PreprintCC BY 4.0arXiv
Dissipative Hamiltonian representations of linear evolution equations
Abstract
We investigate explicit and implicit dissipative Hamiltonian representations of generators of strongly continuous semigroups on real or complex Hilbert spaces. For a prescribed positive definite selfadjoint energy operator H satisfying dom(A)⊂dom(H), we show that the Lyapunov dissipation inequality is equivalent to an exact factorization A=DH, where D can be chosen maximally dissipative. For bounded H, the criterion also yields an implicit representation. We further study energy constructions based on closed intertwiners and output trajectories, distinguishing contractive energy-space dynamics from operator factorizations on the original state space. Examples illustrate the roles of noncoercivity, unboundedness, and domain compatibility.
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BibTeX
@misc{mehrmann2026dissipative,
title = {{Dissipative Hamiltonian representations of linear evolution equations}},
author = {Volker Mehrmann and Timo Reis},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05063},
eprint = {2611.05063},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05063},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dissipative Hamiltonian representations of linear evolution equations
AU - Volker Mehrmann
AU - Timo Reis
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05063
UR - https://arxiv.org/abs/2611.05063
AB - We investigate explicit and implicit dissipative Hamiltonian representations of generators of strongly continuous semigroups on real or complex Hilbert spaces. For a prescribed positive definite selfadjoint energy operator H satisfying dom(A)⊂dom(H), we show that the Lyapunov dissipation inequality is equivalent to an exact factorization A=DH, where D can be chosen maximally dissipative. For bounded H, the criterion also yields an implicit representation. We further study energy constructions based on closed intertwiners and output trajectories, distinguishing contractive energy-space dynamics from operator factorizations on the original state space. Examples illustrate the roles of noncoercivity, unboundedness, and domain compatibility.
ER -