  Motivated by recent experimental observations (C.V. Parker {\it et al.},
Nature Physics, {\bf 9}, 769 (2013)), we analyze the stability of a
Bose-Einstein condensate (BEC) in a one-dimensional lattice subjected to
periodic shaking. In such a system there is no thermodynamic ground state, but
there may be a long-lived steady-state, described as an eigenstate of a
"Floquet Hamiltonian". We calculate how scattering processes lead to a decay of
the Floquet state. We map out the phase diagram of the system and find regions
where the BEC is stable and regions where the BEC is unstable against atomic
collisions. We show that Parker et al. perform their experiment in the stable
region, which accounts for the long life-time of the condensate ($\sim$ 1
second). We also estimate the scattering rate of the bosons in the region where
the BEC is unstable.
