# An $L^2_T$-error bound for time-limited balanced truncation

## TL;DR

This paper establishes an $L^2_T$-error bound for time-limited balanced truncation in model order reduction, linking it to the classical $\\mathcal{H}_\infty$-error bound as the time interval extends to infinity.

## Contribution

It provides the first $L^2_T$-error bound for time-limited balanced truncation and connects it to the classical $\\mathcal{H}_\infty$-error bound, with techniques applicable to unrestricted cases.

## Key findings

- Proves an $L^2_T$-error bound based on truncated singular values.
- Shows the error bound converges to the $\\mathcal{H}_\infty$-error bound as $T \to \infty$.
- Provides a short-time domain proof of the $\\mathcal{H}_\infty$-error bound.

## Abstract

Model order reduction (MOR) is often applied to spatially-discretized partial differential equations to reduce their order and hence decrease computational complexity. A reduced system can be obtained, e.g., by time-limited balanced truncation, a method that aims to construct an accurate reduced order model on a given finite time interval $[0, T]$. This particular balancing related MOR technique is studied in this paper. An $L^2_T$-error bound based on the truncated time-limited singular values is proved and is the main result of this paper. The derived error bound converges (as $T\rightarrow \infty$) to the well-known $\mathcal H_\infty$-error bound of unrestricted balanced truncation, a scheme that is used to construct a good reduced system on the entire time line. The techniques within the proofs of this paper can also be applied to unrestricted balanced truncation so that a relatively short time domain proof of the $\mathcal H_\infty$-error bound is found here.