Dual Seminorms, Ergodic Coefficients and Semicontraction Theory

• 成果类型: Article
• 署名作者: De Pasquale, Giulia; Smith, Kevin D.; Bullo, Francesco; Valcher, Maria Elena
• 署名单位: University of Padua; 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.2023.3302788
• 发表日期: 2024
• 页码: 3040-3053
• 关键词: duality; ergodic coefficients; induced matrix seminorm; logarithmic norm; semicontraction theory

摘要

Dynamical systems that are contracting on a subspace are said to be semicontracting. Semicontraction theory is a useful tool in the study of consensus algorithms and dynamical flow systems such as Markov chains. To develop a comprehensive theory of semicontracting systems, we investigate seminorms on vector spaces and define two canonical notions: Projection and distance seminorms. We show that the well-known l(p) ergodic coefficients are induced matrix seminorms and play a central role in stability problems. In particular, we formulate a duality theorem that explains why the Markov-Dobrushin coefficient is the rate of contraction for both averaging and conservation flows in discrete time. Moreover, we obtain parallel results for induced matrix logarithmic seminorms. Finally, we propose comprehensive theorems for strong semicontractivity of linear and nonlinear time-varying dynamical systems with invariance and conservation properties both in discrete and continuous time.