We prove polynomial energy decay for polynomially controlled singular damping
on the torus. This decay rate extends a result for bounded damping. We
construct a semigroup that provides energy decay information in the singular
case and use it to reduce the problem to resolvent estimates for the stationary
damped wave equation. We then prove sufficiently good resolvent estimates using
a version of the Morawetz multiplier method. We also establish exponential
energy decay for such dampings on the circle, demonstrating that overdamping
does not occur.