We establish the local input-to-state stability of a large class of disturbed nonlinear reaction–diffusion equations w.r.t. the global attractor of the respective undisturbed system.
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Bergh J, Löfström J (1976) Interpolation spaces. Springer, Berlin
Chepyzhov VV, Vishik MI (1996) Trajectory attractors for reaction–diffusion systems. Topol Meth Nonlinear Anal 8:49–76
Chepyzhov VV, Vishik MI (2002) Attractors for equations of mathematical physics, vol 49. American Mathematical Society, Colloquium Publications
Dashkovskiy S, Mironchenko A (2013) Input-to-state stability of infinite-dimensional control systems. Math Control Signals Syst 25:1–35
Dashkovskiy S, Kapustyan OV, Romaniuk I (2017) Global attractors of impulsive parabolic inclusions. Discr Contin Dyn Syst Ser B 22:1875–1886
Diestel J, Uhl JJ (1977) Vector measures. Mathematical surveys and monographs, vol 15. American Mathematical Society
Gorban NV, Kapustyan OV, Kasyanov PO, Paliichuk LS (2014) On global attractors for autonomous damped wave equation with discontinuous nonlinearity. Solid Mech Appl 211:221–237
Gorban NV, Kapustyan AV, Kapustyan EA, Khomenko OV (2015) Strong global attractor for the three-dimensional Navier–Stokes system of equations in unbounded domain of channel type. J Autom Inf Sci 47:48–59
Grüne L (2002) Asymptotic behavior of dynamical and control systems under perturbation and discretization. Springer, Berlin
Henry D (1981) Geometric theory of semilinear parabolic equations. Springer, Berlin
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 (2018) Input-to-state stability of unbounded bilinear control systems. arXiv:1811.08470
Kapustyan AV, Valero J (2009) On the Kneser property for the complex Ginzburg–Landau equation and the Lotka–Volterra system with diffusion. J Math Anal Appl 357:254–272
Kapustyan OV, Kasyanov PO, Valero J (2015) Structure of the global attractor for weak solutions of a reaction–diffusion equation. Appl Math Inf Sci 9:2257–2264
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 (2017) Uniform weak attractivity and criteria for practical global asymptotic stability. Syst Control Lett 105:92–99
Mironchenko A, Wirth F (2018) Characterizations of input-to-state stability for infinite-dimensional systems. IEEE Trans Autom Control 63:1692–1707
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. Translations of mathematical monographs 109. American Mathematical Society
Robinson JC (2001) Infinite-dimensional dyanamical systems. Cambridge University Press, Cambridge
Schmid J, Zwart H (2018) Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances. arXiv:1804.10598, Accepted provisionally in ESAIM Contr Optim Calc, Var
Schmid J (2019) Weak input-to-state stability: characterizations and counterexamples. Math Control Signals Syst 31:433–454
Schmid J, Kapustyan O, Dashkovskiy S (2019) Asymptotic gain results for attractors of semilinear systems. arXiv:1909.06302
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
Valero J, Kapustyan AV (2006) On the connectedness and asymptotic behaviour of solutions of reaction–diffusion equations. J Math Anal Appl 323:614–633
Zheng J, Zhu G (2018) Input-to-state stability with respect to boundary disturbances for a class of semi-linear parabolic equations. Automatica 97:271–277
Zheng J, Zhu G (2019) 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. IEEE Trans Autom Control 64:3476–3483
S. Dashkovskiy and O. Kapustyan are partially supported by the German Research Foundation (DFG) and the State Fund for Fundamental Research of Ukraine (SFFRU) through the joint German-Ukrainian grant “Stability and robustness of attractors of nonlinear infinite-dimensional systems with respect to disturbances” (DA 767/12-1).
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Published in the topical collection Input-to-state stability for infinite-dimensional systems.
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Dashkovskiy, S., Kapustyan, O. & Schmid, J. A local input-to-state stability result w.r.t. attractors of nonlinear reaction–diffusion equations. Math. Control Signals Syst. 32, 309–326 (2020). https://doi.org/10.1007/s00498-020-00256-w
- Local input-to-state stability
- Global attractor
- Nonlinear reaction–diffusion equations