# Model Reduction of Semistable Distributed Parameter Systems

## TL;DR

This paper develops a new approach for reducing models of semistable infinite-dimensional control systems using a generalized Gramian, providing error bounds under certain conditions.

## Contribution

It introduces the semistability Gramian for such systems and derives error formulas in the -norm under a commutativity assumption.

## Key findings

- The semistability Gramian generalizes the controllability Gramian for semistable systems.
- Error bounds in -norm are derived based on the trace of the Gramian.
- The approach is valid when the original and reduced systems commute and semistability is maintained.

## Abstract

The model reduction problem for semistable infinite-dimensional control systems is studied in this paper. In relation to these systems, we study an object we call the semistability Gramian, which serves as a generalization of the ordinary controllability Gramian valid for semistable systems. This Gramian is then given geometric as well as algebraic characterization via a Lyapunov equation. We then proceed to show that under a commutativity assumption relating the original and reduced systems, and as long as the semistability is preserved, we may derive a priori error formulas in $\mathcal{H}_2$-norm in terms of the trace of this Gramian.