Stability of infinitely many interconnected systems

In this paper we consider countable couplings of finite-dimensional input-to-state stable systems. We consider the whole interconnection as an infinite-dimensional system on the lIF state space. We develop stability conditions of the small-gain type to guarantee that the whole system remains ISS and highlight the differences between finite and infinite couplings by means of examples. We show that using our methodology it is possible to study uniform global asymptotic stability of nonlinear spatially invariant systems by solving a finite number of nonlinear algebraic inequalities.