Verifying k-Contraction Without Computing k-Compounds

• 成果类型: Article
• 署名作者: Dalin, Omri; Ofir, Ron; Bar-Shalom, Eyal; Ovseevich, Alexander; Bullo, Francesco; Margaliot, Michael
• 署名单位: Tel Aviv University; University of California System; University of California Santa Barbara
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2023.3326058
• 发表日期: 2024
• 页码: 1492-1506
• 关键词: Contracting systems; Hopfield networks; k-shifted logarithmic norm; matrix measure; STABILITY

摘要

Compound matrices have found applications in many fields of science including systems and control theory. In particular, a sufficient condition fork-contraction is that a logarithmic norm (also called matrix measure)of the k-additive compound of the Jacobian is uniformly negative. However, this computation may be difficult to perform analytically and expensive numerically because the k-additive compound of an n x n matrix has dimensions((n)(k))x((n)(k)). This article establishes a duality relation between the k and(n-k)compounds of an n x n matrix A. This duality relation is used to derive a sufficient condition fork-contraction that does not require the computation of any k-compounds. These theoretical results are demonstrated by deriving a sufficient condition for k-contraction of an n-dimensional Hopfield network that does not require to compute any compounds. In particular, fork=2this sufficient condition implies that the network is 2-contracting and thus admits a strong asymptotic property: every bounded solution of the network converges to an equilibrium point, that may not be unique. This is relevant, for example, when using the Hopfield network as an associative memory that stores patterns as equilibrium points of the dynamics.