Safety Verification of Hybrid Systems Using Certified Multiple Lyapunov-Like Functions

Abstract

In this paper, we present an efficient, hybrid numeric-symbolic method for safety verification of hybrid systems. To start with, we introduce a function of state, defined as a multiple Lyapunov-like function, whose time derivative along the trajectories is non-negative only outside of the initial set, such that its zero level set separates the unsafe region from all possible trajectories starting from the given initial set. Then, a numerical multiple Lyapunov-like function is computed by using sum of squares decomposition and semi-definite programming. Afterwards, in order to recover the possible unreliability of our numerical solution, we apply a continued fractions based rational recovery technique to this floating result and then obtain a certified one with rational coefficients, such that exact verification can be assured by this certified multiple Lyapunov-like function. Finally, several examples, together with discussions, are provided to illustrate the tractability and advantages of our method.