A numerical solution for fractional optimal control problems via Bernoulli polynomials

Abstract

This paper presents a new numerical method for solving fractional optimal control problems (FOCPs). The fractional derivative in the dynamic system is described in the Caputo sense. The method is based upon Bernoulli polynomials. The operational matrices of fractional Riemann–Liouville integration and multiplication for Bernoulli polynomials are derived. The error upper bound for the operational matrix of the fractional integration is also given. The properties of Bernoulli polynomials are utilized to reduce the given optimization problems to the system of algebraic equations. By using Newton’s iterative method, this system is solved and the solution of FOCPs are achieved. Illustrative examples are included to demonstrate the validity and applicability of the technique.

An efficient method to solve a fractional differential equation by using linear programming and its application to an optimal control problem

“…Next, in this example, we use the numerical results of linear programming. The exact solution of equation (39) for ¼ 1 is equal to: Keshavarz et al (2015) Number of solution by Tohidi and Nik (2015) Number of solution by Jafari and Tajadodi (2014) Number of solution by Lotfi et al (2011) 0.1 2.39 Â 10 À6 -1.34 Â 10 À5 1.34 Â 10 À5 0.2 1.21 Â 10 À6 6.2315. 10 À8 2.12 Â 10 À5 2.12 Â 10 À5 0.3 1.72 Â 10 À6 -3.24 Â 10 À5 3.24 Â 10 À5 0.4 6.82 Â 10 À7 2.4565.…”

Novel orthogonal functions for solving differential equations of arbitrary order

“…It was found that various applications can be modeled with the help of the fractional derivatives [6,7]. For example, the nonlinear oscillation of earthquake [8], the fractional optimal control problems for dynamic systems [9,10,11,12], and the fluid-dynamic models with fractional derivatives can eliminate the deficiency arising from the assumption of continuous traffic flow [13,14,15]. During the last decades, several methods have been used to solve fractional differential equations, fractional partial differential equations, fractional integro-differential equations, the initial and boundary value problems, and dynamic systems containing fractional derivatives, such as Adomian's decomposition method [16,17], fractional-order Legendre functions [18], fractional-order Chebyshev functions of the second kind [19], Homotopy analysis method [20], Bessel functions and spectral methods [21], Legendre and Bernstein polynomials [22], finite element methods [23], Legendre collocation [24], modified spline collocation [25], multiquadratic radial basis functions [26], and other methods [27,28,29,30,31,32,33].…”

On Genocchi operational matrix of fractional integration for solving fractional differential equations

“…Lemma 3 Keshavarz, Ordokhani, and Razzaghi [16] Suppose that f ∈ L 2 [0, 1] is approximated by f N such that…”