On an Abstraction of Lyapunov and Lagrange Stability

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.