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.

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 Method for Solving Fractional Optimal Control Problems Using Ritz Method

2016 ◽  
Vol 11 (5) ◽  
Author(s):  
Ali Nemati ◽  
Sohrab Ali Yousefi
Keyword(s):  
Optimal Control ◽  
Fractional Derivative ◽  
Optimal Control Problems ◽  
Ritz Method ◽  
Operational Matrix ◽  
New Method ◽  
Control Problems ◽  
State Function ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

Our paper presents a new method to solve a class of fractional optimal control problems (FOCPs) based on the numerical polynomial approximation. In the proposed method, the fractional derivative in the dynamical system is considered in the Caputo sense. The approach used here is to approximate the state function by the Legendre orthonormal basis by using the Ritz method. Next, we apply a new constructed operational matrix to approximate fractional derivative of the basis. After transforming the problem into a system of algebraic equations, the problem is solved via the Newton's iterative method. Finally, the convergence of the new method is investigated and some examples are included to illustrate the effectiveness and applicability of the proposed methodology.

The approximate solution of non-linear time-delay fractional optimal control problems by embedding process

2018 ◽  
Vol 36 (3) ◽  
pp. 713-727 ◽  
Author(s):  
E Ziaei ◽  
M H Farahi
Keyword(s):  
Optimal Control ◽  
Time Delay ◽  
Fractional Derivative ◽  
Optimal Control Problems ◽  
Linear Time ◽  
Control Problems ◽  
Linear Dynamical Systems ◽  
Fractional Optimal Control ◽  
Non Linear ◽  
Fractional Optimal Control Problems

Abstract In this paper, a class of time-delay fractional optimal control problems (TDFOCPs) is studied. Delays may appear in state or control (or both) functions. By an embedding process and using conformable fractional derivative as a new definition of fractional derivative and integral, the class of admissible pair (state, control) is replaced by a class of positive Radon measures. The optimization problem found in measure space is then approximated by a linear programming problem (LPP). The optimal measure which is representing optimal pair is approximated by the solution of a LPP. In this paper, we have shown that the embedding method (embedding the admissible set into a subset of measures), successfully can be applied to non-linear TDFOCPs. The usefulness of the used idea in this paper is that the method is not iterative, quite straightforward and can be applied to non-linear dynamical systems.

scholarly journals Numerical Solution of Some Types of Fractional Optimal Control Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Nasser Hassan Sweilam ◽  
Tamer Mostafa Al-Ajami ◽  
Ronald H. W. Hoppe
Keyword(s):  
Optimal Control ◽  
Numerical Solution ◽  
Optimal Control Problems ◽  
Ritz Method ◽  
Necessary Optimality Conditions ◽  
State Equation ◽  
The State ◽  
Control Problems ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

We present two different approaches for the numerical solution of fractional optimal control problems (FOCPs) based on a spectral method using Chebyshev polynomials. The fractional derivative is described in the Caputo sense. The first approach follows the paradigm “optimize first, then discretize” and relies on the approximation of the necessary optimality conditions in terms of the associated Hamiltonian. In the second approach, the state equation is discretized first using the Clenshaw and Curtis scheme for the numerical integration of nonsingular functions followed by the Rayleigh-Ritz method to evaluate both the state and control variables. Two illustrative examples are included to demonstrate the validity and applicability of the suggested approaches.

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.

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.

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