  Motivated by the need to simultaneously guarantee safety and stability of
safety-critical dynamical systems, we construct permissive barrier certificates
in this paper that explicitly maximize the region where the system can be
stabilized without violating safety constraints. An optimization strategy is
developed to search for the maximum volume barrier certified region of safe
stabilization. The barrier certified region, which is allowed to take any
arbitrary shape, is proved to be strictly larger than safe regions generated
with Lyapunov sublevel set based methods. The proposed approach effectively
unites a Lyapunov function with multiple barrier functions that might not be
compatible with each other. Iterative search algorithms are developed using
sum-of-squares to compute the most permissive, that is, the maximum volume,
barrier certificates. Simulation results of the iterative search algorithm
demonstrate the effectiveness of the proposed method.
