Generalization of the Multiplicative and Additive Compounds of Square Matrices and Contraction Theory in the Hausdorff Dimension

Chengshuai Wu, Raz Pines, Michael Margaliot*, Jean Jacques Slotine

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

The k multiplicative and k additive compounds of a matrix play an important role in geometry, multilinear algebra, the asymptotic analysis of nonlinear dynamical systems, and in bounding the Hausdorff dimension of fractal sets. These compounds are defined for the integer values of k. Here, we introduce generalizations called the α multiplicative and α additive compounds of a square matrix, with α real. We study the properties of these new compounds and demonstrate an application in the context of the Douady and Oesterlé theorem. Our results lead to a generalization of contracting systems to α-contracting systems, with α real. Roughly speaking, the dynamics of such systems contracts any set with the Hausdorff dimension larger than α. For α =1, they reduce to standard contracting systems. We demonstrate our theoretical results by designing a state-feedback controller for a classical chaotic system, guaranteeing the well-ordered behavior of the closed-loop system.

Original languageEnglish
Pages (from-to)4629-4644
Number of pages16
JournalIEEE Transactions on Automatic Control
Volume67
Issue number9
DOIs
StatePublished - 1 Sep 2022

Keywords

  • Additive compound matrix
  • contraction theory
  • fractal sets
  • multiplicative compound matrix
  • nonlinear dynamical systems
  • ribosome flow model
  • Thomas' cyclically symmetric attractor

Fingerprint

Dive into the research topics of 'Generalization of the Multiplicative and Additive Compounds of Square Matrices and Contraction Theory in the Hausdorff Dimension'. Together they form a unique fingerprint.

Cite this