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

Given a body subject to quadratic drag forces so that the position and the applied control thrust are related by , the control is found which forces the body to a desired position, and stops it there, and which minimizes the cost . The response time is not fixed, , and 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 if , then singular controls exist and they are optimal; if , 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.