Optimal feedback control for undamped wave equations by solving a HJB equation

Axel Kröner; Karl Kunisch; Hasnaa Zidani

doi:10.1051/cocv/2014033

Optimal feedback control for undamped wave equations by solving a HJB equation

Axel Kröner
, Karl Kunisch
, Hasnaa Zidani

Austrian Academy of Sciences
Institute of Mathematics and Scientific Computing
University of Graz
INRIA

Research output: Contribution to journal › Article › peer-review

7 Citations (Scopus)

Abstract

An optimal finite-time horizon feedback control problem for (semi-linear) wave equations is presented. The feedback law can be derived from the dynamic programming principle and requires to solve the evolutionary Hamilton?Jacobi Bellman (HJB) equation. Classical discretization methods based on finite elements lead to approximated problems governed by ODEs in high dimensional spaces which makes the numerical resolution by the HJB approach infeasible. In the present paper, an approximation based on spectral elements is used to discretize the wave equation. The effect of noise is considered and numerical simulations are presented to show the relevance of the approach.

Original language	English
Pages (from-to)	442-464
Number of pages	23
Journal	ESAIM - Control, Optimisation and Calculus of Variations
Volume	21
Issue number	2
DOIs	https://doi.org/10.1051/cocv/2014033
Publication status	Published - 1 Apr 2015
Externally published	Yes

Keywords

Hamilton-Jacobi Bellman equation
Optimal control
Spectral elements
Wave equation

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10.1051/cocv/2014033

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abstract = "An optimal finite-time horizon feedback control problem for (semi-linear) wave equations is presented. The feedback law can be derived from the dynamic programming principle and requires to solve the evolutionary Hamilton?Jacobi Bellman (HJB) equation. Classical discretization methods based on finite elements lead to approximated problems governed by ODEs in high dimensional spaces which makes the numerical resolution by the HJB approach infeasible. In the present paper, an approximation based on spectral elements is used to discretize the wave equation. The effect of noise is considered and numerical simulations are presented to show the relevance of the approach.",

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AU - Kröner, Axel

AU - Kunisch, Karl

AU - Zidani, Hasnaa

N1 - Publisher Copyright: © EDP Sciences, SMAI 2015.

PY - 2015/4/1

Y1 - 2015/4/1

N2 - An optimal finite-time horizon feedback control problem for (semi-linear) wave equations is presented. The feedback law can be derived from the dynamic programming principle and requires to solve the evolutionary Hamilton?Jacobi Bellman (HJB) equation. Classical discretization methods based on finite elements lead to approximated problems governed by ODEs in high dimensional spaces which makes the numerical resolution by the HJB approach infeasible. In the present paper, an approximation based on spectral elements is used to discretize the wave equation. The effect of noise is considered and numerical simulations are presented to show the relevance of the approach.

AB - An optimal finite-time horizon feedback control problem for (semi-linear) wave equations is presented. The feedback law can be derived from the dynamic programming principle and requires to solve the evolutionary Hamilton?Jacobi Bellman (HJB) equation. Classical discretization methods based on finite elements lead to approximated problems governed by ODEs in high dimensional spaces which makes the numerical resolution by the HJB approach infeasible. In the present paper, an approximation based on spectral elements is used to discretize the wave equation. The effect of noise is considered and numerical simulations are presented to show the relevance of the approach.

KW - Hamilton-Jacobi Bellman equation

KW - Optimal control

KW - Spectral elements

KW - Wave equation

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JO - ESAIM - Control, Optimisation and Calculus of Variations

JF - ESAIM - Control, Optimisation and Calculus of Variations

IS - 2

ER -

Optimal feedback control for undamped wave equations by solving a HJB equation

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Keywords

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