Carathéodory–Pesin structures associated with control systems

“…Kawan and Da Silva [11] and [6] analyze invariance entropy of partially hyperbolic controlled invariant sets and chain control sets. Huang and Zhong [8] show dimension-like characterizations of invariance entropy. Measuretheoretic versions of invariance entropy have been considered in Colonius [4] and Wang et al [15].…”

Section: Introductionmentioning

confidence: 99%

Controllability Properties and Invariance Pressure for Linear Discrete-Time Systems

2021

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“…and packing topological entropy and measure-theoretical lower and upper topological entropies of subsets respectively. Motivated by the works of Huang-Zhong [10], Feng-Huang [9], and Colonius [2,3], Wang, Huang, and Sun [18] introduced packing invariance entropy and gave variational principles between Bowen and packing invariance entropies and measure-theoretical lower and upper invariance entropies in some special situations respectively.…”

Section: (Communicated By Xiangdong Ye)mentioning

confidence: 99%

Variational principles of invariance pressures on partitions

Zhong¹

2020

Discrete &Amp; Continuous Dynamical Systems - A

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1. Introduction. Topological feedback entropy was first introduced by Nair et al. [14] by using invariant open covers to characterize the minimal data rate for making a subset of the state space invariant. Later, Colonius and Kawan [5] introduced invariance entropy, which is defined via spanning sets, to describe the exponential growth rate of the minimal number of different control functions sufficient for orbits to stay in a given set when starting in a subset of this set. The fact that these two entropies are equivalent was shown by Colonius, Kawan, and Nair [6]. Recently, Huang and Zhong [10] use the theory of Carathéodory-Pesin structure to obtain a dimension-like characterization for invariance entropy, which is called Bowen invariance entropy. We refer the reader to the monograph written by Kawan [11] for more about invariance entropy.By choosing conditionally invariant measures and quasi-stationary measures, Colonius [2, 3] first introduced four notions of metric invariance entropy in analogy to the topological notion of invariance entropy of deterministic control systems. In [2], Colonius showed that the metric entropy of a given controlled invariant set coincides with the minimal entropy of coder-controllers associated with a quasi-stationary measure rendering that set invariant. Variational principle [17] for topological entropy [1] and measure theoretic entropy [12] in classical dynamical systems states that topological entropy is determined by the supremum of measure theoretic entropies. Feng and Huang gave variational principles between Bowen 2010 Mathematics Subject Classification. Primary: 37A35, 37B40, 37C45; Secondary: 93C55.

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“…Kawan and Da Silva [10] and [11] analyze invariance entropy of partially hyperbolic controlled invariant sets and chain control sets. Huang and Zhong [7] show dimension-like characterizations of invariance entropy. Measure-theoretic versions of invariance entropy have been considered in Colonius [4] and Wang, Huang, and Sun [15].…”

Section: Introductionmentioning

confidence: 99%