Stabilisation avec retard de l'équation de Korteweg-de Vries sur un réseau de type étoile.

International audienceIn this work we deal with the exponential stability of the nonlinear Kortewegde Vries (KdV) equation on a finite star-shaped network in the presence of delayed internal feedback. We start by proving the well-posedness of the system and some regularity results. Then we state an exponential stabilization result using a Lyapunov function by imposing small initial data and a restriction over the lengths. In this part also, we are able to obtain an explicit expression for the rate of decay. Then we prove the exponential stability of the solutions without restriction on the lengths and for small initial data, this result is based on an observability inequality. After that, we obtain a semi-global stabilization result working directly with the nonlinear system. Next we study the case where it may happen that a control domain with delay is outside of the control domain without delay. In that case, we obtain also a local exponential stabilization result. Finally, we present some numerical simulations in order to illustrate the stabilization