$\mathcal{H}_2(t_f)$ Optimality Conditions for a Finite-time Horizon

TL;DR

This paper develops an interpolatory model reduction framework for optimal approximation of MIMO systems over a finite-time horizon using the $ ext{H}_2(t_f)$ norm, including necessary optimality conditions and an algorithm for local optimality.

Contribution

It introduces the first-order optimality conditions and an algorithm for $ ext{H}_2(t_f)$ optimal model reduction, extending existing methods to finite-time horizon scenarios.

Findings

The algorithm achieves locally optimal reduced models satisfying the $ ext{H}_2(t_f)$ optimality conditions.

Numerical examples demonstrate the effectiveness of the proposed reduction method.

The framework generalizes classical $ ext{H}_2$ model reduction to finite-time horizons.

Abstract

In this paper we establish the interpolatory model reduction framework for optimal approximation of MIMO dynamical systems with respect to the $\mathcal{H}_2$ norm over a finite-time horizon, denoted as the $\mathcal{H}_2(t_f)$ norm. Using the underlying inner product space, we derive the interpolatory first-order necessary optimality conditions for approximation in the $\mathcal{H}_2(t_f)$ norm. Then, we develop an algorithm, which yields a locally optimal reduced model that satisfies the established interpolation-based optimality conditions. We test the algorithm on various numerical examples to illustrate its performance.