A new numerical approach based on piecewise interpolation for solving a class of fractional optimal control problems

摘要

This paper presents a numerical approach based on piecewise interpolation for solving a class of fractional optimal control problems (FOCPs). The fractional derivative is described in the terms of Caputo. The performance index is considered as an integral cost function of both the state and control variables, and the dynamic constraint is described by a fractional differential equation of arbitrary order more than zero. The control equations in addition to a transversality condition have been obtained using the calculus of variations, the Lagrange multiplier, and the formula for fractional integration by parts. We introduce one piecewise interpolation method to obtain the numerical approximations of the fractional derivatives of the resulting equations. Then, the resulting equations are reduced to a set of algebraic equations which can be solved directly using a linear equation solver. Illustrative examples are provided to show the validity and high accuracy of the method. It is observed that the solutions approach to the classical solutions as the order of the fractional derivatives approach to 1. In comparison with another approximation method using the Grunwald-Letnikov definition, it has a higher accuracy on the same step size of the entire time domain.