Solving Optimal Control and Pursuit-Evasion Game Problems of High Complexity

Optimal control problems which describe realistic technical applications exhibit various features of complexity. First, the consideration of inequality constraints leads to optimal solutions with highly complex switching structures including bang-bang, singular, and control-and state-constrained sub-arcs. In addition, also isolated boundary points may occur. Techniques are surveyed for the computation of optimal trajectories with multiple subarcs. If the precise computation of the switching structure holds the spotlight, the indirect multiple shooting method is top quality. Second, the differential equations describing the dynamics may be so complicated that they have to be generated by a computer program. In this case, direct methods such as direct collocation are generally superior. Third, the task is often given in applications to solve many optimal control problems, either for parameter homotopies in the course of the solution process itself or for sensitivity investigations of the solutions with respectto various design parameters. Closely related to optimal control problems, pursuit-evasion game problems require, in a natural way, the solution of often thousands of boundary-value problems, in order to synthesize the open-loop controls by feedback strategies. In these cases, efficient homotopy methods must be used in connection with vectorized or parallelized versions of the aforementioned methods.These three degrees of complexity in the solution of optimal control or pursuit-evasion game problems, respectively, are discussed in this survey paper by means of three examples: the abort landing of a passenger aircraft in the presence of a varying down burst, the time-and energy-optimal control of an industrial robot, and a pursuit-evasion game problem between a missile and a fighter aircraft.