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.
Input-to-state stability (weak, strong, uniform) Robust stability Infinite-dimensional dynamical systems with inputs
This is a preview of subscription content, log in to check access.
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).
Cazenave T, Haraux A (1998) An introduction to semilinear evolution equations. Oxford University Press, OxfordzbMATHGoogle Scholar
Curtain R, Zwart H (1995) An introduction to infinite-dimensional linear systems theory, 1st edn. Springer, BerlinCrossRefGoogle Scholar
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
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
Mironchenko A, Wirth F (2018) Characterizations of input-to-state stability for infinite-dimensional systems. IEEE Trans Autom Control 63:1692–1707MathSciNetCrossRefGoogle Scholar
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
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
Pazy A (1983) Semigroups of linear operators and applications to partial differential equations. Springer, BerlinCrossRefGoogle Scholar
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
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 2:265–285MathSciNetCrossRefGoogle Scholar