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

• 成果类型: Article
• 署名作者: Wu, Chengshuai; Pines, Raz; Margaliot, Michael; Slotine, Jean-Jacques
• 署名单位: Tel Aviv University; Tel Aviv University; Massachusetts Institute of Technology (MIT); Massachusetts Institute of Technology (MIT)
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2022.3162547
• 发表日期: 2022
• 页码: 4629-4644
• 关键词: Additive compound matrix; Contraction theory; fractal sets; multiplicative compound matrix; nonlinear dynamical systems; ribosome flow model; Thomas' cyclically symmetric attractor

摘要

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 alpha multiplicative and alpha additive compounds of a square matrix, with alpha real. We study the properties of these new compounds and demonstrate an application in the context of the Douady and Oesterle theorem. Our results lead to a generalization of contracting systems to alpha-contracting systems, with alpha real. Roughly speaking, the dynamics of such systems contracts any set with the Hausdorff dimension larger than alpha. For alpha = 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.