# Solution of Some Types for Composition Fractional Order Differential Equations Corresponding to Optimal Control Problems

Authors: Sameer Qasim Hasan; Moataz Abbas Holel
Journal: Journal of Control Science and Engineering
 (2018)

---

## Abstract

The approximate solution for solving a class of composition fractional order optimal control problems (FOCPs) is suggested and studied in detail. However, the properties of Caputo and Riemann-Liouville derivatives are also given with complete details on Chebyshev approximation function to approximate the solution of fractional differential equation with different approach. Also, the relation between Caputo and Riemann-Liouville of fractional derivative took a big role for simplifying the fractional differential equation that represents the constraints of optimal control problems. The approximate solutions are defined on interval[0,1] and are compared with the exact solution of order one which is an important condition to support the working method. Finally, illustrative examples are included to confirm the efficiency and accuracy of the proposed method.


---

## Body

## 1. Introduction

The idea of fractional derivative dates back to a conversation between two mathematicians: Leibniz and L’Hopital. In 1695, they exchanged about the meaning of a derivative of order 1/2. Their correspondence has been well documented and is stated as the foundation of fractional calculus [1].A fractional optimal control problem (FOCPs) is an optimal control problem focused on the performance index or the fractional differential equations governing the dynamics of the system or both contain at least one fractional order derivative term.The formulation and solution of state and variables FOCPs were first established by Agrawal, where the applied fractional variational calculus (FVC) presented a general formulation and solution scheme for FOCPs in the Riemann-Liouville (RL) sense; it was based on variational virtual work coupled with the Lagrange multiplier technique. Since the Caputo fractional derivatives (CFDs) seems more natural and allows incorporating the usual initial conditions, it becomes a popular choice for researchers [2, 3].The Chebyshev polynomials are used to solve composition FOCPs and Chebyshev polynomialsTnt of degree n are important in approximation theory, since the roots of the Chebyshev polynomials of the first kind, which are also called Chebyshev nodes, are used as nodes in polynomial interpolation.Agrawal and Baleanu [4] obtained necessary conditions for FOCPs with Riemann-Liouville derivative and then were able to solve the problem numerically.Elbarbary introduced Chebyshev finite difference approximation for the boundary value problems of integer derivatives [5]. In [6], Khader and Hendy studied an efficient numerical scheme for solving fractional optimal control problems. In [7], Akbarian and Keyanpour studied a new approach to the numerical solution of fractional order optimal control Problems.This paper presents a new approach for computing the approximate optimal control function for optimal control problems with some types of fractional order differential equation such as multi-fractional order differential equation, and composite fractional order differential equations are investigated with first kind Chebyshev approximate polynomial which works very efficiently with fractional orders and optimal control problems. All the discussion and details in addition to the transformations of the steps are given to compute all the useful of the new work. The obtained solutions of the method show that the technique of the approach is very convenient and efficient, and many calculations give high accuracy and may lead to near to exact solution by changing up the order of fractional positive integer numbers.The multi-fractional differential equations corresponding to optimal control problems and basic theorems have been given with algorithm for multi-composite fractional order optimal control problems. Also, we give illustrative examples for the solution of the approximate systems.The value ofxt is observed in detail to explain the activity of solution approximation with mixed boundary conditions. It is important to notice that the calculations are written by using the mathematical software MATHCAD by version (14.0.3.332).This paper consists of the following sections: in Section2, some basic definitions and properties of fractional order calculus are introduced (R-L and Caputo fractional derivatives). In Section 3, the shifted Chebyshev polynomials and numerical approximations of CFD and RLFD using Chebyshev polynomials are introduced. In Section 4, we derive the necessary optimality conditions of composition order fractional optimal control problems. In Section 5, we give numerical examples to solve composition FOCPs and show the accuracy of the presented method. Finally, conclusions are presented in Section 6.


## 2. Preliminaries

In this section, we give some definitions and properties of fractional derivatives that will be needed later on.


### 2.1. Fractional Order Calculus

Definition 1 (see [8]).
The left (LRLFD) and right (RRLFD) Riemann-Liouville fractional derivatives of a functionft are defined as(1)Dtαaft=1Гn-αdndtn∫att-τn-α-1fτdτ,t>a(2)Dbαtft=1Гn-α-1ndndtn∫tbτ-tn-α-1fτdτ,t<b.where the order of the derivative α satisfies n-1<α≤n, and Гn is the gamma function.Definition 2 (see [8]).
The left (LCFD) and right (RCFD) Caputo fractional derivatives of a functionft are defined as (3)DtαaCft=1Гn-α∫att-τn-α-1fnτdτ.α>0,t>0(4)DbαtCft=-1nГn-α∫tbτ-tn-α-1fnτdτ,where n-1<α≤n,n∈N.


### 2.2. Some Properties of the Fractional Calculus Are Presented in Detail Which Will Be Needed Later on

(i) The fractional operators are linear [9]:(5)Pλft+μgt=λPft+μPgt,where P is one of Dtα,aDbαt,DtαaC,DbαtC,Dt-αa or Db-αt and λ, μ are real numbers.(ii) Let0<α<1,n-1<α+β<n,n-1<β<n,n is a positive integer, f∈Cna,t, and if x(i)0=0, i=0,1,……n-1.) [9], then(6)DtαaCDtβaCxt=DtβaCDtαaCxt=Dtα+βaCxt.(iii) If0<α<1, we obtain the following [2]: (7)∫abgt·DtαaCftdt=∫abft·Dbαtgtdt+tIb1-αgt·ftab∫abgt·DbαtCftdt=∫abft·Dtαagtdt+aIt1-αgt·ftab.Moreover, iff is a function such that fa=fb=0, we have simpler formulas:(8)∫abgt·DtαaCftdt=∫abft·Dbαtgtdt,(9)∫abgt·DbαtCftdt=∫abft·aDtαgtdt.(iv) The relation between the RLFD and the CFD is as follows [2].Forα>0 and n=α+1, the Riemann-Liouville and Caputo fractional derivatives are related by the following formulas:(10)Dtαaft=DtαaCft+∑k=0n-1fkaГk+1-αt-ak-α,(11)Dbαtft=DbαtCft+∑k=0n-1fkbГk+1-αb-tk-α.(v) The constant function and power function of Caputo’s derivative are as follows:(12)1DtαaCC=0,whereCisaconstant.In particular,DtαaC1=0, and, also from right integral, DbαtC1=0. (13)2DtαaCt-aβ=0,forβ∈N0andβ<α,Гβ+1Гβ-α+1t-aβ-α,forβ∈N0andβ≥α,where α denotes the smallest integer greater than or equal to α and N0={0,1,2,...}. Recall that for α∈N, [10].


## 2.1. Fractional Order Calculus

Definition 1 (see [8]).
The left (LRLFD) and right (RRLFD) Riemann-Liouville fractional derivatives of a functionft are defined as(1)Dtαaft=1Гn-αdndtn∫att-τn-α-1fτdτ,t>a(2)Dbαtft=1Гn-α-1ndndtn∫tbτ-tn-α-1fτdτ,t<b.where the order of the derivative α satisfies n-1<α≤n, and Гn is the gamma function.Definition 2 (see [8]).
The left (LCFD) and right (RCFD) Caputo fractional derivatives of a functionft are defined as (3)DtαaCft=1Гn-α∫att-τn-α-1fnτdτ.α>0,t>0(4)DbαtCft=-1nГn-α∫tbτ-tn-α-1fnτdτ,where n-1<α≤n,n∈N.


## 2.2. Some Properties of the Fractional Calculus Are Presented in Detail Which Will Be Needed Later on

(i) The fractional operators are linear [9]:(5)Pλft+μgt=λPft+μPgt,where P is one of Dtα,aDbαt,DtαaC,DbαtC,Dt-αa or Db-αt and λ, μ are real numbers.(ii) Let0<α<1,n-1<α+β<n,n-1<β<n,n is a positive integer, f∈Cna,t, and if x(i)0=0, i=0,1,……n-1.) [9], then(6)DtαaCDtβaCxt=DtβaCDtαaCxt=Dtα+βaCxt.(iii) If0<α<1, we obtain the following [2]: (7)∫abgt·DtαaCftdt=∫abft·Dbαtgtdt+tIb1-αgt·ftab∫abgt·DbαtCftdt=∫abft·Dtαagtdt+aIt1-αgt·ftab.Moreover, iff is a function such that fa=fb=0, we have simpler formulas:(8)∫abgt·DtαaCftdt=∫abft·Dbαtgtdt,(9)∫abgt·DbαtCftdt=∫abft·aDtαgtdt.(iv) The relation between the RLFD and the CFD is as follows [2].Forα>0 and n=α+1, the Riemann-Liouville and Caputo fractional derivatives are related by the following formulas:(10)Dtαaft=DtαaCft+∑k=0n-1fkaГk+1-αt-ak-α,(11)Dbαtft=DbαtCft+∑k=0n-1fkbГk+1-αb-tk-α.(v) The constant function and power function of Caputo’s derivative are as follows:(12)1DtαaCC=0,whereCisaconstant.In particular,DtαaC1=0, and, also from right integral, DbαtC1=0. (13)2DtαaCt-aβ=0,forβ∈N0andβ<α,Гβ+1Гβ-α+1t-aβ-α,forβ∈N0andβ≥α,where α denotes the smallest integer greater than or equal to α and N0={0,1,2,...}. Recall that for α∈N, [10].


## 3. The Shifted Chebyshev Polynomials

The well-known Chebyshev polynomials are defined on the interval-1,1 and can be determined by the following recurrence formula [11]:(14)Tnt=2tTn-1t-Tn-2t,T0t=1,T1t=t,n=2,3….The Chebyshev polynomials can be expanded in power series as follows: (15)Tnt=n2∑m=0n/2-1mn-m-1!m!n-2m!2tn-2m,n=0,1,2,…, m=0,1,2,…,n/2 and n/2 denotes the biggest integral less than or equal to n/2.The Chebyshev polynomialsTnt(n=0,1,2,…) are orthogonal under integration over -1,1 with the weighting function wt=1/1-t2, with orthogonal condition:(16)∫-11TitTjt1-t2dt=0ifi≠j,π2ifi=j≠0,πifi=j=0.where i,j=0,1,2….In order to use these polynomials on the interval0,L we define the so-called shifted pseudo-spectral Chebyshev polynomials by introducing the change of variable z=2t/L-1.The shifted Chebyshev polynomials are defined as follows [12]:(17)Tnpt=Tn2tL-1,whereT0pt=1,T1pt=2tL-1.The analytic form of the shifted pseudo-spectral Chebyshev polynomial Tnpt of degreenis given by the following [10]:(18)Tnpt=n∑k=0n-1n-k22kn+k-1!2k!n-k!Lktk,n=1,2…,where Tnp0=-1n, and TnpL=1.The orthogonality relation is as follows:(19)∫0LTnptTmptwptdt=δm,with the weight function wpt=1/Lt-t2,δm=bk/2π,b0=2,bk=1 for k≥1.The functionxt∈L20,L can be expressed in terms of shifted Chebyshev polynomials as follows:(20)xt=∑j=0∞cnTnpt,where the coefficients cn are given by (21)cn=1δn∫0LxtTnptwptdt,n=0,1,….


### 3.1. The Chebyshev-Gauss-Lobatto Points

We choose the Chebyshev-Gauss-Lobatto points associated with the interval0,L, as follows:(22)tr=L2-L2cos⁡πrN,r=0,1,…N.These grids can be written as L<xN<xN-1…<x1<x0=0.Clenshaw and Curtis [13] introduced an approximation of the function xt, as follows:(23)xNt=∑n=0N′′anTnptan=2N∑r=0N′′xtrTnptr.where (′′) on the summation means that the first and last terms are to be taken with a factor 1/2.Theorem 3 (see [2]).
The fractional derivative of orderα in the Caputo sense for the function xt at the point ts is given by(24)Dtα0CxNts=∑r=0Nds,rαxtr,α>0,Such that (25)ds,rα=4θrN∑n=αN∑j=0N∑k=αnnθnbj-1n-kn+k-1!Гk-α+1/2TnptrTjptsLαГk+1/2n-k!Гk-α-j+1Гk-α+j+1,where (26)s,r=0,1,2,…,Nandθ0=θN=12,θi=1∀i=1,2,…,N-1.Proof.
The fractional derivative of the approximate formula for the functionxt in (22) is given by(27)Dtα0CxNt=∑n=0N′′anDtα0CTnptDtα0CxNt=∑n=0N′′anDtα0Cn∑k=0n-1n-k22kn+k-1!2k!n-k!LktkUsing (12) and (13), in (18) we have(28)Dtα0CTnpt=0,n=0,1,…,α-1,then (29)Dtα0CxNt=∑n=αN′′anDtα0CTnpt,Therefore, for n=α,α+1,…,N and by (12) and (13), in for formula (18), we get(30)Dtα0CTnpt=n∑k=0n-1n-k22kn+k-1!2k!n-k!LkDtα0Ctk=n∑k=αn-1n-k22kn+k-1!k!2k!n-k!LkГk-α+1tk-α.Now, tk-α can be expressed approximately in terms of shifted pseudo-spectral Chebyshev polynomial series, so we have(31)tk-α≅∑j=0NakjTjpt,where akj is obtained from (21) with xt=tk-α. If only the first (N+1)-terms from pseudo-spectral shifted Chebyshev polynomials in (23) are considered, the approximate formula for the fractional derivative of the shifted pseudo-spectral Chebyshev polynomials is as follows:(32)Dtα0CTnpt=∑j=0N∑k=αn-1n-k2nn+k-1!Гk-α+1/2TjptbjLαГk+1/2n-k!Гk-α-j+1Гk-α+j+1.From (29) and (32), we have(33)Dtα0CxNt=4N∑n=αN′′∑r=0N′′∑j=0N∑k=αn-1n-knn+k-1!Гk-α+1/2xtrTnptrTjptbjLαГk+1/2n-k!Гk-α-j+1Гk-α+j+1.From (33), the fractional derivative of order α for the function xt at the point ts leads to the desired result.Theorem 4 (see [2]).
LetDtα0CxNt be the approximation of the fractional derivative Dtα0C of the function x(t) as given by (24). Then it holds(34)Dtα0Cxt-Dtα0CxNt2≤∑n=0N′′anΩnFtk-α;T0p,…,TNpFT0p,…,TNp1/2,where (35)Ωn=∑k=αn-1n-k2nn+k-1!Гk-α+1/2bjLαГk+1/2n-k!Гk-α-j+1Гk-α+j+1,(36)Fx;y1,y2,…yn=x,xx,y1⋯x,yny1,xy1,y1⋯y1,yn⋮⋮⋱⋮yn;xyn;y1⋯yn;yn.


### 3.2. Approximation of the Right Riemann-Liouville Fractional Derivative

Using the right Caputo fractional derivative (RCFD) of a functionft is defined in (4) when 0<α<1 and ft∈AСa,b,(37)Dbαscfs=-1Г1-α∫sbt-s-αf′tdt,From relation between the RLFD and the CFD (11), when 0<α<1,(thatisn=1), we have(38)sDbαfs=fbГ1-αb-s-α+Dbαscfs,Use (37) in (38) to obtain(39)sDbαfs=fbГ1-αb-s-α+1Г1-α∫sbt-s-αf′tdtLet f be a sufficiently smooth function in 0,b and let J(s;f) be defined as follows: (40)Js;f=∫sbt-s-αf′tdt,0<s<b.Substituting (40) in (39), we deduce (41)sDbαfs=fbГ1-αb-s-α+Js;fГ1-α.Now, we approximate ft,0≤t≤b, by a sum of shifted Chebyshev polynomials Tk2t/b-1 according to (42)ft≈pNt=∑k=0N′′akTk2tb-1,ak=2N∑j=0N′′ftjTk2tjb-1, where (43)tj=b2-b2cos⁡πjN,j=0,…,N,andobtainJs;f≈Js;pN=∫sbpN′tt-s-αdt.Lemma 5 (see [7]).
LetpN be the polynomial of degree N as given by (42); then there exists a polynomial FN-1 of degree N-1 such that (44)∫sxpN′t-pN′st-s-αdt=FN-1x-FN-1sx-s1-α.Proof.
LetpN′t-pN′s be expanded in a Taylor series at t=s(45)pN′t-pN′s=∑k=1N-1Akst-sk.Then, (46)∫sxpN′t-pN′st-s-αdt=∑k=1N-1Aks∫sxt-sk-αdt=t-s1-α∑k=1N-1Akst-skk-α+1sx.The assertion follows, if we choose FN-1x=∑k=0N-1Aksx-sk/k-α+1, with an arbitrary constant A0s. From (44), we have (47)Js;pN=∫sbpN′tt-s-αdt=pN′s1-α+FN-1b-FN-1sb-s1-αMoreover, sDbαfs can be approximated by means of (48)sDbαfs≈fbГ1-αb-s-α+Js;pNГ1-α.We express FN-1t in (47) by a sum of Chebyshev polynomials and provide the recurrence relation satisfied by the Chebyshev coefficients. Differentiating both sides of (44) with respect to x yields (49)pN′x-pN′sx-s-α=FN-1′xx-s1-α+FN-1x-FN-1s1-αx-s-α.(50)pN′x-pN′s=FN-1′xx-s+FN-1x-FN-1s1-α.To evaluate FN-1s in (47), we expand FN-1′x in terms of the shifted Chebyshev polynomials: (51)FN-1′x=∑k=0N-2′bkTk2xb-1,0≤x≤b,where (′) on the summation means that the first term is to be taken with a factor 1/2.(52)FN-1x-FN-1s=b4∑k=1N-1bk-1-bk+1kTk2xb-1-Tk2sb-1,where bN-1=bN=0. On the other hand, we have (53)x-sFN-1′x=b2FN-1′x2xb-1-2sb-1.By using the relation Tk+1v+Tk-1v=2vTkv and from (51), it follows that (54)x-sFN-1′x=b4∑k=1N-1′bk+1-22sb-1bk+bk-1Tk2xb-1,Such that b-1=b1.
Let(55)pN′x=∑k=0N-1′ckTk2xb-1.Inserting FN-1x-FN-1s and x-sFN-1′x as given (52) and (54) into (50) and taking (55) into account, we get (56)1-1-αkbk+1-22sb-1bk+1+1-αkbk-1=4bck,1≤k.The Chebyshev coefficients ck of pN′x as given by (55) can be evaluated by integrating and comparing it with (42):(57)ck-1=ck+1+4kbak,k=N,N-1,…,1,with starting values cN=cN+1=0, where ak are the Chebyshev coefficients of pNx.


## 3.1. The Chebyshev-Gauss-Lobatto Points

We choose the Chebyshev-Gauss-Lobatto points associated with the interval0,L, as follows:(22)tr=L2-L2cos⁡πrN,r=0,1,…N.These grids can be written as L<xN<xN-1…<x1<x0=0.Clenshaw and Curtis [13] introduced an approximation of the function xt, as follows:(23)xNt=∑n=0N′′anTnptan=2N∑r=0N′′xtrTnptr.where (′′) on the summation means that the first and last terms are to be taken with a factor 1/2.Theorem 3 (see [2]).
The fractional derivative of orderα in the Caputo sense for the function xt at the point ts is given by(24)Dtα0CxNts=∑r=0Nds,rαxtr,α>0,Such that (25)ds,rα=4θrN∑n=αN∑j=0N∑k=αnnθnbj-1n-kn+k-1!Гk-α+1/2TnptrTjptsLαГk+1/2n-k!Гk-α-j+1Гk-α+j+1,where (26)s,r=0,1,2,…,Nandθ0=θN=12,θi=1∀i=1,2,…,N-1.Proof.
The fractional derivative of the approximate formula for the functionxt in (22) is given by(27)Dtα0CxNt=∑n=0N′′anDtα0CTnptDtα0CxNt=∑n=0N′′anDtα0Cn∑k=0n-1n-k22kn+k-1!2k!n-k!LktkUsing (12) and (13), in (18) we have(28)Dtα0CTnpt=0,n=0,1,…,α-1,then (29)Dtα0CxNt=∑n=αN′′anDtα0CTnpt,Therefore, for n=α,α+1,…,N and by (12) and (13), in for formula (18), we get(30)Dtα0CTnpt=n∑k=0n-1n-k22kn+k-1!2k!n-k!LkDtα0Ctk=n∑k=αn-1n-k22kn+k-1!k!2k!n-k!LkГk-α+1tk-α.Now, tk-α can be expressed approximately in terms of shifted pseudo-spectral Chebyshev polynomial series, so we have(31)tk-α≅∑j=0NakjTjpt,where akj is obtained from (21) with xt=tk-α. If only the first (N+1)-terms from pseudo-spectral shifted Chebyshev polynomials in (23) are considered, the approximate formula for the fractional derivative of the shifted pseudo-spectral Chebyshev polynomials is as follows:(32)Dtα0CTnpt=∑j=0N∑k=αn-1n-k2nn+k-1!Гk-α+1/2TjptbjLαГk+1/2n-k!Гk-α-j+1Гk-α+j+1.From (29) and (32), we have(33)Dtα0CxNt=4N∑n=αN′′∑r=0N′′∑j=0N∑k=αn-1n-knn+k-1!Гk-α+1/2xtrTnptrTjptbjLαГk+1/2n-k!Гk-α-j+1Гk-α+j+1.From (33), the fractional derivative of order α for the function xt at the point ts leads to the desired result.Theorem 4 (see [2]).
LetDtα0CxNt be the approximation of the fractional derivative Dtα0C of the function x(t) as given by (24). Then it holds(34)Dtα0Cxt-Dtα0CxNt2≤∑n=0N′′anΩnFtk-α;T0p,…,TNpFT0p,…,TNp1/2,where (35)Ωn=∑k=αn-1n-k2nn+k-1!Гk-α+1/2bjLαГk+1/2n-k!Гk-α-j+1Гk-α+j+1,(36)Fx;y1,y2,…yn=x,xx,y1⋯x,yny1,xy1,y1⋯y1,yn⋮⋮⋱⋮yn;xyn;y1⋯yn;yn.


## 3.2. Approximation of the Right Riemann-Liouville Fractional Derivative

Using the right Caputo fractional derivative (RCFD) of a functionft is defined in (4) when 0<α<1 and ft∈AСa,b,(37)Dbαscfs=-1Г1-α∫sbt-s-αf′tdt,From relation between the RLFD and the CFD (11), when 0<α<1,(thatisn=1), we have(38)sDbαfs=fbГ1-αb-s-α+Dbαscfs,Use (37) in (38) to obtain(39)sDbαfs=fbГ1-αb-s-α+1Г1-α∫sbt-s-αf′tdtLet f be a sufficiently smooth function in 0,b and let J(s;f) be defined as follows: (40)Js;f=∫sbt-s-αf′tdt,0<s<b.Substituting (40) in (39), we deduce (41)sDbαfs=fbГ1-αb-s-α+Js;fГ1-α.Now, we approximate ft,0≤t≤b, by a sum of shifted Chebyshev polynomials Tk2t/b-1 according to (42)ft≈pNt=∑k=0N′′akTk2tb-1,ak=2N∑j=0N′′ftjTk2tjb-1, where (43)tj=b2-b2cos⁡πjN,j=0,…,N,andobtainJs;f≈Js;pN=∫sbpN′tt-s-αdt.Lemma 5 (see [7]).
LetpN be the polynomial of degree N as given by (42); then there exists a polynomial FN-1 of degree N-1 such that (44)∫sxpN′t-pN′st-s-αdt=FN-1x-FN-1sx-s1-α.Proof.
LetpN′t-pN′s be expanded in a Taylor series at t=s(45)pN′t-pN′s=∑k=1N-1Akst-sk.Then, (46)∫sxpN′t-pN′st-s-αdt=∑k=1N-1Aks∫sxt-sk-αdt=t-s1-α∑k=1N-1Akst-skk-α+1sx.The assertion follows, if we choose FN-1x=∑k=0N-1Aksx-sk/k-α+1, with an arbitrary constant A0s. From (44), we have (47)Js;pN=∫sbpN′tt-s-αdt=pN′s1-α+FN-1b-FN-1sb-s1-αMoreover, sDbαfs can be approximated by means of (48)sDbαfs≈fbГ1-αb-s-α+Js;pNГ1-α.We express FN-1t in (47) by a sum of Chebyshev polynomials and provide the recurrence relation satisfied by the Chebyshev coefficients. Differentiating both sides of (44) with respect to x yields (49)pN′x-pN′sx-s-α=FN-1′xx-s1-α+FN-1x-FN-1s1-αx-s-α.(50)pN′x-pN′s=FN-1′xx-s+FN-1x-FN-1s1-α.To evaluate FN-1s in (47), we expand FN-1′x in terms of the shifted Chebyshev polynomials: (51)FN-1′x=∑k=0N-2′bkTk2xb-1,0≤x≤b,where (′) on the summation means that the first term is to be taken with a factor 1/2.(52)FN-1x-FN-1s=b4∑k=1N-1bk-1-bk+1kTk2xb-1-Tk2sb-1,where bN-1=bN=0. On the other hand, we have (53)x-sFN-1′x=b2FN-1′x2xb-1-2sb-1.By using the relation Tk+1v+Tk-1v=2vTkv and from (51), it follows that (54)x-sFN-1′x=b4∑k=1N-1′bk+1-22sb-1bk+bk-1Tk2xb-1,Such that b-1=b1.
Let(55)pN′x=∑k=0N-1′ckTk2xb-1.Inserting FN-1x-FN-1s and x-sFN-1′x as given (52) and (54) into (50) and taking (55) into account, we get (56)1-1-αkbk+1-22sb-1bk+1+1-αkbk-1=4bck,1≤k.The Chebyshev coefficients ck of pN′x as given by (55) can be evaluated by integrating and comparing it with (42):(57)ck-1=ck+1+4kbak,k=N,N-1,…,1,with starting values cN=cN+1=0, where ak are the Chebyshev coefficients of pNx.


## 4. The Necessary Optimality Conditions of Composition Order Fractional Optimal Control ProblemsMDtαaCt+NDtβaCDtμaCxt=ft,xt,ut

In this problem, the constraint has multiorder and composition for the three fractional derivativesα,β, and μ, where α,β and μ>0. The multi-order fractional optimal control problems refers to the minimization of an objective functional subject to dynamical constraints on the state and the control which have three fractional derivatives order models.The necessary optimality conditions of this type are introduced as follows.Letα,β and μ be real numbers and α,β and μ∈0,1, and let f0 and f be two differentiable functions with domain a,∞×R2. We consider a general form of multicomposition fractional optimal control problem: (58)Jx,u,T=∫aTf0t,xt,utdt,which subject to the multi derivative-composite fractional dynamic control system,(59)MDtαaCt+NDtβaCDtμaCxt=ft,xt,ut,and the boundary conditions,(60)xa=xa,xT=xT.where M,N≠0, T,xaandxT are fixed real numbers.Theorem 6.
Ifx,u,T is a minimizer of (58), (59), and (60), then there exists a function λ(t) for which (x,u,T) satisfies the multi-composite fractional optimality conditions:(i) TheHamiltonian system is defined as (61)MtDTαλt+NtDTβtDTμλt=∂H∂xt,xt,ut,λtMDtαxtaC+NDtβaCDtμaCxt=∂H∂λt,xt,ut,λt.(ii) The stationary condition is(62)∂H∂ut,xt,ut,λt=0Forallt∈a,T.Proof.
We consider the following multi-composite fractional optimal control problem:(63)minimizeJx,u,T=∫aTf0t,xt,utdt,MDtαaCt+NDtβaCDtμaCxt=ft,xt,ut,Thus, (64)J∗x,u,T,λ=∫aTf0t,xt,ut+λtft,xt,ut-MDtαaCtxt+NDtβaCDtμaCxtdtSuppose that (65)F=f0t,xt,ut+λtft,xt,ut-MλtDtαaCtxt-NλtDtβaCDtμaCxtUsing (8) in (65), we get(66)F=f0t,xt,ut+λtft,xt,ut-MxttDTαλt-NxttDTβtDTμλt,Evaluating the Euler equation for xt of (66),(67)∂F∂xt-ddt∂F∂x·t=0,(68)∂f0∂xt+λt∂f∂xt-MtDTαλt-NtDTβtDTμλt-ddt∂F∂x·t=0From (67), we get d/dt∂F/∂x·t=0, as follows:(69)∂f0∂xt+λt∂f∂xt-MtDTαλt-NtDTβtDTμλt=0Now, evaluating Euler's equation for λt of (65):(70)∂F∂λt-ddt∂F∂λ·t=0,(71)∂f0∂λt+λt∂f∂λt-MDtαaCtxt-NDtβaCDtμaCxt=0,Finally, the Euler's equation for ut of (65) is(72)∂F∂ut-ddt∂F∂u·t=0.From (65) it is clear that d/dt∂F/∂u·t=0, then we get (73)∂F∂ut=∂f0∂ut+λt∂f∂ut=0From (69), (71), and (73) we have that(74)MtDTαλt+NtDTβtDTμλt=∂H∂xt,xt,ut,λtMDtαaCtxt+NDtβaCDtμaCxt=∂H∂λt,xt,ut,λt0=∂H∂ut,xt,ut,λtForallt∈a,T.


## 5. Illustrative Example

In this section, we consider the following linear–quadratic of multi-composite fractional optimal control problem:(75)min⁡Jx,u=∫01ut-xt2dt,which is subject to the multi-composite fractional dynamical system, (76)Dtα0Cxt+Dtβ0CDtμ0Cxt=ut-xt+6tα+1Гα+2+6tβ+μ+1Гβ+μ+2,and the boundary conditions,(77)x0=0,x1=6Гα+β+μ+2.The exact solution for α=β=μ=1 is given by the following:(78)x¯t=u¯t=6tα+β+μ+1Гα+β+μ+2.Now, we develop algorithm for solution (75), (76), and (77). It is based on the necessary optimality conditions from Theorem 6 and implements the following steps.Step  1.Compute the Hamiltonian function:(79)H=ut-xt2+λtut-xt+6tα+1Гα+2+6tβ+μ+1Гβ+μ+2.Step2.Derive the necessary optimality conditions from Theorem 6:(80)tD1αλt+tD1βtD1μλt=∂H∂x=-2ut-xt+λt,(81)Dtαxt0C+Dtβ0CDtμ0Cxt=∂H∂λ=ut-xt+6tα+1Гα+2+6tβ+μ+1Гβ+μ+2,(82)2ut-xt+λt=0.Step3.Find the coupled system.(i) From substitution (82) in (80), we get (83)tD1αλt+tD1βtD1μλt=0.(ii) Find control variableut from (82) and substitution in (80) to obtain (84)Dtαxt0C+Dtβ0CDtμ0Cxt=-12λt+6tα+1Гα+2+6tβ+μ+1Гβ+μ+2,Using property (6) for (83) and (84), we get the coupled system:(85a)tD1αλt+tD1β+μλt=0,(85b)Dtαxt0C+Dtβ+μ0Cxt=-12λt+6tα+1Гα+2+6tβ+μ+1Гβ+μ+2.Step  4a.Solve (85a) by Chebyshev expansion method, and use (48) to approximateλt.(86)tD1αλt≈λ1Г1-α1-ts-α+Jts;pNГ1-α,(87)tD1β+μλt≈λ1Г1-β+μ1-ts-β+μ+Jts;pNГ1-β+μ,s=1,2,…,N-1, where Jts;pN1 and Jts;pN2 is defined in (47).Sum both (86) and (87) to get(88)λ11-ts-αГ1-α+1-ts-β+μГ1-β+μ+Jts;pN1Г1-α+Jts;pN2Г1-β+μ=0.s=1,2,…,N-1, where Jts;pN1 and Jts;pN2 is defined in (47).Step  4b.Solve (85b) by Chebyshev expansion method, and use (24) to approximate xt.(89)Dtα0CxNts=∑r=0Nds,rαxtr,(90)Dtβ+μ0CxNts=∑r=0Nds,rβ+μxtr,α,β+μ>0.s,r=0,1,2,…,N, where ds,rα is defined in (25).Using (89) and (90) in (85a) to get (91)∑r=0Nds,rαxtr+∑r=0Nds,rβ+μxtr=-12λts+6tsα+1Гα+2+6tsβ+μ+1Гβ+μ+2.where ds,rα and ds,rβ+μ are defined in (25).Step  5.Find the results at the shifted Gauss-Lobatto nodes ts, from (22) when s=1,2,…,N-1, and α,β and μ∈0,1, in two cases.Case 1.
LetN=2, α=0.4,β=0.2 and μ=0.1→β+μ=0.3.
s = 1,2 , … , N - 1, and since N=2, then s=1.  t1=1/2-1/2cos⁡π/2=0.5.Now, substituting the value ofα,β,μ and t1, in (88), we get (92)λ11.8346+0.6715Jts;pN1+0.7704Jts;pN2=0where s=1,2,…,N-1, where Jts;pN1 and Jts;pN2 are defined in (47).After calculating (92) at the shifted Gauss-Lobatto nodes ts=t1=0.5 and substitution in (88) we get λt1=0.Now, we calculate (91) and, using the boundary conditions (78), we have (93)d1,00.4xt0+d1,10.4xt1+d1,20.4xt2+d1,00.3xt0+d1,10.3xt1+d1,20.3xt2=-12λt1+6t11.4Г2.4+6ts1.3Г2.3.Use (25) to evaluate the computation of d1,10.4,d1,20.4,d1,10.3 and d1,20.3, which is shown in Table 1.Table 1
The values of Caputo fractional derivative when (α=0.4, β=0.2 and μ=0.1 in Case 1) and (α=0.7,β=0.4andμ=0.3 in Case 2 needed later on) are shown.
n j k d 1,1 0.4 d 1,1 0.3 d 1,2 0.4 d 1,2 0.3 d 1,1 0.7 d 1,2 0.7 1 0 1 0 0 0.6722 0.6274 0 0.8154 1 1 1 0 0 0 0 0 0 1 2 1 0 0 0.0387 0.0287 0 0.0572 2 0 1 0.6722 0.6274 -0.6722 -0.6274 0.8154 -0.8154 2 1 1 0 0 0 0 0 0 2 2 1 0.0387 0.0287 -0.0387 -0.0287 0.0572 -0.0572 2 0 2 -0.5777 -0.5210 0.5777 0.5210 -0.7720 0.7720 2 1 2 0 0 0 0 0 0 2 2 2 0.0592 0.0620 -0.0592 -0.0620 0.0396 -0.0396 s u m 0.1924 0.1971 0.5185 0.459 0.1402 0.7324 d s , r ( α ) = 4 θ r / N . s u m 0.3848 0.3942 0.5185 0.459 0.2804 0.7324whenr=1→4θ1/N=4(1)/2=2 and r=2→4θ2/N=40.5/2=1.The results of Table1 and boundary conditions from (78) can be substituted in (93); we get (94)d1,00.4x0+0.3848xt1+0.5185x1+d1,00.3x0+0.3942xt1+0.459x1=6t11.4Г2.4+6t11.3Г2.3,(95)d1,00.40+0.3848xt1+0.51856Г2.7+d1,00.30+0.3942xt1+0.4596Г2.7=60.51.4Г2.4+60.51.3Г2.3,(96)xt1=0.1566.Step  6.Compute approximation solution of x(t) and u(t) at N=2.(97)x2t=2N∑n=0N′′∑r=0N′′xtrTnptrTnpt,(98)x2t=12a0T0pt+a1T1pt+12a2T2pt.To find a0,a1 and a2 from (23), we get (99)Ifn=0→a0=∑r=02′′xtrT0ptr,i.e.T0pt=1.=2.0204.Ifn=1→a1=∑r=02′′xtrT1ptr,i.e.T1pt=2tb-1=1.9421.Ifn=2→a2=∑r=02′′xtrT2ptr,=1.8638.Now, we substitute a0,a1, and a2, in (98), to get(100)x2t=7.4552t2-3.571t.To compute the approximation solution of the control ut from (76), we get (101)ut=Dtα0Cxt+Dtβ0CDtμ0Cxt+xt-6tα+1Гα+2-6tβ+μ+1Гβ+μ+2,(102)Dt0.4x2t0C=10.4290t1.6-3.9963t0.6.(103)Dt0.20CDt0.10Cx2t=9.6527t1.7-3.9301t0.7.Using (102), (103), and (100) in (101), we have:(104)ut=10.4290t1.6-3.9963t0.6+9.6527t1.7-3.9301t0.7+7.4552t2-3.571t-4.8303t1.4-5.1427t1.3.Case 2.
LetN=2, α=0.7,β=0.4 and μ=0.3→β+μ=0.7.
s = 1,2 , … , N - 1, and since N=2, then s=1.t1=1/2-1/2cos⁡(π/2)=0.5.Now, we calculate (91) and using the boundary conditions (78), we have (105)d1,00.7xt0+d1,10.7xt1+d1,20.7xt2+d1,00.7xt0+d1,10.7xt1+d1,20.7xt2=-12λt1+6t11.7Г2.7+6ts1.7Г2.7.To find d1,10.7 and d1,20.7 from Table 1 and we get λt1=0, in the same steps of Case 1.(106)d1,00.70+0.2804xt1+0.73242.0126+d1,00.70+0.2804xt1+0.73242.0126=2.3911,,xt1=0.9932.Now, compute approximation solution of x(t) and u(t) at N=2, in Case 2:(107)x2t=2N∑n=0N′′∑r=0N′′xtrTnptrTnpt,(108)x2t=12a0T0pt+a1T1pt+12a2T2pt.To find a0,a1 and a2, from (23), we get(109)a0=1.9995,a1=1.0063anda2=0.0131.We substitute a0,a1, and a2, in (108), to get(110)x2t=0.0528t2+1.9598t+0.0001.To compute the approximation solution of the control ut, we get (111)ut=Dtα0Cxt+Dtβ0CDtμ0Cxt+xt-6tα+1Гα+2-6tβ+μ+1Гβ+μ+2,(112)Dt0.7x2t0C=0.0905t1.3+2.1837t0.3.Now, using (110) and (112) in (111), we get (113)ut=0.0905t1.3+2.1837t0.3+0.0528t2+1.9598t+0.0001-3.8843t1.7.In Table 2, we give the computation of xt and ut when values t = 0…1 with different values of orders α,βandμ, for fractional derivative (α=0.4,β=0.2,andμ=0.1.); see Figures 1(b) and 2(b) and (α=0.7,β=0.4,andμ=0.3); see Figures 1(c) and 2(c) for N=2, which converge to exact values with (α=β=μ=1); see Figures 1(a) and 2(a).Table 2
The numerical values of the statext and control ut with N=2 which converged to exact values are shown.
t α=0.4 α=0.7 α = 1 α = 0.4 α = 0.7 α = 1 β = 0.2 β = 0.4 β = 1 β = 0.2 β = 0.4 β = 1 μ = 0.1 μ = 0.3 μ = 1 μ = 0.1 μ = 0.3 μ = 1 0 0 0.0001 0 0 0.0001 0 0.1 -0.283 0.1966 0.078 -2.0660 0.5152 0.78 0.2 -0.416 0.3942 0.0252 -2.9336 0.1920 0.252 0.3 -0.4 0.5928 0.502 -2.9336 0.2078 0.502 0.4 -0.236 0.7925 0.818 -3.0727 0.6427 0.818 0.5 0.078 0.9932 1.196 -2.4852 1.0983 1.196 0.6 0.541 1.1950 1.63 -1.5024 1.5673 1.63 0.7 1.153 1.3978 2.118 -0.1432 2.0459 2.118 0.8 1.915 1.6017 2.658 -0.1432 2.5314 2.658 0.9 2.825 1.8067 3.247 -0.1432 3.0221 3.247 1 3.884 2.0127 3.884 6.0665 3.5170 3.884Figure 1
Display the exact and approximate values of state x(t) with different types ofα,β, μ.
(a)
Exact solution ofxt for (N=2) and μ=α=β=1. (b)
Approximate solutions ofxt with (N=2) and α=0.4,β=0.2andμ=0.1 (c)
Approximate solutions ofxt with (N=2)α=0.7,β=0.4andμ=0.3.Figure 2
Display the exact and approximate values of control input u(.) with different types ofα,β, μ.
(a)
Exact solution ofut with (N=2) and α=β=μ=1. (b)
Approximate solutions ofut for (N=2) and α=0.4,β=0.2andμ=0.1 (c)
Approximate solutions ofut with (N=2)α=0.7,β=0.4andμ=0.3.


## 6. Conclusions

In this paper, we have presented algorithm for solution a class of multi-composite order fractional optimal control problems, in two cases one whenα=0.4,β=0.2 and μ=0.1, and the other one when α=0.7,β=0.4andμ=0.3 at (N=2). In both cases, the solution is approximated by Chebyshev series. Numerical results for illustrative example show that the algorithm converges from the exact solution when α=β=μ=1, and we note that the convergent to the exact solution is dependent on increasing of the fractional order of derivative.



---

*Source: 3767263-2018-08-01.xml*