Permissive Barrier Certificates for Safe Stabilization Using Sum-of-squares

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.