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 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 Method for Solving Fractional Optimal Control Problems

2009 ◽  
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.

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 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.

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