# Output-Feedback Boundary Control of a Heat PDE Sandwiched Between Two ODEs

## TL;DR

This paper develops an output-feedback boundary control method for stabilizing a coupled ODE-heat PDE-ODE system using PDE and ODE backstepping, with proofs of exponential stability and numerical validation.

## Contribution

It introduces a novel combined PDE-ODE backstepping approach for boundary control and observer design in a coupled system with limited boundary measurements.

## Key findings

- Proves exponential stability of the closed-loop system.
- Designs an observer estimating PDE and ODE states from boundary measurements.
- Validates the control method through numerical simulations.

## Abstract

We present designs for exponential stabilization of an ODE-heat PDE-ODE coupled system where the control actuation only acts in one ODE. The combination of PDE backstepping and ODE backstepping is employed in a state-feedback control law and in an observer that estimates PDE and two ODE states only using one PDE boundary measurement. Based on the state-feedback control law and the observer, the output-feedback control law is then proposed. The exponential stability of the closed-loop system and the boundedness and exponential convergence of the control law are proved via Lyapunov analysis. Finally, numerical simulations validate the effectiveness of this method for the `sandwiched' system.