[1] 
J. Garcke and A. Kröner.
Suboptimal Feedback Control of PDEs by Solving HJB Equations on
Adaptive Sparse Grids.
Journal of Scientific Computing, 70(1):128, 2017.
also available as INS Preprint No. 1518. [ bib  DOI  http  .pdf 1 ] 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 semidiscrete 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 HamiltonJacobi Bellman (HJB) equation which is defined over a state space whose dimension is equal to the dimension of the underlying semidiscrete system. Besides a low dimensional semidiscretization it is important to solve the HJB equation efficiently to address the curse of dimensionality. We propose to apply a semiLagrangian 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{ö}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.
