ3∫0tt-sek+1sds.Then, one has(77)1+dκ2ek+1t≤ekt+dκ2ek+10+ξΔxkt+dκ1∫0tek+1sds+dκ3∫0tt-sek+1sds≤ekt+dκ2ek+10+ξΔxk0+bIΓα∫0tt-sα-1ΔuksdsecTα/Γα+1+dκ1∫0tek+1sds+dκ3∫0tt-sek+1sds≤ekt+dκ2+ξζecTα/Γα+1ek+10+ξ·ecTα/Γα+1bIΓα∫0tt-sα-1eλsαdsΔukλ,α+dκ1∫0teλsαdsek+1λ,α+dκ3∫0tt-seλsαdsek+1λ,α≤ekt+dκ2+ξζecTα/Γα+1ek+10+bIξecTα/Γα+1Γα·eλtαtαα1+λαΔukλ,α+dκ1teλtαek+1λ,α+dκ3·t22eλtαek+1λ,α,λ=0,ekt+dκ2+ξζecTα/Γα+1ek+10+bIξecTα/Γα+1Γα·eλtα1+tααλαΔukλ,α+dκ1teλtαek+1λ,α+dκ3·t22eλtαek+1λ,α,λ≥1,where ξ=max⁡{β1,β2}.
Multiplyinge-λtα on both sides of (77) and taking λ,α-norm, we can derive (78)1+dκ2ek+1λ,α≤ekλ,α+dκ2+ξζecTα/Γα+1ek+10λ,α+bIξecTα/Γα+1Γα·Tαα1+λαΔukλ,α+dκ1Tek+1λ,α+dκ3·T22ek+1λ,α,λ=0,ekλ,α+dκ2+ξζecTα/Γα+1ek+10λ,α+bIξecTα/Γα+1Γα·1+TααλαΔukλ,α+dκ1Tek+1λ,α+dκ3·T22ek+1λ,α,λ≥1.
Set(79)μ^=max⁡bIξecTα/Γα+1Γα·Tαα1+λα,bIξecTα/Γα+1Γα·1+Tααλα.
There exists a sufficiently largeλ such that μ^ is very small and using (71) and (75) we can derive limk→∞ekλ,α=0. This completes the proof of Theorem 18.


## 4.1. Open-Loop Case

For system (2), consider the open-loop PID-type ILC updating law with initial state learning:(51)xk+10=xk0+ϱek0,uk+1t=ukt+η1ekt+η2e˙kt+η3∫0teksds,where ϱ and ηi(i=1,2,3) are unknown parameters to be determined.For the sake of brevity, the following notations will be used:(52)fkt=ft,xkt,xkt-τ1,…,xkt-τm,Ikt=It,xkt,xkt-ξ1,…,xkt-ξn,gkt=gt,xkt.And we denote that Δuk(t)=uk+1(t)-uk(t), Δxk(t)=xk+1(t)-xk(t), and ek(t)=yd(t)-yk(t) which is called tracking error at kth repetition, where yd(t) denotes desired output bounded trajectory satisfying (53)Dc0+αxdt=ft,xdt,xdt-τ1,…,xdt-τm+It,xdt,xdt-ξ1,…,xdt-ξnudt,t∈0,T,0<α<1,xdt=ψdt,t∈-a,0,a=max⁡τ1,…,τm,ξ1,…,ξn,ydt=gt,xdt+d∫0tudsds,t∈0,T,and ud is a desired control.In this section, we imposed the following assumptions on the class of system described by (2).(
H
4
) The function g is continuous and differentiable for all x and t with partial derivatives gx and gt. For the constant βj,j=1,2, set(54)0<β1≤gx·,x≔∂g·,x∂x≤β2.(
H
5
) The function I(t,x(t),x(t-ξ1),…,x(t-ξm)) is uniformly bounded on [0,T]; that is, there exists bI>0 such that It,xt,xt-ξ1,…,xt-ξm≤bI for any t∈[0,T].Theorem 17.
Assume that(H1)–(H5) hold. If(55)max⁡1-β1ϱ,1-β2ϱ≤r0<1,(56)r1≔1-dγ2+dγ1T+dγ3T22<1,then the nonlinear fractional multiple time-delays differential system (2) with the open-loop PID-type ILC updating law (51) guarantees that yk tends to yd as k→∞ in the sense of (λ,α)-norm for all t∈[0,T], where ψd(t) is the desired initial state on [-a,0] and yd(t) is the desired output trajectory.Proof.
Note that(57)ek+1t=ekt+gkt-gk+1t=ekt-gxμkt,tΔxkt-d∫0tΔuksds,where μk(t) lies in the segment with end point xk(t) and xk+1(t) for t∈[0,T].
In what follows, we show thatekλ,α→0 as k→∞ for a.e. t∈[0,T]. By using mean value theorem we have(58)ek+10=ek0+yk0-yk+10=ek0-gxμk0,0Δxk0.
Substituting (51) into (58) and taking the absolute value, we have (59)ek+10=ek0+yk0-yk+10=ek0-gxμk0,0Δxk0=ek0-gxμk0,0ϱek0≤1-ϱgxμk0,0ek0≤r0ek0.
It follows from (55) that(60)limk→∞⁡ek0λ,α=0.
Now we turn to give an estimation for the upper bound ofΔxk. One has(61)Δxkt=xk+1t-xkt=Δxk0+1Γα∫0tt-sα-1fk+1s+Ik+1suk+1s-fks+Iksuksds=Δxk0+1Γα∫0tt-sα-1fk+1s-fks+Ik+1s-Iksuk+1s+Iksuk+1s-uksds.
Repeating the same procedure in Theorem9, we can get (62)Δxkt=xk+1t-xkt≤Δxk0+1Γα∫0tt-sα-1fk+1s+Ik+1suk+1s-fks-Iksuksds≤Δxk0+1Γα∫0tt-sα-1fk+1s-fks+Ik+1s-Iksuk+1s+Iksuk+1s-uksds≤Δxk0+LfΓα∫0tt-sα-1xk+1s-xks+∑l=1mxk+1s-τl-xks-τlds+LIbuΓα∫0tt-sα-1xk+1s-xks+∑l=1nxk+1s-ξl-xks-ξlds+bIΓα∫0tt-sα-1Δuksds≤Δxk0+LfΓα∫0tt-sα-1Lfm+1+LIbun+1Δxksds+bIΓα∫0tt-sα-1Δuksds,where bI,bu are defined in (H6).
Using Lemma5, we get (63)Δxkt≤Δxk0+bIΓα∫0tt-sα-1Δuksdsec/Γα∫0tt-sα-1ds≤Δxk0+bIΓα∫0tt-sα-1ΔuksdsecTα/Γα+1,where c=Lf(m+1)+LIbu(n+1).
We know(64)ek+1t=ekt-yk+1t-ykt=ekt-gxμkt,tΔxkt-d∫0tΔuksds.
Taking (51) into (64), one obtains (65)ek+1t=ekt-yk+1t-ykt=ekt-gxμkt,tΔxkt-d∫0tΔuksds=ekt-gxμkt,tΔxkt-dη1∫0teksds+η2∫0te˙ksds+η3∫0t∫0sekτdτds=ekt-gxμkt,tΔxkt-dη1∫0teksds+η2ekt-ek0+η3∫0t∫τtekτdsdτ=1-dη2ekt-gxμkt,tΔxkt-dη1∫0teksds-η2ek0+η3∫0tt-seksds=1-dη2ekt+dη2ek0-gxμkt,tΔxkt-dη1∫0teksds-dη3∫0tt-seksds.
Combining with (64) and Lemma 8, we can get(66)ek+1t≤1-dη2ekt+dη2ek0+ξΔxkt+dη1∫0teksds+dη3∫0tt-seksds≤1-dη2ekt+dη2ek0+ξΔxk0+bIΓα∫0tt-sα-1ΔuksdsecTα/Γα+1+dη1∫0teksds+dη3∫0tt-seksds≤1-dη2ekt+dη2+ξϱecTα/Γα+1ek0+bIξecTα/Γα+1Γα·eλtαtααλ+1αΔukλ,α+dη1teλtαekλ,α+dη3·t22eλtαekλ,α,λ=0,1-dη2ekt+dη2+ξϱecTα/Γα+1ek0+bIξecTα/Γα+1Γα·eλtα1+tααλαΔukλ,α+dη1teλtαekλ,α+dη3·t22eλtαekλ,α,λ≥1,where ξ=max⁡{β1,β2}.
Multiplyinge-λtα on both sides of (66) and taking the maximum value on [0,T], we can get (67)ek+1λ,α≤1-dη2+dη1T+dη3·T22ekλ,α+dη2+ξϱecTα/Γα+1ek0λ,α+bIξecTα/Γα+1Γα·Tαα1+λαΔukλ,α,λ=0,1-dη2+dη1T+dη3·T22ekλ,α+dη2+ξϱecTα/Γα+1ek0λ,α+bIξecTα/Γα+1Γα·1+TααλαΔukλ,α,λ≥1.
Set(68)μ=max⁡bIξecTα/Γα+1Γα·Tαα1+λα,bIξecTα/Γα+1Γα·1+Tααλα.
There exists a sufficiently largeλ such that μ is very small and using (56) and (60) we can derive limk→∞ekλ,α=0. 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:(69)xk+10=xk0+ζek+10,uk+1t=ukt+κ1ek+1t+κ2e˙k+1t+κ3∫0tek+1sds,where ϱ and κi(i=1,2,3) are unknown parameters to be determined.Theorem 18.
Assume that(H1)–(H5) hold. If(70)min⁡1+β1ζ,1+β2ζ≥r¯0>1,(71)r2≔1+dκ2-dκ1-dκ3T22>1,then the nonlinear multiple time-delays system (2) with the close-loop PID-type ILC updating law (69) guarantees that yk tends to yd as k→∞ in the sense of λ,α-norm for all t∈[0,T], where ψd(t) is the desired initial state on [-a,0] and yd(t) is the desired output trajectory.Proof.
Note that(72)ek+1t=ekt+gkt-gk+1t=ekt-gxμkt,tΔxkt-d∫0tΔuksds,where μk(t) lies in the segment with end points xk(t) and xk+1(t) for t∈[0,T].
In what follows, we show thatekλ,α→0 as k→∞ for a.e. t∈[0,T]. By using mean value theorem we have(73)ek+10=ek0+yk0-yk+10=ek0-gxμk0,0Δxk0.
Substituting (69) into (73) and taking the absolute value, we have (74)1+gxμk0,0ζek+10=ek0.It follows from (70) that(75)limk→∞⁡ek0λ,α=0.
Similar to the proof of Theorem17, we can get(76)ek+1t=ekt-yk+1t-ykt=ekt-gxμkt,tΔxkt-d∫0tΔuksds=ekt-gxμkt,tΔxkt-dκ1∫0tek+1sds+κ2∫0te˙k+1sds+κ3∫0t∫0sek+1τdτds=ekt-gxμkt,tΔxkt-dκ1∫0tek+1sds+κ2ek+1t-ek+10+κ3∫0t∫τtekτdsdτ=ekt-dκ2ek+1t+dκ2ek+10-gxμkt,tΔxkt-dκ1∫0tek+1sds-dκ3∫0tt-sek+1sds.Then, one has(77)1+dκ2ek+1t≤ekt+dκ2ek+10+ξΔxkt+dκ1∫0tek+1sds+dκ3∫0tt-sek+1sds≤ekt+dκ2ek+10+ξΔxk0+bIΓα∫0tt-sα-1ΔuksdsecTα/Γα+1+dκ1∫0tek+1sds+dκ3∫0tt-sek+1sds≤ekt+dκ2+ξζecTα/Γα+1ek+10+ξ·ecTα/Γα+1bIΓα∫0tt-sα-1eλsαdsΔukλ,α+dκ1∫0teλsαdsek+1λ,α+dκ3∫0tt-seλsαdsek+1λ,α≤ekt+dκ2+ξζecTα/Γα+1ek+10+bIξecTα/Γα+1Γα·eλtαtαα1+λαΔukλ,α+dκ1teλtαek+1λ,α+dκ3·t22eλtαek+1λ,α,λ=0,ekt+dκ2+ξζecTα/Γα+1ek+10+bIξecTα/Γα+1Γα·eλtα1+tααλαΔukλ,α+dκ1teλtαek+1λ,α+dκ3·t22eλtαek+1λ,α,λ≥1,where ξ=max⁡{β1,β2}.
Multiplyinge-λtα on both sides of (77) and taking λ,α-norm, we can derive (78)1+dκ2ek+1λ,α≤ekλ,α+dκ2+ξζecTα/Γα+1ek+10λ,α+bIξecTα/Γα+1Γα·Tαα1+λαΔukλ,α+dκ1Tek+1λ,α+dκ3·T22ek+1λ,α,λ=0,ekλ,α+dκ2+ξζecTα/Γα+1ek+10λ,α+bIξecTα/Γα+1Γα·1+TααλαΔukλ,α+dκ1Tek+1λ,α+dκ3·T22ek+1λ,α,λ≥1.
Set(79)μ^=max⁡bIξecTα/Γα+1Γα·Tαα1+λα,bIξecTα/Γα+1Γα·1+Tααλα.
There exists a sufficiently largeλ such that μ^ is very small and using (71) and (75) we can derive limk→∞ekλ,α=0. This completes the proof of Theorem 18.


## 5. Examples

In this section, we give two examples to illustrate our results above.Example 1.
Letα=1/2, Lf=1/8, LI=1/12, m=2, n=1, and T=0.2. We consider the following nonlinear fractional multiple time-delays differential equations(80)Dc0+1/2xt=18sin⁡5xt+18x2t-11+x2t-1+18cos⁡xt-12+112sin⁡3xt-13ut,t∈0,0.2,x0=1,t∈-1,0and the inequalities(81)Dct1/2zt-ft,ztzt-1,zt-12+It,zt-13ut≤ϵ,Dct1/2zt-ft,ztzt-1,zt-12+It,zt-13ut≤ϵE1/2t1/2.Definef(t,x(t),x(t-1),x(t-1/2))=1/8sin⁡5xt+(1/8)(x2t-1/(1+x2t-1))+1/8cos⁡(x(t-1/2)) and I(t,x(t-1/3))=1/12sin⁡(3x(t-1/3)) and set bu=1,λ=4.2. Thus c=0.5417; then c(1+Tα)/Γ(α+1)[λ]α=0.4423<1. 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 zt-xt≤cϵ,t∈[-1,0.2] and Ulam-Hyers-Mittag-Leffler stable with (82)zt-xt≤cE1/2ϵE1/2t1/2,t∈-1,0.2,where c=26.9701,cE1/2=1.3447.Example 2.
Letα=1/2, Lf=1/8, LI=1/12, m=2, n=1, and T=0.2. We consider the following nonlinear fractional multiple time-delays differential equations(83)Dc0+1/2xkt=18sin⁡5xkt+18xk2t-11+xk2t-1+18cos⁡xkt-12+112sin⁡3xkt-13ukt,t∈0,0.2,x0=1,t∈-1,0,ykt=2xkt+32sin⁡xkt+2∫0tuksds,t∈0,0.2and the desired reference trajectory yd(t)=10sin⁡(10πt)(1+cos⁡(10πt))+2,t∈[0,0.2].
Consider problem (83) and the open-loop PID-type ILC updating law with initial state learning: (84)xk+10=xk0+15ek0,uk+1t=ukt+110ekt+12e˙kt+15∫0teksds.Obviously,r0=0.9<1 and r1=0.048<1. 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: (85)xk+10=xk0+32ek+10,uk+1t=ukt+32ek+1t+52e˙k+1t+6∫0tek+1sds.Thus r¯0=1.75>1 and r2=2.76>1. All the conditions of Theorem 18 are satisfied.Numerical simulation diagram for the open-loop control is shown in Figure1(a).Figure 1
The system outputyk(t) (blue line), the desired trajectory yd(t) (red line), and the tracking error for (83).
(a)
(b)Figure1(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].



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*Source: 9050289-2017-10-15.xml*