Stabilization of a cyclic network of strings by nodal control

We consider a network of coupled strings that contains a cycle. We show that boundary feedback stabilization that uses only data of the state at boundary nodes that are not contained in the cycle is in general impossible. We demonstrate that also with control actuators that act at the interior nodes contained in the cycle with certain standard control laws that use only Neumann control action the situation does not improve and the system is still not stable. We prove that with an actuator that uses also Dirichlet control action and is located within the cycle, exponential stabilization is possible.