# American Institute of Mathematical Sciences

January  2019, 24(1): 83-107. doi: 10.3934/dcdsb.2018162

## Balanced truncation model reduction of a nonlinear cable-mass PDE system with interior damping

 1 School of Mechanical, Industrial, & Manufacturing Engineering, Oregon State University, Corvallis, OR 97331-6011, USA 2 Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, USA

* Corresponding author

† H. Shoori and J. Singler were supported in part by National Science Foundation grant DMS-1217122; B. Batten was supported in part by the Department of Energy under Award Number DE-FG36-08GO18179.

Received  February 2017 Revised  January 2018 Published  January 2019 Early access  June 2018

We consider model order reduction of a nonlinear cable-mass system modeled by a 1D wave equation with interior damping and dynamic boundary conditions. The system is driven by a time dependent forcing input to a linear mass-spring system at one boundary. The goal of the model reduction is to produce a low order model that produces an accurate approximation to the displacement and velocity of the mass in the nonlinear mass-spring system at the opposite boundary. We first prove that the linearized and nonlinear unforced systems are well-posed and exponentially stable under certain conditions on the damping parameters, and then consider a balanced truncation method to generate the reduced order model (ROM) of the nonlinear input-output system. Little is known about model reduction of nonlinear input-output systems, and so we present detailed numerical experiments concerning the performance of the nonlinear ROM. We find that the ROM is accurate for many different combinations of model parameters.

Citation: Belinda A. Batten, Hesam Shoori, John R. Singler, Madhuka H. Weerasinghe. Balanced truncation model reduction of a nonlinear cable-mass PDE system with interior damping. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 83-107. doi: 10.3934/dcdsb.2018162
##### References:

show all references

H. Shoori and J. Singler were supported in part by National Science Foundation grant DMS-1217122; B. Batten was supported in part by the Department of Energy under Award Number DE-FG36-08GO18179.

##### References:
The cable mass system
Eigenvalues of the linear system and energy decay for the nonlinear system with $\gamma = \alpha_{l} = 0.1$, $k_{0} = k_{l} = 1$, and $\alpha_{0} = \alpha = 0$
Eigenvalues of the linear system and energy decay for the nonlinear system with $\gamma = 0$ and $\alpha = \alpha_{0} = \alpha_{l} = k_{0} = k_{l} = 0.01$
Example 1, Input 2: Output of the ROM and FOM for $\alpha_{0} = \alpha = 0$, $\alpha_{l} = k_{0} = k_{l} = 0.1$, and $\gamma = 0.001$
Example 5, Input 4: Output of the nonlinear ROM and FOM for $\alpha = \alpha_{0} = \alpha_{l} = 0$, $\gamma = 0.001$, and $k_{0} = k_{l} = 0.1$
Example 2, Input 1: Output of the ROM and FOM for $\gamma = 0$, $\alpha = \alpha_{0} = \alpha_{l} = 0.1$, and $k_{0} = k_{l} = 0.001$
Example 1, Input 4: Output of the ROM and FOM for $\alpha = \alpha_0 = 0$, $\gamma = \alpha_{l} = 0.1$, and $k_{0} = k_{l} = 0.001$
Example 5, Input 4: Output of the ROM and FOM for $\gamma = 0.1$, $\alpha = \alpha_{0} = \alpha_{l} = 0$, and $k_{0} = k_{l} = 0.001$
Example 3: Output of the nonlinear ROM and FOM for $\alpha_{0} = \alpha_{l} = 0$ and $\alpha = \gamma = k_{0} = k_{l} = 0.001$
Fixed simulation parameters
 $l$ $m_0$ $m_l$ $k_3$ $\beta$ 1 1 1.5 1 1
 $l$ $m_0$ $m_l$ $k_3$ $\beta$ 1 1 1.5 1 1
 [1] Yaru Xie, Genqi Xu. Exponential stability of 1-d wave equation with the boundary time delay based on the interior control. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 557-579. doi: 10.3934/dcdss.2017028 [2] Igor Chueshov, Irena Lasiecka, Daniel Toundykov. Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 459-509. doi: 10.3934/dcds.2008.20.459 [3] Lorena Bociu, Petronela Radu. Existence of weak solutions to the Cauchy problem of a semilinear wave equation with supercritical interior source and damping. Conference Publications, 2009, 2009 (Special) : 60-71. doi: 10.3934/proc.2009.2009.60 [4] Huafei Di, Yadong Shang, Jiali Yu. Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source. Electronic Research Archive, 2020, 28 (1) : 221-261. doi: 10.3934/era.2020015 [5] Enzo Vitillaro. Blow–up for the wave equation with hyperbolic dynamical boundary conditions, interior and boundary nonlinear damping and sources. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4575-4608. doi: 10.3934/dcdss.2021130 [6] Nicolas Fourrier, Irena Lasiecka. Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions. Evolution Equations and Control Theory, 2013, 2 (4) : 631-667. doi: 10.3934/eect.2013.2.631 [7] Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 693-722. doi: 10.3934/dcdss.2011.4.693 [8] Emanuela R. S. Coelho, Valéria N. Domingos Cavalcanti, Vinicius A. Peralta. Exponential stability for a transmission problem of a nonlinear viscoelastic wave equation. Communications on Pure and Applied Analysis, 2021, 20 (5) : 1987-2020. doi: 10.3934/cpaa.2021055 [9] A. Kh. Khanmamedov. Long-time behaviour of wave equations with nonlinear interior damping. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1185-1198. doi: 10.3934/dcds.2008.21.1185 [10] Chris Guiver, Mark R. Opmeer. Bounded real and positive real balanced truncation for infinite-dimensional systems. Mathematical Control and Related Fields, 2013, 3 (1) : 83-119. doi: 10.3934/mcrf.2013.3.83 [11] Serge Nicaise, Cristina Pignotti. Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 791-813. doi: 10.3934/dcdss.2016029 [12] Martin Redmann, Melina A. Freitag. Balanced model order reduction for linear random dynamical systems driven by Lévy noise. Journal of Computational Dynamics, 2018, 5 (1&2) : 33-59. doi: 10.3934/jcd.2018002 [13] Qiong Zhang. Exponential stability of a joint-leg-beam system with memory damping. Mathematical Control and Related Fields, 2015, 5 (2) : 321-333. doi: 10.3934/mcrf.2015.5.321 [14] 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 [15] 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 [16] Vando Narciso. On a Kirchhoff wave model with nonlocal nonlinear damping. Evolution Equations and Control Theory, 2020, 9 (2) : 487-508. doi: 10.3934/eect.2020021 [17] Lorena Bociu, Irena Lasiecka. Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping. Discrete and Continuous Dynamical Systems, 2008, 22 (4) : 835-860. doi: 10.3934/dcds.2008.22.835 [18] Stéphane Gerbi, Belkacem Said-Houari. Exponential decay for solutions to semilinear damped wave equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 559-566. doi: 10.3934/dcdss.2012.5.559 [19] Genni Fragnelli, Dimitri Mugnai. Stability of solutions for nonlinear wave equations with a positive--negative damping. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 615-622. doi: 10.3934/dcdss.2011.4.615 [20] Lei Wang, Zhong-Jie Han, Gen-Qi Xu. Exponential-stability and super-stability of a thermoelastic system of type II with boundary damping. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2733-2750. doi: 10.3934/dcdsb.2015.20.2733

2021 Impact Factor: 1.497