  Coronagraph instruments on future space telescopes will enable the direct
detection and characterization of Earth-like exoplanets around Sun-like stars
for the first time. The quest for the optimal optical coronagraph designs has
made rapid progress in recent years thanks to the Segmented Coronagraph Design
and Analysis (SCDA) initiative led by the Exoplanet Exploration Program at
NASA's Jet Propulsion Laboratory. As a result, several types of
high-performance designs have emerged that make use of dual deformable mirrors
to (1) correct for optical aberrations and (2) suppress diffracted starlight
from obstructions and discontinuities in the telescope pupil. However, the
algorithms used to compute the optimal deformable mirror surface tend to be
computationally intensive, prohibiting large scale design surveys. Here, we
utilize the Fast Linearized Coronagraph Optimizer (FALCO), a tool that allows
for rapid optimization of deformable mirror shapes, to explore trade-offs in
coronagraph designs for obstructed and segmented space telescopes. We compare
designs for representative shaped pupil Lyot and vortex coronagraphs, two of
the most promising concepts for the LUVOIR space mission concept. We analyze
the optical performance of each design, including their throughput and ability
to passively suppress light from partially resolved stars in the presence of
low-order aberrations. Our main result is that deformable mirror based
apodization can sufficiently suppress diffraction from support struts and
inter-segment gaps whose widths are on the order of $\sim$0.1% of the primary
mirror diameter to detect Earth-sized planets within a few tens of
milliarcseconds from the star.
