Suboptimal Feedback Control of PDEs by Solving HJB Equations on Adaptive Sparse Grids

An approach to solve nite time horizon sub-optimal feedback control problems for partial di erential equations is proposed by solving dynamic programming equations on adaptive sparse grids. The approach is illustrated for the wave equation and an extension to equations of Schrödinger type is indicated. 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 HamiltonJacobi Bellman (HJB) equation which is de ned 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 e ciently 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 e ciency an adaptive grid re nement procedure is explored. We present several numerical examples studying the e ect the parameters characterizing the sparse grid have on the accuracy of the value function and the optimal trajectory.