# Stabilization of infinite-dimensional linear control systems by POD reduced-order Riccati feedback

## TL;DR

This paper demonstrates that POD-based reduced-order Riccati feedback can effectively stabilize infinite-dimensional linear control systems under certain spectral conditions, offering a computationally feasible approach.

## Contribution

It establishes the stability of infinite-dimensional systems using POD-reduced Riccati feedback derived from finite-dimensional approximations, under spectral assumptions.

## Key findings

- POD reduction preserves stabilization properties.
- Riccati feedback from reduced models stabilizes the original system.
- Spectral assumptions are key for stability transfer.

## Abstract

There exist many ways to stabilize an infinite-dimensional linear autonomous control systems when it is possible. Anyway, finding an exponentially stabilizing feedback control that is as simple as possible may be a challenge. The Riccati theory provides a nice feedback control but may be computationally demanding when considering a discretization scheme. Proper Orthogonal Decomposition (POD) offers a popular way to reduce large-dimensional systems. In the present paper, we establish that, under appropriate spectral assumptions, an exponentially stabilizing feedback Riccati control designed from a POD finite-dimensional approximation of the system stabilizes as well the infinite-dimensional control system.