An approximate method for solving fractional optimal control problems by the hybrid of block‐pulse functions and Taylor polynomials

Yonthanthum, Winita; Rattana, Amornrat; Razzaghi, Mohsen

doi:10.1002/oca.2383

Cited by 20 publications

(18 citation statements)

References 56 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Furthermore, the approximate and exact state and control functions obtained by the present method with n = 50 and n = 100, together with the corresponding error functions, are plotted in Figure 1. [19,20,21]:…”

Section: Illustrative Examplesmentioning

confidence: 99%

“…It is seen that the present method gives the exact solution of J, with seven significant digits, with n = 32 and that the convergence order of the state function is O(h 4 ). In Table 3, the method of [21] refers to the hybrid block-pulse and Taylor polynomials method, the method of [20] denotes the method of hybrid block-pulse and Bernoulli polynomials, and the method of [19] refers to the Bernstein polynomials method.…”

Section: Illustrative Examplesmentioning

confidence: 99%

See 1 more Smart Citation

A numerical approach for solving fractional optimal control problems using modified hat functions

Nemati

Lima

Torres

2019

Communications in Nonlinear Science and Numerical Simulation

View full text Add to dashboard Cite

We introduce a numerical method, based on modified hat functions, for solving a class of fractional optimal control problems. In our scheme, the control and the fractional derivative of the state function are considered as linear combinations of the modified hat functions. The fractional derivative is considered in the Caputo sense while the Riemann-Liouville integral operator is used to give approximations for the state function and some of its derivatives. To this aim, we use the fractional order integration operational matrix of the modified hat functions and some properties of the Caputo derivative and Riemann-Liouville integral operators. Using results of the considered basis functions, solving the fractional optimal control problem is reduced to the solution of a system of nonlinear algebraic equations. An error bound is proved for the approximate optimal value of the performance index obtained by the proposed method. The method is then generalized for solving a class of fractional optimal control problems with inequality constraints. The most important advantages of our method are easy implementation, simple operations, and elimination of numerical integration. Some illustrative examples are considered to demonstrate the effectiveness and accuracy of the proposed technique.Recently, numerical methods based on modified hat functions have shown to provide an effective way for solving fractional differential equations [15,16]. Here, for the first time in the literature, we introduce such a method for solving a general class of fractional optimal control problems. More precisely, in this work we begin by considering the following fractional optimal control problem (FOCP):

show abstract

Section: Illustrative Examplesmentioning

confidence: 99%

Section: Illustrative Examplesmentioning

confidence: 99%

A numerical approach for solving fractional optimal control problems using modified hat functions

Nemati

Lima

Torres

2019

Communications in Nonlinear Science and Numerical Simulation

View full text Add to dashboard Cite

show abstract

“…Nonstandard finite difference method used to obtain numerical solutions of variable order FOCPs was presented in Reference 30. Finally, numerical schemes for solving FOCPs were carried out in References 31‐36.…”

Section: Introductionmentioning

confidence: 99%

Optimal control problems with Atangana‐Baleanu fractional derivative

Tajadodi

Khan

Gómez‐Aguilar

et al. 2020

Optim Control Appl Methods

View full text Add to dashboard Cite

SummaryIn this paper, we study fractional‐order optimal control problems (FOCPs) involving the Atangana‐Baleanu fractional derivative. A computational method based on B‐spline polynomials and their operational matrix of Atangana‐Baleanu fractional integration is proposed for the numerical solution of this class of problems. With this numerical technique, the FOCPs are reduced to a system of equations which are solved for the unknown parameters with the help of Mathematica® software. Our results show the applicability and usefulness of the numerical technique.

show abstract

“…In recent the years, several numerical methods have been devoted to solve of one‐dimensional FOCPs. We list here some of these numerical methods, as follows: Eigen functions method (Agrawal); Quadratic numerical scheme (Agrawal); Rational approximation method (Tricaud and Chen); Legendre multiwavelet collocation method (Yousefi et al); Legendre orthonormal basis method (Lotfi et al); Bessel collocation method (Tohidi and Nik); Legendre operational technique (Bhrawy and Ezz‐Eldien); Bernoulli polynomials method (Keshavarz et al); Bernoulli wavelet method (Rahimkhani et al); Boubaker polynomials method (Rabiei et al); Hybrid of block‐pulse functions and Bernoulli polynomials (Mashayekhi and Razzaghi); Bernstein operational matrix method (Nemati et al); The hybrid of block‐pulse functions and Taylor polynomials method (Yonthanthum et al); A Legendre collocation method (Zaky). …”

Section: Introductionmentioning

confidence: 99%

Numerical solution a class of 2D fractional optimal control problems by using 2D Müntz‐Legendre wavelets

Rahimkhani

Ordokhani

2018

Optim Control Appl Methods

View full text Add to dashboard Cite

In this paper, a method for finding an approximate solution of a class of 2D fractional optimal control problems with fractional-order dynamical system is discussed. In the proposed method, the fractional derivative is expressed in the Caputo sense. The method consists of expanding unknown functions as the elements of two-dimensional (2D) Müntz-Legendre wavelets. The 2DMüntz-Legendre wavelets are constructed and their properties are presented.The operational matrix of fractional-order integration for these wavelets is utilized to reduce the solution of 2D fractional optimal control problem to an optimization problem, which can then be solved easily. Some results concerning the error analysis are obtained. Finally, two illustrative test problems are included to demonstrate the validity and applicability of the technique. Moreover, our achievements are compared with the previous results to show the superiority of the proposed method. KEYWORDSCaputo derivative, fractional optimal control problem, numerical solution, Riemann-Liouville fractional integral operator, wavelet method INTRODUCTIONNowadays, fractional calculus have a wide range of applications in fields of fluid-dynamic traffic, 1 frequency dependent damping behavior of various viscoelastic materials, 2 colored noise, 3 solid mechanics, 4 economics, 5 signal processing, 6 and control theory. 7 Therefore, many researchers have been interested in finding the approximate solutions of these problems, and some of these methods include spectral Tau method, 8 finite difference method, 9 wavelet method, 10 least square approximation method, 11 finite element method, 12 spline collocation method, 13 fractional wavelet method, 14 shifted fractional-order Jacobi orthogonal functions, 15 Jacobi Tau method, 16,17 and Legendre Tau method. 18 An optimal control problem (OCP) is defined to minimize the functional over an admissible set of control functions subject to dynamic constraints on the state and control variables. A fractional OCP (FOCP) is an optimal control problem in which the performance index or the differential equations governing the dynamics of the system or both contains at least one fractional-order derivative term. In recent the years, several numerical methods have been devoted to solve of one-dimensional FOCPs. We list here some of these numerical methods, as follows: • Eigen functions method (Agrawal 19 ); • Quadratic numerical scheme (Agrawal 20 ); • Rational approximation method (Tricaud and Chen 21 ); • Legendre multiwavelet collocation method (Yousefi et al 22 ); 1916

show abstract

An approximate method for solving fractional optimal control problems by the hybrid of block‐pulse functions and Taylor polynomials

Cited by 20 publications

References 56 publications

A numerical approach for solving fractional optimal control problems using modified hat functions

A numerical approach for solving fractional optimal control problems using modified hat functions

Optimal control problems with Atangana‐Baleanu fractional derivative

Numerical solution a class of 2D fractional optimal control problems by using 2D Müntz‐Legendre wavelets

Contact Info

Product

Resources

About