Variational principles of invariance pressures on partitions

Abstract

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.

Invariance entropy for uncertain control systems

“…For better understanding of invariance entropy, various notions relating to invariance entropy have been proposed by several groups of researchers from different views, such as invariance pressure [15,16,17,18,19,20], measure-theoretic versions of invariance entropy [21,22,23,24], dimension types of invariance entropy [25], complexity of invariance entropy [26,27,28]. Note that Kawan and Yüksel [29] introduced a notion of stabilization entropy which is a variant of invariance entropy.…”

Equi-invariability, bounded invariance complexity and L-stability for control systems

“…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.…”