Optinal rocket trajectories: Singular control

This is a brief review of singular optimal rocket trajectories and necessary conditions for their optimality. There are two forms of the compact Goh-Legendre-Clebsch necessary conditions for singular extremals in optimal control theory. The first is convenient to test the optimality of a singular extremal. The second compact form is very important but it seems to be unknown in the literature. When it is satisfied in the strengthened manner it provides sufficient conditions for the control variable to be computed in terms of the state and costate variables. We illustrate the usefulness of these compact forms of the Goh-Legendre-Clebsch conditions by a detailed analysis of a simple example. Singular extremals can be very important subarcs of optimal trajectories of rockets and aircraft in the atmosphere and in many other applications of optimal control in fisheries, economics and theoretical physics.