# Application of Transcendental Bernstein Polynomials for Solving Two-Dimensional Fractional Optimal Control Problems

Authors: Fateme Ghomanjani; Samad Noeiaghdam; Sanda Micula
Journal: Complexity
 (2022)

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## Abstract

The aim of this study is to introduce a novel method to solve a class of two-dimensional fractional optimal control problems. Since there are some difficulties solving these problems using analytical methods, thus finding numerical methods to approximate their solution is a challenging topic. In this study, we use transcendental Bernstein series. In fact, for solving the problem, we generalize the Bernstein polynomials to a larger class of functions which can provide more accurate approximate solutions. The convergence theorem is proved. Some examples are solved to demonstrate the validity and applicability of this technique. Comparing the results with other methods, we can find the efficiency and applicability of the scheme.


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## Body

## 1. Introduction

Fractional differential equations (FDEs) provide some advantages in the simulation of problems arising in system biology [1], physics [2], hydrology [3], chemistry and biochemistry [1] and finance [4], economic growth models with memory effect [5], and many more. Constructing analytical and numerical methods for solving various types of FDEs has become an ongoing research topic. We mention here the finite element method, the wavelet method [6], the spectral tau method [7], the Gegenbauer spectral method [8], the iterative method [9], the fractional-order wavelet method [6], etc. Also, there are many mathematical and engineering problems which can be modelled in the form of fractional-order differential equations [10–16].Generally, two approaches exist for solving fractional-order optimal control problems (FOCPs) such as integer order optimal control problems. First, the use of Pontryagin’s maximum principle leads to a two-point boundary value problem, while the second approach involves solving the problem directly by discretizing and approximating the state and control functions [17]. To obtain the optimal solution, one can solve the Hamiltonian system of FOCPs. When constructing the Hamiltonian system using integration by parts, the left and right fractional derivatives appear simultaneously, which makes it very difficult to find the exact solution. Hence, some researchers have focused on the numerical solution of FOCPs.A central difference numerical scheme for FOCPs was given in [18]. In [19], the given optimization problem is reduced to a system of algebraic equations utilizing polynomial basis functions. For the fractional variational problems, an approximate solution is achieved by solving the system. In [20], FOCPs and their solutions were analyzed by means of rational approximation. In [21], the shifted Legendre-tau method was presented to solve a class of initial-boundary value problems for the fractional diffusion equations with variable coefficients on a finite domain.The Bezier curves method (BCM) is discussed in some papers. In [22, 23], the BCM was utilized for solving delay differential equation (DDE) and optimal control of switched systems (OCSSs). In [24], the BCM was proposed for some linear optimal control systems (LOCPs) with pantograph delays. Also, to solve the quadratic Riccati differential equation and the Riccati differential-difference equation, the BCM is utilized (see [25]). Other uses of the BCM are found in [26].In the present work, FDEs are utilized as the dynamic constraints, leading to the fractional optimal control problem (FOCP). However, very little work has been done in the area of FOCP, although FOCPs have gained much attention for some of their applications in engineering and physics [27]. In this paper, we solve this problem using the transcendental Bernstein series (TBS). Numerical examples demonstrate the efficiency of the stated technique in solving FOCP.In what follows, we want to obtain a numerical technique for solving two-dimensional fractional optimal control problems (TFOCPs) of the form(1)minI1ζ,u=∫0a′∫0b′Gx,s,ζx,s,ux,s,∂ζ∂xx,s,∂ζ∂sx,sdxds,s.t.∂2ζ∂x∂sx,s=c1∂vζ∂xvx,s+c2∂βζ∂sβx,s+c3ζx,s+c4ux,s,ζ0,s=g1s,ζx,0=f1x,0<v,β≤1,c1,c2,c3,c4∈R,c4≠0,a′,b′∈R,where G, ζ, and u are smooth functions, and f1x and g1s are the given functions.The paper is organized as follows. In Section2, the TBS are defined. Also, the convergence theorem is proved in this section. In Section 3, we discuss several numerical examples and present some comparative results in some tables. Finally, conclusions and future work ideas are given in Section 4.


## 2. The TBS

Definition 1.
The Bezier polynomial of degreen is defined over the interval tj−1,tj as follows:(2)xjt=∑r=0narjBr,nt−tj−1h,where h=tj−tj−1, and(3)Br,nt−tj−1h:=nr1hntj−tn−rt−tj−1r,is the Bernstein polynomial of degree n over the interval tj−1,tj, and arj, r=0,1,…,n, and they are unknown control points.
BCMs can be used to approximate some problems (see [22, 23]). One may note that the BCM is similar to the TBS.Definition 2.
The Caputo fractional derivatives of orderv>0 of the function ux,s are defined as (see [28])(4)∂vu∂xvx,s=1Γn−v∫0sx−ηn−v−1∂nu∂ηnη,sdη,∂vu∂svx,s=1Γn−v∫0ss−θn−v−1∂nu∂θnx,θdθ,n−1<v≤n.
Now, the TBS of degreem is defined as(5)Bi,ms=mi+γisi1−sm−i,0≤i≤m,s∈0,1.
Notice thatγi=0, then the TBS coincides with the Bernstein polynomial of degree m.
The expansions of the functionζx,s and ux,s in terms of TBS can be written in the following forms:(6)ζx,s≃ϕm1xTP′ψm2s,ux,s≃θn1xTQ′Ωn2s,where P′=pij, i=0,1,…,m1, j=0,1,…,m2 and Q′=qij, i=0,1,…,n1, j=0,1,…,n2 are unknown matrices, and m1,m2,n1,n2∈ℤ+=1,2,3,…. We have(7)ϕm1x=11B2,m1x…Bm1,m1xT=A′Tm1x,ψm2s=1B1,m2sB2,m1s…Bm2,m2sT=B′Tm2s,θn1x=1B1,n1xB2,n1x…Bn1,n1xT=C′Tn1x,Ωn2s=1B1,n2sB2,n2s…Bn1,n1sT=D′Tn2s,with(8)A′=100…0010…0a20a21a22…a2m1⋮⋮⋮…⋮am10am11am13…am1m1,B′=10…0b10b11…b1m2b20b21…a2m2⋮⋮…⋮bm20bm21…bm2m2,C′=10…0c10c11…c1n1c20c21…c2n1⋮⋮…⋮cm20cm21…cn1n1,D′=10…0d10d11…d1n2d20d21…d2n2⋮⋮…⋮dn20dn21…dn2n2,aij=−1j−im1im1−ij−i,i≤j0,i>j,bij=−1j−im2im2−ij−i,i≤j0,i>j,cij=−1j−in1in1−ij−i,i≤j0,i>j,dij=−1j−in2in2−ij−i,i≤j0,i>j,Tm1x=ϕ0xϕ1x…ϕm1xT,Tm2x=ψ0xψ1x…ψm2xT,Tn1x=θ0xθ1x…θn1xT,Tn2x=w0xw1x…wn2xT,ϕix=xii=0,1,xi+kii=2,3,…,m1,ψis=1j=0,si+qjj=1,2,3,…,m2,θix=1i=0,xi+rii=1,2,3,…,n1,wjs=1j=0,sj+ljj=1,2,3,…,n2,where the symbols ki, qj, ri, and lj represent the control parameters.


### 2.1. Convergence Analysis

Theorem 1.
Suppose thatV:0,1×0,1↔R is n1+n2+1 times continuously differentiable, V∈Cn+n2+10,1×0,1, and Y=xβitγj,0≤i≤n1,0≤j≤n2. If θn1xTQΩn2t is the good approximation, then we have(9)V−θn1TQΩn22≤Nn1+n2+2k!n1+n2+1−k!,k∈0,1,2,…,n1+n2+1,where(10)N≔sup∂n1+n2+1∂sn1+n2+1−iVs,t:s,t∈0,1,i=0,1,…,n1+n2+1.Proof.
One may write(11)Vs,t=qs,t+1n1+n2+1!s∂∂s+t∂∂tn1+n2+1Vλ0s,λ0t,λ0∈0,1,qs,t=∑i=0n1+n21i!s∂∂s+t∂∂tiV0,0.
Hence,(12)Vs,t−qs,t=1n1+n2+1!s∂∂s+t∂∂tn1+n2+1,Vλ0s,λ0t,λ0∈0,1,which leads to(13)V−θn1TQΩn22=V−q2.
Therefore,(14)s∂∂s+t∂∂tn1+n2+1,Vλ0s,λ0t=sn1+n2+1∂n1+n2+1∂sn1+n2+1Vλ0s,λ0t+sn1+n2tn1+n2+11∂n1+n2+1∂sn1+n2∂tVλ0s,λ0t+⋮+stn1+n2n1+n2+1n1+n2∂n1+n2+1∂s∂tn1+n2Vλ0s,λ0t+tn1+n2+1∂n1+n2+1∂tn1+n2+1Vλ0s,λ0t.
Since(15)n1+n2+1i=n1+n2+1n1+n2+1−i,i=0,1,…,n1+n2+1,there exists a value k∈0,1,2,…,n1+n2+1 such that(16)maxn1+n2+1i,i=0,1,…,n1+n2+1=n1+n2+1k=n1+n2+1!k!n1+n2+1−k!.
Now, one may define(17)N:=sup∂n1+n2+1∂sn1+n2+1−iVs,t:s,t∈0,1,i=0,1,…,n1+n2+1.
For finding an upper bound in equation (13), we have(18)∫01∫01s2n1+2n2+2−2it2idsdt=12i+12n1+2n2+3−2i,i=0,1,…,n1+n2+1,∫01∫01s2n1+2n2+1−iti+1dsdt=1i+22n1+2n2+2−i,i=0,1,…,n1+n2,∫01∫01s2n1+2n2−1−iti+3dsdt=1i+42n1+2n2−i,i=0,1,…,n1+n2−1,∫01∫01s2n1+2n2−3−iti+5dsdt=1i+62n1+2n2−2−i,i=0,1,…,n1+n2−2,∫01∫01si+1t2n1+2n2−i+1dsdt=1i+22n1+2n2+2−i,i=0.
Then, we get(19)V−q2∫01∫011n1+n2+1!s∂∂s+t∂∂tn1+n2+1Vλ0s,λ0t2dsdt1/2,=≤n1+n2+1!k!n1+n2+1−k!×Mn1+n2+1!∫01∫01∑i=0n1+n2+1sn1+n2+1−iti2dsdt1/2=ψ∑i=0n1+n2+112i+12n1+2n2+3−2i+…+∑i=011i+32n1+2n2−i+1+14n1+4n2+41/2≤ψn1+n2+2+n1+n2+1+⋯+2+11/2=ψn1+n2+221/2=ψn1+n2+2,whereψ=Mkn1+n2+1−k!.


## 2.1. Convergence Analysis

Theorem 1.
Suppose thatV:0,1×0,1↔R is n1+n2+1 times continuously differentiable, V∈Cn+n2+10,1×0,1, and Y=xβitγj,0≤i≤n1,0≤j≤n2. If θn1xTQΩn2t is the good approximation, then we have(9)V−θn1TQΩn22≤Nn1+n2+2k!n1+n2+1−k!,k∈0,1,2,…,n1+n2+1,where(10)N≔sup∂n1+n2+1∂sn1+n2+1−iVs,t:s,t∈0,1,i=0,1,…,n1+n2+1.Proof.
One may write(11)Vs,t=qs,t+1n1+n2+1!s∂∂s+t∂∂tn1+n2+1Vλ0s,λ0t,λ0∈0,1,qs,t=∑i=0n1+n21i!s∂∂s+t∂∂tiV0,0.
Hence,(12)Vs,t−qs,t=1n1+n2+1!s∂∂s+t∂∂tn1+n2+1,Vλ0s,λ0t,λ0∈0,1,which leads to(13)V−θn1TQΩn22=V−q2.
Therefore,(14)s∂∂s+t∂∂tn1+n2+1,Vλ0s,λ0t=sn1+n2+1∂n1+n2+1∂sn1+n2+1Vλ0s,λ0t+sn1+n2tn1+n2+11∂n1+n2+1∂sn1+n2∂tVλ0s,λ0t+⋮+stn1+n2n1+n2+1n1+n2∂n1+n2+1∂s∂tn1+n2Vλ0s,λ0t+tn1+n2+1∂n1+n2+1∂tn1+n2+1Vλ0s,λ0t.
Since(15)n1+n2+1i=n1+n2+1n1+n2+1−i,i=0,1,…,n1+n2+1,there exists a value k∈0,1,2,…,n1+n2+1 such that(16)maxn1+n2+1i,i=0,1,…,n1+n2+1=n1+n2+1k=n1+n2+1!k!n1+n2+1−k!.
Now, one may define(17)N:=sup∂n1+n2+1∂sn1+n2+1−iVs,t:s,t∈0,1,i=0,1,…,n1+n2+1.
For finding an upper bound in equation (13), we have(18)∫01∫01s2n1+2n2+2−2it2idsdt=12i+12n1+2n2+3−2i,i=0,1,…,n1+n2+1,∫01∫01s2n1+2n2+1−iti+1dsdt=1i+22n1+2n2+2−i,i=0,1,…,n1+n2,∫01∫01s2n1+2n2−1−iti+3dsdt=1i+42n1+2n2−i,i=0,1,…,n1+n2−1,∫01∫01s2n1+2n2−3−iti+5dsdt=1i+62n1+2n2−2−i,i=0,1,…,n1+n2−2,∫01∫01si+1t2n1+2n2−i+1dsdt=1i+22n1+2n2+2−i,i=0.
Then, we get(19)V−q2∫01∫011n1+n2+1!s∂∂s+t∂∂tn1+n2+1Vλ0s,λ0t2dsdt1/2,=≤n1+n2+1!k!n1+n2+1−k!×Mn1+n2+1!∫01∫01∑i=0n1+n2+1sn1+n2+1−iti2dsdt1/2=ψ∑i=0n1+n2+112i+12n1+2n2+3−2i+…+∑i=011i+32n1+2n2−i+1+14n1+4n2+41/2≤ψn1+n2+2+n1+n2+1+⋯+2+11/2=ψn1+n2+221/2=ψn1+n2+2,whereψ=Mkn1+n2+1−k!.


## 3. Numerical Examples

In this section, some examples are approximated using the mentioned method. The results are presented in tables and are compared with the results obtained in [29–31]. Also, the graphs of the approximate solutions are plotted for different values of ν and β.Example 1.
Consider the following problem [29],(20)I1=min∫03∫03∂ζ∂sx,s+ζx,s2+ζ2x,s+u2x,sdxds,s.t.∂2ζ∂x∂sx,s=−∂vζ∂xvx,s−3∂βζ∂sβx,s+0.2ζx,s+0.3ux,s,ζ0,s=e−2s,ζx,0=e−3xcos2πx.
This example is solved using our method withγi=1 (see Figures 1–4). Applying the method, we have(21)I1,approx=0.0355047374520341296,n1=n2=m1=m2=3,v=β=0.5,I1,approx=0.00199061404713575028,n1=n2=m1=m2=4,v=β=0.5,I1,approx=0.0219108300379033950,n1=n2=m1=m2=4,v=β=1.
Note that these results are better than the obtained in [29]. The comparative study can be found in Table 1. We note that in Table 1, parameters M, M′ are from the method used in [29]. The CPU time for solving the problem is 4.13 s using a Core i3 laptop.(22)ζapproxx,s=1−1.45240394376420t+0.194460761429751x3+0.694697572298107t2−0.106958345297291t3−0.694479683208798x2+1.04133400405558x2t−0.505832311967745x2t2+0.0784741699699019x2t3−0.297681404809978x3t+0.146001746220140x3t2−0.0227563106212003x3t3,forn1=n2=m1=m2=3,v=0.5,β=0.5.Figure 1
The approximate solution ofζx,s with n1=n2=m1=m2=3 for Example 1.Figure 2
The approximate solution ofux,s with n1=n2=m1=m2=3 for Example 1.Figure 3
The approximate solution ofζx,s with n1=n2=m1=m2=4 for Example 1.Figure 4
The approximate solution ofux,s with n1=n2=m1=m2=4 for Example 1.Table 1
The value ofI1 in some references.
Method in [29]Value ofI1[29]M=1,M′=16.3883M=2,M′=35.0251M=6,M′=42.2775M=7,M′=50.5997M=8,M′=50.1906M=9,M′=50.1770M=10,M′=50.0951M=10,M′=60.0947Method in [30]Value ofI1 [30]M=6,M′=14.0203M=6,M′=82.2905M=9,M′=20.8024M=7,M′=80.6202M=8,M′=30.2792M=8,M′=80.2026Method in [31]Value ofI1 [31]X=0.3,T=0.31.4979X=0.2,T=0.21.0953X=0.1,T=0.10.7348X=0.05,T=0.050.5510X=0.03,T=0.030.4760Value ofI1 in method with n1=n2=m1=m2=30.0355047374520341296Value ofI1 in method with n1=n2=m1=m2=40.00199061404713575028Example 2.
Next, let us consider the following problem [30]:(23)I1=min12∫05∫05107ζx,s−sinx+s2+u2x,sdxds,s.t.∂2ζ∂x∂sx,s=−∂vζ∂xvx,s−3∂βζ∂sβx,s+0.2ζx,s+0.3ux,s,ζ0,s=e−2s,ζx,0=e−3xcos2πx.
This example is solved using our method withγi=1. The graphs of the approximate solutions are shown in Figures 5 and 6. Also, we get(24)I1,approx=1.00374989684×106,n1=n2=m1=m2=3,v=0.5,β=0.5.
A comparative study is presented in Table2.Figure 5
The approximate solution ofζx,s with n1=n2=m1=m2=3 for Example 2.Figure 6
The approximate solution ofux,s with n1=n2=m1=m2=3 for Example 2.Table 2
The value ofI1 in some references.
Method in [30]Value ofI1[30]M=7,M′=62.03080×106M=6,M′=81.82721×106M=7,M′=71.60773×106M=6,M′=91.54056×106M=7,M′=81.45170×106M=7,M′=91.33534×106M=8,M′=81.30907×106Value ofI1 in method with n1=n2=m1=m2=31.00374989684×106


## 4. Conclusion

In this paper, an efficient algorithm based on TBS was presented to solve 2-D FOCPs. The main idea of the method is to use the TBS as a new approximation instrument. The validity of the stated method based on TBS was verified in Section3. Furthermore, one may note the 2-D FOCP can be reduced to a system of algebraic equations. Finding the control parameters provides the approximate solution of the problem. The efficiency of the method was confirmed by several numerical examples. Solving the problem based on the Caputo-Hadamard fractional derivative and considering infinite horizon optimal control for nonlinear interconnected large-scale dynamical systems with an application to optimal attitude control will be studied in future.



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