  We study the well-known minimum-energy control of the double integrator,
along with the simultaneous minimization of the total variation in the control
variable. We derive the optimality conditions and obtain the unique optimal
solution to the combined problem, where the initial and terminal boundary
points are specified. We study the problem from a multi-objective optimal
control viewpoint, constructing the Pareto front. We show that the unique
asymptotic optimal control function, for the minimization of the total
variation alone, is piecewise constant with one switching at the midpoint of
the time horizon. For any instance of the boundary conditions of the problem,
we prove that the asymptotic optimal total variation is exactly $2/3$ of the
total variation of the minimum-energy control. We illustrate the results for a
particular instance of the problem and include a link to a video which animates
the solutions while moving along the Pareto front.
