Uniform Exponential Stabilization for Flexural Vibrations of a Solar Panel

ABSTRACT

Here we study a problem of stabilization of the flexural vibrations or transverse vibrations of a rectangular solar panel. The dynamics of vibrations is governed by the fourth order Euler-Bernoulli beam equation. One end of the panel is held by a rigid hub and other end is totally free. Due to attachment of the hub, its dynamics leads to a non-standard equation. The exponential stabilization of the whole system is achieved by applying an active boundary control force only on the rigid hub. The result of uniform stabilization is obtained by means of an explicit form of exponential energy decay estimate.

Here we study a problem of stabilization of the flexural vibrations or transverse vibrations of a rectangular solar panel. The dynamics of vibrations is governed by the fourth order Euler-Bernoulli beam equation. One end of the panel is held by a rigid hub and other end is totally free. Due to attachment of the hub, its dynamics leads to a non-standard equation. The exponential stabilization of the whole system is achieved by applying an active boundary control force only on the rigid hub. The result of uniform stabilization is obtained by means of an explicit form of exponential energy decay estimate.

KEYWORDS

Solar Panel, Hybrid System, Flexural Vibrations, Uniform Exponential Stabilization, Energy Decay Estimate

Solar Panel, Hybrid System, Flexural Vibrations, Uniform Exponential Stabilization, Energy Decay Estimate

Cite this paper

nullP. Nandi, G. Gorain and S. Kar, "Uniform Exponential Stabilization for Flexural Vibrations of a Solar Panel,"*Applied Mathematics*, Vol. 2 No. 6, 2011, pp. 661-665. doi: 10.4236/am.2011.26087.

nullP. Nandi, G. Gorain and S. Kar, "Uniform Exponential Stabilization for Flexural Vibrations of a Solar Panel,"

References

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[9] A. M. Krall, “Asymptotic Stability of the Euler-Bernoulli Beam with Boundary Control,” Journal of Mathematical Analysis and Applications, Vol. 137, No. 1, 1989, pp. 288-295. doi:10.1016/0022-247X(89)90289-8

[10] G. Chen, M. C. Delfour, A. M. Krall and G. Payre, “Modelling, Stabilization and Control of Serially Connected Beams,” SIAM Journal on Control and Optimization, Vol. 25, No. 3, 1987, pp. 526-546. doi:10.1137/0325029

[11] W. Littman and L. Markus, “Stabilization of a Hybrid System of Elasticity by Feedback Boundary Damping,” Annali Di Matematica Pura Ed Applicata, Vol. 152, No. 1, 1988, pp. 281-330. doi:10.1007/BF01766154

[12] B. Rao, “Uniform Stabilization of a Hybrid System of Elasticity,” SIAM Journal on Control and Optimization, Vol. 33, No. 2, 1995, pp. 440-454. doi:10.1137/S0363012992239879

[13] V. Komornik and E. Zuazua, “A Direct Method for Boundary Stabilization of the Wave Equation,” Journal de Mathématiques Pures et Appliquées, Vol. 69, No. 2, 1990, pp. 33-54.

[14] G. C. Gorain, “Exponential Energy Decay Estimate for the Solutions of Internally Damped Wave Equation in a Bounded Domain,” Journal of Mathematical Analysis and Applications, Vol. 216, No. 2, 1997, pp. 510-520. doi:10.1006/jmaa.1997.5678

[15] T. Fukuda, F. Arai, H. Hosogai and N. Yajima, “Torsional Vibrations Control of Flexible Space Structures,” Theoretical and Applied Mechanics, Vol. 36, No. 2, 1988, pp. 285-294.

[16] K. Ammari and M. Tuesnak, “Stabilization of Bernoulli-Euler Beams by Means of a Point Feedback Force,” SIAM Journal on Control and Optimization, Vol. 39, No. 4, 2000, pp. 1160-1181. doi:10.1137/S0363012998349315

[17] K. Liu and Z. Liu, “Exponential Decay of Energy of the Euler-Bernoulli Beam with Locally Distributed Kelvin-Voigt Damping,” SIAM Journal on Control and Optimization, Vol. 36, No. 3, 1998, pp. 1096-1098. doi:10.1137/S0363012996310703

[18] K. Nagaya, “Method of Control of Flexible Beams Subject to Forced Vibrations by Use of Inertia Force Cancellations,” Journal of Sound and Vibration, Vol. 184, No. 2, 1995, pp. 184-194. doi:10.1006/jsvi.1995.0311

[19] R. Rebarbery, “Exponential Stability of Coupled Beams with Dissipative Joints: A Frequency Domain Approach,” SIAM Journal on Control and Optimization, Vol. 33, No. 1, 1995, pp. 1-28. doi:10.1137/S0363012992240321

[20] G. C. Gorain, “Exponential Energy Decay Estimate for the Solutions of N-Dimensional Kirchhoff Type Wave Equation,” Applied Mathematics and Computation, Vol. 177, No. 1, 2006, 235-242. doi:10.1016/j.amc.2005.11.003

[21] G. C. Gorain and S. K. Bose, “Exact Controllability and Boundary Stabilization of Torsional Vibrations of an Internally Damped Flexible Space Structures,” Journal of Optimization Theory and Applications, Vol. 99, No. 2, 1998, pp. 423-442. doi:10.1023/A:1021778428222

[1] G. Chen, “Energy Decay Estimates and Exact Boundary-Value Controllability for the Wave Equation in a Bounded Domain,” Journal de Mathématiques Pures et Appliquées, Vol. 58, No. 3, 1979, pp. 249-273.

[2] G. Chen, “A Note on the Boundary Stabilization of the Wave Equation,” SIAM Journal on Control and Optimization, Vol. 19, No. 1, 1981, pp. 106-113. doi:10.1137/0319008

[3] J. Lagnese, “Note on Boundary Stabilization of Wave Equations,” SIAM Journal on Control and Optimization, Vol. 26, No. 5, 1988, pp. 1250-1256. doi:10.1137/0326068

[4] J. Lagnese, “Decay of Solutions of Wave Equations in a Bounded Region with Boundary Dissipation,” Journal of Differential Equations, Vol. 50, No. 2, 1983, pp. 163-182. doi:10.1016/0022-0396(83)90073-6

[5] J. L. Lions, “Exact Controllability, Stabilization and Perturbations for Distributed Systems,” SIAM Review, Vol. 30, No. 1, 1988, pp. 1-68. doi:10.1137/1030001

[6] V. Komornik, “Rapid Boundary Stabilization of Wave Equations,” SIAM Journal on Control and Optimization, Vol. 29, 1991, pp. 197-208. doi:10.1137/0329011

[7] G. Chen and J. Zhou, “The Wave Propagation Method for the Analysis of Boundary Stabilization in Vibrating Structures,” SIAM Journal on Applied Mathematics, Vol. 50, No. 5, 1990, pp. 1254-1283. doi:10.1137/0150076

[8] O. Morgul, “Dynamic Boundary Control of a Euler-Ber- noulli Beam,” IEEE Transactions on Automatic Control, Vol. 37, No. 5, 1992, pp. 639-642. doi:10.1109/9.135504

[9] A. M. Krall, “Asymptotic Stability of the Euler-Bernoulli Beam with Boundary Control,” Journal of Mathematical Analysis and Applications, Vol. 137, No. 1, 1989, pp. 288-295. doi:10.1016/0022-247X(89)90289-8

[10] G. Chen, M. C. Delfour, A. M. Krall and G. Payre, “Modelling, Stabilization and Control of Serially Connected Beams,” SIAM Journal on Control and Optimization, Vol. 25, No. 3, 1987, pp. 526-546. doi:10.1137/0325029

[11] W. Littman and L. Markus, “Stabilization of a Hybrid System of Elasticity by Feedback Boundary Damping,” Annali Di Matematica Pura Ed Applicata, Vol. 152, No. 1, 1988, pp. 281-330. doi:10.1007/BF01766154

[12] B. Rao, “Uniform Stabilization of a Hybrid System of Elasticity,” SIAM Journal on Control and Optimization, Vol. 33, No. 2, 1995, pp. 440-454. doi:10.1137/S0363012992239879

[13] V. Komornik and E. Zuazua, “A Direct Method for Boundary Stabilization of the Wave Equation,” Journal de Mathématiques Pures et Appliquées, Vol. 69, No. 2, 1990, pp. 33-54.

[14] G. C. Gorain, “Exponential Energy Decay Estimate for the Solutions of Internally Damped Wave Equation in a Bounded Domain,” Journal of Mathematical Analysis and Applications, Vol. 216, No. 2, 1997, pp. 510-520. doi:10.1006/jmaa.1997.5678

[15] T. Fukuda, F. Arai, H. Hosogai and N. Yajima, “Torsional Vibrations Control of Flexible Space Structures,” Theoretical and Applied Mechanics, Vol. 36, No. 2, 1988, pp. 285-294.

[16] K. Ammari and M. Tuesnak, “Stabilization of Bernoulli-Euler Beams by Means of a Point Feedback Force,” SIAM Journal on Control and Optimization, Vol. 39, No. 4, 2000, pp. 1160-1181. doi:10.1137/S0363012998349315

[17] K. Liu and Z. Liu, “Exponential Decay of Energy of the Euler-Bernoulli Beam with Locally Distributed Kelvin-Voigt Damping,” SIAM Journal on Control and Optimization, Vol. 36, No. 3, 1998, pp. 1096-1098. doi:10.1137/S0363012996310703

[18] K. Nagaya, “Method of Control of Flexible Beams Subject to Forced Vibrations by Use of Inertia Force Cancellations,” Journal of Sound and Vibration, Vol. 184, No. 2, 1995, pp. 184-194. doi:10.1006/jsvi.1995.0311

[19] R. Rebarbery, “Exponential Stability of Coupled Beams with Dissipative Joints: A Frequency Domain Approach,” SIAM Journal on Control and Optimization, Vol. 33, No. 1, 1995, pp. 1-28. doi:10.1137/S0363012992240321

[20] G. C. Gorain, “Exponential Energy Decay Estimate for the Solutions of N-Dimensional Kirchhoff Type Wave Equation,” Applied Mathematics and Computation, Vol. 177, No. 1, 2006, 235-242. doi:10.1016/j.amc.2005.11.003

[21] G. C. Gorain and S. K. Bose, “Exact Controllability and Boundary Stabilization of Torsional Vibrations of an Internally Damped Flexible Space Structures,” Journal of Optimization Theory and Applications, Vol. 99, No. 2, 1998, pp. 423-442. doi:10.1023/A:1021778428222