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 numerical approach for solving fractional optimal control problems with mittag-leffler kernel

2021 ◽  
pp. 107754632110169
Author(s):  
Hossein Jafari ◽  
Roghayeh M Ganji ◽  
Khosro Sayevand ◽  
Dumitru Baleanu
Keyword(s):  
Optimal Control ◽  
Approximate Solution ◽  
Optimal Control Problems ◽  
Fractional Integration ◽  
Numerical Approach ◽  
Control Problems ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

In this work, we present a numerical approach based on the shifted Legendre polynomials for solving a class of fractional optimal control problems. The derivative is described in the Atangana–Baleanu derivative sense. To solve the problem, operational matrices of AB-fractional integration and multiplication, together with the Lagrange multiplier method for the constrained extremum, are considered. The method reduces the main problem to a system of nonlinear algebraic equations. In this framework by solving the obtained system, the approximate solution is calculated. An error estimate of the numerical solution is also proved for the approximate solution obtained by the proposed method. Finally, some illustrative examples are presented to demonstrate the accuracy and validity of the proposed scheme.

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.

Formulation of Euler–Lagrange Equations for Multidelay Fractional Optimal Control Problems

2018 ◽  
Vol 13 (6) ◽  
Author(s):  
Sohrab Effati ◽  
Seyed Ali Rakhshan ◽  
Samane Saqi
Keyword(s):  
Optimal Control ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Optimal Control Problems ◽  
Fractional Integration ◽  
Control Problems ◽  
Algebraic Equations ◽  
Lagrange Equations ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

In this paper, a new numerical scheme is proposed for multidelay fractional order optimal control problems where its derivative is considered in the Grunwald–Letnikov sense. We develop generalized Euler–Lagrange equations that results from multidelay fractional optimal control problems (FOCP) with final terminal. These equations are created by using the calculus of variations and the formula for fractional integration by parts. The derived equations are then reduced into system of algebraic equations by using a Grunwald–Letnikov approximation for the fractional derivatives. Finally, for confirming the accuracy of the proposed approach, some illustrative numerical examples are solved.

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.

A direct numerical solution of time-delay fractional optimal control problems by using Chelyshkov wavelets

2018 ◽  
Vol 25 (2) ◽  
pp. 310-324 ◽  
Author(s):  
L Moradi ◽  
F Mohammadi ◽  
D Baleanu
Keyword(s):  
Optimal Control ◽  
Time Delay ◽  
Optimal Control Problems ◽  
Operational Matrix ◽  
The State ◽  
Control Problems ◽  
Operational Matrices ◽  
State Function ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

The aim of the present study is to present a numerical algorithm for solving time-delay fractional optimal control problems (TDFOCPs). First, a new orthonormal wavelet basis, called Chelyshkov wavelet, is constructed from a class of orthonormal polynomials. These wavelet functions and their properties are implemented to derive some operational matrices. Then, the fractional derivative of the state function in the dynamic constraint of TDFOCPs is approximated by means of the Chelyshkov wavelets. The operational matrix of fractional integration together with the dynamical constraints is used to approximate the control function directly as a function of the state function. Finally, these approximations were put in the performance index and necessary conditions for optimality transform the under consideration TDFOCPs into an algebraic system. Moreover, some illustrative examples are considered and the obtained numerical results were compared with those previously published in the literature.

Research on a collocation approach and three metaheuristic techniques based on MVO, MFO, and WOA for optimal control of fractional differential equation

2021 ◽  
pp. 107754632110514
Author(s):  
Asiyeh Ebrahimzadeh ◽  
Raheleh Khanduzi ◽  
Samaneh P A Beik ◽  
Dumitru Baleanu
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Focal Point ◽  
Control Problems ◽  
Operational Matrices ◽  
State Variables ◽  
Caputo Fractional Derivatives ◽  
Fractional Optimal Control ◽  
Whale Optimization ◽  
Fractional Optimal Control Problems

Exploiting a comprehensive mathematical model for a class of systems governed by fractional optimal control problems is the significant focal point of the current paper. The efficiency index is a function of both control and state variables and the dynamic control system relies on Caputo fractional derivatives. The attributes of Bernoulli polynomials and their operational matrices of fractional Riemann–Liouville integrations are applied to convert the optimization problem to the nonlinear programing problem. Executing multi-verse optimizer, moth-flame optimization, and whale optimization algorithm terminate to the most excellent solution of fractional optimal control problems. A study on the advantage and performance between these approaches is analyzed by some examples. Comprehensive analysis ascertains that moth-flame optimization significantly solves the example. Furthermore, the privilege and advantage of preference with its accuracy are numerically indicated. Finally, results demonstrate that the objective function value gained by moth-flame optimization in comparison with other algorithms effectively decreased.

A Quadratic Numerical Scheme for Fractional Optimal Control Problems

2007 ◽  
Vol 130 (1) ◽  
Author(s):  
Om P. Agrawal
Keyword(s):  
Optimal Control ◽  
Control Theory ◽  
Numerical Scheme ◽  
Optimal Control Problems ◽  
Fractional Integration ◽  
Classical Solutions ◽  
Control Problems ◽  
Lagrange Equations ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

This paper presents a quadratic numerical scheme for a class of fractional optimal control problems (FOCPs). The fractional derivative is described in the Caputo sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of fractional differential equations. The calculus of variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler–Lagrange equations for the FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic systems. The formulation is used to derive the control equations for a quadratic linear fractional control problem. For a linear system, this method results into a set of linear simultaneous equations, which can be solved using a direct or an iterative scheme. Numerical results for a FOCP are presented to demonstrate the feasibility of the method. It is shown that the solutions converge as the number of grid points increases, and the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs.

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