Optimal Synthesis in a Model Problem with Two-Dimensional Control Lying in an Arbitrary Convex Set

We consider a model nilpotent convex problem with two-dimensional control from an arbitrary convex set $\Omega$. For the case in which $\Omega$ is a polygon, the problem is solved explicitly. For the case of an arbitrary set $\Omega$, we completely describe the asymptotics of optimal trajectories and the geometric properties of the optimal synthesis.