  An optimal control problem for longitudinal motions of a thin elastic rod is
considered. We suppose that a normal force, which changes piecewise constantly
along the rod's length, is applied to the cross-section so that the positions
of force jumps are equidistantly placed along the length. Additionally,
external loads act at the rod ends. These distributed force and boundary loads
are considered as control functions of the dynamic system. Given initial and
terminal states at fixed time instants, the problem is to minimize the mean
mechanical energy stored in the rod during its motion. We replace the classical
wave equation with a variational problem solved via traveling waves defined on
a special time-space mesh. For a uniform rod, the shortest admissible time
horizon is estimated exactly, and the exact optimal control law is symbolically
found in a recurrent way.
