Stabilization of a Wave-Heat Cascade System

TL;DR

This paper develops a finite-dimensional output-feedback control strategy to stabilize a coupled wave-heat system, achieving exponential decay using spectral reduction and Lyapunov methods, with practical measurement configurations.

Contribution

It introduces a novel stabilization approach for a coupled wave-heat system using explicit spectral conditions and finite-dimensional controllers, extending to pointwise measurements.

Findings

Achieved exponential stabilization of the coupled system.

Designed a finite-dimensional controller based on spectral reduction.

Extended control design to pointwise temperature measurements.

Abstract

We consider the output-feedback stabilization of a one-dimensional cascade coupling a reaction-diffusion equation and a wave equation through an internal term, with Neumann boundary control acting at the wave endpoint. Two measurements are available: the wave velocity at the controlled boundary and a temperature-type observation of the reaction-diffusion component, either distributed or pointwise. Under explicit, necessary and sufficient conditions on the coupling and observation profiles, we show that the generator of the open-loop system is a Riesz-spectral operator. Exploiting this structure, we design a finite-dimensional dynamic output-feedback law, based on a finite number of parabolic modes, which achieves arbitrary exponential decay in both the natural energy space and a stronger parabolic norm. The construction relies on a spectral reduction and a Lyapunov argument in Riesz bases. We also extend the design to pointwise temperature or heat-flux measurements.