Optimization and Control of Nonlinear Systems Using the Second Variation

A feedback control scheme is described that maximizes a terminal quantity while satisfying specified terminal conditions, in the presence of small disturbances. The scheme can also be used in a rapidly converging computation technique to find exact solutions to the nonlinear two-point boundary value problems occurring in the calculus of variations. The scheme is based on a linear perturbation from a nominal optimum path and, as such, involves the second variation of the calculus of variations. A simple analytical example is given for thrust direction control to place a vehicle in orbit. Numerical examples of both the control scheme and the optimization technique are given for a lifting vehicle re-entering the earth’s atmosphere at parabolic speed.