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 $\RR^{ d}$. 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 $\Cc^{ 2}(\RR^{ d})$ to the solution of the Hamilton--Jacobi--Bellman (HJB) equation associated with the ergodic control problem.
Categories: math.OC cs.SY math.AP