## Multilevel techniques for the solution of HJB minimum-time control problems

2018
##### Publication type
Working Paper/Technical Report
Submitted
##### Abstract
The approximation of feedback control via the Dynamic Programming approach is a challenging problem. The computation of the feedback requires the knowledge of the value function, which can be characterized as the unique viscosity solution of a nonlinear Hamilton-Jacobi-Bellman (HJB) equation. The major obstacle is that the numerical methods known in literature strongly suffer when the dimension of the discretized problem becomes large. This is a strong limitation to the application of classical numerical schemes for the solution of the HJB equation in real applications. To tackle this problem, a new multi-level numerical framework is proposed. Numerical evidences show that classical methods have good smoothing properties, which allow one to use them as smoothers in a multilevel strategy. Moreover, a new smoother iterative scheme based on the Anderson acceleration of the classical value function iteration is introduced. The effectiveness of our new framework is proved by several numerical experiments focusing on minimum-time control problems.
510 Mathematics
##### Keywords
Hamilton-Jacobi equation, minimum-time problem, value iteration, policy iteration, Anderson acceleration, multilevel acceleration methods, FAS
##### Cite This
ISO 690CIARAMELLA, Gabriele, Giulia FABRINI, 2018. Multilevel techniques for the solution of HJB minimum-time control problems
BibTex
@techreport{Ciaramella2018Multi-44827,
year={2018},
title={Multilevel techniques for the solution of HJB minimum-time control problems},
author={Ciaramella, Gabriele and Fabrini, Giulia}
}

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<dcterms:abstract xml:lang="eng">The approximation of feedback control via the Dynamic Programming approach is a challenging problem. The computation of the feedback requires the knowledge of the value function, which can be characterized as the unique viscosity solution of a nonlinear Hamilton-Jacobi-Bellman (HJB) equation. The major obstacle is that the numerical methods known in literature strongly suffer when the dimension of the discretized problem becomes large. This is a strong limitation to the application of classical numerical schemes for the solution of the HJB equation in real applications. To tackle this problem, a new multi-level numerical framework is proposed. Numerical evidences show that classical methods have good smoothing properties, which allow one to use them as smoothers in a multilevel strategy. Moreover, a new smoother iterative scheme based on the Anderson acceleration of the classical value function iteration is introduced. The effectiveness of our new framework is proved by several numerical experiments focusing on minimum-time control problems.</dcterms:abstract>
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Yes