Singular regimes in controlled systems with multidimensional control in a polyhedron

We study Hamiltonian systems that are affine in a multidimensional control varying in a polyhedron $\Omega$. Quite often, a crucial role in the study of the global behaviour of solutions of such systems is played by special trajectories and the geometry of their neighbourhoods. We prove a theorem on the structure of the output of optimal trajectories to a first-order singular trajectory in a neighbourhood of this trajectory (and of the exit from it) for systems with holonomic control. We also prove that in a neighbourhood of a first-order singular trajectory, a Lagrangian surface is woven in a special way from the trajectories of the system that are singular with respect to the faces of $\Omega$. We suggest a simple way to find explicitly first-order special trajectories with respect to the faces of $\Omega$. As a result, we describe a complete picture of the optimal synthesis obtained by the successive conjugation of first-order singular extremals.