Well-posedness and stability for a nonlinear Euler-Bernoulli beam equation

Panyu Deng; Jun Zheng; Guchuan Zhu; Panyu Deng; Jun Zheng; Guchuan Zhu

doi:10.3934/cam.2024009

Communications in Analysis and Mechanics

2024, Volume 16, Issue 1: 193-216. doi: 10.3934/cam.2024009

Previous Article Next Article

Research article

Well-posedness and stability for a nonlinear Euler-Bernoulli beam equation

1.
School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan, 611756, P. R. of China
2.
Department of Electrical Engineering, Polytechnique Montréal, 6079, P. O. Box, Station Centre-Ville, Montreal, QC, Canada H3T 1J4

Received: 14 November 2023 Revised: 17 December 2023 Accepted: 23 January 2024 Published: 05 February 2024
35B35, 93D05, 93C73, 93C10

We study the well-posedness and stability for a nonlinear Euler-Bernoulli beam equation modeling railway track deflections in the framework of input-to-state stability (ISS) theory. More specifically, in the presence of both distributed in-domain and boundary disturbances, we prove first the existence and uniqueness of a classical solution by using the technique of lifting and the semigroup method, and then establish the $ L^r $-integral input-to-state stability estimate for the solution whenever $ r\in [2, +\infty] $ by constructing a suitable Lyapunov functional with the aid of Sobolev-like inequalities, which are used to deal with the boundary terms. We provide an extensive extension of relevant work presented in the existing literature.
- Euler-Bernoulli beam equation,
- input-to-state stability,
- integral input-to-state stability,
- boundary disturbance,
- Lyapunov method
Citation: Panyu Deng, Jun Zheng, Guchuan Zhu. Well-posedness and stability for a nonlinear Euler-Bernoulli beam equation[J]. Communications in Analysis and Mechanics, 2024, 16(1): 193-216. doi: 10.3934/cam.2024009

Related Papers:

Abstract

We study the well-posedness and stability for a nonlinear Euler-Bernoulli beam equation modeling railway track deflections in the framework of input-to-state stability (ISS) theory. More specifically, in the presence of both distributed in-domain and boundary disturbances, we prove first the existence and uniqueness of a classical solution by using the technique of lifting and the semigroup method, and then establish the $ L^r $-integral input-to-state stability estimate for the solution whenever $ r\in [2, +\infty] $ by constructing a suitable Lyapunov functional with the aid of Sobolev-like inequalities, which are used to deal with the boundary terms. We provide an extensive extension of relevant work presented in the existing literature.

References

[1]	E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control, 34 (1989), 435–443. https://doi.org/10.1109/9.28018 doi: 10.1109/9.28018
[2]	E. D. Sontag, Comments on integral variants of ISS, Systems Control Lett., 34 (1998), 93–100. https://doi.org/10.1016/S0167-6911(98)00003-6 doi: 10.1016/S0167-6911(98)00003-6
[3]	A. Mironchenko, Input-to-State Stability: Theory and Applications, Springer, 2023. https://doi.org/10.1007/978-3-031-14674-9
[4]	H. Damak, Input-to-state stability and integral input-to-state stability of non-autonomous infinite-dimensional systems, Internat. J. Systems Sci., 52 (2021), 2100–2113. https://doi.org/10.1080/00207721.2021.1879306 doi: 10.1080/00207721.2021.1879306
[5]	H. Damak, Input-to-state stability of non-autonomous infinite-dimensional control systems, Math. Control Relat. Fields, 13 (2023), 1212–1225. https://doi.org/10.3934/mcrf.2022035 doi: 10.3934/mcrf.2022035
[6]	S. Dashkovskiy, A. Mironchenko, Input-to-state stability of infinite-dimensional control systems, Math. Control Signals Systems, 25 (2013), 1–35. https://doi.org/10.1007/s00498-012-0090-2 doi: 10.1007/s00498-012-0090-2
[7]	B. Jacob, A. Mironchenko, J. R. Partington, F. Wirth, Noncoercive Lyapunov functions for input-to-state stability of infinite-dimensional systems, SIAM J. Control Optim., 58 (2020), 2952–2978. https://doi.org/10.1137/19M1297506 doi: 10.1137/19M1297506
[8]	B. Jacob, R. Nabiullin, J. R. Partington, F. L. Schwenninger, Infinite-dimensional input-to-state stability and Orlicz spaces, SIAM J. Control Optim., 56 (2018), 868–889. https://doi.org/10.1137/16M1099467 doi: 10.1137/16M1099467
[9]	A. Mironchenko, C. Prieur, Input-to-state stability of infinite-dimensional systems: recent results and open questions, SIAM Rev., 62 (2020), 529–614. https://doi.org/10.1137/19M1291248 doi: 10.1137/19M1291248
[10]	A. Mironchenko, F. Wirth, Characterizations of input-to-state stability for infinite-dimensional systems, IEEE Trans. Automat. Control, 63 (2018), 1692–1707. https://doi.org/10.1109/tac.2017.2756341 doi: 10.1109/tac.2017.2756341
[11]	B. Jayawardhana, H. Logemann, E. P. Ryan, Infinite-dimensional feedback systems: the circle criterion and input-to-state stability, Commun. Inf. Syst., 8 (2008), 413–444. https://doi.org/10.4310/CIS.2008.v8.n4.a4 doi: 10.4310/CIS.2008.v8.n4.a4
[12]	I. Karafyllis, M. Krstic, ISS with respect to boundary disturbances for 1-D parabolic PDEs, IEEE Trans. Automat. Control, 61 (2016), 3712–3724. https://doi.org/10.1109/TAC.2016.2519762 doi: 10.1109/TAC.2016.2519762
[13]	I. Karafyllis, M. Krstic, ISS in different norms for 1-D parabolic PDEs with boundary disturbances, SIAM J. Control Optim., 55 (2017), 1716–1751. https://doi.org/10.1137/16M1073753 doi: 10.1137/16M1073753
[14]	I. Karafyllis, M. Krstic, Input-to-state stability for PDEs, Springer-Verlag, Cham, 2019. https://doi.org/10.1007/978-3-319-91011-6
[15]	H. Lhachemi, D. Saussié, G. C. Zhu, R. Shorten, Input-to-state stability of a clamped-free damped string in the presence of distributed and boundary disturbances, IEEE Trans. Automat. Control, 65 (2020), 1248–1255. https://doi.org/10.1109/tac.2019.2925497 doi: 10.1109/tac.2019.2925497
[16]	H. Lhachemi, R. Shorten, ISS property with respect to boundary disturbances for a class of Riesz-spectral boundary control systems, Automatica, 109 (2019), 108504. https://doi.org/10.1016/j.automatica.2019.108504 doi: 10.1016/j.automatica.2019.108504
[17]	A. Mironchenko, H. Ito, Construction of Lyapunov functions for interconnected parabolic systems: an iISS approach, SIAM J. Control Optim., 53 (2015), 3364–3382. https://doi.org/10.1137/14097269X doi: 10.1137/14097269X
[18]	F. L. Schwenninger, Input-to-state stability for parabolic boundary control: linear and semilinear systems, in Kerner J., Laasri H., and Mugnolo D. (eds) Control Theory of Infinite-Dimensional Systems. Operator Theory: Advances and Applications, Birkhäuser, Cham, 2020, 83–116. https://doi.org/10.1007/978-3-030-35898-3_4
[19]	J. Zheng, H. Lhachemi, G. C. Zhu, D. Saussié, ISS with respect to boundary and in-domain disturbances for a coupled beam-string system, Math. Control Signals Systems, 30 (2018), 21. https://doi.org/10.1007/s00498-018-0228-y doi: 10.1007/s00498-018-0228-y
[20]	J. Zheng, G. C. Zhu, Input-to-state stability with respect to boundary disturbances for a class of semi-linear parabolic equations, Automatica, 97 (2018), 271–277. https://doi.org/10.1016/j.automatica.2018.08.007 doi: 10.1016/j.automatica.2018.08.007
[21]	J. Zheng, G. C. Zhu, Input-to-state stability for a class of one-dimensional nonlinear parabolic PDEs with nonlinear boundary conditions, SIAM J. Control Optim., 58 (2020), 2567–2587. https://doi.org/10.1137/19M1283720 doi: 10.1137/19M1283720
[22]	J. Zheng, G. C. Zhu, ISS-like estimates for nonlinear parabolic PDEs with variable coefficients on higher dimensional domains, Systems Control Lett., 146 (2020), 104808. https://doi.org/10.1016/j.sysconle.2020.104808 doi: 10.1016/j.sysconle.2020.104808
[23]	L. Aguilar, Y. Orlov, A. Pisano, Leader-follower synchronization and ISS analysis for a network of boundary-controlled wave PDEs, IEEE Control Syst. Lett., 5 (2021), 683–688. https://doi.org/10.1109/LCSYS.2020.3004505 doi: 10.1109/LCSYS.2020.3004505
[24]	M. S. Edalatzadeh, K. A. Morris, Stability and well-posedness of a nonlinear railway track model, IEEE Control Syst. Lett., 3 (2019), 162–167. https://doi.org/10.1109/LCSYS.2018.2849831 doi: 10.1109/LCSYS.2018.2849831
[25]	Y. Y. Jiang, J. Li, Y. Liu, J. Zheng, Input-to-state stability of a variable cross-section beam bridge under moving loads (in Chinese), Math. Pract. Theory, 51 (2021), 177–185.
[26]	F. B. Argomedo, C. Prieur, E. Witrant, S. Bremond, A strict control Lyapunov function for a diffusion equation with time-varying distributed coefficients, IEEE Trans. Automat. Control, 58 (2013), 290–303. https://doi.org/10.1109/TAC.2012.2209260 doi: 10.1109/TAC.2012.2209260
[27]	F. Mazenc, C. Prieur, Strict Lyapunov functions for semilinear parabolic partial differential equations, Math. Control Relat. Fields, 1 (2011), 231–250. https://doi.org/10.3934/mcrf.2011.1.231 doi: 10.3934/mcrf.2011.1.231
[28]	J. Zheng, G. C. Zhu, Approximations of Lyapunov functionals for ISS analysis of a class of higher dimensional nonlinear parabolic PDEs, Automatica, 125 (2021), 109414. https://doi.org/10.1016/j.automatica.2020.109414 doi: 10.1016/j.automatica.2020.109414
[29]	A. Mironchenko, I. Karafyllis, M. Krstic, Monotonicity methods for input-to-state stability of nonlinear parabolic PDEs with boundary disturbances, SIAM J. Control Optim., 57 (2019), 510–532. https://doi.org/10.1137/17M1161877 doi: 10.1137/17M1161877
[30]	J. Zheng, G. C. 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. Control, 64 (2019), 3476–3483. https://doi.org/10.1109/TAC.2018.2880160 doi: 10.1109/TAC.2018.2880160
[31]	I. Karafyllis, M. Krstic, ISS estimates in the spatial sup-norm for nonlinear 1-D parabolic PDEs, ESAIM Control Optim. Calc. Var., 27 (2021), 57. https://doi.org/10.1051/cocv/2021053 doi: 10.1051/cocv/2021053
[32]	R. Díaz, O. Vera, Asymptotic behaviour for a thermoelastic problem of a microbeam with thermoelasticity of type III, Electron. J. Qual. Theory Differ. Equ., 2017 (2017), 1–13. https://doi.org/10.14232/ejqtde.2017.1.74 doi: 10.14232/ejqtde.2017.1.74
[33]	B. W. Feng, B. Chentouf, Exponential stabilization of a microbeam system with a boundary or distributed time delay, Math. Methods Appl. Sci., 44 (2021), 11613–11630. https://doi.org/10.1002/mma.7518 doi: 10.1002/mma.7518
[34]	F. F. Jin, B. Z. Guo, Boundary output tracking for an Euler-Bernoulli beam equation with unmatched perturbations from a known exosystem, Automatica, 109 (2019), 108507. https://doi.org/10.1016/j.automatica.2019.108507 doi: 10.1016/j.automatica.2019.108507
[35]	Z. H. Luo, B. Z. Guo, Stability and stabilization of infinite dimensional systems with applications, Springer-Verlag, London, 1999. https://doi.org/10.1007/978-1-4471-0419-3
[36]	M. Ansari, E. Esmailzadeh, D. Younesian, Frequency analysis of finite beams on nonlinear Kelvin-Voight foundation under moving loads, J. Sound Vib., 330 (2011), 1455–1471. https://doi.org/10.1016/j.jsv.2010.10.005 doi: 10.1016/j.jsv.2010.10.005
[37]	H. Ding, L. Q. Chen, S. P. Yang, Convergence of Galerkin truncation for dynamic response of finite beams on nonlinear foundations under a moving load, J. Sound Vib., 331 (2012), 2426–2442. https://doi.org/10.1016/j.jsv.2011.12.036 doi: 10.1016/j.jsv.2011.12.036
[38]	R. F. Curtain, H. Zwart, An introduction to infinite-dimensional linear systems theory, Springer-Verlag, New York, 1995. https://doi.org/10.1007/978-1-4612-4224-6
[39]	F. F. Jin, B. Z. Guo, Lyapunov approach to output feedback stabilization for the Euler-Bernoulli beam equation with boundary input disturbance, Automatica, 52 (2015), 95–102. https://doi.org/10.1016/j.automatica.2014.10.123 doi: 10.1016/j.automatica.2014.10.123
[40]	Ö. Civalek, Ç. Demir, Bending analysis of microtubules using nonlocal Euler-Bernoulli beam theory, Appl. Math. Model., 35 (2011), 2053–2067. https://doi.org/10.1016/j.apm.2010.11.004 doi: 10.1016/j.apm.2010.11.004
[41]	H. Lhachemi, D. Saussié, G. C. Zhu, Boundary feedback stabilization of a flexible wing model under unsteady aerodynamic loads, Automatica, 97 (2018), 73–81. https://doi.org/10.1016/j.automatica.2018.07.029 doi: 10.1016/j.automatica.2018.07.029
[42]	A. Barari, H. D. Kaliji, M. Ghadimi, G. Domairry, Non-linear vibration of Euler-Bernoulli beams, Latin Amer. J. Solids Struct., 8 (2011), 139–148. https://doi.org/10.1590/S1679-78252011000200002 doi: 10.1590/S1679-78252011000200002
[43]	A. D. Senalp, A. Arikoglu, I. Ozkol, V. Z. Dogan, Dynamic response of a finite length Euler-Bernoulli beam on linear and nonlinear viscoelastic foundations to a concentrated moving force, J. Mech. Sci. Technol., 24 (2010), 1957–1961. https://doi.org/10.1007/s12206-010-0704-x doi: 10.1007/s12206-010-0704-x
[44]	A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983. https://doi.org/10.1007/978-1-4612-5561-1
[45]	M. S. Edalatzadeh, K. A. Morris, Optimal actuator design for semilinear systems, SIAM J. Control Optim., 57 (2019), 2992–3020. https://doi.org/10.1137/18M1171229 doi: 10.1137/18M1171229

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)