Stabilization of a quasi-linear parabolic Cauchy problem associated with ergodic control of diffusions

Abstract

We study the relative value iteration for the ergodic control problem under a near-monotone running cost structure for a nondegenerate diffusion controlled through its drift. This algorithm takes the form of a quasilinear parabolic Cauchy initial value problem in ℝd. We show that this Cauchy problem stabilizes, or in other words, that the solution of the quasilinear parabolic equation converges for every bounded initial condition in C2(ℝd) to the solution of the Hamilton-Jacobi-Bellman (HJB) equation associated with the ergodic control problem.