Generalization of the Multiplicative and Additive Compounds of Square Matrices and Contraction Theory in the Hausdorff Dimension
Name
2012.13441v1.pdf
Description
Accepted version
Size
846.77 KB
Format
Adobe PDF
Checksum (MD5)
8930a0176cd8520e1fac61eef1cc9637
Author(s)
Wu, Chengshuai
Pines, Raz
Margaliot, Michael
Slotine, Jean-Jacques
Date Issued
September 2022
Journal
IEEE Transactions on Automatic Control
Publisher
Institute of Electrical and Electronics Engineers
Citation
C. Wu, R. Pines, M. Margaliot and J. -J. Slotine, "Generalization of the Multiplicative and Additive Compounds of Square Matrices and Contraction Theory in the Hausdorff Dimension," in IEEE Transactions on Automatic Control, vol. 67, no. 9, pp. 4629-4644, Sept. 2022, doi: 10.1109/TAC.2022.3162547.
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mechanical Engineering
Massachusetts Institute of Technology. Department of Brain and Cognitive Sciences
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Creative Commons Attribution-Noncommercial-ShareAlike
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DOI of Published Version
https://doi.org/10.1109/tac.2022.3162547
