The COVID-19  pandemic  has  highlighted  the  need  for  controlmeasures  that  reduce  the epidemic peak ("flattening the curve").
Here we derive the optimal time-limited intervention for reducing peak epidemic prevalence in the standard Susceptible-Infectious-Recovered (SIR) model. We show that alternative,  more practical interventions can perform nearly as well as the provably optimal strategy.  However, none of these strategies are robust to implementation errors:  mistiming the start of the intervention by even a single week can be enormously costly,  for realistic epidemic parameters.  Sustained control measures,  though less effcient than optimal and near-optimal time-limited interventions, can be used in combination with time-limited strategies to mitigate the
catastrophic risks of mistiming.