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 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.

A Quadratic Numerical Scheme for Fractional Optimal Control Problems

2007 ◽  
Vol 130 (1) ◽  
Author(s):  
Om P. Agrawal
Keyword(s):  
Optimal Control ◽  
Control Theory ◽  
Numerical Scheme ◽  
Optimal Control Problems ◽  
Fractional Integration ◽  
Classical Solutions ◽  
Control Problems ◽  
Lagrange Equations ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

This paper presents a quadratic numerical scheme for a class of fractional optimal control problems (FOCPs). The fractional derivative is described 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 set of fractional differential 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 FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic systems. The formulation is used to derive the control equations for a quadratic linear fractional control problem. For a linear system, this method results into a set of linear simultaneous equations, which can be solved using a direct or an iterative scheme. Numerical results for a FOCP are presented to demonstrate the feasibility of the method. It is shown that the solutions converge as the number of grid points increases, and the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs.

Comparison on wavelets techniques for solving fractional optimal control problems

2016 ◽  
Vol 24 (6) ◽  
pp. 1185-1201 ◽  
Author(s):  
PK Sahu ◽  
S Saha Ray
Keyword(s):  
Optimal Control ◽  
Performance Index ◽  
Optimal Control Problems ◽  
Control Problems ◽  
Algebraic Equations ◽  
Fractional Optimal Control ◽  
Dynamic Constraints ◽  
Orthonormal Wavelets ◽  
Fractional Optimal Control Problems ◽  
Wavelet Methods

This paper presents efficient numerical techniques for solving fractional optimal control problems (FOCP) based on orthonormal wavelets. These wavelets are like Legendre wavelets, Chebyshev wavelets, Laguerre wavelets and Cosine And Sine (CAS) wavelets. The formulation of FOCP and properties of these wavelets are presented. The fractional derivative considered in this problem is in the Caputo sense. The performance index of FOCP has been considered as function of both state and control variables and the dynamic constraints are expressed by fractional differential equation. These wavelet methods are applied to reduce the FOCP as system of algebraic equations by applying the method of constrained extremum which consists of adjoining the constraint equations to the performance index by a set of undetermined Lagrange multipliers. These algebraic systems are solved numerically by Newton's method. Illustrative examples are discussed to demonstrate the applicability and validity of the wavelet methods.

Sign in / Sign up

Export Citation Format

Share Document