We consider a mobile connected to a base station, and study how to optimally schedule shutting off its transceiver. First, we study the model from optimal control perspective. We consider off-times (periods of inactivity) of (controlled) duration. We study the question of scheduling "waking up" instants in which the mobile communicates with the base station and checks whether the inactivity period is over. There is a cost proportional to the delay from the moment the off-time ends until the mobile discovers it, a (small) running cost while the mobile is sleeping and a cost for waking up. We present conditions for optimal sleep periods to be constant and derive the optimal period. For the case that the conditions do not hold, we obtain suboptimal solutions which perform strictly better than the optimal constant one. We then investigate optimality restricted to classes of policies with specific constraints. We adopt the parametric optimization approach which entails cost minimization for a given parameterized policy and selection of the best policy among a class. We then compare the performance of optimal policies, of the proposed suboptimal policies as well as that of standard policies like IEEE 802.16e.