Infinite-horizon controllability scores for linear time-invariant systems

Abstract

We introduce a numerically stable reformulation of controllability scoring based on a scaled controllability Gramian, which remains reliably computable even for unstable systems. The resulting optimization problems define dynamics-aware network centrality measures, referred to as the volumetric controllability score (VCS) and the average energy controllability score (AECS). Building on this stable reformulation, we derive the corresponding infinite-horizon problems, develop an algorithm to solve them, and highlight computational advantages over their finite-horizon counterparts. Under suitable assumptions, we prove that the infinite-horizon VCS and AECS are unique and that the finite-horizon scores converge to them. We further show that VCS and AECS can differ markedly in this limit, because VCS enforces controllability of the full system, whereas AECS accounts only for the stable modes. Finally, numerical experiments on Laplacian dynamics illustrate this convergence.