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, l(j)) or the half-line (0,infinity). 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.