Monotone dynamical systems, graphs, and stability of large-scale interconnected systems

For a class of monotone operators T on the positive orthant of n-dimensional Euclidean space we introduce the concept of decay sets. These consist of points x satisfying T(x)<x. Considering the induced dynamical system x(k 1)=T(x(k)), we establish results relating stability properties of the origin, order conditions on T, and topological properties of decay sets. In particular, we construct paths in the decay sets and derive a quasi-invertibility result of the operator (Id-T).These results are applied to derive generalized small-gain type conditions for the input-to-state stability(ISS) of large-scale interconnections of (individually input-to-state stable) control systems: The interconnection topology together with the ISS gains yields a monotone operator with an inherent graph structure. We provide trajectory estimate based small-gain theorems and also construct ISS Lyapunov functions for the composite system. It is also shown how an algorithm due to Eaves can be used to numerically verify the small-gain condition.