Using the Generalize Laguerre Operational Matrix for The Caputo Fractional Derivatives to Solve Some Classes of 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.

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

Transactions of the Institute of Measurement and Control ◽

10.1177/0142331220987537 ◽

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.

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A reliable numerical approach for nonlinear fractional optimal control problems

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2019-0272 ◽

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.

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

Journal of Vibration and Control ◽

10.1177/1077546317705041 ◽

2017 ◽

Vol 24 (15) ◽

pp. 3370-3383 ◽

Cited By ~ 15

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.

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Numerical solution of 2D fractional optimal control problems by the spectral method along with Bernstein operational matrix

International Journal of Control ◽

10.1080/00207179.2017.1334267 ◽

2017 ◽

Vol 91 (12) ◽

pp. 2632-2645 ◽

Cited By ~ 8

Author(s):

Ali Nemati

Keyword(s):

Optimal Control ◽

Numerical Solution ◽

Spectral Method ◽

Optimal Control Problems ◽

Operational Matrix ◽

Control Problems ◽

Fractional Optimal Control ◽

Fractional Optimal Control Problems

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Numerical Method for Solving Fractional Optimal Control Problems

Volume 4: 7th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B and C ◽

10.1115/detc2009-87008 ◽

2009 ◽

Cited By ~ 9

Author(s):

Raj Kumar Biswas ◽

Siddhartha Sen

Keyword(s):

Optimal Control ◽

Optimal Control Problems ◽

Numerical Technique ◽

Control Problems ◽

Caputo Fractional Derivatives ◽

Fractional Optimal Control ◽

Liouville Derivative ◽

Fractional Differential ◽

Fractional Optimal Control Problems ◽

The Right

A numerical technique for the solution of a class of fractional optimal control problems has been proposed in this paper. The technique can used for problems defined both in terms of Riemann-Liouville and Caputo fractional derivatives. In this technique a Reflection Operator is used to convert the right Riemann-Liouville derivative into an equivalent left Riemann-Liouville derivative, and then the two point boundary value problem is solved numerically. The proposed method is straightforward and it uses an available numerical technique to solve fractional differential equations resulting from the formulation. Examples considered here show that the numerical results obtained using this and other techniques agree very well.

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Numerical solution for fractional optimal control problems by Hermite polynomials

Journal of Vibration and Control ◽

10.1177/1077546320933129 ◽

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.

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

Transactions of the Institute of Measurement and Control ◽

10.1177/01423312211047033 ◽

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.

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

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4039900 ◽

2018 ◽

Vol 13 (6) ◽

Cited By ~ 5

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.

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