Autonomy Talks - Kevin D. Smith: Semicontracting Dynamical Systems

Autonomy Talks - 17/01/2023

Speaker: Dr. Kevin D. Smith, University of California, Santa Barbara

Title: Semicontracting Dynamical Systems

Abstract: Contraction theory is a powerful tool to prove robust exponential stability properties of nonlinear systems. Unfortunately, systems with an invariance property (e.g., consensus algorithms) or a conservation property (e.g., Markov chains) can only exhibit strong contraction on a subspace—in other words, they are semicontracting. In this talk, we present new directions towards a comprehensive theory of semicontracting systems. We propose two canonical classes of seminorms, one that is particularly useful for systems with an invariance property and one for systems with a conservation property. We show that these two classes of seminorms are dual, and using this result, we explain why the Markov-Dobrushin coefficient appears in the discrete-time analysis of both averaging systems and Markov chains. Leveraging semicontraction theory, we are able to generalize this classical analysis to nonlinear time-varying dynamical systems, both in discrete and continuous time. Finally, we discuss future applications of semicontraction to power grids, population games, and other systems.
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