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

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.