Solving the Hamilton-Jacobi-Bellman equation for a stochastic system with state constraints

We present a method for finding a stationary solution to the Hamilton-Jacobi-Bellman (HJB) equation for a stochastic system with state constraints. A variable transformation is introduced which turns the HJB equation into a combination of an eigenvalue problem, a set of partial differential equations (PDEs), and a point-wise equation. As a result the difficult infinite boundary conditions of the original HJB becomes homogeneous. To illustrate, we numerically solve for the optimal control of a Linear Quadratic Gaussian (LQG) system with state constraints. A reasonably accurate solution is obtained even with a very small number of collocation points (three in each dimension), which suggests that the method could be used on high order systems, mitigating the curse of dimensionality. Source code for the example is available online.