A new fractional homotopy method for solving nonlinear optimal control problems

Abstract

The homotopy methods have long served as powerful tools in solving nonlinear optimal control problems, particularly for which the solutions are highly sensitive to the unknown initial conditions. The principle of homotopy methods is that a homotopic parameter is embedded into the formulations of the optimal control problems, and the original problem can be solved by tracing the optimal solutions of the embedded problems. The existing homotopy methods typically introduce the homotopic parameter into the time variable, the necessary conditions, the performance index or the rightside of the differential equations. In this paper, a new fractional homotopy method is presented, the homotopic parameter of which is embedded into the derivative of the differential equations. By using the proposed method, the optimal solution of the target homotopy problem can be found by solving a series of fractional two-point-boundary-value-problems. Numerical demonstrations in a nonlinear optimal control problem and a three-dimensional minimum-time low-thrust orbital transfer problem are presented to illustrate the applications of the method.