Dissipation Bounds and Hardness in Globally Asymptotically Stable Quadratic Systems
Abstract
Approximating the maximum retained energy fraction in globally asymptotically stable quadratic systems of growing dimension is NP-hard at every fixed positive additive error below 1/2, even when strict quartic Lyapunov functions and strictness proofs are constructed in polynomial time and included in the input. Polynomial-factor approximation is also NP-hard for the least dissipated fraction, energy-halving time and structured coefficient distance to loss of asymptotic stability. For a partially damped family, minimum unit-sphere coupling norm equals that distance, with two-sided inverse-square halving-time bounds at distances in (0,1/2] and explicit coefficient factors. Additive hardness persists under structured coefficient perturbations and direct damping, even with global exponential stability, a strict quadratic Lyapunov function and a Hurwitz Jacobian.
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BibTeX
@misc{bhattacharjee2026dissipation,
title = {{Dissipation Bounds and Hardness in Globally Asymptotically Stable Quadratic Systems}},
author = {Subhransu S. Bhattacharjee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29512},
eprint = {2610.29512},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29512},
}
RIS
TY - UNPB
TI - Dissipation Bounds and Hardness in Globally Asymptotically Stable Quadratic Systems
AU - Subhransu S. Bhattacharjee
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.29512
UR - https://arxiv.org/abs/2610.29512
AB - Approximating the maximum retained energy fraction in globally asymptotically stable quadratic systems of growing dimension is NP-hard at every fixed positive additive error below 1/2, even when strict quartic Lyapunov functions and strictness proofs are constructed in polynomial time and included in the input. Polynomial-factor approximation is also NP-hard for the least dissipated fraction, energy-halving time and structured coefficient distance to loss of asymptotic stability. For a partially damped family, minimum unit-sphere coupling norm equals that distance, with two-sided inverse-square halving-time bounds at distances in (0,1/2] and explicit coefficient factors. Additive hardness persists under structured coefficient perturbations and direct damping, even with global exponential stability, a strict quadratic Lyapunov function and a Hurwitz Jacobian.
ER -