# 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  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 & Continuous Dynamical Systems - B, 2019, 24 (1) : 83-107. doi: 10.3934/dcdsb.2018162
##### References:

show all references

##### 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 & 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 & Continuous Dynamical Systems - A, 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] Nicolas Fourrier, Irena Lasiecka. Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions. Evolution Equations & Control Theory, 2013, 2 (4) : 631-667. doi: 10.3934/eect.2013.2.631 [5] Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 693-722. doi: 10.3934/dcdss.2011.4.693 [6] A. Kh. Khanmamedov. Long-time behaviour of wave equations with nonlinear interior damping. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1185-1198. doi: 10.3934/dcds.2008.21.1185 [7] Serge Nicaise, Cristina Pignotti. Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 791-813. doi: 10.3934/dcdss.2016029 [8] Chris Guiver, Mark R. Opmeer. Bounded real and positive real balanced truncation for infinite-dimensional systems. Mathematical Control & Related Fields, 2013, 3 (1) : 83-119. doi: 10.3934/mcrf.2013.3.83 [9] 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 [10] Dalibor Pražák. On the dimension of the attractor for the wave equation with nonlinear damping. Communications on Pure & Applied Analysis, 2005, 4 (1) : 165-174. doi: 10.3934/cpaa.2005.4.165 [11] Qiong Zhang. Exponential stability of a joint-leg-beam system with memory damping. Mathematical Control & Related Fields, 2015, 5 (2) : 321-333. doi: 10.3934/mcrf.2015.5.321 [12] Lorena Bociu, Irena Lasiecka. Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 835-860. doi: 10.3934/dcds.2008.22.835 [13] Genni Fragnelli, Dimitri Mugnai. Stability of solutions for nonlinear wave equations with a positive--negative damping. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 615-622. doi: 10.3934/dcdss.2011.4.615 [14] Stéphane Gerbi, Belkacem Said-Houari. Exponential decay for solutions to semilinear damped wave equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 559-566. doi: 10.3934/dcdss.2012.5.559 [15] Lei Wang, Zhong-Jie Han, Gen-Qi Xu. Exponential-stability and super-stability of a thermoelastic system of type II with boundary damping. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2733-2750. doi: 10.3934/dcdsb.2015.20.2733 [16] Kim Dang Phung. Decay of solutions of the wave equation with localized nonlinear damping and trapped rays. Mathematical Control & Related Fields, 2011, 1 (2) : 251-265. doi: 10.3934/mcrf.2011.1.251 [17] Jiayun Lin, Kenji Nishihara, Jian Zhai. Critical exponent for the semilinear wave equation with time-dependent damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4307-4320. doi: 10.3934/dcds.2012.32.4307 [18] Aníbal Rodríguez-Bernal, Enrique Zuazua. Parabolic singular limit of a wave equation with localized boundary damping. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 303-346. doi: 10.3934/dcds.1995.1.303 [19] Tae Gab Ha. On viscoelastic wave equation with nonlinear boundary damping and source term. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1543-1576. doi: 10.3934/cpaa.2010.9.1543 [20] Moez Daoulatli. Energy decay rates for solutions of the wave equation with linear damping in exterior domain. Evolution Equations & Control Theory, 2016, 5 (1) : 37-59. doi: 10.3934/eect.2016.5.37

2018 Impact Factor: 1.008