  The study is devoted to mathematical modeling and optimal control design of
longitudinal motions of a rectilinear elastic rod. The control inputs are a
force, which is normal to the cross section and distributed piecewise
constantly along the rod's axis, as well as two external lumped loads at the
ends. It is assumed that the intervals of constancy in the normal force have
equal length. Given initial and terminal states with a fixed time horizon, the
optimal control problem is to minimize the mean mechanical energy stored in the
rod. To solve the problem, two unknown functions are introduced: the dynamical
potential and the longitudinal displacements. As a result, the initial-boundary
value problem is reformulated in a weak form, in which constitutive relations
are given as an integral quadratic equation. The unknown functions are both
continuous in the new statement. For the uniform rod, they are found as linear
combinations of traveling waves. In this case, all conditions on continuity as
well as boundary, initial, and terminal constraints form a linear algebraic
system with respect to the traveling waves and control functions. The minimal
controllability time is found from the solvability condition for this algebraic
system. After resolving the system, remaining free variables are used to
optimize the cost functional. Thus, the original control problem is reduced to
a one-dimensional variational problem. The Euler-Lagrange necessary condition
yields a linear system of ordinary differential equations with constant
coefficients supplemented by essential and natural boundary conditions.
Therefore, the exact optimal control law and the corresponding dynamic and
kinematic fields are found explicitly. Finally, the energy properties of the
optimal solution are analyzed.
