Polynomial approximation of high-dimensional Hamilton–Jacobi–Bellman equations and applications to feedback control of semilinear parabolic PDES

Kalise, Dante; Kunisch, Karl

Abstract

© 2018 Society for Industrial and Applied Mathematics. A procedure for the numerical approximation of high-dimensional Hamilton–Jacobi–Bellman (HJB) equations associated to optimal feedback control problems for semilinear parabolic equations is proposed. Its main ingredients are a pseudospectral collocation approximation of the PDE dynamics and an iterative method for the nonlinear HJB equation associated to the feedback synthesis. The latter is known as the successive Galerkin approximation. It can also be interpreted as Newton iteration for the HJB equation. At every step, the associated linear generalized HJB equation is approximated via a separable polynomial approximation ansatz. Stabilizing feedback controls are obtained from solutions to the HJB equations for systems of dimension up to fourteen.

Citation

Kalise, D., & Kunisch, K. (2018). Polynomial approximation of high-dimensional Hamilton–Jacobi–Bellman equations and applications to feedback control of semilinear parabolic PDES. SIAM Journal on Scientific Computing, 40(2), A629-A652. https://doi.org/10.1137/17M1116635

• Journal Article Type: Article
• Acceptance Date: Oct 26, 2017
• Online Publication Date: Mar 1, 2018
• Publication Date: Jan 1, 2018
• Deposit Date: Nov 12, 2019
• Publicly Available Date: Jan 16, 2020
• Journal: SIAM Journal on Scientific Computing
• Print ISSN: 1064-8275
• Electronic ISSN: 1095-7197
• Publisher: Society for Industrial and Applied Mathematics
• Peer Reviewed: Peer Reviewed
• Volume: 40
• Issue: 2
• Pages: A629-A652
• DOI: https://doi.org/10.1137/17M1116635
• Keywords: Applied Mathematics; Computational Mathematics
• Public URL: https://nottingham-repository.worktribe.com/output/3219310
• Publisher URL: https://epubs.siam.org/doi/10.1137/17M1116635