Invariance Pressure for Control Systems

Colonius, Fritz; Santana, Alexandre J.; Cossich, João A. N.

doi:10.1007/s10884-018-9646-2

Cited by 20 publications

(19 citation statements)

References 16 publications

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“…Invariance pressure. In this subsection we recall the concepts of invariance and outer invariance pressure introduced in Colonius, Cossich and Santana [6,7] and some of their properties.…”

Section: Preliminariesmentioning

confidence: 99%

“…For the inner invariance pressure of discrete time systems, one considers sets which are (τ, K, intQ)-spanning. Here again finite spanning sets are sufficient (the proof given in [6,Propositon 5] for the case K = Q easily extends to admissible pairs (K, Q)).…”

Section: Preliminariesmentioning

confidence: 99%

“…The notion of invariance pressure generalizes invariance entropy by adding potentials f on the control range. It has been introduced and analyzed in Colonius, Cossich, Santana [6,7]. Zhong and Huang [20] show that invariance pressure can be characterized as a dimension-like notion within the framework due to Pesin.…”

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confidence: 99%

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Bounds for invariance pressure

Colonius

Cossich

Santana

2020

Journal of Differential Equations

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“…Invariance pressure. In this subsection we recall the concepts of invariance and outer invariance pressure introduced in Colonius, Cossich and Santana [6,7] and some of their properties.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

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Bounds for invariance pressure

Colonius

Cossich

Santana

2020

Journal of Differential Equations

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“…Invariance pressure, as a generalization of invariance entropy, was first introduced by Colonius, Santana, and Cossich [8,4]. Recently, the same authors in [7] obtain some bounds of invariance pressure and get an explicit formula for hyperbolic linear control systems.…”

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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“…Measure-theoretic versions of invariance entropy have been considered in Colonius [4] and Wang, Huang, and Sun [15]. Invariance pressure has been analyzed in Colonius, Cossich, and Santana [1,2,3]. In Zhong and Huang [18] it is shown that several generalized notions of invariance pressure fit into the dimension-theoretic framework due to Pesin.…”

Section: Introductionmentioning

confidence: 99%