一个具有度界的平方和逆李雅普诺夫结果
A Converse Sum of Squares Lyapunov Result with a Degree Bound
摘要：
在过去的十年中，平方和规划已被广泛用于非线性系统的稳定性分析，但仍有一些问题没有得到解决。本文证明了有界集上多项式向量场的指数稳定性蕴含着多项式平方和的李雅普诺夫函数的存在。特别地，主要结果表明，如果系统在有界非空集上是指数稳定的，则存在SOS李雅普诺夫函数在该有界集上指数下降。证明是建设性的，并使用Picard迭代。还给出了该逆李雅普诺夫函数的次数的一个界。这一结果意味着半定规划可以用来回答多项式向量场的稳定性问题。
英文摘要：
Sum of Squares programming has been used extensively over the past decade for the stability analysis of nonlinear systems but several questions remain unanswered. In this paper, we show that exponential stability of a polynomial vector field on a bounded set implies the existence of a Lyapunov function which is a sum-of-squares of polynomials. In particular, the main result states that if a system is exponentially stable on a bounded nonempty set, then there exists an SOS Lyapunov function which is exponentially decreasing on that bounded set. The proof is constructive and uses the Picard iteration. A bound on the degree of this converse Lyapunov function is also given. This result implies that semidefinite programming can be used to answer the question of stability of a polynomial vector field with a bound on complexity.
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