Orthonormal shifted discrete Hahn polynomials for a new category of nonlinear variable‐order fractional 2D optimal control problems

2021 ◽  
Author(s):  
Mohammad Hossein Heydari ◽  
Zakieh Avazzadeh ◽  
Carlo Cattani
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Control Problems ◽  
Variable Order ◽  
Hahn Polynomials

A discrete polynomials approach for optimal control of fractional Volterra integro-differential equations

2020 ◽  
pp. 107754632097115
Author(s):  
Fakhrodin Mohammadi ◽  
Leila Moradi ◽  
José António Tenreiro Machado
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Optimal Control Problems ◽  
Computational Cost ◽  
Control Problems ◽  
Operational Matrices ◽  
Hahn Polynomials ◽  
Discrete Norm ◽  
New Type ◽  
Discrete Polynomials

This study develops an efficient numerical method for optimal control problems governed by fractional Volterra integro-differential equations. A new type of polynomials orthogonal with respect to a discrete norm, namely discrete Hahn polynomials, is introduced and its properties investigated. Fractional operational matrices for the orthogonal polynomials are also derived. A direct numerical algorithm supported by the discrete Hahn polynomials and operational matrices is used to approximate solution of optimal control problems governed by fractional Volterra integro-differential equations. Several examples are analyzed and the results compared with those of other methods. The required CPU time assesses the computational cost and complexity of the proposed method.

Solving Two-Dimensional Variable-Order Fractional Optimal Control Problems With Transcendental Bernstein Series

2019 ◽  
Vol 14 (6) ◽  
Author(s):  
Hossein Hassani ◽  
Zakieh Avazzadeh ◽  
José António Tenreiro Machado
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Control Problems ◽  
Operational Matrices ◽  
Variable Order ◽  
Two Dimensional ◽  
Control Functions ◽  
Fractional Optimal Control Problems ◽  
Dimensional Variable ◽  
And Control

This paper studies two-dimensional variable-order fractional optimal control problems (2D-VFOCPs) having dynamic constraints contain partial differential equations such as the convection–diffusion, diffusion-wave, and Burgers' equations. The variable-order time fractional derivative is described in the Caputo sense. To overcome computational difficulties, a novel numerical method based on transcendental Bernstein series (TBS) is proposed. In fact, we generalize the Bernstein polynomials to the larger class of functions which can provide more accurate approximate solutions. In this paper, we introduce the TBS and their properties, and subsequently, the privileges and effectiveness of these functions are demonstrated. Furthermore, we describe the approximation procedure which shows for solving 2D-VFOCPs how the needed basis functions can be determined. To do this, first we derive a number of new operational matrices of TBS. Second, the state and control functions are expanded in terms of the TBS with unknown free coefficients and control parameters. Then, based on these operational matrices and the Lagrange multipliers method, an optimization method is presented to an approximate solution of the state and control functions. Additionally, the convergence of the proposed method is analyzed. The results for several illustrative examples show that the proposed method is efficient and accurate.

Sign in / Sign up

Export Citation Format

Share Document