Solving the Hamilton-Jacobi-Bellman Equation for a Stochastic System with State Constraints
We present a method for solving 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 a linear eigenvalue problem, a set of partial differential
equations (PDE:s), and a point-wise equation. For a fixed solution to the eigenvalue problem, the PDE:s are linear and the point-wise equation is quadratic, indicating that the problem can be solved efficiently using an iterative scheme. As an example, 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.