  For many stochastic dynamic systems, the Mean First Passage Time (MFPT) is a
useful concept, which gives expected time before a state of interest. This work
is an extension of MFPT in several ways. (1) We show that for some systems the
system-wide MFPT, calculated using the second largest eigenvalue only, captures
most of the fundamental dynamics, even for quite complex, high-dimensional
systems. (2) We generalize MFPT to Mean First Passage Value (MFPV), which gives
a more general value of interest, e.g., energy expenditure, distance, or time.
(3) We provide bounds on First Passage Value (FPV) for a given confidence
level. At the heart of this work, we emphasize that for our goals, many hybrid
systems can be approximated as Markov Decision Processes. So, many systems can
be controlled effectively using this framework. However, our framework is
particularly useful for metastable systems. Such systems exhibit interesting
long-living behaviors from which they are guaranteed to inevitably escape
(e.g., eventually arriving at a distinct failure or success state). Our goal is
then either minimizing or maximizing the value until escape, depending on the
application.
