Exponential stabilization of cascade ode-reaction-diffusion pde by pointwise actuation

TL;DR

This paper develops a backstepping control method for stabilizing a coupled ODE-PDE system with pointwise actuation inside the domain, achieving exponential stability.

Contribution

It introduces a novel control design for ODE-PDE cascade systems with interior actuation, extending boundary control techniques to internal points.

Findings

Proves exponential stability of the controlled system.

Demonstrates invertibility of the backstepping transformation.

Establishes well-posedness of the closed-loop system.

Abstract

In this paper, we are concerned with the state feedback stabilization of ODE-PDE cascade systems governed by a linear ordinary differential equation and the 1-d reaction-diffusion equation posed on a bounded interval. In contrast to the previous works in the literature where the control acts at the boundary, the control for the entire system acts at an inside point of the PDE domain whereas the PDE acts in the linear ODE by a Neumann connection. We use the infinite dimensional backstepping design to convert system under consideration to an exponentially target system. By invertibility of the design and Lyapunov analysis, we prove the well posedness and exponential stability of such system.