Boundary feedback stabilization of a chain of serially connected strings

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.