  In this work, we propose a compositional scheme for the safety controller
synthesis of interconnected discrete-time stochastic systems with Markovian
switching signals. Our proposed approach is based on a notion of so-called
control storage certificates computed for individual subsystems, by leveraging
which, one can synthesize state-feedback controllers for interconnected systems
to enforce safety specifications over finite time horizons. To do so, we employ
a sum-of-squares (SOS) optimization approach to search for multiple storage
certificates of each switching subsystem while synthesizing its corresponding
safety controller. We then utilize dissipativity theory to compositionally
construct barrier certificates for interconnected systems based on storage
certificates of individual subsystems. The proposed dissipativity-type
compositional conditions can leverage the structure of the interconnection
topology and be fulfilled independently of the number or gains of subsystems.
We eventually employ the constructed barrier certificate and quantify upper
bounds on the probability that the interconnected system reaches certain unsafe
regions in a finite time horizon. We apply our results to a room temperature
network of 200 rooms with Markovian switching signals while accepting multiple
storage certificates. We compositionally synthesize safety controllers to
maintain the temperature of each room in a comfort zone for a bounded time
horizon.
