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Optimal control of thermoelastic beam vibrations by piezoelectric actuation

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Abstract

This paper presents the vibrations suppression of a thermoelastic beam subject to sudden heat input by a single piezoelectric actuator. An optimization problem is formulated as the minimization of a quadratic functional in terms of displacement and velocity at a given time and with the least control effort. The solution method is based on a combination of modal expansion and variational approaches. The modal expansion approach is used to convert the optimal control of distributed parameter system into the optimal control of lumped parameter system. By utilizing the variational approach, an explicit optimal control law is derived and the determination of the corresponding displacement and velocity is reduced to solving a set of ordinary differential equations. Numerical results are presented to demonstrate the effectiveness and the applicability of the proposed method.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, American University of Sharjah, Sharjah, U.A.E.

    Ibrahim Sadek & Marwan Abukhaled

Authors
  1. Ibrahim Sadek
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  2. Marwan Abukhaled
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Correspondence to Marwan Abukhaled.

Additional information

Ibrahim SADEK (1949–July 2012) received his Ph.D. from the Mathematics Department of University of California at Santa Barbara, U.S.A. He subsequently joined the Mathematics Departments of Queen’s University, Canada, and University of North Carolina (UNCW), U.S.A., and King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia, before joining the Department of Mathematics and Statistics at the American University of Sharjah, Sharjah, U.A.E. in 1997, where was a professor of Mathematics and Associate Dean of the College of Arts and Sciences. He was the recipient of 1996 UNCW Award for excellence in Scholarship, Research and Creativity. His research interests included control theory and applications, approximation and computational methods for control, identification and optimization of distributed parameter systems, control of piezoelectric structures and time-delayed systems, feedback control of microscale structures, and variational principles applicable to carbon Nano-tubes. He published more than 100 journal papers in these fields and presented papers at more than 60 conferences around the world. He was a regular co-organizer of conferences in the fields of mathematical and applied optimization control and modeling.

Marwan ABUKHALED received his Ph.D. degree in Mathematics from Texas Tech University, U.S.A. in 1995. He held academic position at higher institutions in U.S.A., Jordan, and the United Arab Emirates. Currently, he is a professor of Mathematics at the American University of Sharjah, U.A.E. His research interests are in the area of computational and applied mathematics. He published several articles in the areas of stochastic differential equations and control theory. He is an active participant in international and regional conferences.

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Sadek, I., Abukhaled, M. Optimal control of thermoelastic beam vibrations by piezoelectric actuation. J. Control Theory Appl. 11, 463–467 (2013). https://doi.org/10.1007/s11768-013-1204-1

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  • DOI: https://doi.org/10.1007/s11768-013-1204-1

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