Nontangency-Based Lyapunov Tests for Convergence and Stability in Systems Having a Continuum of Equilibria
Nontangency-Based Lyapunov Tests for Convergence and Stability in Systems Having a Continuum of Equilibria
复制标题
DOI：
10.1137/s0363012902407119
复制
发表时间：
2003-05
期刊：
影响因子：
--
通讯作者：
S. Bhat;D. Bernstein
中科院分区：
文献类型：
--
作者：
S. Bhat;D. Bernstein
本文主要研究具有连续平衡点的系统的稳定性分析。与这种系统特别相关的两个概念是收敛性和半稳定性。收敛性是每个解都收敛到一个极限点的性质，该极限点可能取决于初始条件。半稳定性是所有解收敛到李雅普诺夫稳定的极限点的附加要求。我们给出了新的基于李雅普诺夫函数的非线性系统的收敛性和半稳定性的结果。这些结果并没有对李雅普诺夫函数的符号确定性作出假设。相反，我们的研究结果使用了一种新的条件，基于向量场和不变或负不变的水平或子水平集的李雅普诺夫函数或其衍生物的子集之间的非相切性，并表示扩展以前已知的稳定性结果，涉及半定李雅普诺夫函数。为了说明我们的结果，我们推导出收敛和半稳定性的Michaelis-Menten化学反应的动力学和闭环动力学的标量系统下的通用自适应稳定反馈控制器。
This paper focuses on the stability analysis of systems having a continuum of equilibria. Two notions that are of particular relevance to such systems are convergence and semistability. Convergence is the property whereby every solution converges to a limit point that may depend on the initial condition. Semistability is the additional requirement that all solutions converge to limit points that are Lyapunov stable. We give new Lyapunov-function-based results for convergence and semistability of nonlinear systems. These results do not make assumptions of sign definiteness on the Lyapunov function. Instead, our results use a novel condition based on nontangency between the vector field and invariant or negatively invariant subsets of the level or sublevel sets of the Lyapunov function or its derivative and represent extensions of previously known stability results involving semidefinite Lyapunov functions. To illustrate our results we deduce convergence and semistability of the kinetics of the Michaelis--Menten chemical reaction and the closed-loop dynamics of a scalar system under a universal adaptive stabilizing feedback controller.