  This work considers the infinite-time discounted optimal control problem for
continuous time input-affine polynomial dynamical systems subject to polynomial
state and box input constraints. We propose a sequence of sum-of-squares (SOS)
approximations of this problem obtained by first lifting the original problem
into the space of measures with continuous densities and then restricting these
densities to polynomials. These approximations are tightenings, rather than
relaxations, of the original problem and provide a sequence of rational
controllers with value functions associated to these controllers converging
(under some technical assumptions) to the value function of the original
problem. In addition, we describe a method to obtain polynomial approximations
from above and from below to the value function of the extracted rational
controllers, and a method to obtain approximations from below to the optimal
value function of the original problem, thereby obtaining a sequence of
asymptotically optimal rational controllers with explicit estimates of
suboptimality. Numerical examples demonstrate the approach.
