  Motivated by the scalability problem in large networks, we study stability of
a network of infinitely many finite-dimensional subsystems. We develop a
so-called relaxed small-gain theorem for input-to-state stability (ISS) with
respect to a closed set and show that every exponentially input-to-state stable
system necessarily satisfies the proposed small-gain condition. Following our
bottom-up approach, we study the well-posedness of the interconnection based on
the behavior of the individual subsystems. Finally, we over-approximate
large-but-finite networks by infinite networks and show that all the stability
properties and the performance indices obtained for the infinite system can be
transferred to the original finite one if each subsystem of the infinite
network is individually ISS. Interestingly, the size of the truncated network
does not need to be known. The effectiveness of our small-gain theorem is
verified by application to an urban traffic network.
