A Finite-Time Small-Gain Theorem for Infinite Networks and Its Applications

Abstract

We prove a small-gain theorem which addresses the problem of global uniform finite-time stability of infinite networks. The network 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 assumed to be finite-time input-to-state stable with respect to its finite-dimensional inputs produced by this finite set of the neighbors. As a corollary we obtain a new result on decentralized finite-time stabilization of infinite networks composed of a countable set of strict-feedback form systems of ordinary differential equations. For this, we combine our new small-gain theorem with the method proposed 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 stability and finite-time stabilization and redesign the technique proposed in recent work by S. Dashkovskiy and S. Pavlichkov (IFAC World Congress-2017) for asymptotic stabilization of infinite networks.