Explicit solutions for a series of optimization problems with 2-dimensional control via convex trigonometry

We consider a number of optimal control problems with 2-dimensional control lying in an arbitrary convex compact set $\Omega$. Solutions to these problems are obtained using methods of convex trigonometry. The paper includes (1) geodesics in the Finsler problem on the Lobachevsky hyperbolic plane; (2) left-invariant sub-Finsler geodesics on all unimodular 3D Lie groups (SU(2), SL(2), SE(2), SH(2)); (3) the problem of a ball rolling on a plane with a distance function given by $\Omega$; and (4) a series of “yacht problems” generalizing Euler’s elastic problem, the Markov–Dubins problem, the Reeds–Shepp problem, and a new sub-Riemannian problem on SE(2).