A Transformation Approach to Singular Subarcs in Optimal Trajectory and Control Problems

Mayer variational problems in which the control variable appears linearly are considered and a canonical form sought for the system equations which is somewhat analogous to that adopted by Wonham and Johnson for linear constant coefficient systems with cost functional quadratic in the state variables. A means of synthesizing a transformation to the canonical form in terms of the mutually independent solutions of a first order linear homogeneous partial differential equation is described. It is then shown how the Legendre-Clebsch necessary condition applied in the transformed system of variables may be employed to obtain information on the singular extremals of the problem and the possible appearance of singular subarcs in the solution. Two examples are employed for illustration, one a simple servomechanism problem and the other Goddard's problem of optimal thrust programming for a sounding rocket.