Boundary feedback stabilization of a chain of serially connected strings

TL;DR

This paper proves exponential energy decay in a chain of connected strings using frequency domain methods, also analyzing transfer functions and Schrödinger system stability.

Contribution

It introduces a novel frequency domain approach to establish exponential stabilization for a network of connected strings, including transfer function analysis.

Findings

Energy decays exponentially regardless of string densities

Frequency domain method effectively proves stabilization

Transfer function analysis extends to Schrödinger systems

Abstract

We consider N strings connected to one another and forming a particular network which is a chain of strings. We study a stabilization problem and precisley we prove that the energy of the solutions of the dissipative system decay exponentially to zero when the time tends to infinity, independently of the densities of the strings. Our technique is based on a frequency domain method and a special analysis for the resolvent. Moreover, by same appraoch, we study the transfert function associated to the chain of strings and the stability of the Schr\"odinger system.