Global Well-Posedness of the KdV Equation on a Star-Shaped Network and Stabilization by Saturated Controllers

Abstract

In this work, we deal with the global well-posedness and stability of the linear and nonlinear Korteweg-de Vries equations on a finite star-shaped network by acting with saturated controls. We obtain the global well-posedness by using the Kato smoothing property for the linear case and then using some estimates and a fixed point argument we deal with the nonlinear system.Finally, we obtain the exponential stability using two different kinds of saturation by proving an observability inequality via a contradiction argument.

Stability results for the KdV equation with time-varying delay

“…In Menzala, Vasconcellos and Zuazua (2002) the authors showed that if the length 𝐿 is critical, an internal damping allows the exponential stability. This idea was then applied in several works for instance Pazoto (2005); Linares and Pazoto (2009); Parada, Crépeau and Prieur (2022b). We also refer to Cerpa (2014); Rosier and Zhang (2009) for a complete introduction about control of KdV equation.…”