A new efficient numerical scheme for solving fractional optimal control problems via a Genocchi operational matrix of integration

Abstract

In this paper, a new operational matrix of integration is derived using Genocchi polynomials, which is one of the Appell polynomials. By using the matrix, we develop an efficient, direct and new numerical method for solving a class of fractional optimal control problems. The fractional derivative in the dynamic constraints was replaced with the Genocchi polynomials with unknown coefficients and a Genocchi operational matrix of fractional integration. Then, the equation derived from the dynamic constraints was put into the performance index. Hence, the fractional optimal control problems will be reduced to fractional variational problems. By finding a necessary condition for the optimality for the performance index, we will obtain a system of algebraic equations that can be easily solved by using any numerical method. Hence, we obtain the value of unknown coefficients of Genocchi polynomials. Lastly, the solution of the fractional optimal control problems will be obtained. In short, the properties of Genocchi polynomials are utilized to reduce the given problems to a system of algebraic equations. The approximation approach is simple to use and computer oriented. Illustrative examples are given to show the simplicity, accuracy and applicability of the method.

New collocation scheme for solving fractional partial differential equations

“…Other semi-orthogonal polynomials such as Appell polynomials may also had great potential to solve fractional calculus problems effectively. On top of that, we introduced operational matrix of derivative via Genocchi polynomials by Loh et al [10] for the solution of fractional integro-differential equations, (FIDEs), Genocchi operational matrix of integration for solving fractional optimal control problems, (FOCPs) by Phang et al [17] and pantograph equation [7]. Different than these recently developed schemes, here we solve the fractional partial differential equations (FPDEs) by using this Genocchi operational matrix of derivative in conjunction of shifted CGL.…”

The optimal displacement control for fractional incommensurate mass-spring oscillators by Shifted Legendre polynomials

“…It is a common practice to obtain the approximate solutions by various numerical methods. Some researchers use various methods to construct operational matrices to obtain approximate solution (Alipour et al, 2013; Singh, 2018; Singh et al, 2017; Phang et al, 2018). Tohidi and Saberi Nik (2015) used a Bessel collocation method for solving fractional optimal control problems without adopting operational matrices.…”

An efficient numerical method for the optimal control of fractional-order dynamic systems

“…(2016), and Sahu and Saha Ray (2016). In more recent works, Phang et al. (2017) proposed an efficient numerical scheme for solving FOCPs via a Genocchi operational matrix of integration.…”