Numerical solution for a class of fractional optimal control problems using the fractional-order Bernoulli functions

2021 ◽  
pp. 014233122110470
Author(s):  
Forugh Valian ◽  
Yadollah Ordokhani ◽  
Mohammad Ali Vali
Keyword(s):  
Optimal Control ◽  
Fractional Order ◽  
Optimal Control Problems ◽  
Operational Matrix ◽  
Control Problems ◽  
Algebraic Equations ◽  
Fractional Optimal Control ◽  
Newton’S Iterative Method ◽  
Bernoulli Functions ◽  
Fractional Optimal Control Problems

The main purpose of this paper is to provide an efficient method for solving some types of fractional optimal control problems governed by integro-differential and differential equations, and because finding the analytical solutions to these problems is usually difficult, a numerical method is proposed. In this study, the fractional-order Bernoulli functions (F-BFs) are applied as basis functions and a new operational matrix of fractional integration is constructed for these functions. In the first step, the problem is transformed into an equivalent variational problem. Then the F-BFs, the constructed operational matrix, the Gauss quadrature formula, and necessary conditions for optimization are used to convert the problem into a system of algebraic equations. Finally, with the aid of Newton’s iterative method, the system of algebraic equations is solved and the approximate solution of the problem is obtained. Several numerical examples have been analysed for illustrating the efficiency and accuracy of the proposed method, and the results have been compared with the exact solutions and the results of other methods. The results show that the method provides accurate solutions.

Fractional-order Boubaker functions and their applications in solving delay fractional optimal control problems

2017 ◽  
Vol 24 (15) ◽  
pp. 3370-3383 ◽  
Author(s):  
Kobra Rabiei ◽  
Yadollah Ordokhani ◽  
Esmaeil Babolian
Keyword(s):  
Optimal Control ◽  
Fractional Order ◽  
Optimal Control Problems ◽  
Operational Matrix ◽  
Nonlinear Problems ◽  
Control Problems ◽  
Fractional Optimal Control ◽  
Newton’S Iterative Method ◽  
Quadratic Performance ◽  
Fractional Optimal Control Problems

In this paper, a new set of functions called fractional-order Boubaker functions is defined for solving the delay fractional optimal control problems with a quadratic performance index. To solve the problem, first we obtain the operational matrix of the Caputo fractional derivative of these functions and the operational matrix of multiplication to solve the nonlinear problems for the first time. Also, a general formulation for the delay operational matrix of these functions has been achieved. Then we utilized these matrices to solve delay fractional optimal control problems directly. In fact, the delay fractional optimal control problem converts to an optimization problem, which can then be easily solved with the aid of the Gauss–Legendre integration formula and Newton’s iterative method. Convergence of the algorithm is proved. The applicability of the method is shown by some examples; moreover, a comparison with the existing results shows the preference of this method.

Numerical solution for fractional optimal control problems by Hermite polynomials

2020 ◽  
pp. 107754632093312
Author(s):  
Ayatollah Yari
Keyword(s):  
Optimal Control ◽  
Numerical Method ◽  
Optimal Control Problems ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Control Problems ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

In this study, a numerical method based on Hermite polynomial approximation for solving a class of fractional optimal control problems is presented. The order of the fractional derivative is taken as less than one and described in the Caputo sense. Operational matrices of integration by using such known formulas as Caputo and Riemann–Liouville operators for computing fractional derivatives and integration of polynomials is introduced and used to reduce the problem of a system of algebraic equations. The convergence of the proposed method is analyzed, and the error upper bound for the operational matrix of the fractional integration is obtained. To confirm the validity and accuracy of the proposed numerical method, three numerical examples are presented along with a comparison between our numerical results and those obtained using Legendre polynomials. Illustrative examples are included to demonstrate the validity and applicability of the new technique.

A new efficient numerical scheme for solving fractional optimal control problems via a Genocchi operational matrix of integration

2017 ◽  
Vol 24 (14) ◽  
pp. 3036-3048 ◽  
Author(s):  
Chang Phang ◽  
Noratiqah Farhana Ismail ◽  
Abdulnasir Isah ◽  
Jian Rong Loh
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Operational Matrix ◽  
Control Problems ◽  
Algebraic Equations ◽  
Fractional Optimal Control ◽  
Dynamic Constraints ◽  
Genocchi Polynomials ◽  
Fractional Optimal Control Problems ◽  
Operational Matrix Of Integration

In this paper, a new operational matrix of integration is derived using Genocchi polynomials, which is one of the Appell polynomials. By using the matrix, we develop an efficient, direct and new numerical method for solving a class of fractional optimal control problems. The fractional derivative in the dynamic constraints was replaced with the Genocchi polynomials with unknown coefficients and a Genocchi operational matrix of fractional integration. Then, the equation derived from the dynamic constraints was put into the performance index. Hence, the fractional optimal control problems will be reduced to fractional variational problems. By finding a necessary condition for the optimality for the performance index, we will obtain a system of algebraic equations that can be easily solved by using any numerical method. Hence, we obtain the value of unknown coefficients of Genocchi polynomials. Lastly, the solution of the fractional optimal control problems will be obtained. In short, the properties of Genocchi polynomials are utilized to reduce the given problems to a system of algebraic equations. The approximation approach is simple to use and computer oriented. Illustrative examples are given to show the simplicity, accuracy and applicability of the method.

Fibonacci wavelets and Galerkin method to investigate fractional optimal control problems with bibliometric analysis

2020 ◽  
pp. 107754632094834 ◽  
Author(s):  
Sedigheh Sabermahani ◽  
Yadollah Ordokhani
Keyword(s):  
Optimal Control ◽  
Galerkin Method ◽  
Optimal Control Problems ◽  
Operational Matrix ◽  
Inequality Constraints ◽  
Computational Method ◽  
Control Problems ◽  
Algebraic Equations ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

This study presents a computational method for the solution of the fractional optimal control problems subject to fractional systems with equality and inequality constraints. The proposed procedure is based upon Fibonacci wavelets. The fractional derivative is described in the Caputo sense. The Riemann–Liouville operational matrix for Fibonacci wavelets is obtained. Then, we use this operational matrix and the Galerkin method to reduce the given problem into a system of algebraic equations. We discuss the convergence of the algorithm. Several numerical examples are included to observe the validity, effectiveness, and accuracy of the suggested scheme. Moreover, fractional optimal control problems are studied through a bibliometric viewpoint.

Shifted Chebyshev schemes for solving fractional optimal control problems

2019 ◽  
Vol 25 (15) ◽  
pp. 2143-2150 ◽  
Author(s):  
M Abdelhakem ◽  
H Moussa ◽  
D Baleanu ◽  
M El-Kady
Keyword(s):  
Optimal Control ◽  
Fractional Order ◽  
Chebyshev Polynomials ◽  
Optimal Control Problems ◽  
Optimization Problems ◽  
Control Problems ◽  
Numerical Examples ◽  
Fractional Optimal Control ◽  
Functional Approximation ◽  
Fractional Optimal Control Problems

Two schemes to find approximated solutions of optimal control problems of fractional order (FOCPs) are investigated. Integration and differentiation matrices were used in these schemes. These schemes used Chebyshev polynomials in the shifted case as a functional approximation. The target of the presented schemes is to convert such problems to optimization problems (OPs). Numerical examples are included, showing the strength of the schemes.

An efficient approximate method for solving two-dimensional fractional optimal control problems using generalized fractional order of Bernstein functions

2020 ◽  
Author(s):  
Ali Ketabdari ◽  
Mohammad Hadi Farahi ◽  
Sohrab Effati
Keyword(s):  
Optimal Control ◽  
Fractional Order ◽  
Operational Matrix ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Control Variables ◽  
Fractional Optimal Control ◽  
Bernstein Functions ◽  
Fractional Optimal Control Problems ◽  
And Control

Abstract We define a new operational matrix of fractional derivative in the Caputo type and apply a spectral method to solve a two-dimensional fractional optimal control problem (2D-FOCP). To acquire this aim, first we expand the state and control variables based on the fractional order of Bernstein functions. Then we reduce the constraints of 2D-FOCP to a system of algebraic equations through the operational matrix. Now, one can solve straightforward the problem and drive the approximate solution of state and control variables. The convergence of the method in approximating the 2D-FOCP is proved. We demonstrate the efficiency and superiority of the method by comparing the results obtained by the presented method with the results of previous methods in some examples.

Study of B-spline collocation method for solving fractional optimal control problems

2021 ◽  
pp. 014233122098753
Author(s):  
Yousef Edrisi-Tabriz ◽  
Mehrdad Lakestani ◽  
Mohsen Razzaghi
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Direct Method ◽  
Operational Matrix ◽  
Control Problems ◽  
Spline Collocation ◽  
B Spline ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems ◽  
A Direct Method

In this article, a class of fractional optimal control problems (FOCPs) are solved using a direct method. We present a new operational matrix of the fractional derivative in the sense of Caputo based on the B-spline functions. Then we reduce the solution of fractional optimal control problem to a nonlinear programming (NLP) one, where some existing well-developed algorithms may be applied. Numerical results demonstrate the efficiency of the presented technique.

A reliable numerical approach for nonlinear fractional optimal control problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Harendra Singh ◽  
Rajesh K. Pandey ◽  
Devendra Kumar
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Jacobi Polynomials ◽  
Operational Matrix ◽  
Numerical Approach ◽  
Numerical Results ◽  
Control Problems ◽  
Fractional Optimal Control ◽  
Special Cases ◽  
Fractional Optimal Control Problems

AbstractIn this work, we study a numerical approach for studying a nonlinear model of fractional optimal control problems (FOCPs). We have taken the fractional derivative in a dynamical system of FOCPs, which is in Liouville–Caputo sense. The presented scheme is a grouping of an operational matrix of integrations for Jacobi polynomials and the Ritz method. The proposed approach converts the FOCP into a system of nonlinear algebraic equations, which significantly simplify the problem. Convergence analysis of the scheme is also provided. The presented method is verified on the two illustrative examples to show its accuracy and applicability. Distinct special cases of Jacobi polynomials are considered as a basis to solve the FOCPs for comparison purpose. Further, tables and figures are employed to demonstrate the derived numerical results. The numerical results by the present method are also compared with some other techniques.

Sign in / Sign up

Export Citation Format

Share Document