The necessary conditions for the solution of a continuous optimal control problem are well known from the calculus of variations. They (the Euler-Lagrange equations) constitute a two-point-boundary-value problem. Unfortunately, for problems of any sophistication this problem is nonlinear and difficult to solve using conventional methods, for example "shooting" methods. The method of direct collocation with nonlinear programming (DCNLP) has become an accepted and useful alternative. In this method the necessary conditions are not explicitly satisfied. Instead the problem is discretized and converted into a nonlinear programming problem.
There are many forms in which this method may be applied; for example there are several different implicit integration formulas that may be used, there are several choices for how the control variables of the system may be parameterized, and there is a choice of solvers for the nonlinear programming problem. We have found that best results are obtained when the method is tailored to the characteristics of the problem.
The DCNLP method will be described and examples of "tailoring" this method to specific problems will be shown. These problems will include optimal trajectories for Earth orbit to Lunar orbit transfer, for low-thrust interception of a dangerous asteroid, for LEO to GEO orbit transfer, and for unpowered landing of the HL-20 lifting body vehicle. It will also be shown how we have recently extended the method to allow solution of differential games (two-player) problems, exemplified by the case of optimal strategies for two F-16 fighters in combat.