# Weak input-to-state stability: characterizations and counterexamples

Original Article

First Online:

- 7 Downloads

## 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.

## Keywords

Input-to-state stability (weak, strong, uniform) Robust stability Infinite-dimensional dynamical systems with inputs## Notes

### 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).

## References

- 1.Cazenave T, Haraux A (1998) An introduction to semilinear evolution equations. Oxford University Press, OxfordzbMATHGoogle Scholar
- 2.Curtain R, Zwart H (1995) An introduction to infinite-dimensional linear systems theory, 1st edn. Springer, BerlinCrossRefGoogle Scholar
- 3.Curtain R, Zwart H (2016) Stabilization of collocated systems by nonlinear boundary control. Syst Control Lett 96:11–14MathSciNetCrossRefGoogle Scholar
- 4.Dashkovskiy S, Mironchenko A (2013) Input-to-state stability of infinite-dimensional control systems. Math Control Signal Syst 25:1–35MathSciNetCrossRefGoogle Scholar
- 5.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
- 6.Edalatzadeh MS, Morris K (2019) Stability and well-posedness of a nonlinear railway track model. IEEE Control Syst Lett 3:162–167 CrossRefGoogle Scholar
- 7.Engel K-J, Nagel R (2000) One-parameter semigroups for linear evolution equations. Springer, BerlinzbMATHGoogle Scholar
- 8.Jacob B, Partington J (2004) Admissibility of control and observation operators for semigroups: a survey. Oper Theory Adv Appl 149:199–221MathSciNetzbMATHGoogle Scholar
- 9.Jacob B, Nabiullin R, Partington J, Schwenninger F (2018) Infinite-dimensional input-to-state stability and Orlicz spaces. SIAM J Control Optim 56:868–889MathSciNetCrossRefGoogle Scholar
- 10.Jacob B, Schwenninger F, Zwart H (2019) On the continuity of solutions for parabolic control systems and input-to-state stability. J Differ Equ 266:6284–6306MathSciNetCrossRefGoogle Scholar
- 11.Jacob B, Schwenninger F (2018) Input-to-state stability of unbounded bilinear control systems. arXiv:1811.08470
- 12.Karafyllis I, Krstic M (2016) ISS with respect to boundary disturbances for 1-D parabolic PDEs. IEEE Trans Autom Control 61:3712–3724MathSciNetCrossRefGoogle Scholar
- 13.Karafyllis I, Krstic M (2017) ISS in different norms for 1-D parabolic PDEs with boundary disturbances. SIAM J Control Optim 55:1716–1751MathSciNetCrossRefGoogle Scholar
- 14.Mazenc F, Prieur C (2011) Strict Lyapunov functions for semilinear parabolic partial differential equations. Math Control Relat Fields 1:231–250MathSciNetCrossRefGoogle Scholar
- 15.Mironchenko A (2016) Local input-to-state stability: characterizations and counterexamples. Syst Control Lett 87:23–28MathSciNetCrossRefGoogle Scholar
- 16.Mironchenko A, Ito H (2016) Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions. Math Control Relat Fields 6:447–466MathSciNetCrossRefGoogle Scholar
- 17.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 3719–3724Google Scholar
- 18.Mironchenko A, Wirth F (2018) Characterizations of input-to-state stability for infinite-dimensional systems. IEEE Trans Autom Control 63:1692–1707MathSciNetCrossRefGoogle Scholar
- 19.Mironchenko A, Wirth F (2018) Lyapunov characterization of input-to-state stability for semilinear control systems over Banach spaces. Syst Control Lett 119:64–70MathSciNetCrossRefGoogle Scholar
- 20.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 57:510–532MathSciNetCrossRefGoogle Scholar
- 21.Miyadera I (1992) Nonlinear semigroups. American Mathematical Society, ProvidenceCrossRefGoogle Scholar
- 22.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 MathSciNetCrossRefzbMATHGoogle Scholar
- 23.Pazy A (1983) Semigroups of linear operators and applications to partial differential equations. Springer, BerlinCrossRefGoogle Scholar
- 24.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 570–575Google Scholar
- 25.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
- 26.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
- 27.Schmid J (2019) Well-posedness of non-autonomous semilinear input-output systems. arXiv:1904.10376
- 28.Schmid J, Kapustyan OV, Dashkovskiy S (2019) Asymptotic gain results for attractors of semilinear systems. arXiv:1909.06302
- 29.Slemrod M (1989) Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control. Math Control Signal Syst 2:265–285MathSciNetCrossRefGoogle Scholar
- 30.Sontag ED, Wang Y (1996) New characterizations of input-to-state stability. IEEE Trans Autom Control 24:1283–1294MathSciNetCrossRefGoogle Scholar
- 31.Tanwani A, Prieur C, Tarbouriech S (2017) Disturbance-to-state stabilization and quantized control for linear hyperbolic systems. arXiv:1703.00302
- 32.Temam R (1998) Infinite-dimensional dynamical systems in mechanics and physics, 2nd edn. Springer, BerlinGoogle Scholar
- 33.Tucsnak M, Weiss G (2009) Observation and control for operator semigroups. Birkhäuser, BaselCrossRefGoogle Scholar
- 34.Weiss G (1989) Admissibility of unbounded control operators. SIAM J Control Optim 27:527–545MathSciNetCrossRefGoogle Scholar
- 35.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
- 36.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

## Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019