Stabilization for a Cascaded ODE-Wave Equation with Boundary Nonlinear Disturbances

TL;DR

This paper develops a boundary feedback control strategy for a coupled ODE-wave PDE system, achieving exponential stabilization and disturbance rejection through novel transformations and observer design.

Contribution

It introduces a new transformation for boundary control integration and designs a disturbance estimator and observer-based controller for stabilization.

Findings

Achieved exponential stability for the coupled system.

Designed a disturbance estimator for boundary disturbances.

Validated results through numerical simulations.

Abstract

In this article, we investigate the problem of exponential stabilization via output feedback for a cascaded system composed of an ordinary differential equation (ODE) and a wave partial differential equation (PDE) under boundary control. Four types of boundary interconnections are considered. In the absence of disturbances, a novel transformation is introduced to incorporate the PDE boundary control input into the ODE subsystem. Based on this transformation, a state feedback controller is designed to achieve exponential stability for both the ODE and PDE components. When internal uncertainties and external disturbances that match the control structure are present, a disturbance estimator is constructed. Utilizing this estimator, a Luenberger-type state observer is developed to reject the disturbances and exponentially stabilize the original system via an observer-based control scheme. Furthermore, the boundedness of the entire closed-loop system is rigorously established. Numerical simulations are provided to illustrate the theoretical results.