  Control Barrier Functions (CBF) have provided a very versatile framework for
the synthesis of safe control architectures for a wide class of nonlinear
dynamical systems. Typically, CBF-based synthesis approaches apply to systems
that exhibit nonlinear -- but smooth -- relationship in the state of the system
and linear relationship in the control input. In contrast, the problem of safe
control synthesis using CBF for hybrid dynamical systems, i.e., systems which
have a discontinuous relationship in the system state, remains largely
unexplored. In this work, we build upon the progress on CBF-based control to
formulate a theory for safe control synthesis for hybrid dynamical systems.
Under the assumption that local CBFs can be synthesized for each mode of
operation of the hybrid system, we show how to construct CBF that can guarantee
safe switching between modes. The end result is a switching CBF-based
controller which provides global safety guarantees. The effectiveness of our
proposed approach is demonstrated on two simulation studies.
