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

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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.

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References

  1. Colonius F. Metric invariance entropy and conditionally invariant measures. Ergodic Theory Dynam Systems, 2018, 38: 921–939

    Article  MathSciNet  Google Scholar 

  2. Colonius F. Invariance entropy, quasi-stationary measures and control sets. Discrete Contin Dyn Syst, 2018, 38: 2093–2123

    Article  MathSciNet  Google Scholar 

  3. Colonius F, Cossich J A N, Santana A J. Invariance pressure of control sets. SIAM J Control Optim, 2018, 56: 4130–4147

    Article  MathSciNet  Google Scholar 

  4. Colonius F, Cossich JAN, Santana A J. Bounds for invariance pressure. J Differential Equations, 2020, 268: 7877–7896

    Article  MathSciNet  Google Scholar 

  5. Colonius F, Fukuoka R, Santana A J. Invariance entropy for topological semigroup actions. Proc Amer Math Soc, 2013, 141: 4411–4423

    Article  MathSciNet  Google Scholar 

  6. Colonius F, Kawan C. Invariance entropy for control systems. SIAM J Control Optim, 2009, 48: 1701–1721

    Article  MathSciNet  Google Scholar 

  7. Colonius F, Kawan C, Nair G. A note on topological feedback entropy and invariance entropy. Systems Control Lett, 2013, 62: 377–381

    Article  MathSciNet  Google Scholar 

  8. Colonius F, Kliemann W. Some aspects of control systems as dynamical systems. J Dynam Differential Equations, 1993, 5: 469–494

    Article  MathSciNet  Google Scholar 

  9. Colonius F, Kliemann W. The Dynamics of Control. Boston: Birkhauser, 2000

    Book  Google Scholar 

  10. Colonius F, Santana A J, Cossich J A N. Controllability properties and invariance pressure for linear discrete-time systems. arXiv:1909.04382, 2019

  11. Colonius F, Santana A J, Cossich JAN. Invariance pressure for control systems. J Dynam Differential Equations, 2019, 31: 1–23

    Article  MathSciNet  Google Scholar 

  12. Da Silva A. Invariance entropy for random control systems. Math Control Signals Systems, 2013, 25: 491–516

    Article  MathSciNet  Google Scholar 

  13. Da Silva A. Outer invariance entropy for linear systems on Lie groups. SIAM J Control Optim, 2014, 52: 3917–3934

    Article  MathSciNet  Google Scholar 

  14. Da Silva A, Kawan C. Invariance entropy of hyperbolic control sets. Discrete Contin Dyn Syst, 2015, 36: 97–136

    Article  MathSciNet  Google Scholar 

  15. Huang W, Li J, Thouvenot J P, et al. Bounded complexity, mean equicontinuity and discrete spectrum. Ergodic Theory Dynam Systems, 2020, in press

  16. Huang Y, Zhong X. Caratheodory-Pesin structures associated with control systems. Systems Control Lett, 2018, 112: 36–41

    Article  MathSciNet  Google Scholar 

  17. Kawan C. Upper and lower estimates for invariance entropy. Discrete Contin Dyn Syst, 2011, 30: 169–186

    Article  MathSciNet  Google Scholar 

  18. Kawan C. Invariance Entropy for Deterministic Control Systems. Lecture Notes in Mathematics, vol. 2089. New York: Springer, 2013

    Book  Google Scholar 

  19. Kawan C, Da Silva A. Invariance entropy for a class of partially hyperbolic sets. Math Control Signals Systems, 2018, 30: 18

    Article  MathSciNet  Google Scholar 

  20. Li J, Tu S, Ye X. Mean equicontinuity and mean sensitivity. Ergodic Theory Dynam Systems, 2015, 35: 2587–2612

    Article  MathSciNet  Google Scholar 

  21. Nadler S B. Hyperspaces of Sets: A Text with Research Questions. Monographs and Textbooks in Pure and Applied Mathematics, vol. 49. New York-Basel: Marcel Dekker, 1978

    MATH  Google Scholar 

  22. Nair G N, Evans R J, Mareels I M Y, et al. Topological feedback entropy and nonlinear stabilization. IEEE Trans Automat Control, 2004, 49: 1585–1597

    Article  MathSciNet  Google Scholar 

  23. Qiu J, Zhao J. A note on mean equicontinuity. J Dynam Differential Equations, 2020, 32: 101–116

    Article  MathSciNet  Google Scholar 

  24. Wang T, Huang Y, Chen Z. Dichotomy theorem for control sets. Systems Control Lett, 2019, 129: 10–16

    Article  MathSciNet  Google Scholar 

  25. Wang T, Huang Y, Sun H. Measure-theoretic invariance entropy for control systems. SIAM J Control Optim, 2019, 57: 310–333

    Article  MathSciNet  Google Scholar 

  26. Zhong X F. Variational principles of invariance pressures on partitions. Discrete Contin Dyn Syst, 2020, 40: 491–508

    Article  MathSciNet  Google Scholar 

  27. Zhong X F, Huang Y. Invariance pressure dimensions for control systems. J Dynam Differential Equations, 2019, 31: 2205–2222

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The first and third authors were supported by National Natural Science Foundation of China (Grant No. 11771459). The first author was supported by Research Funds of Guangdong University of Foreign Studies (Grant Nos. 299-X5218165 and 299-X5219222) and Guangdong Basic and Applied Basic Research Foundation (Grant No. 2019A1515110932). The second author was supported by National Natural Science Foundation of China (Grant Nos. 11701584 and 11871228) and the Natural Science Research Project of Guangdong Province (Grant No. 2018KTSCX122). The authors thank Professor Jian Li for sharing his research on dynamical systems. The authors thank the anonymous referees for their critical comments and suggestions that led to the improvement of this manuscript.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Guangdong University of Foreign Studies, Guangzhou, 510006, China

    Xing-fu Zhong

  2. School of Mathematics and Systems Science, Guangdong Polytechnic Normal University, Guangzhou, 510665, China

    Zhi-jing Chen

  3. School of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China

    Yu Huang

Authors
  1. Xing-fu Zhong
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  2. Zhi-jing Chen
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  3. Yu Huang
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Correspondence to Zhi-jing Chen.

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Zhong, Xf., Chen, Zj. & Huang, Y. Equi-invariability, bounded invariance complexity and L-stability for control systems. Sci. China Math. 64, 2275–2294 (2021). https://doi.org/10.1007/s11425-020-1693-7

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  • DOI: https://doi.org/10.1007/s11425-020-1693-7

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