Solutions of a Hamiltonian system with two-dimensional control in a neighbourhood of a singular second-order extremal

Abstract

was considered. If the domain U in (1) is a triangle, then the optimal synthesis can be constructed completely (see [1]). Partial synthesis, including synthesis for the problem with a triangle, has also been constructed for a Hamiltonian system of general form with U having the shape of a convex polygon [1]. In the case when U has a smooth boundary, the question of complete optimal synthesis is still open. For the problem (1) where U is a disc, trajectories with chattering and logarithmic spirals have explicitly been found for certain classes of initial conditions (see [2] and [3]). We consider the Hamiltonian system of Pontryagin’s maximum principle