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 overc...