  For a general entropy-regularized stochastic control problem on an infinite
horizon, we prove that a policy iteration algorithm (PIA) converges to an
optimal relaxed control. Contrary to the standard stochastic control
literature, classical H\"{o}lder estimates of value functions do not ensure the
convergence of the PIA, due to the added entropy-regularizing term. To
circumvent this, we carry out a delicate estimation by moving back and forth
between appropriate H\"{o}lder and Sobolev spaces. This requires new Sobolev
estimates designed specifically for the purpose of policy iteration and a
nontrivial technique to contain the entropy growth. Ultimately, we obtain a
uniform H\"{o}lder bound for the sequence of value functions generated by the
PIA, thereby achieving the desired convergence result. Characterization of the
optimal value function as the unique solution to an exploratory
Hamilton-Jacobi-Bellman equation comes as a by-product. The PIA is numerically
implemented in an example of optimal consumption.
