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Stabilization of Non-Linear Systems: Information Theoretic vs. Geometric Analysis

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Stochastic Teams, Games, and Control under Information Constraints

Part of the book series: Systems & Control: Foundations & Applications ((SCFA))

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Abstract

In this chapter, we consider non-linear models controlled under information constraints. More specifically, this chapter is concerned with necessary and sufficient conditions on information channels in a networked non-linear stochastic control system for which there exist coding and control policies such that the controlled system is stochastically stable in one or more of the following senses: (i) The state \(\{x_t\}\) and the coding and control parameters lead to a stochastically stable process (in senses to be made precise), and (ii) \(\{x_t\}\) is asymptotically stationary, or asymptotically mean stationary (AMS), and (iii) \(\{x_t\}\) is (asymptotically) ergodic.

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Notes

  1. 1.

    Two technical intricacies here are as follows: For differential entropy (unlike discrete entropy) the relationship \(h(x+y) \leq h(x) + h(y)\) does not in general hold for random variables \(x, y\); this is why first a conditioning on \(\mathcal {S}\) is taken in the proof. Furthermore, we cannot obtain an upper bound by taking out the conditioning on the event \(|x_{T}| > b(T)\), since conditioning on a single event may decrease or increase entropy; note that conditioning on a random variable, however, does not increase the entropy.

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Author information

Authors and Affiliations

  1. Mathematics & Statistics, Queen’s University, Kingston, ON, Canada

    Serdar Yüksel

  2. Coordinated Science Laboratory, University of Illinois Urbana-Champaign, Urbana, IL, USA

    Tamer Başar

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  1. Serdar Yüksel
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  2. Tamer Başar
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Correspondence to Serdar Yüksel or Tamer Başar .

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Yüksel, S., Başar, T. (2024). Stabilization of Non-Linear Systems: Information Theoretic vs. Geometric Analysis. In: Stochastic Teams, Games, and Control under Information Constraints. Systems & Control: Foundations & Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-54071-4_14

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