Invariance Pressure Dimensions for Control Systems

Zhong, Xingfu; Yu, Huan

doi:10.1007/s10884-018-9701-z

Cited by 15 publications

(7 citation statements)

References 21 publications

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“…In this section we recall the concept of invariance pressure considered in [1,2,18] where potentials are defined on the control range. Furthermore, we introduce the generalized version of total invariance pressure, where the potentials are defined on the product of the state space and the control range.…”

Section: Invariance Pressurementioning

confidence: 99%

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Controllability Properties and Invariance Pressure for Linear Discrete-Time Systems

2021

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Section: Invariance Pressurementioning

confidence: 99%

“…Invariance pressure has been analyzed in Colonius et al [1][2][3]. In Zhong and Huang [18] it is shown that several generalized notions of invariance pressure fit into the dimensiontheoretic framework due to Pesin.…”

Section: Introductionmentioning

confidence: 99%

Controllability Properties and Invariance Pressure for Linear Discrete-Time Systems

2021

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“…Recently, the same authors in [7] obtain some bounds of invariance pressure and get an explicit formula for hyperbolic linear control systems. Zhong and Huang [19] introduced a version of invariance pressure in a way resembling Hausdorff dimension, which is called Bowen invariance pressure. On the other hand, Tang, Cheng, and Zhao [16] extended Feng and Huang's result and gave variational principle between Pesin-Pitskel topological pressure (also called Bowen topological pressure) and measure-theoretic lower pressure.…”

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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“…For controlled invariant sets with zero invariance entropy, it is useful to consider the invariance complexity function first studied by Wang, Huang and Chen [24], which is an analogue in topological dynamical systems (see [12] and the references therein). We refer the readers to [1,2,3,5,6,7,4,10,11,13,14,16,21,22,23,25,26,27] for more details about invariance entropy.…”

Section: Introductionmentioning

confidence: 99%