  Piecewise-deterministic Markov processes (PDMPs) are often used to model
abrupt changes in the global environment or capabilities of a controlled
system. This is typically done by considering a set of "operating modes" (each
with its own system dynamics and performance metrics) and assuming that the
mode can switch stochastically while the system state evolves. Such models have
a broad range of applications in engineering, economics, manufacturing,
robotics, and biological sciences. Here, we introduce and analyze an
"occasionally observed" version of mode-switching PDMPs. We show how such
systems can be controlled optimally if the planner is not alerted to
mode-switches as they occur but may instead have access to infrequent mode
observations. We first develop a general framework for handling this through
dynamic programming on a higher-dimensional mode-belief space. While quite
general, this method is rarely practical due to the curse of dimensionality. We
then discuss assumptions that allow for solving the same problem much more
efficiently, with the computational costs growing linearly (rather than
exponentially) with the number of modes. We use this approach to derive
Hamilton-Jacobi-Bellman PDEs and quasi-variational inequalities encoding the
optimal behavior for a variety of planning horizons (fixed, infinite,
indefinite, random) and mode-observation schemes (at fixed times or on-demand).
We discuss the computational challenges associated with each version and
illustrate the resulting methods on test problems from surveillance-evading
path planning. We also include an example based on robotic navigation: a Mars
rover that minimizes the expected time to target while accounting for the
possibility of unobserved/incremental damages and dynamics-altering breakdowns.
