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

Mitigating the Curse of Dimensionality: Sparse Grid Characteristics Method for Optimal Feedback Control and HJB Equations