  Squeezing is a non-classical feature of quantum states that is a useful
resource, for example in quantum sensing of mechanical forces. Here, we show
how to use optimal control theory to maximize squeezing in an optomechanical
setup with two external drives and determine how fast the mechanical mode can
be squeezed. For the autonomous drives considered here, we find the inverse
cavity decay to lower-bound the protocol duration. At and above this limit, we
identify a family of protocols leveraging a two-stage control strategy, where
the mechanical mode is cooled before it is squeezed. Identification of the
control strategy allows for two important insights - to determine the factors
that limit squeezing and to simplify the time-dependence of the external
drives, making our protocol readily applicable in experiments.
