@Article{ Garcke.Kroener:2017,
abstract = {An approach to solve finite time horizon suboptimal
feedback control problems for partial differential
equations is proposed by solving dynamic programming
equations on adaptive sparse grids. A semi-discrete optimal
control problem is introduced and the feedback control is
derived from the corresponding value function. The value
function can be characterized as the solution of an
evolutionary Hamilton--Jacobi Bellman (HJB) equation which
is defined over a state space whose dimension is equal to
the dimension of the underlying semi-discrete system.
Besides a low dimensional semi-discretization it is
important to solve the HJB equation efficiently to address
the curse of dimensionality. We propose to apply a
semi-Lagrangian scheme using spatially adaptive sparse
grids. Sparse grids allow the discretization of the value
functions in (higher) space dimensions since the curse of
dimensionality of full grid methods arises to a much
smaller extent. For additional efficiency an adaptive grid
refinement procedure is explored. The approach is
illustrated for the wave equation and an extension to
equations of Schr{\{}{\"{o}}{\}}dinger type is indicated.
We present several numerical examples studying the effect
the parameters characterizing the sparse grid have on the
accuracy of the value function and the optimal trajectory.},
author = {Garcke, Jochen and Kr{\"{o}}ner, Axel},
doi = {10.1007/s10915-016-0240-7},
issn = {1573-7691},
note = {also available as INS Preprint No. 1518},
annote = {journal},
inspreprintnum= {1518},
pdf = {http://garcke.ins.uni-bonn.de/research/pub/GarckeKroener.pdf}
,
journal = {Journal of Scientific Computing},
number = {1},
pages = {1--28},
title = {{Suboptimal Feedback Control of PDEs by Solving HJB
Equations on Adaptive Sparse Grids}},
volume = {70},
http = {http://rdcu.be/tGmo},
year = {2017}
}