doi: 10.3934/eect.2021054
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 and 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.

[2]

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

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[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. 

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

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

[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).

[28]

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

[29]

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

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.

[2]

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

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[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. 

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

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

[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).

[28]

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

[29]

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

[1]

Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 821-836. doi: 10.3934/dcdsb.2021066

[2]

Andrii Mironchenko, Hiroshi Ito. Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions. Mathematical Control and Related Fields, 2016, 6 (3) : 447-466. doi: 10.3934/mcrf.2016011

[3]

Ruofeng Rao, Shouming Zhong. Input-to-state stability and no-inputs stabilization of delayed feedback chaotic financial system involved in open and closed economy. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1375-1393. doi: 10.3934/dcdss.2020280

[4]

Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure and Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281

[5]

Renhai Wang, Bixiang Wang. Random dynamics of lattice wave equations driven by infinite-dimensional nonlinear noise. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2461-2493. doi: 10.3934/dcdsb.2020019

[6]

Huijuan Li, Junxia Wang. Input-to-state stability of continuous-time systems via finite-time Lyapunov functions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 841-857. doi: 10.3934/dcdsb.2019192

[7]

Hiroshi Ito. Input-to-state stability and Lyapunov functions with explicit domains for SIR model of infectious diseases. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5171-5196. doi: 10.3934/dcdsb.2020338

[8]

Brahim Alouini. Global attractor for a one dimensional weakly damped half-wave equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2655-2670. doi: 10.3934/dcdss.2020410

[9]

Francesca Bucci, Igor Chueshov, Irena Lasiecka. Global attractor for a composite system of nonlinear wave and plate equations. Communications on Pure and Applied Analysis, 2007, 6 (1) : 113-140. doi: 10.3934/cpaa.2007.6.113

[10]

Björn Augner, Birgit Jacob. Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems. Evolution Equations and Control Theory, 2014, 3 (2) : 207-229. doi: 10.3934/eect.2014.3.207

[11]

Max E. Gilmore, Chris Guiver, Hartmut Logemann. Sampled-data integral control of multivariable linear infinite-dimensional systems with input nonlinearities. Mathematical Control and Related Fields, 2022, 12 (1) : 17-47. doi: 10.3934/mcrf.2021001

[12]

Teng Wang, Yi Wang. Nonlinear stability of planar rarefaction wave to the three-dimensional Boltzmann equation. Kinetic and Related Models, 2019, 12 (3) : 637-679. doi: 10.3934/krm.2019025

[13]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6207-6228. doi: 10.3934/dcdsb.2021015

[14]

Mustapha Yebdri. Existence of $ \mathcal{D}- $pullback attractor for an infinite dimensional dynamical system. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 167-198. doi: 10.3934/dcdsb.2021036

[15]

Masashi Wakaiki, Hideki Sano. Stability analysis of infinite-dimensional event-triggered and self-triggered control systems with Lipschitz perturbations. Mathematical Control and Related Fields, 2022, 12 (1) : 245-273. doi: 10.3934/mcrf.2021021

[16]

Zhiming Liu, Zhijian Yang. Global attractor of multi-valued operators with applications to a strongly damped nonlinear wave equation without uniqueness. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 223-240. doi: 10.3934/dcdsb.2019179

[17]

Dalibor Pražák. On the dimension of the attractor for the wave equation with nonlinear damping. Communications on Pure and Applied Analysis, 2005, 4 (1) : 165-174. doi: 10.3934/cpaa.2005.4.165

[18]

Kazuhiro Kurata, Yuki Osada. Asymptotic expansion of the ground state energy for nonlinear Schrödinger system with three wave interaction. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4239-4251. doi: 10.3934/cpaa.2021157

[19]

Didier Georges. Infinite-dimensional nonlinear predictive control design for open-channel hydraulic systems. Networks and Heterogeneous Media, 2009, 4 (2) : 267-285. doi: 10.3934/nhm.2009.4.267

[20]

Peng Chen, Linfeng Mei, Xianhua Tang. Nonstationary homoclinic orbit for an infinite-dimensional fractional reaction-diffusion system. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021279

2020 Impact Factor: 1.081

Metrics

  • PDF downloads (301)
  • HTML views (226)
  • Cited by (0)

Other articles
by authors

[Back to Top]