On application of an alternating direction method to Hamilton–Jacobin–Bellman equations

This paper presents a numerical method for the approximation of viscosity solutions to a Hamilton–Jacobi–Bellman (HJB) equation governing a class of optimal feedback control problems. The first-order HJB equation is first perturbed by adding a diffusion term with a singular perturbation parameter. The time and spatial variables in the resulting equation are then discretized respectively by an implicit modified method of characteristics and the alternating direction (AD) scheme. We show that the AD procedureʹs perturbation error is virtually negligible due to the small perturbation parameter. And the efficient AD scheme can be applied to our HJB equation without generating splitting error. Numerical results, performed to verify the usefulness of the method, show that the method gives accurate approximate solutions to both of the control and the state variables.