基于平方和松弛和有理向量恢复, 提出了一种符号数值混合计算方法来构造多项式 Lyapunov 函数以判定非线性混成系统的稳定性. 首先, 为Lyapunov函数预定一个给定次数的多项式模板, 则Lyapunov函数构造问题可转化为相应的带参数的多项式优化问题, 然后运用平方和松弛方法求得一个近似的数值多项式Lyapunov函数,
再应用高斯-牛顿精化和有理向量恢复将数值多项式转化为验证的有理多项式Lyapunov函数.

In this paper, we present a symbolic-numeric hybrid method, based on Sum-of-Squares (SOS) relaxation and rational vector recovery, to compute an verified Lyapunov func- tion for analyzing the stability of nonlinear hybrid systems. At first, finding Lyapunov functions of hybrid systems can be converted into the  onstrained polynomial optimization problem with parameters. SOS relaxation method is then used to compute approximate Lyapunov functions with floating point coefficients. And Gauss-Newton refinement and rational vector recovery are applied on the approximate polynomials to obtain candidates, whose coefficients are rational numbers. The existences of SOS representation is used to verify the polynomial which exactly satisfies the conditions of Lyapunov functions. In the end, several examples are given to show that our method can successfully yield Lyapunov functions with rational coefficients.