Stability of KdV equation in a network with bounded and unbounded lengths

Hugo Parada (1) , Emmanuelle Crépeau (2) , Christophe Prieur (3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 EDP - Equations aux Dérivées Partielles
3 GIPSA-INFINITY - GIPSA - Infinite Dimensional Dynamics

In this work, we studied the exponential stability of the nonlinear KdV equation posed in a finite star shaped network. On each branch of the network we define a KdV equation posed on a finite domain (0, ℓ j) or the half-line (0, ∞). We start by proving well-posedness and some regularity results. Then, we state the exponential stability of the linear KdV equation by acting with a damping term on some branches. The main idea is to prove a suitable observability inequality. In the nonlinear case, we obtain two kinds of results: The first result holds for small amplitude solutions, and is proved using a perturbation argument from the linear case but without acting on all edges. The second result is a semiglobal stability result, and it is obtained by proving an observability inequality directly for the nonlinear system, but we need to act with damping terms in all the branches. In this case, we are able to prove the stabilization in weighted spaces.

Keywords: Korteweg-de Vries equation; star-shaped network; stabilization; half-line

Domains: Analysis of PDEs [math.AP]; Systems and Control [cs.SY]; Optimization and Control [math.OC]

Hugo Parada, Emmanuelle Crépeau, Christophe Prieur. Stability of KdV equation in a network with bounded and unbounded lengths. 2024. ⟨hal-04405778⟩