Numerically efficient approximations to the Hamilton-Jacobi-Bellman equation

We present an implementation of the successive Galerkin approximation (SGA) algorithm to the Hamilton-Jacobi-Bellman equation that is less sensitive to Bellman´s curse of dimensionality. The SGA algorithm takes an arbitrary stabilizing control law and improves the performance of the control law. An elementary application of the SGA algorithm results in many multidimensional integrations that increase exponentially as a function of the size of the state space and polynomially as a function of the size of the Galerkin basis. The main result of this paper is the elimination of the exponentially growing number of multidimensional integrals. To do so we make several minimally restrictive assumptions about the dynamics of the system and the Galerkin basis elements. As a result we reduce the problem to the calculation of 1D integrals: the number of these integrations increases polynomially as a function of the size of the state space and linearly as a function of the size of the Galerkin basis