PreprintCC BY 4.0arXiv
L2-L2-gain bounds for quadratic output systems
Abstract
We derive an explicit bound for the L2-L2-gain of linear time-invariant systems whose output is a quadratic function of the state and the input. Such systems appear naturally in many areas, for example for port-Hamiltonian systems, optimal-control, and stochastic problems. In case the output is purely quadratic in the state, the bound is determined by the supremum over ω∈ℝ of the L2-norms of the bivariate transfer function restricted to the anti-diagonals {(s, iω−s)| s∈ iℝ} of the frequency domain $i\mathbb R\times i\mathbb R $. Further, we show how the bound can be computed by solving linear matrix equations. This result provides a practical tool for assessing and reducing quadratic-output models.
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BibTeX
@misc{hillebrecht2026l2l2gain,
title = {{L2-L2-gain bounds for quadratic output systems}},
author = {Birgit Hillebrecht},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.00552},
eprint = {2607.00552},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.00552},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - L2-L2-gain bounds for quadratic output systems
AU - Birgit Hillebrecht
PY - 2026
DA - 2026/10/31
PB - arXiv
DO - 10.48550/arxiv.2607.00552
UR - https://arxiv.org/abs/2607.00552
AB - We derive an explicit bound for the L2-L2-gain of linear time-invariant systems whose output is a quadratic function of the state and the input. Such systems appear naturally in many areas, for example for port-Hamiltonian systems, optimal-control, and stochastic problems. In case the output is purely quadratic in the state, the bound is determined by the supremum over ω∈ℝ of the L2-norms of the bivariate transfer function restricted to the anti-diagonals {(s, iω−s)| s∈ iℝ} of the frequency domain $i\mathbb R\times i\mathbb R $. Further, we show how the bound can be computed by solving linear matrix equations. This result provides a practical tool for assessing and reducing quadratic-output models.
ER -