  In this work, we propose a compositional framework for the construction of
control barrier functions for networks of continuous-time stochastic hybrid
systems enforcing complex logic specifications expressed by finite-state
automata. The proposed scheme is based on a notion of so-called pseudo-barrier
functions computed for subsystems, by employing which one can synthesize hybrid
controllers for interconnected systems enforcing complex specifications over a
finite-time horizon. Particularly, we first leverage sufficient small-gain type
conditions to compositionally construct control barrier functions for
interconnected systems based on the corresponding pseudo-barrier functions
computed for subsystems. Then, using the constructed control barrier functions,
we provide probabilistic guarantees on the satisfaction of given complex
specifications in a bounded time horizon. In this respect, we decompose the
given complex specification to simpler reachability tasks based on automata
representing the complements of original finite-state automata. We then provide
systematic approaches to solve those simpler reachability tasks by computing
corresponding pseudo-barrier functions. Two different systematic techniques are
provided based on (i) the sum-of-squares (SOS) optimization program and (ii)
counter-example guided inductive synthesis (CEGIS) to search for pseudo-barrier
functions of subsystems while synthesizing local controllers. We demonstrate
the effectiveness of our proposed results by applying them to a
fully-interconnected Kuramoto network of 100 nonlinear oscillators with
Markovian switching signals.
