一致弱吸引性与实用全局渐近稳定性准则
Uniform weak attractivity and criteria for practical global asymptotic stability
摘要：
状态空间的子集$A$称为一致全局弱吸引，如果对于$A$的任何邻域$S$和任何有界子集$B$，存在一致有限时间$τ$，使得从$B$开始的任何轨迹在不大于$τ$的时间内与$S$相交。我们证明了实际一致全局渐近稳定性（pUGAS）等价于存在一个有界一致全局弱吸引集。这一结果适用于广泛的一类分布参数系统，包括时滞系统、切换系统、多种偏微分方程和巴拿赫空间中的发展微分方程。我们应用我们的结果表明，存在一个非强制性的李雅普诺夫函数，确保pUGAS这类系统。对于带有一致有界扰动的常微分方程，一致弱吸引性的概念等价于著名的弱吸引性的概念.然而，它本质上是强于弱吸引性的无穷维系统，即使是线性的。
英文摘要：
A subset $A$ of the state space is called uniformly globally weakly attractive if for any neighborhood $S$ of $A$ and any bounded subset $B$ there is a uniform finite time $τ$ so that any trajectory starting in $B$ intersects $S$ within the time not larger than $τ$. We show that practical uniform global asymptotic stability (pUGAS) is equivalent to the existence of a bounded uniformly globally weakly attractive set. This result is valid for a wide class of distributed parameter systems, including time-delay systems, switched systems, many classes of PDEs and evolution differential equations in Banach spaces. We apply our results to show that existence of a non-coercive Lyapunov function ensures pUGAS for this class of systems. For ordinary differential equations with uniformly bounded disturbances, the concept of uniform weak attractivity is equivalent to the well-known notion of weak attractivity. It is however essentially stronger than weak attractivity for infinite-dimensional systems, even for linear ones.
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