Function approximation for the deterministic Hamilton-Jacobi-Bellman equation

Based on Gaussian basis functions, a new method for calculating the Hamilton-Jacobi-Bellman equation for deterministic continuous-time and continuous-valued optimal control problems is proposed. A semi-Lagrangian discretization scheme is used to obtain a discrete-time finite-state approximation of the continuous dynamics. The value function of the discretized system is approximated by a Gaussian network. Limit behavior analysis provides a proof of convergence for the scheme. The performance of the presented approach is demonstrated for an underpowered inverted pendulum as numerical example. Furthermore, a comparison to the approximation by continuous piecewise affine functions (the current state of the art) shows the benefits of the approximation technique proposed here.