TY - JOUR

T1 - Generalization of the Multiplicative and Additive Compounds of Square Matrices and Contraction Theory in the Hausdorff Dimension

AU - Wu, Chengshuai

AU - Pines, Raz

AU - Margaliot, Michael

AU - Slotine, Jean Jacques

N1 - Publisher Copyright:
© 1963-2012 IEEE.

PY - 2022/9/1

Y1 - 2022/9/1

N2 - 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.

AB - 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.

KW - Additive compound matrix

KW - contraction theory

KW - fractal sets

KW - multiplicative compound matrix

KW - nonlinear dynamical systems

KW - ribosome flow model

KW - Thomas' cyclically symmetric attractor

UR - http://www.scopus.com/inward/record.url?scp=85137191756&partnerID=8YFLogxK

U2 - 10.1109/TAC.2022.3162547

DO - 10.1109/TAC.2022.3162547

M3 - מאמר

AN - SCOPUS:85137191756

VL - 67

SP - 4629

EP - 4644

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

SN - 0018-9286

IS - 9

ER -