Müntz–Legendre spectral collocation method for solving delay fractional optimal control problems

Abstract

In this paper, a numerical method is applied for solving delay fractional optimal control problems (DFOCPs). The fractional derivative is described in the Caputo sense. Since the fractional derivative of Müntz polynomials can be expressed in terms of the same polynomials, those polynomials can accurately represent properties of fractional calculus. In some situations such as in the frequency response based analysis of control systems containing a time-delay, it is necessary to substitute exponential function with an approximation in the form of a rational function. The most common approximation is the Padé approximation. At the first step, using Padé approximation, the delay problem is transformed to a non-delay problem. Next, using the operational matrix of the fractional derivative of Müntz polynomials and pseudospectral method, fractional optimal control problem (FOCP) is reduced to a nonlinear programming problem. Some numerical examples are given to illustrate the effectiveness of the proposed scheme.