Necessary optimality conditions for different phase portraits in a neigh-borhood of a singular arc

An optimal control problem with scalar control is characterized by two Hamiltonians related to boundary values of the control parameter. Intermediate (internal) values of the control and the corresponding singular trajectories (arcs) can be constructed in terms of these two Hamiltonians using Poisson brackets. All multiple Poisson brackets using these Hamiltonians two, three, and four times vanish on a singular arc of the second order and the brackets with five Hamiltonians in general differ from zero. There exist six different multiple Poisson brackets in which Hamiltonians are used five times. A regular arc in the optimal phase portrait is linked with a singular arc after one, several, or infinitely many (Fuller phenomenon) switchings. In the paper it is shown that various collections of the signs for these six quantities – multiple Poisson brackets – correspond to the above-mentioned cases. There exist four different collections of the signs for the set consisting of six Poisson brackets. The singularity including a universal surface is investigated for the general case, whereas two other types of singularities are studied in particular examples.