  This paper considers two different problems in trajectory tracking control
for linear systems. First, if the control is not unique which is most input
energy efficient. Second, if exact tracking is infeasible which control
performs most accurately. These are typical challenges for over-actuated
systems and for under-actuated systems, respectively. We formulate both goals
as optimal output regulation problems. Then we contribute two new sets of
regulator equations to output regulation theory that provide the desired
solutions. A thorough study indicates solvability and uniqueness under weak
assumptions. E.g., we can always determine the solution of the classical
regulator equations that is most input energy efficient. This is of great value
if there are infinitely many solutions. We derive our results by a linear
quadratic tracking approach and establish a useful link to output regulation
theory.
