Two-term small-time spectral expansions for controllability Gramians
Trélat E, Zuazua E (2026)
Publication Language: English
Publication Status: Submitted
Publication Type: Unpublished / Preprint
Future Publication Type: Article in Edited Volumes
Publication year: 2026
URI: https://dcn.nat.fau.eu/wp-content/uploads/Gramian.pdf
Open Access Link: https://arxiv.org/abs/2609.37077
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.
Involved external institutions
France (FR)How to cite
APA:
Trélat, E., & Zuazua, E. (2026). Two-term small-time spectral expansions for controllability Gramians. (Unpublished, Submitted).
MLA:
Trélat, Emmanuel, and Enrique Zuazua. Two-term small-time spectral expansions for controllability Gramians. Unpublished, Submitted. 2026.
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