  In this paper, we study the probabilistic stability analysis of a subclass of
stochastic hybrid systems, called the Planar Probabilistic Piecewise Constant
Derivative Systems (Planar PPCD), where the continuous dynamics is
deterministic, constant rate and planar, the discrete switching between the
modes is probabilistic and happens at boundary of the invariant regions, and
the continuous states are not reset during switching. These aptly model
piecewise linear behaviors of planar robots. Our main result is an exact
algorithm for deciding absolute and almost sure stability of Planar PPCD under
some mild assumptions on mutual reachability between the states and the
presence of non-zero probability self-loops. Our main idea is to reduce the
stability problems on planar PPCD into corresponding problems on Discrete Time
Markov Chains with edge weights.
