Safety Verification of Nonlinear Hybrid Systems Based on Bilinear Programming

In safety verification of hybrid systems, barrier certificates are generated by solving the verification conditions derived from non-negative representations of different types. This paper presents a new computational method, sequential linear programming projection, for directly solving the set of verification conditions represented by the Krivine-Vasilescu-Handelman's positivstellensatz. The key idea is to decompose it into two successive optimization problems that refine the desired barrier certificate and those undetermined multipliers, respectively, and solve it in an iterative scheme. The most important benefit of the proposed approach lies in that it is much more effective than the LP relaxation method in producing real barrier certificates, and possesses a much lower computational complexity than the popular sum of square relaxation methods, which is demonstrated by the theoretical analysis on complexity and the experiment on a set of examples gathered from the literature.