  We consider the time-varying actuator placement in continuous time, where the
goal is to maximize the trace of the controllability Grammian. A natural
relaxation of the problem is to allow the binary $\{0,1\}$ variable indicating
whether an actuator is used at a given time to take on values in the closed
interval $[0,1]$. We show that all optimal solutions of both the original and
the relaxed problems can be given via an explicit formula, and that, as long as
the input matrix has no zero columns, the solutions sets of the original and
relaxed problem coincide.
