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首页> 外文会议>45th Annual Allerton Conference on Communication, Control, and Computing 2007 >Exponentially Stable Nonlinear Systems have Polynomial Lyapunov Functions on Bounded Regions
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Exponentially Stable Nonlinear Systems have Polynomial Lyapunov Functions on Bounded Regions

机译：指数稳定非线性系统在有界区域上具有多项式Lyapunov函数

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摘要

This paper presents a proof that existence of a polynomial Lyapunov function is necessary and sufficient for exponential stability of sufficiently smooth nonlinear ordinary differential equations on bounded sets. The main result implies that if there exists an n-times continuously differentiable Lyapunov function which proves exponential stability on a bounded subset of R~n, then there exists a polynomial Lyapunov function which proves exponential stability on the same region. Such a continuous Lyapunov function will exist if, for example, the right-hand side of the differential equation is polynomial or at least n-times continuously differentiable. Our investigation is motivated by the use of polynomial optimization algorithms to construct polynomial Lyapunov functions for systems of nonlinear ordinary differential equations.
机译：本文提出了证明，对于有界集上足够光滑的非线性常微分方程的指数稳定性，必须存在多项式Lyapunov函数。主要结果表明，如果存在一个n次连续可微分的Lyapunov函数，在R〜n的一个有界子集上证明了指数稳定性，那么就存在一个多项式Lyapunov函数，该函数证明了在同一区域上的指数稳定性。例如，如果微分方程的右侧是多项式或至少n倍可连续微分，则将存在这种连续Lyapunov函数。我们的研究是基于使用多项式优化算法为非线性常微分方程组构造多项式Lyapunov函数的。

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