Eigenvalue Clustering, Control Energy, and Logarithmic Capacity

TL;DR

This paper establishes bounds linking eigenvalue clustering in the complex plane to the control energy needed for discrete-time linear systems, highlighting the role of logarithmic capacity in this relationship.

Contribution

It introduces novel bounds connecting eigenvalue distribution, control energy, and logarithmic capacity, providing new insights into system controllability.

Findings

Eigenvalue clustering increases control energy requirements.

Logarithmic capacity quantifies the influence of eigenvalue regions.

Bounds depend on the logarithmic capacity of eigenvalue regions.

Abstract

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.