指数稳定的非线性系统在有界区域上具有多项式李雅普诺夫函数
Exponentially Stable Nonlinear Systems have Polynomial Lyapunov Functions on Bounded Regions
摘要：
证明了在有界集上充分光滑的非线性常微分方程指数稳定的充分必要条件是存在多项式李雅普诺夫函数.主要结果是：如果存在一个n次连续可微的李雅普诺夫函数证明在R^n的一个有界子集上指数稳定，则存在一个多项式李雅普诺夫函数证明在同一区域上指数稳定.例如，如果微分方程的右侧是多项式或至少n次连续可微，则这种连续李雅普诺夫函数将存在。证明是基于推广的维尔斯特拉斯逼近定理的可微函数在几个变量。具体地说，我们展示了如何使用多项式来逼近Sobolev范数W^{1，\infty }中的可微函数以达到任何期望的精度。我们将这个近似结果与二阶泰勒级数展开相结合，发现多项式李雅普诺夫函数可以在有界集上任意好地逼近连续李雅普诺夫函数。我们的调查是出于使用多项式优化算法来构造多项式李雅普诺夫函数。
英文摘要：
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 states 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. The proof is based on a generalization of the Weierstrass approximation theorem to differentiable functions in several variables. Specifically, we show how to use polynomials to approximate a differentiable function in the Sobolev norm W^{1,\infty} to any desired accuracy. We combine this approximation result with the second-order Taylor series expansion to find that polynomial Lyapunov functions can approximate continuous Lyapunov functions arbitrarily well on bounded sets. Our investigation is motivated by the use of polynomial optimization algorithms to construct polynomial Lyapunov functions.
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