Hermite kernel surrogates for the value function of high-dimensional nonlinear optimal control problems

Numerical methods for the optimal feedback control of high-dimensional dynamical systems typically suffer from the curse of dimensionality. In the current presentation, we devise a mesh-free data-based approximation method for the value function of optimal control problems, which partially mitigates...

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Vydané v:	Advances in computational mathematics Ročník 50; číslo 3; s. 36
Hlavní autori:	Ehring, Tobias, Haasdonk, Bernard
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.06.2024 Springer Nature B.V
Predmet:	Approximation Cholesky factorization Computational mathematics Computational Mathematics and Numerical Analysis Computational Science and Engineering Dynamical systems Energy consumption Feedback control Interpolation Mathematical and Computational Biology Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics MORe 2022 Neural networks Nonlinear control Numerical methods Optimal control Ordinary differential equations Partial differential equations Visualization Kernel methods 93C10 Hermite interpolation 93C15 Optimal feedback control 30C40 Surrogate modeling 49N35 93B52
ISSN:	1019-7168, 1572-9044
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Shrnutí:	Numerical methods for the optimal feedback control of high-dimensional dynamical systems typically suffer from the curse of dimensionality. In the current presentation, we devise a mesh-free data-based approximation method for the value function of optimal control problems, which partially mitigates the dimensionality problem. The method is based on a greedy Hermite kernel interpolation scheme and incorporates context knowledge by its structure. Especially, the value function surrogate is elegantly enforced to be 0 in the target state, non-negative and constructed as a correction of a linearized model. The algorithm allows formulation in a matrix-free way which ensures efficient offline and online evaluation of the surrogate, circumventing the large-matrix problem for multivariate Hermite interpolation. Additionally, an incremental Cholesky factorization is utilized in the offline generation of the surrogate. For finite time horizons, both convergence of the surrogate to the value function and for the surrogate vs. the optimal controlled dynamical system are proven. Experiments support the effectiveness of the scheme, using among others a new academic model with an explicitly given value function. It may also be useful for the community to validate other optimal control approaches.
Bibliografia:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ISSN:	1019-7168 1572-9044
DOI:	10.1007/s10444-024-10128-5