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PreprintCC BY 4.0arXiv

A Generalized Stability Theorem for Interconnections of Dissipative Behaviors

  • Michaela Repasova
  • Henk J. van Waarde
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Abstract

Dissipativity theory enables the stability analysis of complex, interconnected systems through the physical properties of their components. Celebrated examples of such stability results include the small gain and passivity theorems. In this paper, we prove a unifying stability theorem for dissipative linear time-invariant (LTI) systems. We use the behavioral approach to express storage functions and supply rates in terms of quadratic differential forms (QDFs). Our main result establishes conditions for the stability of an interconnection of two LTI behaviors that are both dissipative with respect to general, dynamic supply rates. As special cases, we recover existing results for static supply rates, such as those capturing finite L2-gain and passivity, as well as dynamic ones capturing, for example, delta dissipativity. The key idea of the paper is to construct a Lyapunov function as a sum of storage functions and a coupling QDF, thereby allowing for a richer class of Lyapunov functions than existing works dealing with just sums of storage functions.

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Cite

BibTeXRIS
BibTeX
@misc{repasova2026generalized,
  title = {{A Generalized Stability Theorem for Interconnections of Dissipative Behaviors}},
  author = {Michaela Repasova and Henk J. van Waarde},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.15765},
  eprint = {2609.15765},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.15765},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A Generalized Stability Theorem for Interconnections of Dissipative Behaviors
AU  - Michaela Repasova
AU  - Henk J. van Waarde
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.15765
UR  - https://arxiv.org/abs/2609.15765
AB  - Dissipativity theory enables the stability analysis of complex, interconnected systems through the physical properties of their components. Celebrated examples of such stability results include the small gain and passivity theorems. In this paper, we prove a unifying stability theorem for dissipative linear time-invariant (LTI) systems. We use the behavioral approach to express storage functions and supply rates in terms of quadratic differential forms (QDFs). Our main result establishes conditions for the stability of an interconnection of two LTI behaviors that are both dissipative with respect to general, dynamic supply rates. As special cases, we recover existing results for static supply rates, such as those capturing finite L_2-gain and passivity, as well as dynamic ones capturing, for example, delta dissipativity. The key idea of the paper is to construct a Lyapunov function as a sum of storage functions and a coupling QDF, thereby allowing for a richer class of Lyapunov functions than existing works dealing with just sums of storage functions.
ER  -