  In this short note, the problems of frequency-limited and time-limited
H2-optimal model order reduction are considered within the projection
framework. It is shown that it is inherently not possible to satisfy all the
necessary conditions for the local minimizer in the projection framework. The
conditions for the exact satisfaction of the optimality conditions are also
discussed. The equivalence between the tangential interpolation conditions and
the gramians-based necessary condition is established. The stationary point
iteration algorithms that satisfy two out of three necessary conditions for the
local minimizer are also proposed. The efficacy of the proposed algorithms is
validated by considering an illustrative example.
