Small-time local stabilization for a Korteweg–de Vries equation

Abstract

This paper focuses on the (local) small-time stabilization of a Korteweg-de Vries equation on bounded interval, thanks to a time-varying Dirichlet feedback law on the left boundary. Recently, backstepping approach has been successfully used to prove the null controllability of the corresponding linearized system, instead of Carleman inequalities. We use the "adding an integrator" technique to gain regularity on boundary control term which clears the difficulty from getting stabilization in small-time.

Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback

“…This is done in [14], where the authors give an explicit control to bring a heat equation to 0, then a time-varying, periodic feedback to stabilize the equation in small time. In [47], the author obtains the same kind of results for the Korteweg-de Vries equation.…”

“…This was used to prove a host of results on the boundary stabilization of partial dierential equations: let us cite for example [24] and [39] for the wave equation, [46,47] for the Korteweg-de Vries equation, [2]*Chapter 7 for an application to rstorder hyperbolic systems, and also [17], which combines the backstepping method with Lyapunov functions to prove nite-time stabilization in H 2 for a quasilinear 2 × 2 hyperbolic system.…”

Finite-time internal stabilization of a linear 1-D transport equation

“…This was used to prove many results on the boundary stabilization of partial differential equations (see for example [21,31,38,37,15], and also [2, chapter 7]).…”

“…Because backstepping provides explicit feedback laws, it has helped prove null-controllability or small-time stabilization (stabilization in an arbitrarily small time) results for some systems: see [14] for the heat equation, and [37] for the Korteweg-de Vries equation. In this article, we use the explicit feedback laws obtained by the backstepping method in [40] to design an explicit stationary feedback law that achieves finite-time stabilization.…”

“…As we have mentioned in the introduction, one of the advantages of the backstepping method is that it can provide explicit feedback laws for exponential stabilization. This allows the construction of explicit controls for null controllability ( [38,14]) as well as time-varying feedbacks that stabilize the system in finite time T > 0 ( [37,14]).…”