THERMAL SCIENCE

International Scientific Journal

Authors of this Paper

External Links

Arif Engin

,

Yesim Sarac

,

Ercan Celik

ON OPTIMAL CONTROL OF THE INITIAL VELOCITY OF AN EULER-BERNOULLI BEAM SYSTEM

ABSTRACT
In this study, we consider an optimal control problem for an Euler-Bernoulli beam equation. The initial velocity of the system is given by the control function. We give sufficient conditions for the existence of a unique solution of the hyperbolic system and prove that the optimal solution for the considered optimal control problem is exists and unique. After obtaining the Frechet derivative of the cost functional via an adjoint problem, we also give an iteration algorithm for the numerical solution of the problem by using the Gradient method. Finally, we furnish some numerical examples to demonstrate the effectiveness of the result obtained.
KEYWORDS
hyperbolic system, beam equation, frechet derivative, optimal control
PAPER SUBMITTED: 2022-08-10
PAPER REVISED: 2022-09-17
PAPER ACCEPTED: 2022-09-25
PUBLISHED ONLINE: 2023-01-29
DOI REFERENCE: https://doi.org/10.2298/TSCI22S2735E
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2022, VOLUME 26, ISSUE Special issue 2, PAGES [735 - 744]
REFERENCES
  1. Gunakala, S. R., et al., A Finite Element Solution of Beam Equation via Matlab, International Journal of Applied Science and Technology, 2 (2012), 8, pp. 80-88
  2. Chang, J. D., Guo, B. Z., Identification of Variable Spacial Coefficients for a Beam Equation from Boundary Measurements, Automatica, 43 (2007), 4, pp. 732-737
  3. Lesnic, D., Hasanov, A., Determination of the Leading Coefficient in Fourth-Order Sturm-Liouville Operator from Boundary Measurements, Inverse Problems in Science Engineering, 16 (2008), 4, pp. 413-424
  4. Ohsumi, A., Nakano, N., Identification of Physical Parameters of a Flexible Structure from Noisy Measurement Data, Instrumentation and Measurement Technology Conference, IMTC 2001, Proceedings, 18th IEEE, Arlington, Va., USA, 2001, vol. 2, pp. 1354-1359
  5. Sun, B., Optimal Control of Vibrations of a Dynamic Gao Beam in Contact with a Reactive Foundation, International Journal of Systems Science, 48 (2017), 5, pp. 1084-1091
  6. Conrad, F., Morgül Ö., On the Stabilization of a Flexible Beam with a Tip Mass, SIAM J. Control Optim., 36 (1998), 6, pp. 1962-1986
  7. Guo, B. Z., On the Boundary Control of a Hybrid System with Variable Coefficients, Journal of Optimization Theory and Applications, 114 (2002), 2, pp. 373-395
  8. Guo, B. Z., et al., Dynamic Stabilization of an Euler-Bernoulli Beam Under Boundary Control and Non-Collocated Observation, Systems and Control Letters, 57 (2008), 9, pp. 740-749
  9. Guo, B. Z., Kang, W., Lyapunov Approach to Boundary Stabilization of a Beam Equation with Boundary Disturbance, International Journal of Control, 87 (2013), 5, pp. 925-939
  10. Karagiannis, D., Radisavljeviç-Gajic, V., Sliding Mode Boundary Control of an Euler-Bernoulli Beam Subject to Disturbances, IEEE Transaction on Automatic Control, 63 (2018), 10, pp. 3442-344
  11. Shang, Y. F., et al., Stability Analysis of Euler-Bernoulli Beam with Input Delay in the Boundary Control, Asian Journal of Control, 14 (2012), 1, pp. 186-196
  12. Hasanov, A., Kawano, A., Identification of Unknown Spatial Load Distributions in a Vibrating Euler-Bernoulli Beam from Limited Measured Data, Inverse Problems, 32 (2016), 5, pp. 1-31
  13. Kawano, A., Uniqueness in the Identification of Asynchronous Sources and Damage in Vibrating Beams, Inverse Problems, 30 (2014), 6, pp. 1-16
  14. Lin, C., et al., Optimal Multi-Interval Control of a Cantiveler Beam by a Recursive Control Algorithm, Optimal Control Applications and Methods, 30 (2009), 4, pp. 399-414
  15. Liu, C. S. A., Lie-Group Adaptive Differential Quadrature Method to Identify an Unknown Force in an Euler-Bernoulli Beam Equation, Acta Mechanica, 223 (2012), 10, pp. 2207-2223
  16. Marin, F. J., et al., Robust Averaged Control of Vibrations for the Bernoulli-Euler Beam Equation, J. Optim Theory Appl, 174 (2017), 2, pp. 428-454
  17. Hao, D. N., Oanh, N. T. N., Determination of the Initial Condition in Parabolic Equations from Boundary Observations, J. Inverse III-Posed Problems, 24 (2016), 2, pp. 195-220
  18. Hao, D. N., Oanh, N. T. N., Determination of the Initial Condition in Parabolic Equations from Integral Observations, Inverse Problems in Science and Engineering, 25 (2017), 8, pp. 1138-1167
  19. Klibanov, M. V., Estimates of Initial Conditions of Parabolic Equations and Inequalities via Lateral Cauchy Data, Inverse Problems, 22 (2006), 2, pp. 495-514
  20. Kowalewski, A., Optimal Control via Initial state of an Infinite Order Time Delay Hyperbolic System, Proceeding, 18th International Conferences on Process Control, Tatranska Lomnica, Slovakia, 2011
  21. Kowalewski, A., Optimal Control via Initial Conditions of a Time Delay Hyperbolic System, Proceedings, 18th International Conferences on Methods and Models in Automation and Robotics, Miedzyzdroje, Poland, 2011
  22. Sarac, Y., Symbolic and Numeric Computation of Optimal Initial Velocity in A Wave Equation, Journal of Computational and Non-linear Dynamics, 8 (2012), 1, pp. 1-4
  23. Evans, L. C., Partial Differential Equations, American Mathematical Society, Rhode Island, 2002
  24. Hasanov, A., Ituo, H., A Priori Estimates for the General Dynamic Euler-Bernoulli Beam Equation: Supported and Cantilever Beams, Applied Mathematics Letters, 87 (2019), Jan., pp. 141-146
  25. Kundu, B., Ganguli, R., Analysis of Weak Solution of Euler-Bernoulli Beam with Axial Force, Applied Mathematics and Computation, 298 (2017), Apr., pp. 247-260
PDF VERSION [DOWNLOAD]
On optimal control of the initial velocity of an Euler-Bernoulli beam system

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence