  In this paper, by using the Brunovsky normal form, we provide a reformulation
of the problem consisting in finding the actuator design which minimizes the
controllability cost for finite-dimensional linear systems with scalar
controls. Such systems may be seen as spatially discretized linear partial
differential equations with lumped controls. The change of coordinates induced
by Brunovsky's normal form allows us to remove the restriction of having to
work with diagonalizable system dynamics, and does not entail a randomization
procedure as done in past literature on diffusion equations or waves. Instead,
the optimization problem reduces to a minimization of the norm of the inverse
of a change of basis matrix, and allows for an easy deduction of existence of
solutions, and for a clearer picture of some of the problem's intrinsic
symmetries. Numerical experiments help to visualize these artifacts, indicate
further open problems, and also show a possible obstruction of using
gradient-based algorithms - this is alleviated by using an evolutionary
algorithm.
