From Interpolation to $\mathcal{H}_2$ Optimality: Model Reduction for Infinite-Dimensional Linear Control Systems

TL;DR

This paper extends the interpolatory $\\mathcal{H}_2$ optimal model reduction framework to infinite-dimensional linear control systems, providing a data-driven realization method and characterizing optimality conditions.

Contribution

It develops a comprehensive infinite-dimensional interpolatory $\mathcal{H}_2$ optimal model reduction theory, including a data-driven approach and precise subspace construction guidelines.

Findings

Reduced models interpolate the original transfer function at prescribed points.

The classical $\mathcal{H}_2$ optimality conditions extend to infinite-dimensional systems.

A data-driven realization framework analogous to Loewner's method is established.

Abstract

We develop the interpolatory $\mathcal{H}_2$ optimal model reduction framework for linear control systems posed on infinite dimensional state, input and output spaces. Specifically, we consider linear systems formulated as controlled abstract Cauchy problems on a Banach space and approximate them via Petrov-Galerkin projection onto finite dimensional trial and test subspaces. We show that the resulting reduced order transfer function interpolates the original at prescribed points, and we characterize precisely how the projection subspaces must be constructed to enforce this interpolation. Building on this, we develop a data-driven realization framework -- an infinite dimensional analogue of the Loewner approach -- that recovers the system behavior directly from input-output data without requiring access to the underlying operators. Finally, we derive $\mathcal{H}_2$ optimality conditions for the reduced model and show that the classical interpolatory characterization persists in this infinite dimensional setting: first-order optimality requires Hermite interpolation of the transfer function at the mirror images of the reduced model's poles. Taken together, these results establish that the interpolatory $\mathcal{H}_2$ optimal model reduction theory extends naturally and completely to infinite dimensional linear control systems with infinite dimensional input and output spaces.