  We address finding the semi-global solutions to optimal feedback control and
the Hamilton--Jacobi--Bellman (HJB) equation. Using the solution of an HJB
equation, a feedback optimal control law can be implemented in real-time with
minimum computational load. However, except for systems with two or three state
variables, using traditional techniques for numerically finding a semi-global
solution to an HJB equation for general nonlinear systems is infeasible due to
the curse of dimensionality. Here we present a new computational method for
finding feedback optimal control and solving HJB equations which is able to
mitigate the curse of dimensionality. We do not discretize the HJB equation
directly, instead we introduce a sparse grid in the state space and use the
Pontryagin's maximum principle to derive a set of necessary conditions in the
form of a boundary value problem, also known as the characteristic equations,
for each grid point. Using this approach, the method is spatially causality
free, which enjoys the advantage of perfect parallelism on a sparse grid.
Compared with dense grids, a sparse grid has a significantly reduced size which
is feasible for systems with relatively high dimensions, such as the $6$-D
system shown in the examples. Once the solution obtained at each grid point,
high-order accurate polynomial interpolation is used to approximate the
feedback control at arbitrary points. We prove an upper bound for the
approximation error and approximate it numerically. This sparse grid
characteristics method is demonstrated with two examples of rigid body attitude
control using momentum wheels.
