Abstract
We establish characterizations of weak input-to-state stability for abstract dynamical systems with inputs, which are similar to characterizations of uniform and of strong input-to-state stability established in a recent paper by A. Mironchenko and F. Wirth. We also investigate the relation of weak input-to-state stability to other common stability concepts, thus contributing to a better theoretical understanding of input-to-state stability theory.
Similar content being viewed by others
References
Cazenave T, Haraux A (1998) An introduction to semilinear evolution equations. Oxford University Press, Oxford
Curtain R, Zwart H (1995) An introduction to infinite-dimensional linear systems theory, 1st edn. Springer, Berlin
Curtain R, Zwart H (2016) Stabilization of collocated systems by nonlinear boundary control. Syst Control Lett 96:11–14
Dashkovskiy S, Mironchenko A (2013) Input-to-state stability of infinite-dimensional control systems. Math Control Signal Syst 25:1–35
Dashkovskiy S, Kapustyan OV, Schmid J (2019) A local input-to-state stability result w.r.t. attractors of nonlinear reaction–diffusion equations. arXiv:1909.07022
Edalatzadeh MS, Morris K (2019) Stability and well-posedness of a nonlinear railway track model. IEEE Control Syst Lett 3:162–167
Engel K-J, Nagel R (2000) One-parameter semigroups for linear evolution equations. Springer, Berlin
Jacob B, Partington J (2004) Admissibility of control and observation operators for semigroups: a survey. Oper Theory Adv Appl 149:199–221
Jacob B, Nabiullin R, Partington J, Schwenninger F (2018) Infinite-dimensional input-to-state stability and Orlicz spaces. SIAM J Control Optim 56:868–889
Jacob B, Schwenninger F, Zwart H (2019) On the continuity of solutions for parabolic control systems and input-to-state stability. J Differ Equ 266:6284–6306
Jacob B, Schwenninger F (2018) Input-to-state stability of unbounded bilinear control systems. arXiv:1811.08470
Karafyllis I, Krstic M (2016) ISS with respect to boundary disturbances for 1-D parabolic PDEs. IEEE Trans Autom Control 61:3712–3724
Karafyllis I, Krstic M (2017) ISS in different norms for 1-D parabolic PDEs with boundary disturbances. SIAM J Control Optim 55:1716–1751
Mazenc F, Prieur C (2011) Strict Lyapunov functions for semilinear parabolic partial differential equations. Math Control Relat Fields 1:231–250
Mironchenko A (2016) Local input-to-state stability: characterizations and counterexamples. Syst Control Lett 87:23–28
Mironchenko A, Ito H (2016) Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions. Math Control Relat Fields 6:447–466
Mironchenko A, Wirth F (2017) Input-to-state stability of time-delay systems: criteria and open problems. In: Conference of proceedings of the 56th IEEE conference on decision and control, pp 3719–3724
Mironchenko A, Wirth F (2018) Characterizations of input-to-state stability for infinite-dimensional systems. IEEE Trans Autom Control 63:1692–1707
Mironchenko A, Wirth F (2018) Lyapunov characterization of input-to-state stability for semilinear control systems over Banach spaces. Syst Control Lett 119:64–70
Mironchenko A, Karafyllis I, Krstic M (2019) Monotonicity methods for input-to-state stability of nonlinear parabolic PDEs with boundary disturbances. SIAM J Control Optim 57:510–532
Miyadera I (1992) Nonlinear semigroups. American Mathematical Society, Providence
Nabiullin R, Schwenninger F (2018) Strong input-to-state stability for infinite dimensional linear systems. Math Control Signal Syst 30:4. https://doi.org/10.1007/s00498-018-0210-8
Pazy A (1983) Semigroups of linear operators and applications to partial differential equations. Springer, Berlin
Schmid J, Zwart H (2018) Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances. In: Conference proceedings of the 23rd symposium on mathematical theory of networks and systems, pp 570–575
Schmid J, Zwart H (2018) Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances. ESAIM Control Optim Calc Var. arXiv:1804.10598
Schmid J (2019) Infinite-time admissibility under compact perturbations. In: Kerner J, Laasri H, Mugnolo D (eds) Topics in control theory of infinite-dimensional systems. Birkhäuser. arxiv:1904.11380
Schmid J (2019) Well-posedness of non-autonomous semilinear input-output systems. arXiv:1904.10376
Schmid J, Kapustyan OV, Dashkovskiy S (2019) Asymptotic gain results for attractors of semilinear systems. arXiv:1909.06302
Slemrod M (1989) Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control. Math Control Signal Syst 2:265–285
Sontag ED, Wang Y (1996) New characterizations of input-to-state stability. IEEE Trans Autom Control 24:1283–1294
Tanwani A, Prieur C, Tarbouriech S (2017) Disturbance-to-state stabilization and quantized control for linear hyperbolic systems. arXiv:1703.00302
Temam R (1998) Infinite-dimensional dynamical systems in mechanics and physics, 2nd edn. Springer, Berlin
Tucsnak M, Weiss G (2009) Observation and control for operator semigroups. Birkhäuser, Basel
Weiss G (1989) Admissibility of unbounded control operators. SIAM J Control Optim 27:527–545
Zheng J, Zhu G (2017) Input-to-state stability with respect to boundary disturbances for a class of semi-linear parabolic equations. arXiv:1709.01880
Zheng J, Zhu G (2017) A De Giorgi iteration-based approach for the establishment of ISS properties of a class of semi-linear parabolic PDEs with boundary and in-domain disturbances. arXiv:1710.09917
Acknowledgements
I would like to thank the German Research Foundation (DFG) financial support through the grant “Input-to-state stability and stabilization of distributed-parameter systems” (DA 767/7-1).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Schmid, J. Weak input-to-state stability: characterizations and counterexamples. Math. Control Signals Syst. 31, 433–454 (2019). https://doi.org/10.1007/s00498-019-00248-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00498-019-00248-5