Stability of Linear KdV Equation in a Network with Bounded and Unbounded Lengths

In this work, we study the exponential stability of a system of linear Korteweg-de Vries (KdV) equations interconnected through the boundary conditions on a star-shaped network structure. On each branch of the network we define a linear KdV equation defined on a bounded domain
$(0,\ell_{j})$
or the half-line
$(0, \infty)$
. We start by proving well-posedness using semigroup theory and then some hidden regularity results. Then, we state the exponential stability of the linear KdV equation by acting with a damping term on not all the branches. This is proved by using compactness argument deriving a suitable observability inequality.