{"title":"Control sets on maximal compact subgroups","authors":"Mauro Patrão, Laércio dos Santos","doi":"10.1007/s00498-024-00391-8","DOIUrl":null,"url":null,"abstract":"<p>Let <i>G</i> be a noncompact connected semisimple Lie group, with finite center. In this paper we study the action of semigroups <i>S</i> of <i>G</i>, with nonempty interior, acting on maximal compact connected subgroups <i>K</i> of <i>G</i>. When <i>S</i> is connected, it is well known that the invariant control sets on <i>K</i> are the connected components of <span>\\(\\pi ^{-1}(C)\\)</span>, where <span>\\(\\pi \\)</span> is the canonical projection of <i>K</i> onto <i>F</i>, <i>F</i> is the flag type of <i>S</i> and <i>C</i> is the only invariant control set for <i>S</i> on <i>F</i>. One of the main results of the present paper describes the set of transitivity of a control set, not necessarily invariant, of a semigroup <i>S</i>, not necessarily connected, acting on <i>K</i>, as fixed points of regular elements in <i>S</i>. Furthermore, we show that the number of control sets on <i>K</i> is the product of the number of control sets on the maximal flag manifold of <i>G</i> by the number of invariant control sets on <i>K</i>.</p>","PeriodicalId":51123,"journal":{"name":"Mathematics of Control Signals and Systems","volume":"1 1","pages":""},"PeriodicalIF":1.8000,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematics of Control Signals and Systems","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1007/s00498-024-00391-8","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"AUTOMATION & CONTROL SYSTEMS","Score":null,"Total":0}
引用次数: 0
Abstract
Let G be a noncompact connected semisimple Lie group, with finite center. In this paper we study the action of semigroups S of G, with nonempty interior, acting on maximal compact connected subgroups K of G. When S is connected, it is well known that the invariant control sets on K are the connected components of \(\pi ^{-1}(C)\), where \(\pi \) is the canonical projection of K onto F, F is the flag type of S and C is the only invariant control set for S on F. One of the main results of the present paper describes the set of transitivity of a control set, not necessarily invariant, of a semigroup S, not necessarily connected, acting on K, as fixed points of regular elements in S. Furthermore, we show that the number of control sets on K is the product of the number of control sets on the maximal flag manifold of G by the number of invariant control sets on K.
假设 G 是一个非紧凑连通的半简单李群,具有有限中心。当 S 是连通的,众所周知,K 上的不变控制集是 \(\pi ^{-1}(C)\) 的连通分量,其中 \(\pi \) 是 K 在 F 上的规范投影,F 是 S 的旗型,C 是 S 在 F 上的唯一不变控制集。本文的主要结果之一描述了作用于 K 的半群 S 的控制集(不一定是不变的)作为 S 中规则元素的定点的传递性集合。
期刊介绍:
Mathematics of Control, Signals, and Systems (MCSS) is an international journal devoted to mathematical control and system theory, including system theoretic aspects of signal processing.
Its unique feature is its focus on mathematical system theory; it concentrates on the mathematical theory of systems with inputs and/or outputs and dynamics that are typically described by deterministic or stochastic ordinary or partial differential equations, differential algebraic equations or difference equations.
Potential topics include, but are not limited to controllability, observability, and realization theory, stability theory of nonlinear systems, system identification, mathematical aspects of switched, hybrid, networked, and stochastic systems, and system theoretic aspects of optimal control and other controller design techniques. Application oriented papers are welcome if they contain a significant theoretical contribution.