We prove two bounds showing that if the eigenvalues of a matrix are clustered in a region of the complex plane then the corresponding discrete-time linear system requires significant energy to control. A curious feature of one of our bounds is that the dependence on the region is via its logarithmic capacity, which is a measure of how well a unit of mass may be spread out over the region to minimize a logarithmic potential.
Olshevsky, A. (2016). Eigenvalue clustering, control energy, and logarithmic capacity. Systems and Control Letters, 96, 45–50. https://doi.org/10.1016/j.sysconle.2016.06.013