A small gain theorem for finite-time input-to-state stability of infinite networks and its applications

Abstract

We prove a small-gain sufficient condition for (global) finite-time input-to-state stability (FTISS) of infinite networks. The network under consideration is composed of a countable set of finite-dimensional subsystems of ordinary differential equations, each of which is interconnected with a finite number of its “neighbors” only and is affected by some external disturbances. We assume that each node (subsystem) of our network is finite-time input-to-state stable (FTISS) with respect to its finite-dimensional inputs produced by this finite set of the neighbors and with respect to the corresponding external disturbance. As an application we obtain a new theorem on decentralized finite-time input-to-state stabilization with respect to external disturbances for infinite networks composed of a countable set of strict-feedback form systems of ordinary differential equations. For this we combine our small-gain theorem proposed in the current work with the controllers design developed by S. Pavlichkov and C. K. Pang (NOLCOS-2016) for the gain assignment of the strict-feedback form systems in the case of finite networks. The current results address the finite-time input-to-state stability and decentralized finite-time input-to-state stabilization and redesign the technique proposed in recent work S. Dashkovskiy and S. Pavlichkov, Stability conditions for infinite networks of nonlinear systems and their application for stabilization, Automatica. – 2020. – 112. – 108643, in which the case of $\ell_{\infty}$-ISS of infinite networks was investigated. The current paper extends and generalizes its conference predecessor to the case of finite-time ISS stability and decentralized stabilization in presence of external disturbance inputs and with respect to these disturbance inputs. In the special case when all these external disturbances are zeroes (i.e. are abscent), we just obtain finite-time stability and finite-time decentralized stabilization of infinite networks accordingly.