Stability conditions for infinite networks of nonlinear systems and their application for stabilization

We introduce a new concept of ℓ_∞-input-to-state stability for infinite networks composed of a countable set of interconnected nonlinear subsystems of ordinary differential equations. We suppose that the entire state vector is an element of ℓ_∞ and each subsystem is input-to-state stable whereas the dimension of its entire disturbance input including possible interconnections with other subsystems is finite. Our first main result provides conditions for ℓ_∞-input-to-state stability of such infinite-dimensional networks. In our second main result, we solve the problem of decentralized ℓ_∞-ISS stabilization for such networks composed of interconnected lower-triangular form subsystems with uncontrollable linearization. To apply our first main result and obtain the second one, we construct a feedback for each individual agent, which satisfies our new stability conditions. This yields the stabilization of the entire network. Our design is also new for finite networks and this can be considered as an important special case.