Two-term small-time spectral expansions for controllability Gramians
Abstract
For a controllable single-input linear system in dimension n, the controllability Gramian eigenvalues, ordered decreasingly, are known to have the successive small-time orders T, T3, …, T2n−1. We refine this leading-order hierarchy by explicitly computing the next-order coefficient of every eigenvalue and the first-order variation of its eigendirection and of the nested spectral subspaces. The resulting formulas have intrinsic expressions in the orthonormal Krylov basis: eigenvalue corrections describe the action of the dynamics along each Krylov direction, while variations of the spectral subspaces describe coupling to the next direction. We also establish local joint real-analytic dependence on $(T, A, b)$ near T=0 for the eigenvalues divided by their leading powers of T, consistently oriented eigenvectors, and spectral projectors. This provides convergent expansions with remainders uniform on compact families of controllable pairs. These results yield refined asymptotics for the worst-case minimum energy required to reach a unit target from the origin, the Gramian determinant and condition number, and the Ornstein--Uhlenbeck Gaussian profile in moving principal coordinates. The energy equals the reciprocal of the smallest Gramian eigenvalue, whose eigendirection identifies the most energy-demanding targets for sufficiently small times. We further derive an exact Gramian-weighted energy identity for the forward Fokker--Planck equation and sharp anisotropic short-time smoothing asymptotics using, respectively, the Lyapunov equation and an exact Fourier norm formula combined with the graded Gramian factorization. For symmetric dynamics, the spectral data reconstruct A and determine b up to its global sign.
Related
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BibTeX
@misc{trelat2026twoterm,
title = {{Two-term small-time spectral expansions for controllability Gramians}},
author = {Emmanuel Trélat and Enrique Zuazua},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37077},
eprint = {2609.37077},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05768599v1},
}
RIS
TY - UNPB
TI - Two-term small-time spectral expansions for controllability Gramians
AU - Emmanuel Trélat
AU - Enrique Zuazua
PY - 2026
DA - 2026/09/29
PB - HAL
DO - 10.48550/arxiv.2609.37077
UR - https://hal.science/hal-05768599v1
AB - For a controllable single-input linear system in dimension n, the controllability Gramian eigenvalues, ordered decreasingly, are known to have the successive small-time orders T, T^3, …, T^2n−1. We refine this leading-order hierarchy by explicitly computing the next-order coefficient of every eigenvalue and the first-order variation of its eigendirection and of the nested spectral subspaces. The resulting formulas have intrinsic expressions in the orthonormal Krylov basis: eigenvalue corrections describe the action of the dynamics along each Krylov direction, while variations of the spectral subspaces describe coupling to the next direction. We also establish local joint real-analytic dependence on $(T, A, b)$ near T=0 for the eigenvalues divided by their leading powers of T, consistently oriented eigenvectors, and spectral projectors. This provides convergent expansions with remainders uniform on compact families of controllable pairs. These results yield refined asymptotics for the worst-case minimum energy required to reach a unit target from the origin, the Gramian determinant and condition number, and the Ornstein--Uhlenbeck Gaussian profile in moving principal coordinates. The energy equals the reciprocal of the smallest Gramian eigenvalue, whose eigendirection identifies the most energy-demanding targets for sufficiently small times. We further derive an exact Gramian-weighted energy identity for the forward Fokker--Planck equation and sharp anisotropic short-time smoothing asymptotics using, respectively, the Lyapunov equation and an exact Fourier norm formula combined with the graded Gramian factorization. For symmetric dynamics, the spectral data reconstruct A and determine b up to its global sign.
ER -