Extremal Fields and Neighboring Optimal Control of Constrained Systems

Abstract

This work provides first and second-order expressions to approximate neighboring solutions to the m-point boundary value problem. Multi-point problems arise in optimal control because of interior constraints or switching dynamics. Many problems have this form, and so this work fills a void in the study of extremal fields and neighboring optimal control of constrained systems. Only first and second-order terms are written down, but the approach is systematic and higher order expressions can be found similarly. The constraints and their parameters define an extremal field because any solution to the problem must satisfy the constraints. The approach is to build a Taylor series using constraint differentials, state differentials, and state variations. The differential is key to these developments, and it is a unifying element in the optimization of points, optimal control, and neighboring optimal control. The method is demonstrated on several types of problems including lunar descent, which has nonlinear dynamics, bounded thrust, and free final time. The control structure is bang-off-bang, and the method successfully approximates the unknown initial conditions, switch times, and final time. Compared to indirect shooting, computation time decreases by about three orders of magnitude.