  We study the exploratory Hamilton--Jacobi--Bellman (HJB) equation arising
from the entropy-regularized exploratory control problem, which was formulated
by Wang, Zariphopoulou and Zhou (J. Mach. Learn. Res., 21, 2020) in the context
of reinforcement learning in continuous time and space. We establish the
well-posedness and regularity of the viscosity solution to the equation, as
well as the convergence of the exploratory control problem to the classical
stochastic control problem when the level of exploration decays to zero. We
then apply the general results to the exploratory temperature control problem,
which was introduced by Gao, Xu and Zhou (arXiv:2005.04057, 2020) to design an
endogenous temperature schedule for simulated annealing (SA) in the context of
non-convex optimization. We derive an explicit rate of convergence for this
problem as exploration diminishes to zero, and find that the steady state of
the optimally controlled process exists, which is however neither a Dirac mass
on the global optimum nor a Gibbs measure.
