A Formulation and Numerical Scheme for Fractional Optimal Control Problems

This article presents a general formulation and general numerical scheme for a class of fractional optimal control problems (FOCPs). The fractional derivative is described in the Caputo sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of fractional differential equations. The calculus of variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler-Lagrange equations for the FOCP. The formulation presented, and the resulting equations, are very similar to those that appear in classical optimal control theory. Thus, the present formulation essentially extends classical control theory to fractional dynamic systems. The formulation is used to derive the control equations for a quadratic linear fractional control problem. An iterative numerical scheme for finding the approximate numerical solution of the resulting equations is presented. For a linear system, this method results in a set of linear simultaneous equations, which can be solved directly. Numerical results for a FOCP are presented to demonstrate the feasibility of the method. It is shown that the solutions converge as the number of grid points increases, and the solutions approach the classical solutions as the order of the fractional derivatives approaches 1. The formulation presented is simple, and can be extended to other FOCPs.