The following variational problem was solved by \textit{H. H. Johnson} [Proc. Am. Math Soc. 44, 432-435 (1974; Zbl 0295.53001)]: To find, in the Euclidean plane, the curve of minimal length among all curves that join two given points with given tangents at these points and whose curvature is everywhere bounded above by a given constant. Here the author considers the analogous problem in a Minkowski plane (\(=2\)-dimensional flat Finsler space), i.e., in the case that length and curvature are measured with respect to a non-Euclidean norm.