  Markov-Dubins path is the shortest planar curve joining two points with
prescribed tangents, with a specified bound on its curvature. Its structure, as
proved by Dubins in 1957 nearly 70 years after Markov posed the problem of
finding it, is elegantly simple: a selection of at most three arcs are
concatenated, each of which is either a circular arc of maximum (prescribed)
curvature or a straight line. The Markov-Dubins problem and its variants have
since been extensively studied in practical and theoretical settings. A
reformulation of the Markov-Dubins problem as an optimal control problem was
subsequently studied by various researchers using the Pontryagin maximum
principle and additional techniques, to reproduce Dubins' result. In the
present paper, we study the same reformulation, and apply the maximum
principle, with new insights, to derive Dubins' result again. We prove that
abnormal control solutions, which were ruled out in previous studies, do exist.
We characterize them as a concatenation of at most two circular arcs and show
that they are also solutions of the normal problem. More importantly, we prove
that any feasible path of the types mentioned above is a stationary solution,
i.e., that it satisfies the Pontryagin maximum principle. We propose a
numerical method for computing Markov-Dubins path. We illustrate the theory and
the numerical approach by three qualitatively different examples.
