标题： 关于李雅普诺夫和拉格朗日稳定性的抽象概念 显示英文标题

标题： On an Abstraction of Lyapunov and Lagrange Stability

摘要： 本文研究了由集值映射描述的抽象系统中李雅普诺夫和拉格朗日稳定性的集合论推广。 李雅普诺夫稳定性被表征为将滤子逆映射为滤子的性质， 拉格朗日稳定性则被表征为将理想映射为理想的特点。 这些抽象定义揭示了两种稳定性概念之间的深刻对偶性，使得可以对抽象系统定义全局稳定性，并得出基本串联、并联和反馈互联系统的稳定性定理的灵活推广，包括一个小增益定理。 此外，展示了拉格朗日稳定性在抽象意义上与控制理论中其他感兴趣性质相同，例如安全性与正性，由于所发展的稳定性结果，可以对此类性质在互联系统中的保持情况进行研究。 显示英文摘要

摘要： This paper studies a set-theoretic generalization of Lyapunov and Lagrange stability for abstract systems described by set-valued maps. Lyapunov stability is characterized as the property of inversely mapping filters to filters, Lagrange stability as that of mapping ideals to ideals. These abstract definitions unveil a deep duality between the two stability notions, enable a definition of global stability for abstract systems, and yield an agile generalization of the stability theorems for basic series, parallel, and feedback interconnections, including a small-gain theorem. Moreover, it is shown that Lagrange stability is abstractly identical to other properties of interest in control theory, such as safety and positivity, whose preservation under interconnections can be thus studied owing to the developed stability results.