Double-Homotopy Method for Solving Optimal Control Problems

The homotopy method has long served as a useful tool in solving optimal control problems, particularly highly nonlinear and sensitive ones for which good initial guesses are difficult to obtain, such as some of the well-known problems in aerospace trajectory optimization. However, the traditional homotopy method often fails midway: a fact that occasional practitioners are not aware of, and a topic which is rarely investigated in aerospace engineering. This paper first reviews the main reasons why traditional homotopy fails. A new double-homotopy method is developed to address the common failures of the traditional homotopy method. In this approach, the traditional homotopy is employed until it encounters a difficulty and stops moving forward. Another homotopy originally designed for finding multiple roots of nonlinear equations takes over at this point, and it finds a different solution to allow the traditional homotopy to continue on. This process is repeated whenever necessary. The proposed method overcomes some of the frequent difficulties of the traditional homotopy method. Numerical demonstrations in a nonlinear optimal control problem and a three-dimensional low-thrust orbital transfer problem are presented to illustrate the applications of the method.