  We present a novel method to compute unstable periodic orbits (UPOs) that
optimize the infinite-time average of a given quantity for polynomial ODE
systems. The UPO search procedure relies on polynomial optimization to
construct nonnegative polynomials whose sublevel sets approximately localize
parts of the optimal UPO, and that can be used to implement a simple yet
effective control strategy to reduce the UPO's instability. Precisely, we
construct a family of controlled ODE systems parameterized by a scalar $k$ such
that the original ODE system is recovered for $k = 0$, and such that the
optimal orbit is less unstable, or even stabilized, for $k>0$. Periodic orbits
for the controlled system can be more easily converged with traditional methods
and numerical continuation in $k$ allows one to recover optimal UPOs for the
original system. The effectiveness of this approach is illustrated on three
low-dimensional ODE systems with chaotic dynamics.
