Passive and reciprocal linear time-and-space-invariant systems

TL;DR

This paper extends the concept of reciprocity to linear time-and-space-invariant systems, exploring their structural properties and physical realizations, with implications for infinite-dimensional systems in control theory.

Contribution

It introduces the notion of reciprocity for LTSI systems and links it to passivity and realizability, addressing challenges in infinite-dimensional system analysis.

Findings

Reciprocity is extended to LTSI systems.

Combining reciprocity and passivity yields realizable physical models.

Illustrated with a relevant physical example.

Abstract

Reciprocity is a fundamental symmetry property observed across many physical domains, including acoustics, elasticity, electromagnetics, and thermodynamics. In systems and control theory, it provides key insights into the internal structure of linear time-invariant (LTI) systems and is closely linked to properties such as passivity, relaxation, and time-reversibility. This paper extends the concept of reciprocity to linear time-and-space-invariant (LTSI) systems, a class of infinite-dimensional systems with spatio-temporal dynamics. It is suggested that, analogously to the LTI case, combining the internal properties of reciprocity and (impedance) passivity entails physical state-space realizations. This is of particular relevance for infinite-dimensional systems, where issues of unboundedness can be detrimental to the well-posedness of the system. The results are motivated and illustrated with a physical example.