Compact forms of the generalized Legendre-Clebsch conditions and the computation of singular control trajectories

Two compact forms of the generalized Legendre-Clebsch necessary conditions for a singular extremal are described. One is more convenient for testing whether or not the generalized Legendre-Clebsch conditions are satisfied. The strengthened form of the other compact form of the generalized Legendre-Clebsch conditions provides just the right conditions for the control variables to be evaluated as functions of the state and costate variables and thereby providing a canonical formalism for a singular variational problem.