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Transient Behavior and Entropy for Dynamical and Control Systems

动力和控制系统的瞬态行为和熵

基本信息

批准号：
261882120
负责人：
Professor Dr. Fritz Colonius
金额：
--
依托单位：
Lehrstuhl für Angewandte Analysis mit Schwerpunkt Numerische Mathematik
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2014
资助国家：
德国
起止时间：
2013-12-31 至 2016-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/261882120?language=en
关键词：
Transient Behavior Entropy Dynamical Control

项目摘要

The aim of this project is to analyze transient behavior of deterministic control systems and of random dynamical systems. For control systems, the recently introduced notion of (topological) invariance entropy which has been developed in analogy to topological entropy of dynamical systems describes the difficulty to keep a control system in a fixed subset of the state space and gives insight into the required minimal data rates. On the other hand, one may formally replace the control term by a (bounded) random perturbation. Here it is well known that the supports of stationary measures for random dynamical systems can often be described by invariant control sets, i.e., invariant subsets of complete approximate controllability. The proposed project is built on conditionally stationary (probability) measures instead of stationary measures. They describe the transient behavior of random dynamical systems and, via the skew product formalism, they are closely related to conditionally invariant measures of deterministic dynamical systems. It is expected that the supports of conditionally stationary measures can often be described by relatively invariant control sets, instead of invariant control sets. The first part of the project will analyze relatively invariant control sets. Then conditionally stationary measures and the relation of their supports to controllability properties will be analyzed. The third part of the project will be devoted to a metric notion of invariance entropy based on conditionally stationary measures. In particular, the relation to the topological invariance entropy will be explored with a view toward a variational principle. All parts of the project will be considered for systems in discrete and in continuous time. The expected results will be of relevance in the analysis of the connections between the transient behavior of random systems and control systems, as well as in the analysis of minimal data rates for digitally connected control systems.

本课题的目的是分析确定性控制系统和随机动力系统的暂态行为。对于控制系统，最近引入的（拓扑）不变性熵（topological invariance entropy）的概念描述了将控制系统保持在状态空间的固定子集中的困难，并给出了所需的最小数据速率的见解。另一方面，形式上可以用一个（有界的）随机扰动代替控制项。众所周知，随机动力系统的平稳测度的支持通常可以用不变控制集来描述，即完全近似可控性的不变子集。建议的项目是建立在有条件的平稳（概率）措施，而不是平稳措施。它们描述了随机动力系统的瞬态行为，并通过偏积形式，与确定性动力系统的条件不变测度密切相关。期望条件平稳测度的支持通常可以用相对不变的控制集来描述，而不是不变的控制集。项目的第一部分将分析相对不变的控制集。然后分析了条件平稳测度及其支撑与可控性的关系。该项目的第三部分将致力于基于条件平稳度量的不变性熵的度量概念。特别地，将从变分原理的角度来探讨与拓扑不变性熵的关系。项目的所有部分将考虑系统在离散和连续时间。预期结果将在分析随机系统和控制系统的暂态行为之间的联系以及分析数字连接控制系统的最小数据速率方面具有相关性。