Exponential stabilization of variable coefficient wave equations in a generic tree with small time-delays in the nodal feedbacks

In this paper, we study the stability of a general tree network of variable coefficient wave equations with a small delay term in the nodal feedbacks. Using the Lax–Milgram theorem and C0C0-semigroup theory, we obtain the well-posedness of the system. By a detailed spectral analysis, we show that the spectrum of the system operator distributes in a strip parallel to the imaginary axis under certain conditions. Furthermore, we prove that there is a sequence of (generalized) eigenfunctions that forms a Riesz basis with parenthesis for the energy state space. As a consequence, we obtain the exponential stabilization of the closed-loop system under certain conditions.