  This paper studies the problem of steering a linear time-invariant system
subject to state and input constraints towards a goal location that may be
inferred only through partial observations. We assume mixed-observable
settings, where the system's state is fully observable and the environment's
state defining the goal location is only partially observed. In these settings,
the planning problem is an infinite-dimensional optimization problem where the
objective is to minimize the expected cost. We show how to reformulate the
control problem as a finite-dimensional deterministic problem by optimizing
over a trajectory tree. Leveraging this result, we demonstrate that when the
environment is static, the observation model piecewise, and cost function
convex, the original control problem can be reformulated as a Mixed-Integer
Convex Program (MICP) that can be solved to global optimality using a
branch-and-bound algorithm. The effectiveness of the proposed approach is
demonstrated on navigation tasks, where the system has to reach a goal location
identified from partial observations.
