Equi-invariability, bounded invariance complexity and L-stability for control systems
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https://doi.org/10.1007/s11425-020-1693-7
Equi-invariability, bounded invariance complexity and L-stability for control systems Xing-fu Zhong1 , Zhi-jing Chen2,∗ & Yu Huang3 1School
of Mathematics and Statistics, Guangdong University of Foreign Studies, Guangzhou 510006, China; 2School of Mathematics and Systems Science, Guangdong Polytechnic Normal University, Guangzhou 510665, China; 3School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China Email: [email protected], [email protected], [email protected] Received January 11, 2020; accepted May 15, 2020
Abstract
In this paper, we introduce the notions of bounded invariance complexity, bounded invariance
complexity in the mean and mean Lyapunov-stability for control systems. Then we characterize these notions by introducing six types of equi-invariability. As a by-product, two new dichotomy theorems for the control system on the control sets are established. Keywords MSC(2010)
equi-invariability, invariance complexity, dichotomy theorem, control set, invariance entropy 37B05, 93C55, 93D09
Citation: Zhong X-F, Chen Z-J, Huang Y. Equi-invariability, bounded invariance complexity and L-stability for control systems. Sci China Math, 2020, 63, https://doi.org/10.1007/s11425-020-1693-7
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Introduction
In this paper, we mainly consider a discrete-time control system on a metric space X of the following form: xn+1 = F (xn , un ) =: Fun (xn ), n ∈ N0 = {0, 1, . . .}, (1.1) where F is a map from X × U to X, U is a compact set, and Fu (·) ≡ F (·, u) is continuous for every u ∈ U . Given a control sequence ω = (ω0 , ω1 , . . .) in U , the solution of (1.1) can be written as ϕ(k, x, ω) = Fωk−1 ◦ · · · ◦ Fω0 (x). For convenience, we denote the system (1.1) by Σ = (N0 , X, U, U , ϕ), where U = U N0 . Furthermore, we assume that ϕ : N0 × X × U → X is continuous. Invariance entropy introduced by Colonius and Kawan [6] as well as topological feedback entropy introduced by Nair et al. [22] characterizes the minimal data rate for making a subset of the state space invariant. It is a very useful invariant to describe the exponential growth rate of the minimal number of * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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Zhong X-F et al.
Sci China Math
different control functions sufficient for orbits to stay in a given set when starting in a subset of this set. For controlled invariant sets with zero invariance entropy, it is useful to consider the invariance complexity function first studied by Wang et al. [24], which is an analogue in topological dynamical systems (see [15] and the references therein). We refer the readers to [1–7, 10], [11–14, 16, 17, 19, 25] and [26, 27] for more details about invariance entropy. In 1993, Colonius and Kliemann [8] introduced a notion of a control set and obtained a beautiful result that control sets of a given control system coincide with maximal topologically mixing (transitive) sets of the con
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