A Comparative Approach for Time‐Delay Fractional Optimal Control Problems: Discrete Versus Continuous Chebyshev Polynomials

2018 ◽  
Vol 22 (1) ◽  
pp. 204-216 ◽  
Author(s):  
Leila Moradi ◽  
Fakhrodin Mohammadi
Keyword(s):  
Optimal Control ◽  
Time Delay ◽  
Chebyshev Polynomials ◽  
Optimal Control Problems ◽  
Comparative Approach ◽  
Control Problems ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

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.

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.

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.

Modified Adomian decomposition method for solving fractional optimal control problems

2017 ◽  
Vol 40 (6) ◽  
pp. 2054-2061 ◽  
Author(s):  
Ali Alizadeh ◽  
Sohrab Effati
Keyword(s):  
Optimal Control ◽  
Control Problem ◽  
Decomposition Method ◽  
Optimal Control Problems ◽  
Adomian Decomposition Method ◽  
Control Problems ◽  
Fractional Optimal Control ◽  
Modified Adomian Decomposition Method ◽  
Adomian Decomposition ◽  
Fractional Optimal Control Problems

In this study, we use the modified Adomian decomposition method to solve a class of fractional optimal control problems. The performance index of a fractional optimal control problem is considered as a function of both the state and the control variables, and the dynamical system is expressed in terms of a Caputo type fractional derivative. Some properties of fractional derivatives and integrals are used to obtain Euler–Lagrange equations for a linear tracking fractional control problem and then, the modified Adomian decomposition method is used to solve the resulting fractional differential equations. This technique rapidly provides convergent successive approximations of the exact solution to a linear tracking fractional optimal control problem. We compare the proposed technique with some numerical methods to demonstrate the accuracy and efficiency of the modified Adomian decomposition method by examining several illustrative test problems.

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