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We study the convergence rate of moment-sum-of-squares hierarchies ofsemidefinite programs for optimal control problems with polynomial data. It isknown that these hierarchies generate polynomial under-approximations to thevalue function of the optimal control problem and that theseunder-approximations converge in the L1 norm to the value function as theirdegree 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 discountedproblem and describe in brief how the same rate can be obtained for thefinite-horizon continuous-time problem and for the discrete-time counterpartsof both problems.

Convergence rates of moment-sum-of-squares hierarchies for optimal control problems