- Research
- Open Access
- Published:

# Numerical solutions of fractional optimal control with Caputo–Katugampola derivative

*Advances in Difference Equations*
**volume 2021**, Article number: 425 (2021)

## Abstract

In this paper, we present a numerical technique for solving fractional optimal control problems with a fractional derivative called Caputo–Katugampola derivative. This derivative is a generalization of the Caputo fractional derivative. The proposed technique is based on a spectral method using shifted Chebyshev polynomials of the first kind. The Clenshaw and Curtis scheme for the numerical integration and the Rayleigh–Ritz method are used to estimate the state and control variables. Moreover, the error bound of the fractional derivative operator approximation of Caputo–Katugampola is derived. Illustrative examples are provided to show the validity and applicability of the presented technique.

## Introduction

In recent years, many engineering and science problems have arisen in fractional differential equations (FDEs). Torvik and Bagley [6] utilized a fractional derivative to depict the behavior of viscoelastic materials. The fractional logistic model with feedback control has been suggested and analyzed in [17]. Characterization and synthesis of the frequency-band complex noninteger differentiator are studied in [26]. A rational approximation technique is used to approximate the fractional order optimal control problems (FOCPs) in [37].

The main idea of FDEs is the order of the derivative is replaced with a noninteger order. Various definitions of a fractional derivative have been presented in the literature such as Riemann–Liouville, Caputo (see [22, 23, 25, 27, 29]). Recently, several definitions of fractional operators and generalized fractional derivatives have been presented [5, 30, 40]. Katugampola proposed a generalization of the Caputo fractional derivative (see [1, 19, 20]). Furthermore, Abdeljawad et al. evolved them in various research papers [1, 18].

Optimal control problems (OCPs) appear widely in various applications such as aerospace, atmospheric reentry, quantum systems, and space shuttle. Bonnard et al. in [9] used the smooth continuation method in optimal control and performed the analysis of a dissipative two-level quantum system. In [16], the homotopy method combined with the shooting method is used to solve an optimal control problem of the atmospheric reentry of a space shuttle [36].

Nowadays, numerous researchers have widespread OCPs to the fractional case where the dynamic of systems involves a fractional derivative term. Fractional optimal control problems (FOCPs) have been widely investigated in the literature. Agrawal [2] provided a general formulation for a class of FOCPs, and this formulation is similar to the classical OCPs. In [24], the authors used a modified hat function to solve a class of FOCPs. FOCPs with free terminal time are considered in [28]. In [39], time-fractional optimal control problems with Caputo–Fabrizio fractional derivative are presented. In all these papers, FOCPs have been defined with respect to different definitions of fractional derivatives such as Riemann–Liouville and Caputo. However, FOCPs with Caputo–Katugampola derivative have not been investigated yet, and to the best of our knowledge, it is the first time that FOCPs with Caputo–Katugampola derivative are studied in the literature. The Caputo–Katugampola derivative is highly influenced by the value of *α* and another parameter *ρ* which is beneficial in graphical simulations associated with real data, see [14]. The motivation of this paper is to investigate the nature of a class of FOCPs with various fractional-order values and an additional parameter *ρ*. Therefore, in this paper, we deal with the numerical solutions of FOCPs with Caputo–Katugampola derivative, and we derive an estimation of the Caputo–Katugampola definition in terms of Chebyshev polynomials.

In order to find the numerical solutions of FOCPs, the spectral method which tries to approximate the unknown functions by means of orthogonal polynomials, namely Chebyshev polynomials, is used ([8, 21]). Furthermore, the Clenshaw and Curtis technique (see [12, 35]) has been utilized to discretize the objective function, and the optimality conditions have been obtained by the Rayleigh–Ritz method.

The main aim of this paper is to study the numerical solution for the FOCP in the following form:

subject to

where \({}^{C}D_{a^{+}}^{\alpha ,\rho }\) is the Caputo–Katugampola derivative operator, \(\mathcal{Q}_{1},\mathcal{Q}_{2}\neq 0\), \(t_{f}\), \(x_{a}\), \(x_{f}\) are fixed real numbers and \(\chi (t,x(t))\neq 0\), \(\forall t\in [a,t_{f}]\).

The rest of the paper is given as follows: In Sect. 2, the properties of shifted Chebyshev polynomials of the first kind and various definitions of the fractional derivatives and integrals are given. In Sect. 3, the Caputo–Katugampola derivative is approximated in terms of Chebyshev polynomials, and the error bound of the Caputo–Katugampola derivative operator is derived. The structure of the proposed numerical scheme is provided in Sect. 4. Section 5 deals with applying the presented scheme to some FOCPs to show its validity and applicability. The conclusions are stated in Sect. 6.

## Basic notations and preliminaries

This section provides some definitions of fractional derivatives and integrals that have been presented in the literature and some properties of the Chebyshev polynomials of the first kind.

### Fractional derivatives and integrals

In the following, we recall some definitions regarding fractional integrals and derivatives.

### Definition 2.1

The Riemann–Liouville fractional integral of order \(\alpha >0\) is given by

where \(\Gamma (\cdot )\) is the gamma function defined by

The Riemann–Liouville fractional derivative of order \(\alpha >0\) is defined by

### Definition 2.2

([13])

The Caputo fractional of order \(\alpha >0\) has the following form:

### Definition 2.3

Suppose that *y* is an integrable function on \([a,b]\), where \(0 < a < b < \infty \). For \(0<\alpha < 1\) and \(\rho > 0\), the Caputo–Katugampola fractional derivative of order *α* is defined by

Also, we give the following theorem that gives an equivalent form of Caputo–Katugampola fractional derivatives in Definition 2.3, when \(y \in C^{1}[a, b]\).

### Theorem 1

([4])

*Assume that* \(y \in C^{1}[a, b]\), *then*

### Remark 2.1

In case \(\rho = 1\), the Caputo–Katugampola fractional derivative is reduced to the classical Caputo fractional derivative, and for \(\rho \to 0^{+}\), we get the Caputo–Hadamard fractional derivative [15, 18].

### Definition 2.4

The Caputo–Katugampola fractional integral is defined by

### Definition 2.5

Let \(\alpha \in \mathbb{R^{+}}\), \(I=[a,b]\), *y* be an integrable function on *I*, \(\psi \in C^{n}(I)\) be increasing, and \(\psi ^{\prime }(x) \neq 0\) for all \(x \in I\). The *ψ*-Riemann–Liouville fractional integral and fractional derivative are defined by

and

where \(n=[\alpha ]+1\).

### Shifted Chebyshev polynomials of the first kind

The Chebyshev polynomials of the first kind with degree *n*, \(n\geq 0\), are defined as follows:

These polynomials satisfy the following recurrence relation:

with starting values \(T_{0}(x)=1\) and \(T_{1}(x)=x\). The explicit analytic form of \(T_{n}(x)\) is given by

where \(\lfloor n/2 \rfloor \) denotes the greatest integer less than or equal to \(n/2\). It is known that the Chebyshev polynomials of the first kind satisfy the following orthogonality relations:

Some of useful properties of Chebyshev polynomials [33] are the following:

The shifted Chebyshev polynomials of the first kind \(T^{*}_{n}(t)\) are defined on \([0,\tau ]\) as follows:

Again, it is known that the explicit analytic form of the shifted Chebyshev polynomials of the first kind is given by

and \(T_{n}^{*}(t)\) are orthogonal polynomials on the interval \([0, \tau ]\) with respect to the weight function \(\frac{1}{\sqrt{\tau t-t^{2}}}\), that is,

## Numerical approximations

A function \(f\in L^{2}([0,\tau ])\) can be expanded in terms of shifted Chebyshev polynomials as follows:

with \(c_{0}=2\) and \(c_{i}=1\) for all \(i\geq 1\).

Let \(f\in C[0,\tau ]\) and \(\mathcal{N}\) be a positive integer. Denote

where \(f_{\mathcal{N}}(t)\) is the Chebyshev–Gauss–Lobatto (CGL) interpolation of \(f(t)\) on \([0,\tau ]\) and the CGL points ([11, 34])

It is important to mention that the double-primed summation means halving the first and the last terms.

To compute an approximate solution for a definite integral of a continuous function \(g:[-1,1]\rightarrow \mathbb{R}\) over the interval \([-1, 1]\), we use the Clenshaw and Curtis formula (see [12, 35])

where \(\omega _{j}\), \(j=0,1,\dots ,\mathcal{N}\), are weights and \(t_{j}\), \(j=0,1,\dots ,\mathcal{N}\), are the roots of \((1-t^{2})\frac{dT_{\mathcal{N}}(t)}{dt}\). For \(\mathcal{N}\) even, the weights are:

while the weights for \(\mathcal{N}\) odd are determined by

Next, we study the convergence of the proposed technique.

### Theorem 2

*The series in Eq*. (10) *converges to the exact solution*.

### Proof

Consider the exact and the truncated series such that \(\mathcal{M} \geq \mathcal{N}\) are given as follows:

Then we have

In what follows, we show that \(f_{\mathcal{M}}(t)\) is a Cauchy sequence in \(L_{w}^{2}[0,1]\) and consequently it converges. From Eq. (13), we have

Using Bessel’s inequality, \({\sum_{i=\mathcal{N}+1}^{\mathcal{M}} \vert a_{i} \vert ^{2} \to 0}\) as \(\mathcal{M,N} \to \infty \), therefore \(f_{\mathcal{M}}\) is a Cauchy, and by the completeness of \(L_{w}^{2}\), \(f_{\mathcal{M}}\) converges to \(\vartheta (t) \in L_{w}^{2}\). Now, we prove that \(f(t) = \vartheta (t)\),

□

### Approximation of the Caputo–Katugampola fractional derivative

In what follows, we give some fundamental results for the fractional derivative \({}^{C}D_{0^{+}}^{\alpha ,\rho }f(t)\). First, we present a lemma which helps us in the proofs that follow.

### Lemma 3.1

*If* \(f,f^{(1)},\dots ,f^{(\nu )} \in AC[0,1]\) *and* \(|f^{(\nu +1)}(t)|\leq W_{\nu }<\infty \), \(\forall t \in [0,1]\), *for some* \(\nu \geq 0\), *then for each* \(m\geq \nu +1\),

### Proof

Using the substitution \(x=\frac{1}{2}(1+\cos \theta )\), we have

Integrating by parts,

For \(\nu =0\),

Moreover, integration by parts for (15) brings higher derivatives of *f* and shows more cosine terms. It is noticeable that there is a factor *m* in the denominator of (15). The second integration by parts gives factors \(m-1\) and \(m+1\), the third provides factors \(m-2\), *m*, and \(m+2\), and so on. For simplicity, we replace all various denominators with \(m-1\) at the second differentiation, \(m-2\) at the third differentiation, and so on until \(m - \nu \) at the \((\nu + 1)\text{st}\) differentiation, hence we fulfilled the desired result. □

Now, we give an estimation of \({}^{C}D_{0^{+}}^{\alpha , \rho }f_{\mathcal{N}}(t)\) in terms of Chebyshev polynomials.

### Theorem 3

*The Caputo–Katugampola fractional derivatives of the function* \(f_{\mathcal{N}}(t)\) *defined in Eq*. (11) *can be obtained as follows*:

*where the primed summation means that the last term is halved*.

### Proof

Let us assume that \(y(t) = t^{n}\) in (5) and take \(\theta =\frac{s^{\rho }}{t^{\rho }}\), then we get

where \(B(\cdot,\cdot)\) is the beta function. Then, from Eq. (8) and Eq. (11), we have

□

In the following theorem, based on the proof of Theorem 2.1 [38], we show an estimation of the error bound of the fractional derivative operator of Caputo–Katugampola.

### Theorem 4

*Suppose that* *f* *satisfies conditions of Lemma *3.1, \(\nu >2\). *Then*, *for* \(\mathcal{N} \geq \nu +1\),

### Proof

From the Caputo–Katugampola derivative definition given in (5), we obtain

and

Then

since \(\vert T^{*'}_{j}(s) \vert \leq 2 j^{2}\) for all \(j=0,1,\dots \). The relation between \(\tilde{a}_{j}\) and \(a_{j}\) given in [10, 12]

yields the following inequality [38]:

Therefore,

The proof is complete. □

### Remark 3.1

The error bound in Theorem 4 converges to zero as \(\mathcal{N} \to \infty \) for all \(t>0\)

## The structure of the numerical scheme

In what follows, we list the main steps to find the solution of Eqs. (1)–(3). It is important to mention that this technique is generally known as a “direct method”. The basic steps are given as follows:

*Step 1:*:-
Use Eq. (2) to write Eq. (1) without the control function

*u*. *Step 2:*:-
Approximate both the unknown function

*x*and its Caputo–Katugampola fractional derivative by using Eq. (11) and Eq. (16). *Step 3:*:-
In order to use the integration on the interval \([-1,1]\), we take the transformation \(t=a+\frac{t_{f}-a}{2}(\eta +1)\).

*Step 4:*:-
Approximate the integral whose limits are −1 to 1 by using the Clenshaw and Curtis formula given by Eq. (12).

*Step 5:*:-
Determine the critical points of Eq. (1) using the Rayleigh–Ritz approach as follows:

$$\begin{aligned} \frac{\partial \mathcal{J}}{\partial x(t_{i})}=0,\quad i = 1,2, \ldots ,\mathcal{N}-1, \end{aligned}$$which yields a system of nonlinear algebraic equations.

*Step 6:*:-
Use Newton’s iterative method to solve the nonlinear system and get the values \(x(t_{i})\), \(i=1,2,\ldots ,\mathcal{N}-1\).

*Step 7:*:-
Compute the solutions

*x*of the FOPC by$$ x(t) = \frac{2}{\mathcal{N}}\sum_{n=0}^{\mathcal{N}}{}^{\prime \prime } \sum_{r=0}^{ \mathcal{N}}{}^{\prime \prime }x(t_{r})T^{*}_{n}(t_{r})T^{*}_{n}(t), $$where \(x(t_{0})\) and \(x(t_{\mathcal{N}})\) are calculated from the boundary conditions. Moreover, we use Eq. (2) to find the control function

*u*.

## Numerical results

In this section, we apply the proposed numerical approach in Sect. 4 to illustrative examples.

### Example 1

Consider a FOCP as follows:

The exact solution is provided by

Table 1 shows the maximum absolute errors for the proposed technique and other published results in the literature at \(\rho = 1\) and \(\alpha = 0.5\) with different values of \(\mathcal{N}\). It is obvious that the accuracy of the proposed technique is better than that of other methods. In addition, Tables 2, 3, and 4 show the maximum absolute errors \(E_{x}\) and \(E_{u}\) of the state and control with various choices of *α*, *ρ*, and \(\mathcal{N}\), respectively. Moreover, the approximate value of the objective function \(\mathcal{J}\) is given. Figures 1 and 2 elucidate the approximate of the state and control at \(\mathcal{N}=3 \), with different values of *ρ*, \(\alpha = 0.5\) and \(\alpha = 0.9\), respectively. Figure 3 shows the exact and the approximate solutions of \(x(t)\) and \(u(t)\) using \(\rho = 1.5\) and \(\alpha = 0.5 \). Finally, in Table 5, we compute \(E_{x}\) and \(E_{u}\) at \(\alpha = 1\) and \(\rho = 1\) with various choices of \(\mathcal{N}\). It is clear that the numerical results agree with the analytical solutions and the proposed technique gives accurate numerical results.

### Example 2

We consider an FOCP in the form

The exact solution is provided by

In Tables 6, 7, and 8, we obtain the computational results of \(E_{x}\), \(E_{u}\), and \(\mathcal{J}\) with different values of *ρ*, *α* when the value of \(\mathcal{N}\) increases. The approximate state and control with various choices of *ρ* and *α* are displayed in Figs. 4 and 5. Furthermore, Fig. 6 displays the approximate solutions of \(x(t)\) and \(u(t)\) with their exact solutions. It is evident from the results of Examples 5.1 and 5.2 that the approximate solutions converge to the exact solutions by increasing the value of \(\mathcal{N}\).

## Conclusions

In this work, the numerical solutions of FOCPs involving the Caputo–Katugampola derivative have been presented. The numerical scheme used in this paper is called a direct method, and it is based on expanding the unknown function in terms of shifted Chebyshev polynomials of the first kind and using the Clenshaw and Curtis formula to approximate the integration. Then, we have deduced an estimation of the error bound of a fractional derivative operator of Caputo–Katugampola. Finally, some numerical examples are supplied to show the capability and accuracy of the proposed scheme.

## Availability of data and materials

Data sharing is not applicable to this study.

## References

Abdeljawad, T., Madjidi, F., Jarad, F., Sene, N.: On dynamic systems in the frame of singular function dependent kernel fractional derivatives. Mathematics

**7**(10), 946 (2019)Agrawal, O.P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn.

**38**(1), 323–337 (2004)Almeida, R.: A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul.

**44**, 460–481 (2017)Almeida, R., Malinowska, A.B., Odzijewicz, T.: Fractional differential equations with dependence on the Caputo-Katugampola derivative. J. Comput. Nonlinear Dyn.

**11**(6), 061017 (2016)Atangana, A.: Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system. Chaos Solitons Fractals

**102**, 396–406 (2017)Bagley, R.L., Torvik, P.J.: On the appearance of the fractional derivative in the behavior of real materials. Appl. Mech.

**51**, 294–298 (1984)Ben Makhlouf, A., Nagy, A.M.: Finite-time stability of linear Caputo-Katugampola fractional-order time delay systems. Asian J. Control

**22**(1), 297–306 (2020)Bhrawy, A.H., Ezz-Eldien, S.S., Doha, E.H., Abdelkawy, M.A., Baleanu, D.: Solving fractional optimal control problems within a Chebyshev-Legendre operational technique. Int. J. Control

**90**(6), 1230–1244 (2017)Bonnard, B., Shcherbakova, N., Sugny, D.: The smooth continuation method in optimal control with an application to quantum systems. ESAIM Control Optim. Calc. Var.

**17**(1), 267–292 (2011)Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Dover, New York (2000)

Clenshaw, C.W., Curtis, A.R.: A method for numerical integration on an automatic computer. Numer. Math.

**2**(1), 197–205 (1960)Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration. Academic Press, New York (1975)

Diethelm, K.: The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type. Lecture Notes in Mathematics. Springer, Berlin (2010)

Erturk, V.S., Kumar, P.: Solution of a COVID-19 model via new generalized Caputo-type fractional derivatives. Chaos Solitons Fractals

**139**, 110280 (2020)Gambo, Y.Y., Jarad, F., Baleanu, D., Abdeljawad, T.: On Caputo modification of the Hadamard fractional derivatives. Adv. Differ. Equ.

**2014**, 10, 1–12 (2014)Hermant, A.: Optimal control of the atmospheric reentry of a space shuttle by an homotopy method. Optim. Control Appl. Methods

**32**(6), 627–646 (2011)Hoang, M.T., Nagy, A.M.: Uniform asymptotic stability of a logistic model with feedback control of fractional order and nonstandard finite difference schemes. Chaos Solitons Fractals

**123**, 24–34 (2019)Jarada, F., Abdeljawadb, T., Baleanua, D.: On the generalized fractional derivatives and their Caputo modification. J. Nonlinear Sci. Appl.

**10**, 2607–2619 (2017)Katugampola, U.N.: New approach to a generalized fractional integral. Appl. Math. Comput.

**218**, 860–865 (2011)Katugampola, U.N.: A new approach to generalized fractional derivative. Bull. Math. Anal. Appl.

**6**(4), 1–15 (2014)Khalifa, A.K., Elbarbary, E.M.E., Abd-Elrazek, M.A.: Chebyshev expansion method for solving second and fourth-order elliptic equations. Appl. Math. Comput.

**135**, 307–318 (2003)Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

Nemati, S., Lima, P.M., Torres, D.F.M.: A numerical approach for solving fractional optimal control problems using modified hat functions. Commun. Nonlinear Sci. Numer. Simul.

**78**, 104849 (2019)Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)

Oustaloup, A., Levron, F., Mathieu, B., Nanot, F.M.: Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Trans. Circuits Syst.

**47**, 25–39 (2000)Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

Pooseh, S., Almeida, R., Torres, D.F.M.: Fractional order optimal control problems with free terminal time. J. Ind. Manag. Optim.

**10**(2), 363–381 (2014)Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, London (1993)

Seemab, A., Rehman, M.U., Alzabut, J., Hamdi, A.: On the existence of positive solutions for generalized fractional boundary value problems. Bound. Value Probl.

**2019**, 186 (2019)Sene, N.: Generalized Mittag-Leffler input stability of the fractional-order electrical circuits. IEEE Open J. Circuits Syst.

**1**, 233–242 (2020)Sene, N.: Introduction to the fractional-order chaotic system under fractional operator in Caputo sense. Alex. Eng. J.

**60**, 3997–4014 (2021)Shen, J., Tang, T., Wang, L.L.: Spectral Methods Algorithms, Analysis and Applications. Springer Series in Computational Mathematics (2011)

Sweilam, N.H., Al-Ajami, T.M., Hoppe, R.H.W.: Numerical solution of some types of fractional optimal control problems. Sci. World J.

**2013**, Article ID 306237 (2013)Trefethen, L.N.: Spectral Methods in MATLAB. SIAM, Philadelphia (2000)

Trèlat, E.: Optimal control of a space shuttle and numerical simulations. Discrete Contin. Dyn. Syst.

**2003**(suppl), 842–851 (2003)Tricaud, C., Chen, Y.-Q.: An approximation method for numerically solving fractional order optimal control problems of general form. Comput. Math. Appl.

**59**, 1644–1655 (2010)Xiang, S., Chen, X., Wang, H.: Error bounds for approximation in Chebyshev points. Numer. Math.

**116**(3), 463–491 (2010)Yıldız, T.A., Jajarmi, A., Yıldız, B., Baleanu, D.: New aspects of time fractional optimal control problems within operators with nonsingular kernel. Discrete Contin. Dyn. Syst., Ser. S

**13**(3), 407–428 (2020)Zafar, Z.U.A., Sene, N., Rezazadeh, H., Esfandian, N.: Tangent nonlinear equation in context of fractal fractional operators with nonsingular kernel. Math. Sci. (2021). https://doi.org/10.1007/s40096-021-00403-7

## Acknowledgements

Authors would like to thank the anonymous reviewers for their valuable suggestions that improved the quality of the manuscript.

## Funding

No funding sources to be declared.

## Author information

### Authors and Affiliations

### Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

### Corresponding author

## Ethics declarations

### Competing interests

The authors declare that they have no competing interests.

## Rights and permissions

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

## About this article

### Cite this article

Sweilam, N.H., Nagy, A.M. & Al-Ajami, T.M. Numerical solutions of fractional optimal control with Caputo–Katugampola derivative.
*Adv Differ Equ* **2021, **425 (2021). https://doi.org/10.1186/s13662-021-03580-w

Received:

Accepted:

Published:

DOI: https://doi.org/10.1186/s13662-021-03580-w

### MSC

- 65K10
- 26A33
- 49K15

### Keywords

- Caputo–Katugampola fractional derivative
- Fractional optimal control problems
- Chebyshev expansion
- Spectral methods