Explicit Solutions for a Series of Optimization Problems with 2-Dimensional Control via Convex Trigonometry
Andrei Andreevich Ardentov, Lev Vyacheslavovich Lokutsievskiy, Yu. L. Sachkov · Doklady Mathematics · 2020
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 ( $${\text{SU}}(2)$$ , $${\text{SL}}(2)$$ , $${\text{SE}}(2)$$ , $${\text{SH}}(2)$$ ); (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).