# Application of Bernoulli Polynomials for Solving Variable-Order Fractional Optimal Control-Affine Problems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. The Variable-Order Fractional Calculus

**Definition**

**1**

**Definition**

**2**

#### 2.2. Bernoulli Polynomials

## 3. Operational Matrix of Variable-Order Fractional Integration

## 4. Methods of Solution

#### 4.1. Approach I

#### 4.2. Approach II

## 5. Error Bounds

**Lemma**

**1**

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

**Theorem**

**2.**

**Proof.**

**Remark**

**2.**

**Lemma**

**2**

**Theorem**

**3.**

**Proof.**

**Remark**

**3.**

**Remark**

**4.**

## 6. Test Problems

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Machado, J.T.; Kiryakova, V.; Mainardi, F. Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul.
**2011**, 16, 1140–1153. [Google Scholar] [CrossRef] [Green Version] - Keshavarz, E.; Ordokhani, Y.; Razzaghi, M. A numerical solution for fractional optimal control problems via Bernoulli polynomials. J. Vib. Control
**2016**, 22, 3889–3903. [Google Scholar] [CrossRef] - Bhrawy, A.H.; Tohidi, E.; Soleymani, F. A new Bernoulli matrix method for solving high-order linear and nonlinear Fredholm integro-differential equations with piecewise intervals. Appl. Math. Comput.
**2012**, 219, 482–497. [Google Scholar] [CrossRef] - Tohidi, E.; Bhrawy, A.H.; Erfani, K. A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation. Appl. Math. Model.
**2013**, 37, 4283–4294. [Google Scholar] [CrossRef] - Toutounian, F.; Tohidi, E. A new Bernoulli matrix method for solving second order linear partial differential equations with the convergence analysis. Appl. Math. Comput.
**2013**, 223, 298–310. [Google Scholar] [CrossRef] - Bazm, S. Bernoulli polynomials for the numerical solution of some classes of linear and nonlinear integral equations. J. Comput. Appl. Math.
**2015**, 275, 44–60. [Google Scholar] [CrossRef] - Sahu, P.K.; Mallick, B. Approximate solution of fractional order Lane-Emden type differential equation by orthonormal Bernoulli’s polynomials. Int. J. Appl. Comput. Math.
**2019**, 5, 89. [Google Scholar] [CrossRef] - Loh, J.R.; Phang, C. Numerical solution of Fredholm fractional integro-differential equation with right-sided Caputo’s derivative using Bernoulli polynomials operational matrix of fractional derivative. Mediterr. J. Math.
**2019**, 16, 28. [Google Scholar] [CrossRef] - Rosa, S.; Torres, D.F.M. Optimal control of a fractional order epidemic model with application to human respiratory syncytial virus infection. Chaos Solitons Fractals
**2018**, 117, 142–149. [Google Scholar] [CrossRef] [Green Version] - Malinowska, A.B.; Torres, D.F.M. Introduction to the Fractional Calculus of Variations; Imperial College Press: London, UK, 2012. [Google Scholar] [CrossRef]
- Malinowska, A.B.; Odzijewicz, T.; Torres, D.F.M. Advanced methods in the fractional calculus of variations. In Springer Briefs in Applied Sciences and Technology; Springer: Cham, Switzerland, 2015. [Google Scholar]
- Almeida, R.; Pooseh, S.; Torres, D.F.M. Computational Methods in the Fractional Calculus of Variations; Imperial College Press: London, UK, 2015. [Google Scholar] [CrossRef]
- Ali, M.S.; Shamsi, M.; Khosravian-Arab, H.; Torres, D.F.M.; Bozorgnia, F. A space-time pseudospectral discretization method for solving diffusion optimal control problems with two-sided fractional derivatives. J. Vib. Control
**2019**, 25, 1080–1095. [Google Scholar] [CrossRef] - 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.
**2019**, 78, 104849. [Google Scholar] [CrossRef] [Green Version] - Salati, A.B.; Shamsi, M.; Torres, D.F.M. Direct transcription methods based on fractional integral approximation formulas for solving nonlinear fractional optimal control problems. Commun. Nonlinear Sci. Numer. Simul.
**2019**, 67, 334–350. [Google Scholar] [CrossRef] [Green Version] - Rabiei, K.; Ordokhani, Y.; Babolian, E. Numerical solution of 1D and 2D fractional optimal control of system via Bernoulli polynomials. Int. J. Appl. Comput. Math.
**2018**, 4, 7. [Google Scholar] [CrossRef] - Behroozifar, M.; Habibi, N. A numerical approach for solving a class of fractional optimal control problems via operational matrix Bernoulli polynomials. J. Vib. Control
**2018**, 24, 2494–2511. [Google Scholar] [CrossRef] - Rahimkhani, P.; Ordokhani, Y. Generalized fractional-order Bernoulli-Legendre functions: An effective tool for solving two-dimensional fractional optimal control problems. IMA J. Math. Control Inf.
**2019**, 36, 185–212. [Google Scholar] [CrossRef] - Samko, S.G.; Ross, B. Integration and differentiation to a variable fractional order. Integral Transform. Spec. Funct.
**1993**, 1, 277–300. [Google Scholar] [CrossRef] - Lorenzo, C.F.; Hartley, T.T. Variable order and distributed order fractional operators. Nonlinear Dyn.
**2002**, 29, 57–98. [Google Scholar] [CrossRef] - Abdeljawad, T.; Mert, R.; Torres, D.F.M. Variable order Mittag-Leffler fractional operators on isolated time scales and application to the calculus of variations. In Fractional Derivatives with Mittag-Leffler Kernel; Springer: Cham, Switzerland, 2019; Volume 194, pp. 35–47. [Google Scholar]
- Hassani, H.; Naraghirad, E. A new computational method based on optimization scheme for solving variable-order time fractional Burgers’ equation. Math. Comput. Simul.
**2019**, 162, 1–17. [Google Scholar] [CrossRef] - Odzijewicz, T.; Malinowska, A.B.; Torres, D.F.M. Fractional variational calculus of variable order. In Advances in Harmonic Analysis and Operator Theory; Birkhäuser/Springer Basel AG: Basel, Switzerland, 2013; Volume 229, pp. 291–301. [Google Scholar] [CrossRef] [Green Version]
- Yan, R.; Han, M.; Ma, Q.; Ding, X. A spectral collocation method for nonlinear fractional initial value problems with a variable-order fractional derivative. Comput. Appl. Math.
**2019**, 38, 66. [Google Scholar] [CrossRef] - Almeida, R.; Tavares, D.; Torres, D.F.M. The variable-order fractional calculus of variations. In Springer Briefs in Applied Sciences and Technology; Springer: Cham, Switzerland, 2019. [Google Scholar] [CrossRef] [Green Version]
- Heydari, M.H.; Avazzadeh, Z. A new wavelet method for variable-order fractional optimal control problems. Asian J. Control
**2018**, 20, 1804–1817. [Google Scholar] [CrossRef] - Mohammadi, F.; Hassani, H. Numerical solution of two-dimensional variable-order fractional optimal control problem by generalized polynomial basis. J. Optim. Theory Appl.
**2019**, 180, 536–555. [Google Scholar] [CrossRef] - Costabile, F.; Dell’Accio, F.; Gualtieri, M.I. A new approach to Bernoulli polynomials. Rend. Mat. Appl.
**2006**, 26, 1–12. [Google Scholar] - Arfken, G. Mathematical Methods for Physicists; Academic Press: New York, NY, USA; London, UK, 1966. [Google Scholar]
- Shen, J.; Tang, T.; Wang, L.L. Spectral Methods; Springer Series in Computational Mathematics; Springer: Heidelberg, Germany, 2011; Volume 41. [Google Scholar] [CrossRef]
- Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A. Spectral Methods. In Scientific Computation; Springer: Berlin, Germany, 2006. [Google Scholar]
- Lotfi, A.; Yousefi, S.A.; Dehghan, M. Numerical solution of a class of fractional optimal control problems via the Legendre orthonormal basis combined with the operational matrix and the Gauss quadrature rule. J. Comput. Appl. Math.
**2013**, 250, 143–160. [Google Scholar] [CrossRef]

**Figure 1.**(Example 1.) Comparison between the approximate state (

**left**) and control (

**right**) functions obtained by Approach I with $M=5$ and different $\alpha \left(t\right)$ (40).

**Figure 2.**(Example 1.) Comparison between the approximate state (

**left**) and control (

**right**) functions obtained by Approach II with $M=5$ and different $\alpha \left(t\right)$ (40).

**Figure 3.**(Example 2.) Comparison between the exact and approximate state (

**left**) and control (

**right**) functions obtained by Approach I with different values of M.

**Figure 4.**(Example 3.) Comparison between the exact and approximate state (

**left**) and control (

**right**) functions obtained by Approach I with $M=1$.

**Figure 5.**(Example 3.) Comparison between the exact and approximate state (

**left**) and control (

**right**) functions obtained by Approach II with different values of M.

**Table 1.**(Example 1.) Numerical results obtained by Approach II for the performance index with different M and $\alpha \left(t\right)=sin\left(t\right)$.

M | 1 | 2 | 3 | 2 | 5 |
---|---|---|---|---|---|

J | $6.80\times {10}^{-3}$ | $2.33\times {10}^{-3}$ | $1.76\times {10}^{-3}$ | $1.57\times {10}^{-3}$ | $1.56\times {10}^{-3}$ |

**Table 2.**(Example 1.) Numerical results for the performance index with $M=5$ and different $\alpha \left(t\right)$ (40).

Method | ${\mathit{\alpha}}_{1}\left(\mathit{t}\right)$ | ${\mathit{\alpha}}_{2}\left(\mathit{t}\right)$ | ${\mathit{\alpha}}_{3}\left(\mathit{t}\right)$ | ${\mathit{\alpha}}_{4}\left(\mathit{t}\right)$ |
---|---|---|---|---|

Approach I | $3.05\times {10}^{-33}$ | $3.26\times {10}^{-33}$ | $6.89\times {10}^{-33}$ | $2.08\times {10}^{-33}$ |

Approach II | $2.74\times {10}^{-33}$ | $1.56\times {10}^{-3}$ | $1.71\times {10}^{-4}$ | $2.50\times {10}^{-5}$ |

**Table 3.**(Example 2.) Numerical results for the performance index obtained by Approach I with different M.

M | 1 | 3 | 5 | 7 |
---|---|---|---|---|

J | $5.24\times {10}^{-4}$ | $7.59\times {10}^{-6}$ | $4.65\times {10}^{-7}$ | $5.86\times {10}^{-8}$ |

**Table 4.**(Example 3.) Numerical results for the performance index obtained by Approach II with different M.

M | 2 | 4 | 6 | 8 |
---|---|---|---|---|

J | $3.79\times {10}^{-4}$ | $5.42\times {10}^{-7}$ | $1.21\times {10}^{-8}$ | $7.36\times {10}^{-10}$ |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Nemati, S.; Torres, D.F.M.
Application of Bernoulli Polynomials for Solving Variable-Order Fractional Optimal Control-Affine Problems. *Axioms* **2020**, *9*, 114.
https://doi.org/10.3390/axioms9040114

**AMA Style**

Nemati S, Torres DFM.
Application of Bernoulli Polynomials for Solving Variable-Order Fractional Optimal Control-Affine Problems. *Axioms*. 2020; 9(4):114.
https://doi.org/10.3390/axioms9040114

**Chicago/Turabian Style**

Nemati, Somayeh, and Delfim F. M. Torres.
2020. "Application of Bernoulli Polynomials for Solving Variable-Order Fractional Optimal Control-Affine Problems" *Axioms* 9, no. 4: 114.
https://doi.org/10.3390/axioms9040114