Linear time-and-space-invariant relaxation systems
Abstract
This paper generalizes the physical property of relaxation from linear time-invariant (LTI) to linear time-and-space-invariant (LTSI) systems. It is shown that the defining features of relaxation -- complete monotonicity, passivity, and memory-based storage -- carry over seamlessly to the spatio-temporal domain. An LTSI system is shown to be of relaxation type if and only if its associated spatio-temporal Hankel operator is cyclically monotone. This implies the existence of an intrinsic quadratic storage functional defined uniquely by past inputs, independently of any state-space realization. As in the LTI case, LTSI relaxation systems are shown to be those systems for which the state-space concept of storage coincides with the input-output concept of fading memory functional.
Cite
@article{arxiv.2504.06009,
title = {Linear time-and-space-invariant relaxation systems},
author = {Tihol Ivanov Donchev and Brayan M. Shali and Rodolphe Sepulchre},
journal= {arXiv preprint arXiv:2504.06009},
year = {2025}
}