  In this paper, the primary goal is to offer additional insights into the
value iteration through the lens of switching system models in the control
community. These models establish a connection between value iteration and
switching system theory and reveal additional geometric behaviors of value
iteration in solving discounted Markov decision problems. Specifically, the
main contributions of this paper are twofold: 1) We provide a switching system
model of value iteration and, based on it, offer a different proof for the
contraction property of the value iteration. 2) Furthermore, from the
additional insights, new geometric behaviors of value iteration are proven when
the initial iterate lies in a special region. We anticipate that the proposed
perspectives might have the potential to be a useful tool, applicable in
various settings. Therefore, further development of these methods could be a
valuable avenue for future research.
