A numerical method of the Euler-Bernoulli beam with optimal local Kelvin-Voigt damping. (English) Zbl 1463.93118

Summary: This paper deals with the numerical approximation problem of the optimal control problem governed by the Euler-Bernoulli beam equation with local Kelvin-Voigt damping, which is a nonlinear coefficient control problem with control constraints. The goal of this problem is to design a control input numerically, which is the damping and distributes locally on a subinterval of the region occupied by the beam, such that the total energy of the beam and the control on a given time period is minimal. We firstly use the finite element method (FEM) to obtain a finite-dimensional model based on the original PDE system. Then, using the control parameterization method, we approximate the finite-dimensional problem by a standard optimal parameter selection problem, which is a suboptimal problem and can be solved numerically by nonlinear mathematical programming algorithm. At last, some simulation studies will be presented by the proposed numerical approximation method in this paper, where the damping controls act on different locations of the Euler-Bernoulli beam.


93C20 Control/observation systems governed by partial differential equations
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
49M37 Numerical methods based on nonlinear programming
Full Text: DOI


[1] Banks, H. T.; Smith, R. C.; Wang, Y., The modeling of piezoceramic patch interactions with shells, plates, and beams, Quarterly of Applied Mathematics, 53, 2, 353-381 (1995)
[2] Banks, H. T.; Zhang, Y., Computational methods for a curved beam with piezoceramic patches, Journal of Intelligent Material Systems and Structures, 8, 3, 260-278 (1997)
[3] Ip, K. H.; Tse, P. C., Optimal configuration of a piezoelectric patch for vibration control of isotropic rectangular plates, Smart Materials and Structures, 10, 2, 395-403 (2001)
[4] Dyke, S. J.; Spencer, B. F.; Sain, M. K.; Carlson, J. D., Modeling and control of magnetorheological dampers for seismic response reduction, Smart Materials and Structures, 5, 5, 565-575 (1996)
[5] Kim, Y.; Langari, R.; Hurlebaus, S., Semiactive nonlinear control of a building with a magnetorheological damper system, Mechanical Systems and Signal Processing, 23, 2, 300-315 (2009)
[6] Lai, C. Y.; Liao, W. H., Vibration control of a suspension system via a magnetorheological fluid damper, Journal of Vibration and Control, 8, 4, 527-547 (2002)
[7] Yang, G.; Spencer, B. F.; Jung, H. J.; Carlson, J. D., Dynamic modeling of large-scale magnetorheological damper systems for civil engineering applications, Journal of Engineering Mechanics, 130, 9, 1107-1114 (2004)
[8] Liu, Z. Y.; Zheng, S. M., Semigroups Associated with Dissipative Systems, 398 (1999), New York, NY, USA: CRC Press, New York, NY, USA
[9] Zhang, C., Boundary feedback stabilization of the undamped Timoshenko beam with both ends free, Journal of Mathematical Analysis and Applications, 326, 1, 488-499 (2007) · Zbl 1103.74040
[10] Zhao, H. L.; Liu, K. S.; Zhang, C. G., Stability for the Timoshenko beam system with local Kelvin-Voigt damping, Acta Mathematica Sinica (English Series), 21, 3, 655-666 (2005) · Zbl 1084.35013
[11] Liu, K. S.; Liu, Z. Y., Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping, SIAM Journal on Control and Optimization, 36, 3, 1086-1098 (1998) · Zbl 0909.35018
[12] Guo, B. Z., Riesz basis property and exponential stability of controlled Euler-Bernoulli beam equations with variable coefficients, SIAM Journal on Control and Optimization, 40, 6, 1905-1923 (2002) · Zbl 1015.93025
[13] Ammari, K.; Tucsnak, M., Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force, SIAM Journal on Control and Optimization, 39, 4, 1160-1181 (2000) · Zbl 0983.35021
[14] Chen, G.; Krantz, S. G.; Ma, D. W.; Wayne, C. E.; Lee, S. J., The Euler-Bernoulli beam equation with boundary energy dissipation, Operator Methods for Optimal control Problems. Operator Methods for Optimal control Problems, Lecture Notes in Pure and Appl. Math., 108, 67-96 (1987), Baton Rouge, La, USA: Louisiana, Baton Rouge, La, USA
[15] Guo, B.; Yang, K., Dynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observation, Automatica, 45, 6, 1468-1475 (2009) · Zbl 1166.93360
[16] Shen, I. Y., Stability and controllability of Euler-Bernoulli beams with intelligent constrained layer treatments, Journal of Vibration and Acoustics, 118, 1, 70-77 (1996)
[17] Tanaka, N.; Kikushima, Y., Optimal vibration feedback control of an Euler-Bernoulli beam: toward realization of the active sink method, Journal of Vibration and Acoustics, Transactions of the ASME, 121, 2, 174-182 (1999)
[18] Ball, J. M.; Marsden, J. E.; Slemrod, M., Controllability for distributed bilinear systems, SIAM Journal on Control and Optimization, 20, 4, 575-597 (1982) · Zbl 0485.93015
[19] Beauchard, K., Local controllability and non-controllability for a 1D wave equation with bilinear control, Journal of Differential Equations, 250, 4, 2064-2098 (2011) · Zbl 1221.35221
[20] Slemrod, M., Stabilization of bilinear control systems with applications to nonconservative problems in elasticity, SIAM Journal on Control and Optimization, 16, 1, 131-141 (1978) · Zbl 0388.93037
[21] Lin, Q.; Loxton, R.; Teo, K. L., The control parameterization method for nonlinear optimal control: a survey, Journal of Industrial and Management Optimization, 10, 1, 275-309 (2014) · Zbl 1276.49025
[22] Lin, Q.; Loxton, R.; Teo, K. L.; Wu, Y. H., Optimal control computation for nonlinear systems with state-dependent stopping criteria, Automatica, 48, 9, 2116-2129 (2012) · Zbl 1258.49051
[23] Loxton, R. C.; Teo, K. L.; Rehbock, V.; Yiu, K. F., Optimal control problems with a continuous inequality constraint on the state and the control, Automatica, 45, 10, 2250-2257 (2009) · Zbl 1179.49032
[24] Teo, K. L.; Goh, C. J.; Wong, K. H., A Unified Computational Approach to Optimal Control Problems (1991), London, UK: Longman Scientific and Technical, London, UK
[25] Bazaraa, M. S.; Sherali, H. D.; Shetty, C. M., Nonlinear Programming: Theory and Algorithms (2013), New York, NY, USA: John Wiley & Sons, New York, NY, USA
[26] Luenberger, D. G.; Ye, Y. Y., Linear and Nonlinear Programming, 116 (2008), New York, NY, USA: Springer, New York, NY, USA
[27] Loxton, R. C.; Teo, K. L.; Rehbock, V., Optimal control problems with multiple characteristic time points in the objective and constraints, Automatica, 44, 11, 2923-2929 (2008) · Zbl 1160.49033
[28] Loxton, R.; Teo, K. L.; Rehbock, V., Robust suboptimal control of nonlinear systems, Applied Mathematics and Computation, 217, 14, 6566-6576 (2011) · Zbl 1209.93041
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