Exponential stabilization of cascade ode-linearized kdv system by boundary Dirichlet actuation

TL;DR

This paper develops a boundary control method using backstepping to achieve exponential stabilization of a coupled ODE and linearized KdV PDE system on a bounded interval.

Contribution

It introduces a backstepping design for boundary control of a cascade ODE-KdV system, ensuring exponential stability through Lyapunov analysis.

Findings

Successfully stabilizes the cascade system exponentially.

Provides a systematic control design via backstepping.

Validates stability through Lyapunov methods.

Abstract

In this paper, we solve the problem of exponential stabilization for a class of cascade ODE-PDE system governed by a linear ordinary differential equation and a 1-d linearized Korteweg-de Vries equation (KdV) posed on a bounded interval. The control for the entire system acts on the left boundary with Dirichlet condition of the KdV equation whereas the KdV acts in the linear ODE by a Dirichlet connection. We use the socalled backstepping design in infinite dimension to convert the system under consideration into an exponentially stable cascade ODE-PDE system.Then, we use the invertibility of such design to achieve the exponential stability for the ODE-PDE cascade system under consideration by using Lyapunov analysis.