  We consider a driven-dissipative system consisting of an atomic Bose-Einstein
condensates loaded into a two-dimensional Hubbard lattice and coupled to a
single mode of an optical cavity. Due to the interplay between strong,
repulsive atomic interaction and the atom-cavity coupling, the system exhibits
several phases of atoms and photons including the atomic superfluid (SF) and
supersolid (SS). We investigate the dynamical behaviour of the system, where we
include dissipation by means of Lindblad master equation formalism. Due to the
discontinuous nature of the Dicke transition for strong atomic repulsion, we
find extended co-existence region of different phases. We investigate the
resulting switching dynamics, particularly between the coexisting SF and SS
phases, which eventually becomes damped by the dissipation.
