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Numerical Resolution of Robust Optimal Control Problems

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Optimal Control of PDEs under Uncertainty

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Abstract

Both gradient-based methods and methods based on the resolution of first-order optimality conditions may be used for solving numerically the robust optimal control problems presented in the preceding chapters. In both cases, the main difficulty arises in the numerical approximation of statistical quantities of interest associated to solutions of random PDEs. To handle this issue, it is assumed that the random inputs of the underlying PDEs depend on a finite number of random variables.

... a computational solution ... is quite definitely not routine when the number of variables is large. All this may be subsumed under the heading “the curse of dimensionality.” Since this is a curse which has hung over the head of the physicist and astronomer for many a year, there is no need to feel discouraged about the possibility of obtaining significant results despite it.

Richard Bellman.

Dynamic Programming, 1957.

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Notes

  1. 1.

    The sparse grid quadrature rule presented in this chapter was proposed by Smolyak [18]. The main idea underlying this construction is described in Sect. 4.5.

  2. 2.

    For instance, when a(x, z) is expressed as a truncated KL expansion of a log-normal randon field (see [2, Example 3]).

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Authors and Affiliations

  1. Computational Mechanics and Scientific Computing Group, Department of Structures and Construction, Technical University of Cartagena, Cartagena, Murcia, Spain

    Jesús Martínez-Frutos

  2. Department of Applied Mathematics and Statistics, Technical University of Cartagena, Cartagena, Murcia, Spain

    Francisco Periago Esparza

Authors
  1. Jesús Martínez-Frutos
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  2. Francisco Periago Esparza
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Correspondence to Francisco Periago Esparza .

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Martínez-Frutos, J., Periago Esparza, F. (2018). Numerical Resolution of Robust Optimal Control Problems. In: Optimal Control of PDEs under Uncertainty. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-98210-6_4

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