Safety Verification of Discrete-Time Systems via Interpolation-Inspired Barrier Certificates

Barrier certificates provide an effective automated approach to verifying the safety of dynamical systems. A barrier certificate is a real-valued function over states of the system whose zero level set separates the unsafe region from all possible trajectories starting from a given set of initial states. Typically, the system dynamics must be nonincreasing in the value of the barrier certificate with each transition. Thus, the states of the system that are nonpositive with respect to the barrier certificate act as an over-approximation of the reachable states. The search for such certificates is typically automated by first fixing a template of functions and then using optimization and satisfiability modulo theory (SMT) solvers to find them. Unfortunately, it may not be possible to find a single function in this fixed template. To tackle this challenge, we propose the notion of interpolation-inspired barrier certificate. Instead of a single function, an interpolation-inspired barrier certificate consists of a set of functions such that the union of their sublevel sets over-approximate the reachable set of states. We show how one may find interpolation-inspired barrier certificates of a fixed template, even when we fail to find standard barrier certificates of the same template. We present sum-of-squares (SOS) programming as a computational method to find this set of functions and demonstrate effectiveness of this method over a case study.