非线性系统局部渐近和指数镇定的变分方法
A Variational Approach to Local Asymptotic and Exponential Stabilization of Nonlinear Systems
摘要：
局部渐近稳定性是一个具有重要理论意义和实际意义的问题。广义地说，如果一个系统$\dot{x} = f(x,u)$是局部渐近稳定的，我们保证有一个反馈控制器$u(x)$，迫使收敛到一个平衡点的轨迹初始化足够接近它。一个必要条件是由Brockett给出的：这样的控制器存在，只有当$f$是开放的平衡。最近，古普塔，贾法里，基普卡和Mordukhovich考虑修改这一条件，取代布罗克特的拓扑开放性的线性开放性的现代变分分析。在本文中，我们证明了在线性开放性假设下，存在一个局部的邻域的平衡点，系统是指数稳定的连续平稳反馈律。引入一个横截性性质，并将其与上述的自同构联系起来，证明了平衡点的穿孔邻域上的线性开性和横截性是亏秩线性化系统局部指数镇定的充分条件，主要结果超越了通常的Kalman和Hautus关于指数镇定反馈律存在的判据，因为它允许我们处理通过高阶项实现指数稳定的系统。然而，到目前为止，它是在相当严格的行秩条件下实现的。这表明了使用这些属性的双重方法：逐点版本足以通过线性化确保稳定性，而局部版本足以克服线性化中的缺陷。
英文摘要：
Local asymptotic stabilizability is a topic of great theoretical interest and practical importance. Broadly, if a system $\dot{x} = f(x,u)$ is locally asymptotically stabilizable, we are guaranteed a feedback controller $u(x)$ that forces convergence to an equilibrium for trajectories initialized sufficiently close to it. A necessary condition was given by Brockett: such controllers exist only when $f$ is open at the equilibrium. Recently, Gupta, Jafari, Kipka and Mordukhovich considered a modification to this condition, replacing Brockett's topological openness by the linear openness property of modern variational analysis. In this paper, we show that under the linear openness assumption there is a local diffeomorphism of neighborhoods of the equilibirum on which the system is exponentially stabilizable by means of continuous stationary feedback laws. Introducing a transversality property and relating it to the above diffeomorphism, we prove that linear openness and transversality on a punctured neighborhood of an equilibrium is sufficient for local exponential stabilizability of systems with a rank deficient linearization.The main result goes beyond the usual Kalman and Hautus criteria for the existence of exponential stabilizing feedback laws, since it allows us to handle systems for which exponential stabilization is achieved through higher-order terms. However, it is implemented so far under a rather restrictive row-rank conditions. This suggests a twofold approach to the use of these properties: a point-wise version is enough to ensure stability via the linearization, while a local version is enough to overcome deficiencies in the linearization.
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