  We study the convergence rate of moment-sum-of-squares hierarchies of
semidefinite programs for optimal control problems with polynomial data. It is
known that these hierarchies generate polynomial under-approximations to the
value function of the optimal control problem and that these
under-approximations converge in the L1 norm to the value function as their
degree d tends to infinity. We show that the rate of this convergence is O(1/
log log d). We treat in detail the continuous-time infinite-horizon discounted
problem and describe in brief how the same rate can be obtained for the
finite-horizon continuous-time problem and for the discrete-time counterparts
of both problems.
