# American Institute of Mathematical Sciences

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 and 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, COCV., 6 (2001), 361-386. doi: 10.1051/cocv:2001114. [2] K. Ammari and M. Tucsnak, Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force, SIAM, J. Control Optim., 39 (2000), 1160-1181. doi: 10.1137/S0363012998349315. [3] C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780. doi: 10.1007/s00028-008-0424-1. [4] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478. doi: 10.1007/s00208-009-0439-0. [5] H. Brezis, Analyse Fonctionelle, Théorie et Applications, Masson, Paris, 1983. [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), 526-546. doi: 10.1137/0325029. [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), Lecture Notes in Pure and Appl. Math., 108, Dekker, New York, 1987, 67-96. [8] A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series, Math. Z., 41 (1936), 367-379. doi: 10.1007/BF01180426. [9] T. Kato, Perturbaton Theory for Linear Operators, Springer-Verlag, New-York, 1976. [10] J. S. Gibson, A note on stabilization of infinite-dimensional linear oscillators by compact linear feedback, SIAM J. Control Optim., 18 (1980), 311-316. doi: 10.1137/0318022. [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, Control, Optimisation and Calculus Variations, 14 (2008), 632-656. doi: 10.1051/cocv:2008001. [12] J. E. Lagnese, Boundary Stabilization of Thin Plates, SIAM Studies in Applied Mathematics, 10, SIAM, Philadelphia, PA, 1989. doi: 10.1137/1.9781611970821. [13] Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644. doi: 10.1007/s00033-004-3073-4. [14] D. Mercier and V. Régnier, Exponential stability of a network of serially connected Euler-Bernoulli beams, International Journal of Control, 87 (2014), 1266-1281. doi: 10.1080/00207179.2013.874597. [15] Ö. Morgül, Dynamic boundary control of an Euler-Bernoulli beam, IEEE Trans. Automat. Control, 37 (1992), 639-642. doi: 10.1109/9.135504. [16] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equation, Applied Mathematical Sciences, 44, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1. [17] B. Rao, A Compact perturbation method for the boundary stabilization of the Rayleigh beam equation, Appl. Math. Optim., 33 (1996), 253-264. doi: 10.1007/BF01204704. [18] B. Rao, Uniform stabilisation of a hybrid system of elasticity, SIAM J. Control Optim., 33 (1995), 440-454. doi: 10.1137/S0363012992239879. [19] B. Rao, Optimal energy decay rate in a damped Rayleigh beam, Discrete Contin. Dynam. Systems, 4 (1998), 721-734. doi: 10.3934/dcds.1998.4.721. [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), Frontiers in Applied Mathematics, Philadelphia, PA, 1992, 125-169. doi: 10.1137/1.9781611970982.ch4. [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), 33-50. doi: 10.1016/S0167-6911(03)00205-6. [22] A. Wehbe, Optimal energy decay rate in the Rayleigh beam equation with boundary dynamical controls, Bull. Belg. Math. Soc., 13 (2006), 385-400.

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##### References:
 [1] K. Ammari and M. Tucsnak, Stabilizaton of second order evolution equations by class of unbounded feedbacks, ESIAM, COCV., 6 (2001), 361-386. doi: 10.1051/cocv:2001114. [2] K. Ammari and M. Tucsnak, Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force, SIAM, J. Control Optim., 39 (2000), 1160-1181. doi: 10.1137/S0363012998349315. [3] C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780. doi: 10.1007/s00028-008-0424-1. [4] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478. doi: 10.1007/s00208-009-0439-0. [5] H. Brezis, Analyse Fonctionelle, Théorie et Applications, Masson, Paris, 1983. [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), 526-546. doi: 10.1137/0325029. [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), Lecture Notes in Pure and Appl. Math., 108, Dekker, New York, 1987, 67-96. [8] A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series, Math. Z., 41 (1936), 367-379. doi: 10.1007/BF01180426. [9] T. Kato, Perturbaton Theory for Linear Operators, Springer-Verlag, New-York, 1976. [10] J. S. Gibson, A note on stabilization of infinite-dimensional linear oscillators by compact linear feedback, SIAM J. Control Optim., 18 (1980), 311-316. doi: 10.1137/0318022. [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, Control, Optimisation and Calculus Variations, 14 (2008), 632-656. doi: 10.1051/cocv:2008001. [12] J. E. Lagnese, Boundary Stabilization of Thin Plates, SIAM Studies in Applied Mathematics, 10, SIAM, Philadelphia, PA, 1989. doi: 10.1137/1.9781611970821. [13] Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644. doi: 10.1007/s00033-004-3073-4. [14] D. Mercier and V. Régnier, Exponential stability of a network of serially connected Euler-Bernoulli beams, International Journal of Control, 87 (2014), 1266-1281. doi: 10.1080/00207179.2013.874597. [15] Ö. Morgül, Dynamic boundary control of an Euler-Bernoulli beam, IEEE Trans. Automat. Control, 37 (1992), 639-642. doi: 10.1109/9.135504. [16] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equation, Applied Mathematical Sciences, 44, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1. [17] B. Rao, A Compact perturbation method for the boundary stabilization of the Rayleigh beam equation, Appl. Math. Optim., 33 (1996), 253-264. doi: 10.1007/BF01204704. [18] B. Rao, Uniform stabilisation of a hybrid system of elasticity, SIAM J. Control Optim., 33 (1995), 440-454. doi: 10.1137/S0363012992239879. [19] B. Rao, Optimal energy decay rate in a damped Rayleigh beam, Discrete Contin. Dynam. Systems, 4 (1998), 721-734. doi: 10.3934/dcds.1998.4.721. [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), Frontiers in Applied Mathematics, Philadelphia, PA, 1992, 125-169. doi: 10.1137/1.9781611970982.ch4. [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), 33-50. doi: 10.1016/S0167-6911(03)00205-6. [22] A. Wehbe, Optimal energy decay rate in the Rayleigh beam equation with boundary dynamical controls, Bull. Belg. Math. Soc., 13 (2006), 385-400.
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