An example of solving HJB equations using sparse grid for feedback control

It is well known that solving the Hamilton-Jacobi-Bellman (HJB) equation in moderate and high dimensions (d > 3) suffers the curse of dimensionality. In this paper, we introduce and demonstrate an example of solving the 6-D HJB equation for the optimal attitude control of a rigid body equipped with two pairs of momentum wheels. The system is uncontrollable. To mitigate the curse-of-dimensionality, a computational method based on sparse grids is introduced. The method is causality free, which enjoys the advantage of perfect parallelism. The problem is solved using several hundred CPU cores in parallel. In the simulations, the solution of the HJB equation is integrated into a model predictive control for optimal attitude stabilization.