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doi: 10.3934/eect.2021054
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Robustness of global attractors: Abstract framework and application to dissipative wave equations

1. 

University of Würzburg, Institute of Mathematics, Emil-Fischer-Straße 40, 97074 Würzburg, Germany

2. 

Taras Shevchenko National University of Kyiv, Department of Mathematics and Mechanics, Volodymyrska Street 64, 01601 Kyiv, Ukraine

* Corresponding author: Sergey Dashkovskiy

Received  March 2021 Revised  July 2021 Early access October 2021

Fund Project: This work is partially supported by the German Research Foundation (DFG) grant DA 767/12-1 and the National Research Foundation of Ukraine (NRFU) through the joint German-Ukrainian grant "Stability and robustness of attractors of nonlinear infinite-dimensional systems with respect to disturbances"

We establish local input-to-state stability and the asymptotic gain property for a class of infinite-dimensional systems with respect to the global attractor of the respective undisturbed system. We apply our results to a large class of dissipative wave equations with nontrivial global attractors.

Citation: Sergey Dashkovskiy, Oleksiy Kapustyan. Robustness of global attractors: Abstract framework and application to dissipative wave equations. Evolution Equations & Control Theory, doi: 10.3934/eect.2021054
References:
[1]

F. BalibreaT. CaraballoP. E. Kloeden and J. Valero, Recent developments in dynamical systems: Three perspectives, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2591-2636.  doi: 10.1142/S0218127410027246.  Google Scholar

[2]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete and Continuous Dynamical Systems, 10 (2004), 31-52.   Google Scholar

[3]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[4]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, 2002.  Google Scholar

[5]

I. ChueshovM. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), 1901-1951.  doi: 10.1081/PDE-120016132.  Google Scholar

[6]

S. DashkovskiyO. Kapustyan and I. Romaniuk., Global attractors of impulsive parabolic inclusions, Discr. Contin. Dyn. Syst. - Series B, 22 (2017), 1875-1886.  doi: 10.3934/dcdsb.2017111.  Google Scholar

[7]

S. DashkovskiyO. Kapustyan and J. Schmid, A local input-to-state stability result w.r.t. attractors of nonlinear reaction-diffusion equations, Mathematics of Control, Signals, and Systems, 32 (2020), 309-326.  doi: 10.1007/s00498-020-00256-w.  Google Scholar

[8]

S. Dashkovskiy and A. Mironchenko, Input-to-state stability of infinite-dimensional control systems, Math. Contr. Sign. Syst., 25 (2013), 1-35.  doi: 10.1007/s00498-012-0090-2.  Google Scholar

[9]

N. V. GorbanO. V. KapustyanP. O. Kasyanov and L. S. Paliichuk, On global attractors for autonomous damped wave equation with discontinuous nonlinearity, Solid Mechanics and its Applications, 211 (2014), 221-237.  doi: 10.1007/978-3-319-03146-0_16.  Google Scholar

[10]

L. Grüne, Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83677.  Google Scholar

[11]

R. Hosfeld, B. Jacob and F. Schwenninger, Input-to-state stability of unbounded bilinear control systems, arXiv: 1811.08470, (2018). Google Scholar

[12]

L. V. Kapitanski${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$ and I. N. Kostin, Attractors of nonlinear evolution equations and their approximations, Algebra i Analiz, 2 (1990), 114-140.   Google Scholar

[13]

A. V. Kapustyan, Global attractors of a nonautonomous reaction-diffusion equation, Differential Equations, 38 (2002), 1467-1471.  doi: 10.1023/A:1022378831393.  Google Scholar

[14]

I. KarafyllisM. Kontorinaki and M. Krstic, Boundary-to-Displacement asymptotic gains for wave systems with Kelvin-Voigt damping, International J. Control, 94 (2021), 2822-2833.  doi: 10.1080/00207179.2020.1736641.  Google Scholar

[15]

I. Karafyllis and M. Krstic, ISS with respect to boundary disturbances for 1-D parabolic PDEs, IEEE Trans. Automat. Contr., 61 (2016), 3712-3724.  doi: 10.1109/TAC.2016.2519762.  Google Scholar

[16]

I. Karafyllis and M. Krstic, ISS in different norms for 1-D parabolic PDEs with boundary disturbances, SIAM J. Contr. Optim., 55 (2017), 1716-1751.  doi: 10.1137/16M1073753.  Google Scholar

[17]

I. Karafyllis and M. Krstic, Input-to-State Stability for PDEs, Communications and Control Engineering Series. Springer, Cham, 2019. doi: 10.1007/978-3-319-91011-6.  Google Scholar

[18]

P. E. Kloeden and and J. Lorenz, Stable attracting sets in dynamical systems and in their one-step discretizations, SIAM J. Numer. Anal., 23 (1986), 986-995.  doi: 10.1137/0723066.  Google Scholar

[19]

A. Mironchenko, Local input-to-state stability: Characterizations and counterexamples, Systems Control Lett., 87 (2016), 23-28.  doi: 10.1016/j.sysconle.2015.10.014.  Google Scholar

[20]

A. Mironchenko, Criteria for input-to-state practical stability, IEEE Trans. Autom. Contr., 64 (2019), 298-304.  doi: 10.1109/TAC.2018.2824983.  Google Scholar

[21]

A. Mironchenko and H. Ito, Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions, Math. Contr. Rel. Fields, 6 (2016), 447-466.  doi: 10.3934/mcrf.2016011.  Google Scholar

[22]

A. MironchenkoI. Karafyllis and M. Krstic, Monotonicity methods for input-to-state stability of nonlinear parabolic PDEs with boundary disturbances, SIAM J. Contr. Optim., 57 (2019), 510-532.  doi: 10.1137/17M1161877.  Google Scholar

[23]

A. Mironchenko and Ch. Prieur, Input-to-state stability of infinite-dimensional systems: Recent results and open questions, SIAM Review, 62 (2020), 529-614.  doi: 10.1137/19M1291248.  Google Scholar

[24]

A. Mironchenko and F. Wirth, Characterizations of input-to-state stability for infinite-dimensional systems, IEEE Trans. Autom. Contr., 63 (2018), 1602-1617.  doi: 10.1109/tac.2017.2756341.  Google Scholar

[25] J. C. Robinson, Infinite-Dimensional Dyanamical Systems, Cambridge University Press, 2001.  doi: 10.1007/978-94-010-0732-0.  Google Scholar
[26]

J. Schmid, Weak input-to-state stability: Characterizations and counterexamples, Math. Contr. Sign. Syst., 31 (2019), 433-454.  doi: 10.1007/s00498-019-00248-5.  Google Scholar

[27]

J. Schmid, V. Kapustyan and S. Dashkovskiy, Asymptotic gain results for attractors of semilinear systems, arXiv: 1909.06302, (2020). (Accepted provisionally in Mathematical Control & Related Fields). Google Scholar

[28]

E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control, 34 (1989), 435-443.  doi: 10.1109/9.28018.  Google Scholar

[29]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, 2nd edition, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

show all references

References:
[1]

F. BalibreaT. CaraballoP. E. Kloeden and J. Valero, Recent developments in dynamical systems: Three perspectives, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2591-2636.  doi: 10.1142/S0218127410027246.  Google Scholar

[2]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete and Continuous Dynamical Systems, 10 (2004), 31-52.   Google Scholar

[3]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[4]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, 2002.  Google Scholar

[5]

I. ChueshovM. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), 1901-1951.  doi: 10.1081/PDE-120016132.  Google Scholar

[6]

S. DashkovskiyO. Kapustyan and I. Romaniuk., Global attractors of impulsive parabolic inclusions, Discr. Contin. Dyn. Syst. - Series B, 22 (2017), 1875-1886.  doi: 10.3934/dcdsb.2017111.  Google Scholar

[7]

S. DashkovskiyO. Kapustyan and J. Schmid, A local input-to-state stability result w.r.t. attractors of nonlinear reaction-diffusion equations, Mathematics of Control, Signals, and Systems, 32 (2020), 309-326.  doi: 10.1007/s00498-020-00256-w.  Google Scholar

[8]

S. Dashkovskiy and A. Mironchenko, Input-to-state stability of infinite-dimensional control systems, Math. Contr. Sign. Syst., 25 (2013), 1-35.  doi: 10.1007/s00498-012-0090-2.  Google Scholar

[9]

N. V. GorbanO. V. KapustyanP. O. Kasyanov and L. S. Paliichuk, On global attractors for autonomous damped wave equation with discontinuous nonlinearity, Solid Mechanics and its Applications, 211 (2014), 221-237.  doi: 10.1007/978-3-319-03146-0_16.  Google Scholar

[10]

L. Grüne, Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83677.  Google Scholar

[11]

R. Hosfeld, B. Jacob and F. Schwenninger, Input-to-state stability of unbounded bilinear control systems, arXiv: 1811.08470, (2018). Google Scholar

[12]

L. V. Kapitanski${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$ and I. N. Kostin, Attractors of nonlinear evolution equations and their approximations, Algebra i Analiz, 2 (1990), 114-140.   Google Scholar

[13]

A. V. Kapustyan, Global attractors of a nonautonomous reaction-diffusion equation, Differential Equations, 38 (2002), 1467-1471.  doi: 10.1023/A:1022378831393.  Google Scholar

[14]

I. KarafyllisM. Kontorinaki and M. Krstic, Boundary-to-Displacement asymptotic gains for wave systems with Kelvin-Voigt damping, International J. Control, 94 (2021), 2822-2833.  doi: 10.1080/00207179.2020.1736641.  Google Scholar

[15]

I. Karafyllis and M. Krstic, ISS with respect to boundary disturbances for 1-D parabolic PDEs, IEEE Trans. Automat. Contr., 61 (2016), 3712-3724.  doi: 10.1109/TAC.2016.2519762.  Google Scholar

[16]

I. Karafyllis and M. Krstic, ISS in different norms for 1-D parabolic PDEs with boundary disturbances, SIAM J. Contr. Optim., 55 (2017), 1716-1751.  doi: 10.1137/16M1073753.  Google Scholar

[17]

I. Karafyllis and M. Krstic, Input-to-State Stability for PDEs, Communications and Control Engineering Series. Springer, Cham, 2019. doi: 10.1007/978-3-319-91011-6.  Google Scholar

[18]

P. E. Kloeden and and J. Lorenz, Stable attracting sets in dynamical systems and in their one-step discretizations, SIAM J. Numer. Anal., 23 (1986), 986-995.  doi: 10.1137/0723066.  Google Scholar

[19]

A. Mironchenko, Local input-to-state stability: Characterizations and counterexamples, Systems Control Lett., 87 (2016), 23-28.  doi: 10.1016/j.sysconle.2015.10.014.  Google Scholar

[20]

A. Mironchenko, Criteria for input-to-state practical stability, IEEE Trans. Autom. Contr., 64 (2019), 298-304.  doi: 10.1109/TAC.2018.2824983.  Google Scholar

[21]

A. Mironchenko and H. Ito, Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions, Math. Contr. Rel. Fields, 6 (2016), 447-466.  doi: 10.3934/mcrf.2016011.  Google Scholar

[22]

A. MironchenkoI. Karafyllis and M. Krstic, Monotonicity methods for input-to-state stability of nonlinear parabolic PDEs with boundary disturbances, SIAM J. Contr. Optim., 57 (2019), 510-532.  doi: 10.1137/17M1161877.  Google Scholar

[23]

A. Mironchenko and Ch. Prieur, Input-to-state stability of infinite-dimensional systems: Recent results and open questions, SIAM Review, 62 (2020), 529-614.  doi: 10.1137/19M1291248.  Google Scholar

[24]

A. Mironchenko and F. Wirth, Characterizations of input-to-state stability for infinite-dimensional systems, IEEE Trans. Autom. Contr., 63 (2018), 1602-1617.  doi: 10.1109/tac.2017.2756341.  Google Scholar

[25] J. C. Robinson, Infinite-Dimensional Dyanamical Systems, Cambridge University Press, 2001.  doi: 10.1007/978-94-010-0732-0.  Google Scholar
[26]

J. Schmid, Weak input-to-state stability: Characterizations and counterexamples, Math. Contr. Sign. Syst., 31 (2019), 433-454.  doi: 10.1007/s00498-019-00248-5.  Google Scholar

[27]

J. Schmid, V. Kapustyan and S. Dashkovskiy, Asymptotic gain results for attractors of semilinear systems, arXiv: 1909.06302, (2020). (Accepted provisionally in Mathematical Control & Related Fields). Google Scholar

[28]

E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control, 34 (1989), 435-443.  doi: 10.1109/9.28018.  Google Scholar

[29]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, 2nd edition, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

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