Infinite-horizon controllability scores for linear time-invariant systems
We formulate infinite-horizon controllability scores for linear time-invariant systems whose conventional controllability Gramians may diverge. A mode-dependent scaling yields finite-horizon volumetric controllability score (VCS) and average energy controllability score (AECS) problems that are exactly equivalent to the original ones, while the scaled Gramians converge as the horizon tends to infinity. Using epi-convergence, we establish convergence of the finite-horizon optimal allocations to the limiting solution set and obtain convergence to a unique limiting score under explicit rank conditions. The limiting VCS retains the stable, center, and unstable components and preserves full-system controllability, whereas the limiting AECS depends only on the stable component and may neglect an input required for a center or unstable mode. The VCS theory allows arbitrary spectral structures, whereas the AECS limit requires a nonempty stable component. The additional restriction on the center spectrum is imposed only for the real-Schur implementation. A directed-Laplacian example verifies the convergence and uniqueness conditions and illustrates the controllability distinction between the two limiting scores.