An adaptive max-plus eigenvector method for continuous time optimal control problems

Abstract

An adaptive max-plus eigenvector method is proposed for approximating the solution of continuous time nonlinear optimal control problems. At each step of the method, given a set of quadratic basis functions, a standard max-plus eigenvector method is applied to yield an approximation of the value function of interest. Using this approximation, an approximate level set of the back substitution error defined by the Hamiltonian is tessellated according to where each basis function is active in approximating the value function. The polytopes obtained, and their vertices, are sorted according to this back substitution error, allowing “worst-case” basis functions to be identified. The locations of these basis functions are subsequently evolved to yield new basis functions that reduce this worst-case. Basis functions that are inactive in the value function approximation are pruned, and the aforementioned steps repeated. Underlying algebraic properties associated with max-plus linearity, dynamic programming, and semiconvex duality are provided as a foundation for the development, and the utility of the proposed method is illustrated by example.