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arXiv · 2504.06009

Linear time-and-space-invariant relaxation systems

Tihol Ivanov Donchev·Brayan M. Shali·Rodolphe Sepulchre
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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.

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Tihol Ivanov Donchev, Brayan M. Shali, Rodolphe Sepulchre. 2025-04-08. Linear time-and-space-invariant relaxation systems. https://arxiv.org/abs/2504.06009

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