Application of algebraic identities to singular optimal control

High order necessary conditions are given for the optimality of totally and partially singular arcs in the case of nonlinear systems not necessarily linear in the control variables. The term `singular´ is used in optimal control problems in which the Pontryagin maximum principle does not furnish in explicit relationship between the control and the state and costate variables. Singular arcs rise in practical applications in rocket and air vehicle flight, and many techniques have been used. In the present study these results are unified and third and higher order necessary conditions are obtained. The necessary conditions are all expressed in terms of the derivative of a particular function along suitable Lie brackets involving vector fields associated with the control process. It is shown that there is a rich interplay between the Lie bracket structure of a system and the qualitative properties of external controls. The result is a key tool in that it provides some suitable control variations in the proof of the theorems. The remaining results are proved by combining Volterra series expansions, special control variations, and multiple integral identities