Mathematical Problems in Engineering
Volume 2009 (2009), Article ID 530287, 15 pages
Research Article

A Galerkin-Parameterization Method for the Optimal Control of Smart Microbeams

Department of Mathematics and Statistics, American University of Sharjah, P.O. Box 26666, Sharjah, UAE

Received 18 April 2009; Revised 1 September 2009; Accepted 22 September 2009

Academic Editor: Mohammad Younis

Copyright © 2009 Marwan Abukhaled and Ibrahim Sadek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A proposed computational method is applied to damp out the excess vibrations in smart microbeams, where the control action is implemented using piezoceramic actuators. From a mathematical point of view, we wish to determine the optimal boundary actuators that minimize a given energy-based performance measure. The minimization of the performance measure over the actuators is subjected to the full motion of the structural vibrations of the micro-beams. A direct state-control parametrization approach is proposed where the shifted Legendre polynomials are employed to solve the optimization problem. Legendre operational matrix and the properties of Kronecker product are utilized to find the approximated optimal trajectory and optimal control law of the lumped parameter systems with respect to the quadratic cost function by solving linear algebraic equations. Numerical examples are provided to demonstrate the applicability and efficiency of the proposed approach.