Applications of the homotopy analysis method to optimal control problems
Traditionally, trajectory optimization for aerospace applications has been performed using either direct or indirect methods. Indirect methods produce highly accurate solutions but suer from a small convergence region, requiring initial guesses close to the optimal solution. In past two decades, a new series of analytical approximation methods have been used for solving systems of dierential equations and boundary value problems. ^ The Homotopy Analysis Method (HAM) is one such method which has been used to solve typical boundary value problems in nance, science, and engineering. In this investigation, a methodology is created to solve indirect trajectory optimization problems using the Homotopy Analysis Method. Use of the auxiliary convergence control parameter to widen the convergence region and increase the rate of convergence have been demonstrated on multiple optimal control problems. The guaranteed convergence and the ease of selecting the initial guess for trajectory optimization problems makes the method of high signicance. It has been demonstrated that initial guesses for the optimal control problem can be generated using a simple approach based on only the initial boundary conditions. The approach has been demonstrated on the Zermelo's problem and two cases of a 2D ascent problem. It has been established that for free nal-time boundary value problems, nding the convergence region is much harder as compared to xed nal-time cases. To validate the approach, results are compared with those obtained using the MATLAB's bvp4c function. A number of new challenges are discovered and listed during the process.^
Michael J. Grant, Purdue University.