  This paper is concerned with a compositional scheme for the construction of
control barrier certificates for interconnected discrete-time stochastic
systems. The main objective is to synthesize switching control policies against
$\omega$-regular properties that can be described by accepting languages of
deterministic Streett automata (DSA) along with providing probabilistic
guarantees for the satisfaction of such specifications. The proposed framework
leverages the interconnection topology and a notion of so-called \emph{control
sub-barrier certificates} of subsystems, which are used to compositionally
construct control barrier certificates of interconnected systems by imposing
some dissipativity-type compositionality conditions. We propose a systematic
approach to decompose high-level $\omega$-regular specifications into simpler
tasks by utilizing the automata corresponding to the specifications. In
addition, we formulate an alternating direction method of multipliers (ADMM)
optimization problem in order to obtain suitable control sub-barrier
certificates of subsystems while satisfying compositionality conditions. For
systems with polynomial dynamics, we provide a sum-of-squares (SOS)
optimization problem for the computation of control sub-barrier certificates
and local control policies of subsystems. Finally, we demonstrate the
effectiveness of our proposed approaches by applying them to a physical case
study.
