Delayed stabilization of the Korteweg–de Vries equation on a star-shaped network

Abstract

In 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.

Stability Results for the Kdv Equation with Time-Varying Delay

“…The aim of this section is to illustrate the stability results obtained in this work with some numerical simulations that adapt the schemes used in [2,4,19]. We choose a final time T and build a uniform spatial and time discretization of N x + 1 and N t + 1 points, respectively, separated by the steps ∆x = L/N x and ∆t = T /N t .…”

“…We present now the numerical scheme in the case of boundary delay. The internal case follows similar ideas, (see [19] for a similar scheme in the case of constant delay in a network). We choose the delay step ∆ρ = 1/N ρ .…”

“…The problems of stability of systems with delay are of both theoretical and practical interest. Recently, the problem of robustness with respect to constant time-delay of the KdV equation was studied in [2,19,24] using Lyapunov theory or deriving suitable observability inequalities. The stability of PDE's involving time-varying delays was analyzed in [18] for one-dimensional heat and wave equations, in [16,17] for wave equations in domains in R n and in [7] for general second-order evolution equations.…”

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

“…With respect to the KdV equation on networks, we can mention the work [8] where well-posedness of the KdV equation on a star metric graph was studied. In the works [1,10], stabilization and controllability problems were studied, for the KdV equation on a star-shaped network, and recently the problem of stabilization using internal delay was addressed in [16].…”

“…With respect to saturated control in infinite-dimensional systems, we can refer to [19] where a wave equation with distributed and boundary saturated feedback law was studied, [14] where the saturated internal stabilization of a single KdV equation was studied and recently [15] where a saturated feedback control law was derived for a linear reaction-diffusion equation. Our idea closely follows works [14] and [16] to prove the stability of the KdV equation in a star-shaped network with saturated internal control. In this work, we consider a saturation map \fraks \fraka \frakt that could be any of the following cases:…”