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Error Analysis for POD Approximations of Infinite Horizon Problems via the Dynamic Programming Approach

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2017

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Alla, Alessandro
Falcone, Maurizio

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https://doi.org/10.1137/15M1039596
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SIAM Journal on Control and Optimization. 2017, 55(5), pp. 3091-3115. ISSN 0036-1402. eISSN 1095-7138. Available under: doi: 10.1137/15M1039596

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In this paper infinite horizon optimal control problems for nonlinear high-dimensional dynamical systems are studied. Nonlinear feedback laws can be computed via the value function characterized as the unique viscosity solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation which stems from the dynamic programming approach. However, the bottleneck is mainly due to the curse of dimensionality and HJB equations are only solvable in a relatively small dimension. Therefore, a reduced-order model is derived for the dynamical system and for this purpose the method of proper orthogonal decomposition (POD) is used. The resulting errors in the HJB equations are estimated by an a-priori error analysis, which suggests a new sampling strategy for the POD method. Numerical experiments illustrates the theoretical findings.

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510 Mathematik

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ISO 690ALLA, Alessandro, Maurizio FALCONE, Stefan VOLKWEIN, 2017. Error Analysis for POD Approximations of Infinite Horizon Problems via the Dynamic Programming Approach. In: SIAM Journal on Control and Optimization. 2017, 55(5), pp. 3091-3115. ISSN 0036-1402. eISSN 1095-7138. Available under: doi: 10.1137/15M1039596
BibTex
@article{Alla2017Error-31958.2,
  year={2017},
  doi={10.1137/15M1039596},
  title={Error Analysis for POD Approximations of Infinite Horizon Problems via the Dynamic Programming Approach},
  number={5},
  volume={55},
  issn={0036-1402},
  journal={SIAM Journal on Control and Optimization},
  pages={3091--3115},
  author={Alla, Alessandro and Falcone, Maurizio and Volkwein, Stefan}
}
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