March  2015, 4(1): 21-38. doi: 10.3934/eect.2015.4.21

Optimal energy decay rate of Rayleigh beam equation with only one boundary control force

1. 

Université Libanaise, EDST, Equipe EDP-AN, Hadath, Beyrouth, Lebanon

2. 

Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques of Valenciennes, F-59313 - Valenciennes Cedex 9

3. 

Université Libanaise, Sciences 1 et EDST, Equipe EDP-AN, Hadath, Beyrouth, Lebanon

Received  July 2014 Revised  January 2015 Published  February 2015

We consider a clamped Rayleigh beam equation subject to only one boundary control force. Using an explicit approximation, we first give the asymptotic expansion of eigenvalues and eigenfunctions of the undamped underlying system. We next establish a polynomial energy decay rate for smooth initial data via an observability inequality of the corresponding undamped problem combined with a boundedness property of the transfer function of the associated undamped problem. Finally, by a frequency domain approach, using the real part of the asymptotic expansion of eigenvalues of the infinitesimal generator of the associated semigroup, we prove that the obtained energy decay rate is optimal.
Citation: Maya Bassam, Denis Mercier, Ali Wehbe. Optimal energy decay rate of Rayleigh beam equation with only one boundary control force. Evolution Equations & Control Theory, 2015, 4 (1) : 21-38. doi: 10.3934/eect.2015.4.21
References:
[1]

K. Ammari and M. Tucsnak, Stabilizaton of second order evolution equations by class of unbounded feedbacks,, ESIAM, 6 (2001), 361. doi: 10.1051/cocv:2001114. Google Scholar

[2]

K. Ammari and M. Tucsnak, Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force,, SIAM, 39 (2000), 1160. doi: 10.1137/S0363012998349315. Google Scholar

[3]

C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces,, J. Evol. Equ., 8 (2008), 765. doi: 10.1007/s00028-008-0424-1. Google Scholar

[4]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups,, Math. Ann., 347 (2010), 455. doi: 10.1007/s00208-009-0439-0. Google Scholar

[5]

H. Brezis, Analyse Fonctionelle, Théorie et Applications,, Masson, (1983). Google Scholar

[6]

G. Chen, M. C. Delfour, A. M. Krall and G. Payre, Modeling, stabilization and control of serially connected beams,, SIAM J. Control Optim., 25 (1987), 526. doi: 10.1137/0325029. Google Scholar

[7]

G. Chen, S. G. Krantz, D. W. Ma, C. E. Wayne and H. H. West, The Euler-Bernoulli beam equation with boundary energy dissipation,, in operator Methods for Optimal Control Problems (ed. Sung J. Lee), (1987), 67. Google Scholar

[8]

A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series,, Math. Z., 41 (1936), 367. doi: 10.1007/BF01180426. Google Scholar

[9]

T. Kato, Perturbaton Theory for Linear Operators,, Springer-Verlag, (1976). Google Scholar

[10]

J. S. Gibson, A note on stabilization of infinite-dimensional linear oscillators by compact linear feedback,, SIAM J. Control Optim., 18 (1980), 311. doi: 10.1137/0318022. Google Scholar

[11]

B. Z. Guo, J. M. Wang and C. L. Zhou, On the dynamic behavior and stability of controlled connected Rayleigh beams under pointwise output feedback,, ESAIM, 14 (2008), 632. doi: 10.1051/cocv:2008001. Google Scholar

[12]

J. E. Lagnese, Boundary Stabilization of Thin Plates,, SIAM Studies in Applied Mathematics, (1989). doi: 10.1137/1.9781611970821. Google Scholar

[13]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation,, Z. Angew. Math. Phys., 56 (2005), 630. doi: 10.1007/s00033-004-3073-4. Google Scholar

[14]

D. Mercier and V. Régnier, Exponential stability of a network of serially connected Euler-Bernoulli beams,, International Journal of Control, 87 (2014), 1266. doi: 10.1080/00207179.2013.874597. Google Scholar

[15]

Ö. Morgül, Dynamic boundary control of an Euler-Bernoulli beam,, IEEE Trans. Automat. Control, 37 (1992), 639. doi: 10.1109/9.135504. Google Scholar

[16]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equation,, Applied Mathematical Sciences, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar

[17]

B. Rao, A Compact perturbation method for the boundary stabilization of the Rayleigh beam equation,, Appl. Math. Optim., 33 (1996), 253. doi: 10.1007/BF01204704. Google Scholar

[18]

B. Rao, Uniform stabilisation of a hybrid system of elasticity,, SIAM J. Control Optim., 33 (1995), 440. doi: 10.1137/S0363012992239879. Google Scholar

[19]

B. Rao, Optimal energy decay rate in a damped Rayleigh beam,, Discrete Contin. Dynam. Systems, 4 (1998), 721. doi: 10.3934/dcds.1998.4.721. Google Scholar

[20]

D. L. Russell, On the mathematical models for the elastic beam with frequence-proportional damping,, in Control and Estimation in Distributed Parameters Systems (ed. H. T. Bank), (1992), 125. doi: 10.1137/1.9781611970982.ch4. Google Scholar

[21]

J. M. Wang, G. Q. Xu and S. P. Yung, Exponential stability of variable coefcients Rayleigh beams under boundary feedback controls: A Riesz basis approach,, Systems and Control Letters, 51 (2004), 33. doi: 10.1016/S0167-6911(03)00205-6. Google Scholar

[22]

A. Wehbe, Optimal energy decay rate in the Rayleigh beam equation with boundary dynamical controls,, Bull. Belg. Math. Soc., 13 (2006), 385. Google Scholar

show all references

References:
[1]

K. Ammari and M. Tucsnak, Stabilizaton of second order evolution equations by class of unbounded feedbacks,, ESIAM, 6 (2001), 361. doi: 10.1051/cocv:2001114. Google Scholar

[2]

K. Ammari and M. Tucsnak, Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force,, SIAM, 39 (2000), 1160. doi: 10.1137/S0363012998349315. Google Scholar

[3]

C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces,, J. Evol. Equ., 8 (2008), 765. doi: 10.1007/s00028-008-0424-1. Google Scholar

[4]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups,, Math. Ann., 347 (2010), 455. doi: 10.1007/s00208-009-0439-0. Google Scholar

[5]

H. Brezis, Analyse Fonctionelle, Théorie et Applications,, Masson, (1983). Google Scholar

[6]

G. Chen, M. C. Delfour, A. M. Krall and G. Payre, Modeling, stabilization and control of serially connected beams,, SIAM J. Control Optim., 25 (1987), 526. doi: 10.1137/0325029. Google Scholar

[7]

G. Chen, S. G. Krantz, D. W. Ma, C. E. Wayne and H. H. West, The Euler-Bernoulli beam equation with boundary energy dissipation,, in operator Methods for Optimal Control Problems (ed. Sung J. Lee), (1987), 67. Google Scholar

[8]

A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series,, Math. Z., 41 (1936), 367. doi: 10.1007/BF01180426. Google Scholar

[9]

T. Kato, Perturbaton Theory for Linear Operators,, Springer-Verlag, (1976). Google Scholar

[10]

J. S. Gibson, A note on stabilization of infinite-dimensional linear oscillators by compact linear feedback,, SIAM J. Control Optim., 18 (1980), 311. doi: 10.1137/0318022. Google Scholar

[11]

B. Z. Guo, J. M. Wang and C. L. Zhou, On the dynamic behavior and stability of controlled connected Rayleigh beams under pointwise output feedback,, ESAIM, 14 (2008), 632. doi: 10.1051/cocv:2008001. Google Scholar

[12]

J. E. Lagnese, Boundary Stabilization of Thin Plates,, SIAM Studies in Applied Mathematics, (1989). doi: 10.1137/1.9781611970821. Google Scholar

[13]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation,, Z. Angew. Math. Phys., 56 (2005), 630. doi: 10.1007/s00033-004-3073-4. Google Scholar

[14]

D. Mercier and V. Régnier, Exponential stability of a network of serially connected Euler-Bernoulli beams,, International Journal of Control, 87 (2014), 1266. doi: 10.1080/00207179.2013.874597. Google Scholar

[15]

Ö. Morgül, Dynamic boundary control of an Euler-Bernoulli beam,, IEEE Trans. Automat. Control, 37 (1992), 639. doi: 10.1109/9.135504. Google Scholar

[16]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equation,, Applied Mathematical Sciences, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar

[17]

B. Rao, A Compact perturbation method for the boundary stabilization of the Rayleigh beam equation,, Appl. Math. Optim., 33 (1996), 253. doi: 10.1007/BF01204704. Google Scholar

[18]

B. Rao, Uniform stabilisation of a hybrid system of elasticity,, SIAM J. Control Optim., 33 (1995), 440. doi: 10.1137/S0363012992239879. Google Scholar

[19]

B. Rao, Optimal energy decay rate in a damped Rayleigh beam,, Discrete Contin. Dynam. Systems, 4 (1998), 721. doi: 10.3934/dcds.1998.4.721. Google Scholar

[20]

D. L. Russell, On the mathematical models for the elastic beam with frequence-proportional damping,, in Control and Estimation in Distributed Parameters Systems (ed. H. T. Bank), (1992), 125. doi: 10.1137/1.9781611970982.ch4. Google Scholar

[21]

J. M. Wang, G. Q. Xu and S. P. Yung, Exponential stability of variable coefcients Rayleigh beams under boundary feedback controls: A Riesz basis approach,, Systems and Control Letters, 51 (2004), 33. doi: 10.1016/S0167-6911(03)00205-6. Google Scholar

[22]

A. Wehbe, Optimal energy decay rate in the Rayleigh beam equation with boundary dynamical controls,, Bull. Belg. Math. Soc., 13 (2006), 385. Google Scholar

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