A.A. Ardentov, L.V. Lokutsievskiy, Y.L. Sachkov, Explicit Solutions for a Series of Optimization Problems with 2-Dimensional Control via Convex Trigonometry. Dokl. Math. 102, 427–432 (2020).
We consider a number of optimal control problems with 2-dimensional control lying in an arbitrary convex compact set Ω. 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 Ω; 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).