On the fuel-optimal singular control of nonlinear second-order systems

Given a body subject to quadratic drag forces so that the positiony(t)and the applied control thrustu(t)are related by\ddot{y}(t)+a\dot{y}(t)|\dot{y}(t)| = u(t), |u(t)| \leq 1, the controlu(t)is found which forces the body to a desired position, and stops it there, and which minimizes the costJ=\int\liminf{0} \limsup{T}\{k + |u(t)|\}dt. The response timeTis not fixed,k > 0, and|u(t)|is proportional to the rate of flow of fuel. Repeated use of the necessary conditions provided by the Maximum Principle results in the optimum feedback system. It is shown that ifk\leq 1, then singular controls exist and they are optimal; ifk > 1, then singular controls are not optimal. Techniques for the construction of the various switch curves are given, and extensions of the results to other nonlinear systems are discussed.