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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| Published in | Advances in computational mathematics Vol. 50; no. 3 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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01.06.2024
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| Abstract | 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. |
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| AbstractList | 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. |
| ArticleNumber | 36 |
| Author | Ehring, Tobias Haasdonk, Bernard |
| Author_xml | – sequence: 1 givenname: Tobias orcidid: 0009-0000-4378-1757 surname: Ehring fullname: Ehring, Tobias email: ehringts@mathematik.uni-stuttgart.de organization: Institute of Applied Analysis and Numerical Simulation, University of Stuttgart – sequence: 2 givenname: Bernard surname: Haasdonk fullname: Haasdonk, Bernard organization: Institute of Applied Analysis and Numerical Simulation, University of Stuttgart |
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| Cites_doi | 10.1007/978-3-319-78384-0 10.1515/9781400874668 10.2514/1.33117 10.1007/s10444-020-09744-8 10.48550/ARXIV.2302.13122 10.1007/s10444-022-09998-4 10.1002/nla.2463 10.4064/fm-22-1-77-108 10.1137/19M1288802 10.23967/coupled.2021.026 10.1017/CBO9780511617539 10.48550/ARXIV.2208.14120 10.1109/TSMC.1979.4310171 10.1016/j.apnum.2019.11.023 10.1016/j.automatica.2011.01.085 10.1007/BF01442644 10.2307/1912772 10.48550/ARXIV.2302.09878 10.1137/1.9781611973051 10.1137/19M1305136 10.1007/s10589-017-9910-0 10.2307/1910417 10.1137/18M1203900 10.3182/20080706-5-KR-1001.00635 10.1051/cocv/2021009 10.1070/RM9915 10.1137/17M1116635 10.1109/TCYB.2014.2314612 10.1007/s00365-022-09592-3 10.1007/s11006-005-0147-3 10.1137/130932284 10.1016/j.ifacol.2022.09.116 10.1016/j.anihpc.2019.01.001 10.2307/1967124 10.1109/LCSYS.2021.3086697 10.1007/s10915-023-02208-3 10.1137/21m1412190 10.1016/j.ifacol.2018.03.053 10.1007/978-3-030-91745-6 10.1007/978-0-85729-501-9 10.1007/978-3-0348-7964-4 10.1007/s10915-012-9648-x 10.1007/s10444-004-1829-1 10.1137/19M1262139 10.2307/2525753 |
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| References_xml | – reference: Ehring, T., Haasdonk, B.: Feedback control for a coupled soft tissue system by kernel surrogates. In: Proceedings of COUPLED 2021. Scipedia (2021). https://doi.org/10.23967/coupled.2021.026 – reference: Çimen, T.: State-dependent Riccati equation (SDRE) control: a survey. IFAC Proc. Vol. 41(2), 3761–3775 (2008). https://doi.org/10.3182/20080706-5-KR-1001.00635. 17th IFAC World Congress – reference: Bellman, R.E.: A dynamic programming. Princeton University Press (1957) – reference: KirszbraunMÜber die zusammenziehende und Lipschitzsche TransformationenFundam. Math.19342217710810.4064/fm-22-1-77-108 – reference: Khalil, H.: Nonlinear systems: Pearson New, vol. 3, International Pearson Education, Limited, Harlow (2013) – reference: Seierstad, A., Sydsaeter, K.: Sufficient conditions in optimal control theory. International Economic Review 18(2), 367. https://doi.org/10.2307/2525753 – reference: AllaAFalconeMSaluzziLAn efficient DP algorithm on a tree-structure for finite horizon optimal control problemsSIAM J. Sci. Comput.201941423842406398431110.1137/18M1203900 – reference: SchmidtAHaasdonkBReduced basis approximation of large scale parametric algebraic Riccati equationsESAIM: Control Optim. Calculus Var.20182411291513764137 – reference: FahrooFRossIMPseudospectral methods for infinite-horizon nonlinear optimal control problemsJ. Guid. Control. Dyn.200831492793610.2514/1.33117 – reference: BokanowskiOGarckeJGriebelMKlompmakerIAn adaptive sparse grid semi-Lagrangian scheme for first order Hamilton-Jacobi Bellman equationsJ. Sci. Comput.2013553575605304570410.1007/s10915-012-9648-x – reference: KangWWilcoxLCMitigating the curse of dimensionality: sparse grid characteristics method for optimal feedback control and HJB equationsComput. Optim. 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| Title | Hermite kernel surrogates for the value function of high-dimensional nonlinear optimal control problems |
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