Effect of a fourth-order diffusion term on the robustness of the transfer function stability for hyperbolic systems
Abstract
We study a 2×2 hyperbolic system with anti-located control and observation. It is well known that the system can be stabilized with a delayed output-feedback control and that, unfortunately, this design is arbitrarily sensitive to small errors in the transport velocity. It was recently proved that adding a second-order diffusion term to the system restores robustness. We show that the same phenomenon occurs with a fourth-order diffusion term: the maximal spectral abscissa of the closed-loop system converges, as the diffusion parameter goes to zero, to the explicit value −ln(2), and the resulting stability margin is robust to small perturbations of the transport velocity. This additional term leads to a quartic characteristic equation whose roots have no tractable closed-form expression. We develop an asymptotic classification of its four roots and use it to obtain a first-order expansion of the transfer function with an explicit error bound. Combining this expansion with a topological degree argument, we show that −ln(2) is in fact the exact limiting value of the spectral abscissa, extending to the fourth-order case the sharpness result previously available only for second-order diffusive terms.
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BibTeX
@misc{cerpa2026effect,
title = {{Effect of a fourth-order diffusion term on the robustness of the transfer function stability for hyperbolic systems}},
author = {Eduardo Cerpa and Felipe Labra},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05845},
eprint = {2611.05845},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05845},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Effect of a fourth-order diffusion term on the robustness of the transfer function stability for hyperbolic systems
AU - Eduardo Cerpa
AU - Felipe Labra
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.05845
UR - https://arxiv.org/abs/2611.05845
AB - We study a 2×2 hyperbolic system with anti-located control and observation. It is well known that the system can be stabilized with a delayed output-feedback control and that, unfortunately, this design is arbitrarily sensitive to small errors in the transport velocity. It was recently proved that adding a second-order diffusion term to the system restores robustness. We show that the same phenomenon occurs with a fourth-order diffusion term: the maximal spectral abscissa of the closed-loop system converges, as the diffusion parameter goes to zero, to the explicit value −ln(2), and the resulting stability margin is robust to small perturbations of the transport velocity. This additional term leads to a quartic characteristic equation whose roots have no tractable closed-form expression. We develop an asymptotic classification of its four roots and use it to obtain a first-order expansion of the transfer function with an explicit error bound. Combining this expansion with a topological degree argument, we show that −ln(2) is in fact the exact limiting value of the spectral abscissa, extending to the fourth-order case the sharpness result previously available only for second-order diffusive terms.
ER -