# Feedback stabilization Problems on tree-like structures

We consider PDE models on tree-like structures modeling their vibrations. As one knows,tree-structures consist of edges/branches which meet at vertices/nodes. It is known that such structures are interesting and have practical relevance. (e.g., Gas or water network system in a city, air flow inside lungs). For obvious practical reasons, we wish to control vibrations of such structures. To this end, we consider a model which describes vibrations of not only edges/tubes but also nodes/joints and coupling between them. Compared with standard models existing in the literature,the novelty of our model is the dynamics of nodes which represents a type of Kirchoff generalized transmission condition between two edges passing through a particular node under consideration. The objective is to study controllability, observability and stabilization of such complex coupled systems. To stabilize the system using a feedback control, especially, we seek feedback controls supported at isolated points of the structure. As a first step, we prove that the energy of the solutions of the closed loop system decay exponentially to zero when the time tends to infinity. Our technique is based on a frequency domain method and a spectral analysis for the resolvent.

**People**: Kais Ammari and M. Vanninathan

**Status**: Ongoing