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.
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Acknowledgements
Work supported by Projects MTM2013-47053 and DPI2016-77538-R from Ministerio de Economía y Competitividad (Spain) and 19274/PI/14 from Fundación Séneca (Agencia de Ciencia y Tecnología de la Región de Murcia (Spain)). The first author was also supported by a Ph.D. Grant from Fundación Séneca.
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Communicated by Enrique Zuazua.
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Marín, F.J., Martínez-Frutos, J. & Periago, F. Robust Averaged Control of Vibrations for the Bernoulli–Euler Beam Equation. J Optim Theory Appl 174, 428–454 (2017). https://doi.org/10.1007/s10957-017-1128-x
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DOI: https://doi.org/10.1007/s10957-017-1128-x
Keywords
- Bernoulli–Euler beam
- Robust averaged optimal control
- Adaptive anisotropic sparse grid
- Stochastic finite elements