Applications of the homotopy analysis method to optimal control problems

Singh, Shubham, MS, Purdue University, August 2016. Applications of the Homotopy Analysis Method to Optimal Control Problems. Major Professor: Michael J. Grant. Traditionally, trajectory optimization for aerospace applications has been performed using either direct or indirect methods. Indirect methods produce highly accurate solutions but suffer 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 differential 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 finance, 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 significance. 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 final-time boundary value problems, finding the convergence region is much harder as compared to fixed final-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.