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doi: 10.3934/mcrf.2021044
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Asymptotic gain results for attractors of semilinear systems

1. 

Fraunhofer Institute for Industrial Mathematics (ITWM), Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany

2. 

Kiev National Taras Shevchenko University, 01033 Kiev, Ukraine

3. 

Institute for Mathematics, University of Würzburg, 97074 Würzburg, Germany

* Corresponding author: Jochen Schmid

Received  September 2019 Revised  October 2020 Early access September 2021

Fund Project: This article was recruited by Andrii Mironchenko

We establish asymptotic gain along with input-to-state practical stability results for disturbed semilinear systems w.r.t. the global attractor of the respective undisturbed system. We apply our results to a large class of nonlinear reaction-diffusion equations comprising disturbed Chaffee–Infante equations, for example.

Citation: Jochen Schmid, Oleksiy Kapustyan, Sergey Dashkovskiy. Asymptotic gain results for attractors of semilinear systems. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2021044
References:
[1]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, 2nd edition, Birkhäuser, 2011. doi: 10.1007/978-3-0348-0087-7.

[2]

J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, 1990.

[3]

J. Bergh and J. Löfström, Interpolation Spaces, Springer, 1980.

[4]

V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for reaction-diffusion systems, Topol. Methods Nonlinear Anal., 7 (1996), 49-76.  doi: 10.12775/TMNA.1996.002.

[5]

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

[6]

F. H. ClarkeY. S. Ledyaev and R. J. Stern, Asymptotic stability and smooth Lyapunov functions, J. Differential Equations, 149 (1998), 69-114.  doi: 10.1006/jdeq.1998.3476.

[7]

D. L. Cohn, Measure Theory, 2nd edition, Birkhäuser, 2013. doi: 10.1007/978-1-4614-6956-8.

[8]

J. B. Conway, A Course in Functional Analysis, 2nd edition, Springer, 1990.

[9]

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

[10]

S. DashkovskiyO. Kapustyan and J. Schmid, A local input-to-state stability result w.r.t. attractors of nonlinear reaction–diffusion equations, Math. Contr. Sign. Syst., 32 (2020), 309-326.  doi: 10.1007/s00498-020-00256-w.

[11]

S. DashkovskiyO. Kapustyan and J. Schmid, Input-to-state stability results w.r.t. global attractors of semi-linear reaction-diffusion equations, IFAC-PapersOnLine, 53 (2020), 3186-3191.  doi: 10.1016/j.ifacol.2020.12.2536.

[12]

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

[13]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, 2000.

[14]

N. V. GorbanA. V. KapustyanE. A. Kapustyan and O. V. Khomenko, Strong global attractor for the three-dimensional Navier-Stokes system of equations in unbounded domain of channel type, J. Autom. Inform. Sciences, 47 (2015), 48-59.  doi: 10.1615/JAutomatInfScien.v47.i11.40.

[15]

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.

[16]

B. JacobR. NabiullinJ. R. Partington and F. L. Schwenninger, Infinite-dimensional input-to-state stability and Orlicz spaces, SIAM J. Contr. Optim., 56 (2018), 868-889.  doi: 10.1137/16M1099467.

[17]

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

[18]

B. JacobF. L. Schwenninger and H. Zwart, On continuity of solutions for parabolic control systems and input-to-state stability, J. Differential Equations, 266 (2019), 6284-6306.  doi: 10.1016/j.jde.2018.11.004.

[19]

O. V. KapustyanP. O. Kasyanov and J. Valero, Structure of the global attractor for weak solutions of a reaction-diffusion equation, Appl. Math. Inf. Sci., 9 (2015), 2257-2264. 

[20]

A. V. KapustyanV. S. Melnik and J. Valero, Attractors of multivalued dynamical processes generated by phase-field equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 1969-1983.  doi: 10.1142/S0218127403007801.

[21]

A. V. Kapustyan and J. Valero, On the Kneser property for the complex Ginzburg–Landau equation and the Lotka–Volterra system with diffusion, J. Math. Anal. Appl., 357 (2009), 254-272.  doi: 10.1016/j.jmaa.2009.04.010.

[22]

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

[23]

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.

[24]

Y. LinE. D. Sontag and Y. Wang, A smooth converse Lyapunov theorem for robust stability, SIAM J. Contr. Optim., 34 (1996), 124-160.  doi: 10.1137/S0363012993259981.

[25]

F. Mazenc and C. Prieur, Strict Lyapunov functions for semilinear parabolic partial differential equations, Math. Contr. Rel. Fields, 1 (2011), 231-250.  doi: 10.3934/mcrf.2011.1.231.

[26]

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

[27]

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.

[28]

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

[29]

A. Mironchenko and F. Wirth, Characterizations of input-to-state stability for infinite-dimensional systems, IEEE Trans. Autom. Contr., 63 (2018), 1692-1707.  doi: 10.1109/tac.2017.2756341.

[30]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983. doi: 10.1007/978-1-4612-5561-1.

[31] J. C. Robinson, Infinite-Dimensional Dyanamical Systems, Cambridge University Press, 2001. 
[32]

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.

[33]

J. Schmid, Well-posedness and stability of non-autonomous semilinear input-output systems, Accepted provisionally in Evol. Equ. Contr. Th., arXiv: 1904.10376.

[34]

J. Schmid, Infinite-time admissibility under compact perturbations, Control Theory of Infinite-Dimensional Systems, 73–82, Oper. Theory Adv. Appl., 277, Linear Oper. Linear Syst., Birkhäuser, Cham, (2020). doi: 10.1007/978-3-030-35898-3_3.

[35]

J. Schmid and H. Zwart, Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances, Conference proceedings of the 23rd Symposium on Mathematical Theory of Networks and Systems, (2018), 570–575, http://mtns2018.ust.hk/media/files/0144.pdf

[36]

J. SchmidS. DashkovskiyB. Jacob and H. Laasri, Well-posedness of non-autonomous semilinear systems, IFAC-PapersOnLine, 52 (2019), 216-220.  doi: 10.1016/j.ifacol.2019.11.781.

[37]

J. Schmid and H. Zwart, Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances, ESAIM Contr. Optim. Calc. Var., 27 (2021), paper no. 53, 37 pp. doi: 10.1051/cocv/2021051.

[38]

F. L. Schwenninger, Input-to-state stability for parabolic boundary control: Linear and semi-linear systems, Control Theory of Infinite-Dimensional Systems, 83–116, Oper. Theory Adv. Appl., 277, Linear Oper. Linear Syst., Birkhäuser, Cham, (2020). doi: 10.1007/978-3-030-35898-3_4.

[39]

E. D. Sontag and Y. Wang, New characterizations of input-to-state stability, IEEE Trans. Autom. Contr., 41 (1996), 1283-1294.  doi: 10.1109/9.536498.

[40]

A. Tanwani, C. Prieur and S. Tarbouriech, Disturbance-to-state stabilization and quantized control for linear hyperbolic systems, arXiv: 1703.00302, (2017).

[41]

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

[42]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser, 2009. doi: 10.1007/978-3-7643-8994-9.

[43]

J. Valero and A. V. Kapustyan, On the connectedness and asymptotic behaviour of solutions of reaction-diffusion equations, J. Math. Anal. Appl., 323 (2006), 614-633.  doi: 10.1016/j.jmaa.2005.10.042.

[44]

J. Zheng and G. Zhu, Input-to-state stability with respect to boundary disturbances for a class of semi-linear parabolic equations, Automatica J. IFAC, 97 (2018), 271-277.  doi: 10.1016/j.automatica.2018.08.007.

[45]

J. Zheng and G. Zhu, A De Giorgi iteration-based approach for the establishment of ISS properties for Burgers–Equation with boundary and in-domain disturbances, IEEE Trans. Automat. Contr., 64 (2019), 3476-3483.  doi: 10.1109/TAC.2018.2880160.

[46]

J. Zheng and G. Zhu, A weak maximum principle-based approach for input-to-state stability of nonlinear parabolic PDEs with boundary disturbances, Math. Control Signals Syst., 32 (2020), 157-176.  doi: 10.1007/s00498-020-00258-8.

show all references

References:
[1]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, 2nd edition, Birkhäuser, 2011. doi: 10.1007/978-3-0348-0087-7.

[2]

J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, 1990.

[3]

J. Bergh and J. Löfström, Interpolation Spaces, Springer, 1980.

[4]

V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for reaction-diffusion systems, Topol. Methods Nonlinear Anal., 7 (1996), 49-76.  doi: 10.12775/TMNA.1996.002.

[5]

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

[6]

F. H. ClarkeY. S. Ledyaev and R. J. Stern, Asymptotic stability and smooth Lyapunov functions, J. Differential Equations, 149 (1998), 69-114.  doi: 10.1006/jdeq.1998.3476.

[7]

D. L. Cohn, Measure Theory, 2nd edition, Birkhäuser, 2013. doi: 10.1007/978-1-4614-6956-8.

[8]

J. B. Conway, A Course in Functional Analysis, 2nd edition, Springer, 1990.

[9]

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

[10]

S. DashkovskiyO. Kapustyan and J. Schmid, A local input-to-state stability result w.r.t. attractors of nonlinear reaction–diffusion equations, Math. Contr. Sign. Syst., 32 (2020), 309-326.  doi: 10.1007/s00498-020-00256-w.

[11]

S. DashkovskiyO. Kapustyan and J. Schmid, Input-to-state stability results w.r.t. global attractors of semi-linear reaction-diffusion equations, IFAC-PapersOnLine, 53 (2020), 3186-3191.  doi: 10.1016/j.ifacol.2020.12.2536.

[12]

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

[13]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, 2000.

[14]

N. V. GorbanA. V. KapustyanE. A. Kapustyan and O. V. Khomenko, Strong global attractor for the three-dimensional Navier-Stokes system of equations in unbounded domain of channel type, J. Autom. Inform. Sciences, 47 (2015), 48-59.  doi: 10.1615/JAutomatInfScien.v47.i11.40.

[15]

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.

[16]

B. JacobR. NabiullinJ. R. Partington and F. L. Schwenninger, Infinite-dimensional input-to-state stability and Orlicz spaces, SIAM J. Contr. Optim., 56 (2018), 868-889.  doi: 10.1137/16M1099467.

[17]

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

[18]

B. JacobF. L. Schwenninger and H. Zwart, On continuity of solutions for parabolic control systems and input-to-state stability, J. Differential Equations, 266 (2019), 6284-6306.  doi: 10.1016/j.jde.2018.11.004.

[19]

O. V. KapustyanP. O. Kasyanov and J. Valero, Structure of the global attractor for weak solutions of a reaction-diffusion equation, Appl. Math. Inf. Sci., 9 (2015), 2257-2264. 

[20]

A. V. KapustyanV. S. Melnik and J. Valero, Attractors of multivalued dynamical processes generated by phase-field equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 1969-1983.  doi: 10.1142/S0218127403007801.

[21]

A. V. Kapustyan and J. Valero, On the Kneser property for the complex Ginzburg–Landau equation and the Lotka–Volterra system with diffusion, J. Math. Anal. Appl., 357 (2009), 254-272.  doi: 10.1016/j.jmaa.2009.04.010.

[22]

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

[23]

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.

[24]

Y. LinE. D. Sontag and Y. Wang, A smooth converse Lyapunov theorem for robust stability, SIAM J. Contr. Optim., 34 (1996), 124-160.  doi: 10.1137/S0363012993259981.

[25]

F. Mazenc and C. Prieur, Strict Lyapunov functions for semilinear parabolic partial differential equations, Math. Contr. Rel. Fields, 1 (2011), 231-250.  doi: 10.3934/mcrf.2011.1.231.

[26]

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

[27]

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.

[28]

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

[29]

A. Mironchenko and F. Wirth, Characterizations of input-to-state stability for infinite-dimensional systems, IEEE Trans. Autom. Contr., 63 (2018), 1692-1707.  doi: 10.1109/tac.2017.2756341.

[30]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983. doi: 10.1007/978-1-4612-5561-1.

[31] J. C. Robinson, Infinite-Dimensional Dyanamical Systems, Cambridge University Press, 2001. 
[32]

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.

[33]

J. Schmid, Well-posedness and stability of non-autonomous semilinear input-output systems, Accepted provisionally in Evol. Equ. Contr. Th., arXiv: 1904.10376.

[34]

J. Schmid, Infinite-time admissibility under compact perturbations, Control Theory of Infinite-Dimensional Systems, 73–82, Oper. Theory Adv. Appl., 277, Linear Oper. Linear Syst., Birkhäuser, Cham, (2020). doi: 10.1007/978-3-030-35898-3_3.

[35]

J. Schmid and H. Zwart, Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances, Conference proceedings of the 23rd Symposium on Mathematical Theory of Networks and Systems, (2018), 570–575, http://mtns2018.ust.hk/media/files/0144.pdf

[36]

J. SchmidS. DashkovskiyB. Jacob and H. Laasri, Well-posedness of non-autonomous semilinear systems, IFAC-PapersOnLine, 52 (2019), 216-220.  doi: 10.1016/j.ifacol.2019.11.781.

[37]

J. Schmid and H. Zwart, Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances, ESAIM Contr. Optim. Calc. Var., 27 (2021), paper no. 53, 37 pp. doi: 10.1051/cocv/2021051.

[38]

F. L. Schwenninger, Input-to-state stability for parabolic boundary control: Linear and semi-linear systems, Control Theory of Infinite-Dimensional Systems, 83–116, Oper. Theory Adv. Appl., 277, Linear Oper. Linear Syst., Birkhäuser, Cham, (2020). doi: 10.1007/978-3-030-35898-3_4.

[39]

E. D. Sontag and Y. Wang, New characterizations of input-to-state stability, IEEE Trans. Autom. Contr., 41 (1996), 1283-1294.  doi: 10.1109/9.536498.

[40]

A. Tanwani, C. Prieur and S. Tarbouriech, Disturbance-to-state stabilization and quantized control for linear hyperbolic systems, arXiv: 1703.00302, (2017).

[41]

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

[42]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser, 2009. doi: 10.1007/978-3-7643-8994-9.

[43]

J. Valero and A. V. Kapustyan, On the connectedness and asymptotic behaviour of solutions of reaction-diffusion equations, J. Math. Anal. Appl., 323 (2006), 614-633.  doi: 10.1016/j.jmaa.2005.10.042.

[44]

J. Zheng and G. Zhu, Input-to-state stability with respect to boundary disturbances for a class of semi-linear parabolic equations, Automatica J. IFAC, 97 (2018), 271-277.  doi: 10.1016/j.automatica.2018.08.007.

[45]

J. Zheng and G. Zhu, A De Giorgi iteration-based approach for the establishment of ISS properties for Burgers–Equation with boundary and in-domain disturbances, IEEE Trans. Automat. Contr., 64 (2019), 3476-3483.  doi: 10.1109/TAC.2018.2880160.

[46]

J. Zheng and G. Zhu, A weak maximum principle-based approach for input-to-state stability of nonlinear parabolic PDEs with boundary disturbances, Math. Control Signals Syst., 32 (2020), 157-176.  doi: 10.1007/s00498-020-00258-8.

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