  In 1934, Reinhardt conjectured that the shape of the centrally symmetric
convex body in the plane whose densest lattice packing has the smallest density
is a smoothed octagon. This conjecture is still open. We formulate the
Reinhardt Conjecture as a problem in optimal control theory. The smoothed
octagon is a Pontryagin extremal trajectory with bang-bang control. More
generally, the smoothed regular $6k+2$-gon is a Pontryagin extremal with
bang-bang control. The smoothed octagon is a strict (micro) local minimum to
the optimal control problem. The optimal solution to the Reinhardt problem is a
trajectory without singular arcs. The extremal trajectories that do not meet
the singular locus have bang-bang controls with finitely many switching times.
Finally, we reduce the Reinhardt problem to an optimization problem on a
five-dimensional manifold. (Each point on the manifold is an initial condition
for a potential Pontryagin extremal lifted trajectory.) We suggest that the
Reinhardt conjecture might eventually be fully resolved through optimal control
theory. Some proofs are computer-assisted using a computer algebra system.
