# Feedback Stabilization for a coupled PDE-ODE Production System

## TL;DR

This paper develops a Lyapunov-based control approach to stabilize a coupled PDE-ODE production system, ensuring exponential stability even in bottleneck scenarios through a mixed feedback law, supported by theoretical and numerical analysis.

## Contribution

It introduces a novel feedback stabilization method for coupled PDE-ODE systems in production models, combining theoretical analysis with computational validation.

## Key findings

- The mixed feedback law guarantees exponential stability.
- Stability is achieved even in bottleneck conditions.
- Numerical examples confirm theoretical results.

## Abstract

We consider an interlinked production model consisting of conservation laws (PDE) coupled to ordinary differential equations (ODE). Our focus is the analysis of control laws for the coupled system and corresponding stabilization questions of equilibrium dynamics in the presence of disturbances. These investigations are carried out using an appropriate Lyapunov function on the theoretical and numerical level. The discrete $L^2-$stabilization technique allows to derive a mixed feedback law that is able to ensure exponential stability also in bottleneck situations. All results are accompanied by computational examples.