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Robust Averaged Control of Vibrations for the Bernoulli–Euler Beam Equation

Author

Listed:
  • Francisco J. Marín

    (Universidad Politécnica de Cartagena)

  • Jesús Martínez-Frutos

    (Universidad Politécnica de Cartagena)

  • Francisco Periago

    (Universidad Politécnica de Cartagena)

Abstract

This paper proposes an approach for the robust averaged control of random vibrations for the Bernoulli–Euler beam equation under uncertainty in the flexural stiffness and in the initial conditions. The problem is formulated in the framework of optimal control theory and provides a functional setting, which is so general as to include different types of random variables and second-order random fields as sources of uncertainty. The second-order statistical moment of the random system response at the control time is incorporated in the cost functional as a measure of robustness. The numerical resolution method combines a classical descent method with an adaptive anisotropic stochastic collocation method for the numerical approximation of the statistics of interest. The direct and adjoint stochastic systems are uncoupled, which permits to exploit parallel computing architectures to solve the set of deterministic problem that arise from the stochastic collocation method. As a result, problems with a relative large number of random variables can be solved with a reasonable computational cost. Two numerical experiments illustrate both the performance of the proposed method and the significant differences that may occur when uncertainty is incorporated in this type of control problems.

Suggested Citation

  • Francisco J. Marín & Jesús Martínez-Frutos & Francisco Periago, 2017. "Robust Averaged Control of Vibrations for the Bernoulli–Euler Beam Equation," Journal of Optimization Theory and Applications, Springer, vol. 174(2), pages 428-454, August.
  • Handle: RePEc:spr:joptap:v:174:y:2017:i:2:d:10.1007_s10957-017-1128-x
    DOI: 10.1007/s10957-017-1128-x
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    References listed on IDEAS

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    1. S. Rangavajhala & A. A. Mullur & A. Messac, 2009. "Equality Constraints in Multiobjective Robust Design Optimization: Decision Making Problem," Journal of Optimization Theory and Applications, Springer, vol. 140(2), pages 315-337, February.
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    Cited by:

    1. Carlos J. García-Cervera & Mathieu Kessler & Francisco Periago, 2023. "Control of Partial Differential Equations via Physics-Informed Neural Networks," Journal of Optimization Theory and Applications, Springer, vol. 196(2), pages 391-414, February.

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