A Numerical Scheme and an Error Analysis for a Class of Fractional Optimal Control Problems

2009 ◽  
Author(s):  
Om P. Agrawal
Keyword(s):  
Optimal Control ◽  
Error Analysis ◽  
Fractional Derivative ◽  
Performance Index ◽  
Numerical Solutions ◽  
Fractional Power ◽  
The State ◽  
Control Variables ◽  
Fractional Optimal Control ◽  
Linear Algebraic Equations

There has been a growing interest in recent years in the area of Fractional Optimal Control (FOC). In this paper, we present a formulation for a class of FOC problems, in which a performance index is defined as an integral of a quadratic function of the state and the control variables, and a dynamic constraint is defined as a Fractional Differential Equation (FDE) linear in both the state and the control variables. The fractional derivative is defined in the Caputo sense. In this formulation, the FOC problem is reduced to a Fractional Variational Problem (FVP), and the necessary differential equations for the problems are obtained using the recently developed theories for FVPs. For the numerical solutions of the problems, a direct approach is taken in which the solutions are approximated using a truncated Fractional Power Series (FPS). An error analysis is also performed. It is demonstrated that the solution converges from above in the sense that the value of the approximate performance index is always higher than the optimum performance index. An expression for the error in the performance index is also given. The application of a FPS and an optimality criterion reduces the FOC to a set of linear algebraic equations which are solved using a linear solver. It is demonstrated numerically that the solution converges as the number of terms in the series increases, and the approximate solution approaches to the analytical solution as the order of the fractional derivative approaches to an integer order derivative. Numerical results are presented to demonstrate the performance of the Formulation.

A numerical approach based on Legendre orthonormal polynomials for numerical solutions of fractional optimal control problems

2016 ◽  
Vol 23 (1) ◽  
pp. 16-30 ◽  
Author(s):  
SS Ezz-Eldien ◽  
EH Doha ◽  
D Baleanu ◽  
AH Bhrawy
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Fractional Derivative ◽  
Control Problem ◽  
Performance Index ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Fractional Optimal Control Problem ◽  
Fractional Optimal Control ◽  
Orthonormal Polynomials

The numerical solution of a fractional optimal control problem having a quadratic performance index is proposed and analyzed. The performance index of the fractional optimal control problem is considered as a function of both the state and the control variables. The dynamic constraint is expressed as a fractional differential equation that includes an integer derivative in addition to the fractional derivative. The order of the fractional derivative is taken as less than one and described in the Caputo sense. Based on the shifted Legendre orthonormal polynomials, we employ the operational matrix of fractional derivatives, the Legendre–Gauss quadrature formula and the Lagrange multiplier method for reducing such a problem into a problem consisting of solving a system of algebraic equations. The convergence of the proposed method is analyzed. For confirming the validity and accuracy of the proposed numerical method, a numerical example is presented along with a comparison between our numerical results and those obtained using the Legendre spectral-collocation method.

Fractional Optimal Control of a Distributed System Using Eigenfunctions

2007 ◽  
Author(s):  
Om P. Agrawal
Keyword(s):  
Optimal Control ◽  
Distributed Systems ◽  
Dimensional Space ◽  
Numerical Technique ◽  
The State ◽  
Finite Dimensional Space ◽  
Lagrange Equations ◽  
Time Step ◽  
Control Variables ◽  
Fractional Optimal Control

This paper presents a formulation and a numerical scheme for Fractional Optimal Control (FOC) for a class of distributed systems. The fractional derivative is defined in the Caputo sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equations (PFDEs). Eigenfunctions are used to eliminate the space parameter, and to define the problem in terms of a set of state and control variables. This leads to a multi FOCP in which each FOCP could be solved independently. Several other strategies are pointed out to reduce the problem to a finite dimensional space, some of which may not provide a decoupled set of equations. The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler-Lagrange equations for the problem. The numerical technique presented in [1] is used to obtain the state and the control variables. In this technique, the FOC equations are reduced to Volterra type integral equations. The time domain is descretized into several segments and a time marching scheme is used to obtain the response at discrete time points. For a linear case, the numerical technique results into a set of algebraic equations which can be solved using a direct or an iterative scheme. The problem is solved for different number of eigenfunctions and time discretizations. Numerical results show that only a few eigenfunctions are sufficient to obtain good results, and the solutions converge as the size of the time step is reduced. The formulation presented is simple and can be extended to FOC of other distributed systems.

scholarly journals A numerical method for solving a class of fractional optimal control problems using Boubaker polynomial expansion scheme

Filomat ◽  
2018 ◽  
Vol 32 (13) ◽  
pp. 4485-4502 ◽  
Author(s):  
N. Singha ◽  
C. Nahak
Keyword(s):  
Optimal Control ◽  
Performance Index ◽  
Numerical Scheme ◽  
Optimal Control Problems ◽  
Control Variable ◽  
The State ◽  
Polynomial Expansion ◽  
Control Problems ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

We construct a numerical scheme for solving a class of fractional optimal control problems by employing Boubaker polynomials. In the proposed scheme, the state and control variables are approximated by practicingNth-order Boubaker polynomial expansion. With these approximations, the given performance index is transformed to a function of N + 1 unknowns. The objective of the present formulation is to convert a fractional optimal control problem with quadratic performance index into an equivalent quadratic programming problem with linear equality constraints. Thus, the latter problem can be handled efficiently in comparison to the original problem. We solve several examples to exhibit the applicability and working mechanism of the presented numerical scheme. Graphical plots are provided to monitor the nature of the state, control variable and the absolute error function. All the numerical computations and graphical representations have been executed with the help of Mathematica software.

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.

An efficient approximate method for solving two-dimensional fractional optimal control problems using generalized fractional order of Bernstein functions

2020 ◽  
Author(s):  
Ali Ketabdari ◽  
Mohammad Hadi Farahi ◽  
Sohrab Effati
Keyword(s):  
Optimal Control ◽  
Fractional Order ◽  
Operational Matrix ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Control Variables ◽  
Fractional Optimal Control ◽  
Bernstein Functions ◽  
Fractional Optimal Control Problems ◽  
And Control

Abstract We define a new operational matrix of fractional derivative in the Caputo type and apply a spectral method to solve a two-dimensional fractional optimal control problem (2D-FOCP). To acquire this aim, first we expand the state and control variables based on the fractional order of Bernstein functions. Then we reduce the constraints of 2D-FOCP to a system of algebraic equations through the operational matrix. Now, one can solve straightforward the problem and drive the approximate solution of state and control variables. The convergence of the method in approximating the 2D-FOCP is proved. We demonstrate the efficiency and superiority of the method by comparing the results obtained by the presented method with the results of previous methods in some examples.

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.

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