Finding multiple local solutions to optimal control problems via saddle points and its application to the ascent trajectory of a winged rocket

Abstract

Complex nonlinear optimal control problems can have more than a single local solution due to the lack of convexity, and widely-used gradient-based approaches for optimal control have a risk of producing an inferior local solution. Although global optimization methods for optimal control may overcome this difficulty, such existing methods suffer from high computational complexities. This paper presents a novel numerical algorithm for multi-modal optimal control problems based on a saddle-point search method. Starting from an already-found local solution, a saddle-point search algorithm finds a surrounding saddle point of the constrained optimization problem, from which a new local solution is obtained by applying a gradient-based optimizer. By repeating this procedure, multiple locally optimal solutions are explored exhaustively and systematically. Although the developed method is not rigorously guaranteed to produce the global solution, it is applicable to large-scale general optimal control problems with equality and inequality constraints, which makes it a practically beneficial technique. After numerical experiments with an example problem, the ascent trajectory optimization problem of a winged rocket for maximizing apogee altitude is solved by using the proposed method.