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

We study the convergence rate of the moment-sum-of-squares hierarchy of semidefinite programs for optimal control problems with polynomial data. It is known that this hierarchy generates 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∕loglogd). 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.