Optimal singular rocket and aircraft trajectories

We give a brief review of optimal singular control rocket and aircraft trajectories in outer space and in the atmosphere. Singular control rocket and aircraft trajectories occur when the thrust is at intermediate levels. The analysis of a singular control rocket or aircraft trajectory requires the Goh-Legendre-Clebsch necessary conditions. Two sets of compact forms of these conditions are described. The first compact form of these optimality conditions is convenient for testing the optimality of a singular control trajectory. The second compact form is important because its strengthened form provides sufficient conditions for the control variables to be expressed in terms of the state and costate variables.