  This paper presents an interpolatory framework for time-limited $H_2$ optimal
model order reduction named Limited Time Iterative Rational Krylov Algorithm
(LT-IRKA). The algorithm yields high fidelity reduced order models over limited
time intervals of the form, $\begin{bmatrix}0 & \tau \end{bmatrix}$ with $\tau
< \infty$ for linear time invariant (LTI) systems. Using the time limited $H_2$
norm, we derive interpolation based $H_{2,\tau}$ optimality conditions. The
LT-IRKA yields a near optimal $H_2(\tau)$ reduced order system. The nearness to
the exact $H_2(\tau)$ optimal reduced system is quantized in terms of the
errors in the interpolation based $H_2(\tau)$ optimality conditions. We
demonstrate with numerical examples how the proposed algorithm nearly satisfies
the time-limited optimality conditions and also how it performs with respect to
the Time-Limited Two sided Iteration Algorithm (TL-TSIA), the Time-Limited
Balanced Truncation (TL-BT), the Iterative Rational Krylov Algorithm (IRKA) and
the Time-Limited Pseudo Optimal Rational Krylov (TL-PORK) Algorithm over a
finite time interval.
