2000 character limit reached
For time-invariant delay systems, global asymptotic stability does not imply uniform global attractivity (2403.00387v1)
Published 1 Mar 2024 in eess.SY and cs.SY
Abstract: Adapting a counterexample recently proposed by J.L. Mancilla-Aguilar and H. Haimovich, we show here that, for time-delay systems, global asymptotic stability does not ensure that solutions converge uniformly to zero over bounded sets of initial states. Hence, the convergence might be arbitrarily slow even if initial states are confined to a bounded set.
- D. Angeli and E.D. Sontag. Forward completeness, unboundedness observability, and their Lyapunov characterizations. Systems & Control Letters, 38:209–217, 1999.
- Forward completeness implies bounded reachable sets for time-delay systems on the state space of essentially bounded measurable functions. ArXiV preprint, 2024.
- The ISS framework for time-delay systems: a survey. Mathematics of Control, Signals, and Systems, pages 1–70, 2023.
- E. Fridman. Introduction to Time-Delay Systems. Analysis and Control. Springer, September 2014.
- J.P. Gauthier and I. Kupka. Deterministic Observation Theory and Applications. Cambridge University Press, 2001.
- Stability of Time-Delay Systems. Springer, 2012.
- J.K. Hale. Theory of Functional Differential Equations, volume 3 of Applied mathematical sciences. Springer, New York, NY, 1977.
- I. Karafyllis and Z.P. Jiang. Stability and Stabilization of Nonlinear Systems. Communications and Control Engineering Series. Springer-Verlag, London, 2011.
- Is global asymptotic stability necessarily uniform for time-invariant time-delay systems? SIAM Journal on Control and Optimization, 60(6):3237–3261, 2022.
- A smooth converse Lyapunov theorem for robust stability. SIAM J. on Contr. and Opt., 34:124–160, 1996.
- On the representation of switched systems with inputs by perturbed control systems. Nonlinear Analysis: Theory, Methods & Applications, 60(6):1111–1150, 2005.
- J.L. Mancilla-Aguilar and H. Haimovich. Forward completeness does not imply bounded reachability sets and global asymptotic stability is not necessarily uniform for time-delay systems. arXiv preprint arXiv:2308.07130, 2023.
- A. Mironchenko. Uniform weak attractivity and criteria for practical global asymptotic stability. Systems & Control Letters, 105:92–99, 2017.
- A. Mironchenko and C. Prieur. Input-to-state stability of infinite-dimensional systems: recent results and open questions. SIAM Review, 62(3):529–614, 2020.
- A. Mironchenko and F. Wirth. Input-to-state stability of time-delay systems: Criteria and open problems. In Proc. of 56th IEEE Conference on Decision and Control, pages 3719–3724, 2017.
- A. Mironchenko and F. Wirth. Characterizations of input-to-state stability for infinite-dimensional systems. IEEE Transactions on Automatic Control, 63(6):1602–1617, 2018.
- A. Mironchenko and F. Wirth. Non-coercive Lyapunov functions for infinite-dimensional systems. Journal of Differential Equations, 105:7038–7072, 2019.
- S.I. Niculescu. Delay Effects on Stability: A Robust Control Approach, volume 269 of Lecture Notes in Control and Information Sciences. Springer, 2001.
- Stability Theory by Liapunov’s Direct Method, volume 22 of Appl. Math. Sc. Springer-Verlag, New York, 1977.
- A.R. Teel and L. Praly. A smooth Lyapunov function from a class-𝒦L𝒦𝐿{\mathcal{K}L}caligraphic_K italic_L estimate involving two positive semidefinite functions. ESAIM: Control, Optimisation and Calculus of Variations, 5:313–367, 2000.
- T. Yoshizawa. Stability Theory by Lyapunov’s Second Method. The Mathematical Society of Japan, Tokyo, 1966.