High-order numerical solutions to Bellman's equation of optimal control

In this paper we develop a numerical method to compute high-order approximate solutions to Bellman's dynamic programming equation that arises in the optimal regulation of discrete-time nonlinear control systems. The method uses a patchy technique to build Taylor polynomial approximations defined on small domains which are then patched together to create a piecewise-smooth approximation. Using the values of the computed cost function as the step-size, levels of patches are constructed such that their radial boundaries are level sets of the computed cost functions and their lateral boundaries are invariants sets of the closed-loop dynamics. To minimize the computational effort, an adaptive scheme is used to determine the number of patches on each level depending on the relative error of the computed solutions.