A. E. Alshorbagy, M. A. Eltaher, and F. F. Mahmoud, Free vibration characteristics of a functionally graded beam by finite element method, Appl. Math. Model, vol.35, pp.412-425, 2011.

M. Aminbgahai, J. Murin, and V. Hutis, Modal analysis of the FGM-beams with continuous transversal symmetric and longitudinal variation of material properties with effect of large axial force, Eng. Struct, vol.34, pp.314-329, 2012.

A. Arbel, Controllability measures and actuator placement in oscillatory systems, Int. J. Control, vol.33, issue.3, pp.565-574, 1981.

V. Balamurugan and S. Narayanan, Shell finite element for smart piezoelectric composite plate/shell structures and its application to the study of active vibration control, Finite Elem. Anal. Des, vol.37, pp.713-738, 2001.

M. Biglar, H. R. Mirdamadi, and M. Danesh, Optimal locations and orientations of piezoelectric transducers on cylindrical shell based on gramians of contributed and undesired Rayleigh-Ritz modes using genetic algorithm, J. Sound Vib, vol.333, pp.1224-1244, 2014.

I. Bruant, G. Coffignal, F. Léné, and M. Vergé, Optimal location of piezoelectric actuators on a beam, Proceedings of Active 97, pp.635-649, 1997.

I. Bruant, G. Coffignal, F. Léné, and M. Vergé, Active control of beam structures with piezoelectric actuators and sensors: modeling and simulation, Smart Mater. Struct, vol.10, pp.404-408, 2001.
URL : https://hal.archives-ouvertes.fr/hal-01978408

I. Bruant and L. Proslier, Optimal location of actuators and sensors in active vibration control, J. Intell. Mater. Syst. Struct, vol.16, pp.197-206, 2005.
URL : https://hal.archives-ouvertes.fr/hal-01978500

I. Bruant and L. Proslier, Improved active control of a functionally graded material beam with piezoelectric patches, J. Vib. Control, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01421512

S. Caddemi and I. Calio, Exact closed-form solution for the vibration modes of the Euler-Bernoulli beam with multiple open cracks, J. Sound Vib, vol.327, pp.473-489, 2009.

S. Caddemi and I. Calio, The exact explicit dynamic stiffness matrix of multi-cracked Euler-Bernoulli beam and applications to damaged frame structures, J. Sound Vib, vol.332, pp.3049-3063, 2013.

S. Devasia, T. Meressi, B. Paden, and E. Bayo, Piezoelectric actuator design for vibration suppression: placement and sizing, J. Guid. Control Dyn, vol.16, issue.5, pp.859-864, 1993.

A. Dhingra and B. Lee, Multiobjective design of actively controlled structures using a hybrid optimizatoon method, Int. J. Numer. Methods Eng, vol.38, pp.3383-3401, 1995.

K. D. Dhuri and P. Seshu, Piezo actuator placement and sizing for good control effectiveness and minimal change in original system dynamics, Smart Mater. Struct, vol.15, pp.1661-1672, 2006.

V. Fakhari and A. Ohadi, Nonlinear vibration control of functionally graded plate with piezoelectric layers in thermal environment, J. Vib. Control, vol.17, pp.448-469, 2010.

M. Frecker, Recent advances in optimization of smart structures and actuators, J. Intell. Mater. Syst. Struct, vol.14, pp.207-215, 2003.

Y. Fu, J. Wang, and Y. Mao, Nonlinear vibration and active control of functionally graded beams with piezoelectric sensors and actuators, J. Intell. Mater. Syst. Struct, vol.22, pp.2093-2102, 2013.

W. Gawronski, Simultaneous placement of actuators and sensors, J. Sound Vib, vol.228, issue.4, pp.915-922, 1999.

A. Gharib, M. Salehi, and S. Fazeli, Deflection control of functionally graded material beams with bonded piezoelectric sensors and actuators, Mater. Sci. Eng. A, vol.498, pp.110-114, 2008.

G. Giunta, D. Crisafulli, S. Belouettar, and E. Carrera, Hierarchical theories for free vibration analysis of functionally graded beams, Compos. Struct, vol.94, pp.68-74, 2011.

M. Gney and E. Eskinat, Optimal actuator and sensor placement in flexible structures using closed-loop criteria, J. Sound Vib, vol.312, pp.210-233, 2007.

A. Hac and L. Liu, Sensor and actuator location in motion control of flexible structures, J. Sound Vib, vol.167, pp.239-261, 1993.

D. Halim and S. O. Reza-moheimani, An optimization approach to optimal placement of collocated piezoelectric actuators and sensors on a thin plate, Mechatronics, vol.13, pp.27-47, 2003.

X. Q. He, T. Y. Ng, S. Sivashanker, and K. M. Liew, Active control of FGM plates with integrated piezoelectric sensors and actuators, Int. J. Solids Struct, vol.38, pp.1641-1655, 2001.

Y. Huang and X. F. Li, A new approach for free vibration of axially functionally graded beams with non-uniform cross-section, J. Sound Vib, vol.329, pp.2291-2303, 2010.

Y. Huang, L. E. Yang, and Q. Z. Luo, Free vibration of axially functionally graded Timoshenko beams with non-uniform cross-section, Compos. Part B, vol.45, pp.1493-1498, 2013.

K. Hiramoto, H. Doki, and G. Obinata, Optimal sensor/actuator placement for active vibration control using explicit solution of algebraic Riccati equation, J. Sound Vib, vol.229, issue.5, pp.1057-1075, 2000.

A. K. Jha and D. J. Inman, Optimal sizes and placements of piezoelectric actuators and sensors for an inflated torus, J. Intell. Mater. Syst. Struct, vol.14, pp.563-576, 2003.

T. Kailath, Linear Systems, 1980.

M. H. Kargarnovin, M. M. Najafizadeh, and N. S. Viliani, Vibration control of functionally graded material plate patched with piezoelectric actuators and sensors under a constant electric charge, Smart Mater. Struct, vol.16, pp.1252-1259, 2007.

Y. Kiani, M. Sadighi, and M. R. Eslami, Dynamic analysis and active control of smart doubly curved FGM panels, Compos. Struct, vol.102, pp.205-216, 2013.

S. Kondoh, C. Yatomi, and K. Inoue, The positioning of sensors and actuators in the vibration control of flexible systems, JSME Int. J, vol.33, pp.145-152, 1990.

X. F. Li, Y. A. Kang, and J. X. Wu, Exact frequency equations of free vibration of exponentially functionally graded beams, Appl. Acoust, vol.74, pp.413-420, 2013.

K. M. Liew, S. Sivashanker, X. Q. He, and T. Y. Ng, The modeling and design of smart structures using functionally graded materials and piezoelectrical sensor/actuator patches, Smart Mater. Struct, vol.12, pp.647-655, 2003.

W. Liu, Z. Hou, and M. A. Demetriou, A computational scheme for the optimal sensor/actuator placement of flexible structures using spatial H2 measures, Mech. Syst. Signal Process, vol.20, pp.881-895, 2006.

D. Y. Liu, C. Y. Wang, and W. Q. Chen, Free vibration of FGM plates with in-plane material inhomogeneity, Compos. Struct, vol.92, pp.1047-1051, 2010.

R. M. Mahamood and E. T. Akinlabi, Functionally graded material: an overview, Proceedings of the World Congress on Engineering, p.3, 2012.

A. J. Markworth, K. S. Ramesh, and W. P. Parks, Review: modeling studies applied to functionally graded materials, J. Mater. Sci, vol.30, pp.2183-2193, 2012.

R. Mirzaeifar, H. Bahai, and S. Shahab, Active control of natural frequencies of FGM plates by piezoelectric sensor/actuator pairs, Smart Mater. Struct, vol.17, p.8, 2008.

J. Murin, M. Aminbaghai, and V. Kutis, Exact solution of the bending vibration problem of FGM beams with variation of material properties, Eng. Struct, vol.32, pp.1631-1640, 2010.

J. Murin, M. Aminbaghai, J. Hrabovsky, V. Kutis, and S. Kugler, Modal analysis of the FGM beams with the effect of the shear correction function, Compos. Part B, vol.45, pp.1575-1582, 2013.

C. Nam, Y. Kim, and T. Weisshaar, Optimal sizing and placement of piezo-actuators for active flutter suppression, Smart Mater. Struct, vol.5, pp.2216-2224, 1996.

S. Narayanan and V. Balamurugan, Functionally graded shells with distributed piezoelectric sensors and actuators for active vibration control, IUTAM Symposium on Multi Functional Material Structures and Systems, 2010.

F. Peng, A. Ng, and Y. R. Hu, Actuator placement optimization and adaptive vibration control of plate smart structures, J. Intell. Mater. Syst. Struct, vol.16, pp.263-271, 2005.

A. Preumont, Vibration Control of Active Structures, 1999.

Z. C. Qiu, X. M. Zhang, H. X. Wu, and H. H. Zhang, Optimal placement and active vibration control for piezoelctric smart flexible cantilever plate, J. Sound Vib, vol.301, pp.521-543, 2007.

R. Kumar, K. Narayanan, and S. , Active vibration control of beams with optimal placement of piezoelectric sensors/ actuator pairs, Smart Mater. Struct, 2008.

K. Sarkar and R. Ganguli, Closed-form solutions for axially functionally graded Timoshenko beams having uniform cross-section and fixed-fixed boundary condition. Compos, Part B, vol.58, pp.361-370, 2014.

S. L. Schulz, H. M. Gomes, and A. M. Awruch, Optimal discrete piezoelectric patch allocation on composite structures for vibration control based on GA and modal LQR, Comput. Struct, vol.128, pp.101-115, 2013.

A. Shahba, R. Attarnejad, M. Tavanaie-marvi, and S. Hajilar, Free vibration and stability of axially functionally graded trapered Timoshenko beams with classical and non-classical boundary conditions, Compos. Part B, vol.42, pp.801-808, 2011.

A. Shahba and S. Rajasekaran, Free vibration and stability of tapered Euler-Bernoulli beams made of axially functionally graded materials, Appl. Math. Model, vol.36, pp.3094-3111, 2012.

G. G. Sheng and X. Wang, Active control of functionally graded laminated cylindrical shells, Compos. Struct, vol.90, pp.448-457, 2009.

M. Simsek, T. Kocaturk, and S. D. Akbas, Dynamic behavior of an axially functionally graded beam under action of a moving harmonic load, Compos. Struct, vol.94, pp.2358-2364, 2012.

B. Uymaz, M. Aydogdu, and S. Filiz, Vibration analyses of FGM plates with in-plane material inhomogeneity by Ritz method, Compos. Struct, vol.94, pp.1398-1405, 2012.

Q. Wang and C. Wang, A controllability index for optimal design of piezoelectric actuators in vibration control of beam structures, J. Sound Vib, vol.242, issue.3, pp.507-518, 2001.

L. Wu, Q. S. Wang, and I. Elishakoff, Semi-inverse method for axially functionally graded beams with an anti-symmetric vibration mode, J. Sound Vib, vol.284, pp.1190-1202, 2005.

H. J. Xiang and J. Yang, Free and forced vibration of a laminated FGM Timoshenko beam of variable thickness under heat conduction, Compos. Part B, vol.39, pp.292-303, 2008.

S. Yang and Y. Lee, Optimization of noncollocated sensor/actuator location and feedback gain in control systems, Smart Mater. Struct, vol.2, pp.96-102, 1993.

Y. Yang, Z. Jin, and C. Kiong, So Integrated optimal design of vibration control system for smart beams using genetic algorithms, J. Sound Vib, vol.282, pp.1293-1307, 2005.

M. Yiqi and F. Yiming, Nonlinear dynamic response and active vibration control for piezoelectric functionally graded plate, J. Sound Vib, vol.329, pp.2015-2028, 2010.

S. J. Zheng, F. Dai, and Z. Song, Active control of piezothermoelastic FGM shells using integrated piezoelectric sensor/ actuator layers, Int. J. Appl. Electromagn. Mech, vol.30, pp.107-124, 2009.