scholarly journals Numerical solutions of fractional optimal control with Caputo–Katugampola derivative

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
N. H. Sweilam ◽  
A. M. Nagy ◽  
T. M. Al-Ajami
Keyword(s):  
Optimal Control ◽  
Fractional Derivative ◽  
Optimal Control Problems ◽  
Numerical Solutions ◽  
Numerical Technique ◽  
Ritz Method ◽  
Derivative Operator ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems ◽  
And Control

AbstractIn this paper, we present a numerical technique for solving fractional optimal control problems with a fractional derivative called Caputo–Katugampola derivative. This derivative is a generalization of the Caputo fractional derivative. The proposed technique is based on a spectral method using shifted Chebyshev polynomials of the first kind. The Clenshaw and Curtis scheme for the numerical integration and the Rayleigh–Ritz method are used to estimate the state and control variables. Moreover, the error bound of the fractional derivative operator approximation of Caputo–Katugampola is derived. Illustrative examples are provided to show the validity and applicability of the presented 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.

Solving a class of fractional optimal control problems by the Hamilton–Jacobi–Bellman equation

2016 ◽  
Vol 24 (9) ◽  
pp. 1741-1756 ◽  
Author(s):  
Seyed Ali Rakhshan ◽  
Sohrab Effati ◽  
Ali Vahidian Kamyad
Keyword(s):  
Optimal Control ◽  
Fractional Derivative ◽  
Bellman Equation ◽  
Optimal Control Problems ◽  
Control Problems ◽  
Fractional Optimal Control ◽  
Solution Scheme ◽  
Hamilton Jacobi Bellman Equation ◽  
Fractional Optimal Control Problems ◽  
Hamilton Jacobi Bellman

The performance index of both the state and control variables with a constrained dynamic optimization problem of a fractional order system with fixed final Time have been considered here. This paper presents a general formulation and solution scheme of a class of fractional optimal control problems. The method is based upon finding the numerical solution of the Hamilton–Jacobi–Bellman equation, corresponding to this problem, by the Legendre–Gauss collocation method. The main reason for using this technique is its efficiency and simple application. Also, in this work, we use the fractional derivative in the Riemann–Liouville sense and explain our method for a fractional derivative of order of [Formula: see text]. Numerical examples are provided to show the effectiveness of the formulation and solution scheme.

Numerical solutions for solving a class of fractional optimal control problems via fixed-point approach

SeMA Journal ◽  
2017 ◽  
Vol 74 (4) ◽  
pp. 585-603 ◽  
Author(s):  
Samaneh Soradi Zeid ◽  
Ali Vahidian Kamyad ◽  
Sohrab Effati ◽  
Seyed Ali Rakhshan ◽  
Soleiman Hosseinpour
Keyword(s):  
Optimal Control ◽  
Fixed Point ◽  
Optimal Control Problems ◽  
Numerical Solutions ◽  
Control Problems ◽  
Fractional Optimal Control ◽  
Fixed Point Approach ◽  
Fractional Optimal Control Problems

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.

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