Gauss pseudospectral and continuation methods for solving two-point boundary value problems in optimal control theory

摘要

In this study, we propose an efficient pseudospectral method for solving two-point boundary value problems in optimal control theory. In our proposed approach, the Gauss pseudospectral method is utilized to reduce a two-point boundary value problem into the solution of a system of algebraic equations. However, the convergence to the solution of the system of equations obtained may be slow, or it can even fail, if a very good initial estimate of the optimal solution is not available. To overcome this drawback, we employ a numerical continuation method, which resolves the sensitivity of the proposed method to the initial estimate. The main advantages of the present combined method are that good results are obtained even when using a small number of discretization points, while the sensitivity to the initial estimate when solving the final system of algebraic equations is resolved successfully. The proposed method is especially useful when shooting methods fail due to the sensitivity or stiffness of the problem. We present numerical results for two examples to demonstrate the efficiency of the combined method.