论文标题
使用矢量场分解来构建Lyapunov功能
Construction of Lyapunov Functions Using Vector Field Decomposition
论文作者
论文摘要
在本文中，提出了一种基于新型矢量场分解方法来构建Lyapunov函数。对于给定的动力系统，如果定义矢量场将分解为两个相互正交的矢量场，其中一个是无卷曲的，另一个是无差异的，那么前者的潜在功能可以用作Lyapunov函数候选者，因为它的正定性将反映系统的稳定性。此外，在某些其他条件下，其巨型集合将提供系统的确切吸引力域。首先，通过求解部分微分方程，然后将其推广到$ n $维系统，以$ 2 $维系统获得了提议的向量场分解的足够条件。此外，提出的矢量场分解始终存在于线性系统中，并且可以通过求解特定的代数riccati方程来获得。
In the present paper, a novel vector field decomposition based approach for constructing Lyapunov functions is proposed. For a given dynamical system, if the defining vector field admits a decomposition into two mutually orthogonal vector fields, one of which is curl-free and the other is divergence-free, then the potential function of the former can serve as a Lyapunov function candidate, since its positive definiteness will reflect the stability of the system. Moreover, under some additional conditions, its sublevel sets will give the exact attraction domain of the system. A sufficient condition for the existence of the proposed vector field decomposition is first obtained for $2$-dimensional systems by solving a partial differential equation and then generalized to $n$-dimensional systems. Furthermore, the proposed vector field decomposition always exists for linear systems and can be obtained by solving a specific algebraic Riccati equation.
