High-order numerical solutions to Bellman's equation of optimal control
Aguilar, Cesar O.
Krener, Arthur J.
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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.
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