We develop a small-gain theory for systems
described by set-valued maps between topological spaces.
We introduce an abstract notion of stability unifying the
continuity properties behind different existing concepts,
such as Lyapunov stability of equilibria, sets, or motions,
(incremental) input-output stability, asymptotic gain properties,
and continuity with respect to fast-switching inputs.
Then, we prove that a feedback interconnection enjoying
a given abstract small-gain property is stable. While, in
general, the proposed small-gain property cannot be decomposed
as the union of stability of the subsystems and
a contractiveness condition, we show that it is implied
by standard assumptions in the context of input-to-state
stable systems. Finally, we provide application examples
illustrating how the developed theory can be used for the
analysis of interconnected systems and synthesis of control
systems.
