SINGULAR OPTIMAL-CONTROL PROBLEMS - ON THE NECESSARY CONDITIONS OF OPTIMALITY

The purpose of this paper is to give higher-order necessary conditions for the optimality of a totally singular arc in the case of nonlinear systems not necessarily linear in the control variables. These conditions, stated in Theorems 2 and 3, are all expressed in terms of the derivative of a particular function along suitable Lie brackets involving vector fields associated with the control process. These conditions contain most of the early results and in addition some of them are new, especially those of third order. To prove these results two different methods are used, namely, Sussmann's techniques arising in local controllability theory and Volterra series expansion.