An indirect optimization method with improved convergence characteristics



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An improved numerical procedure is developed for solving the nonlinear two-point boundary value problem which results from the formulation of optimal control problems for solution by an indirect optimization method. Several modifications and extensions to previously known iterative methods are implemented which dramatically improve the convergence characteristics of the indirect optimization approach. The method of particular solutions for solving linear boundary value problems in ordinary differential equations is extended to solve systems of linear differential equations with boundary values specified in the form of nonlinear functions of the dependent and independent variables. This capability is exploited in the development of a new indirect method for solving trajectory optimization problems where the final state and final time are not specified explicitly. Because the method uses a Perturbation approach for linearizing the nonlinear system of differential equations, and because particular solutions are used to construct the general solution of the linearized system, the method is called the Particular Solution Perturbation Method (PSPM). A power series numerical integration method is adapted for use in solving the nonlinear two-point boundary value problem and is found to have several characteristics which make it uniquely suited for this purpose. The convergence characteristics of the PSPM are compared to those of previous indirect optimization methods for a problem which considers the minimum time orbit transfer of continuous, low-thrust rocket. Conclusions and recommendations for further study are included.