  Two subsets of a given set are path-disconnected if they lie in different
connected components of the larger set. Verification of path-disconnectedness
is essential in proving the infeasibility of motion planning and trajectory
optimization algorithms. We formulate path-disconnectedness as the
infeasibility of a single-integrator control task to move between an initial
set and a target set in a sufficiently long time horizon. This
control-infeasibility task is certified through the generation of a
time-dependent barrier function that separates the initial and final sets. The
existence of a time-dependent barrier function is a necessary and sufficient
condition for path-disconnectedness under compactness conditions. Numerically,
the search for a polynomial barrier function is formulated using the
moment-sum-of-squares hierarchy of semidefinite programs. The barrier function
proves path-disconnectedness at a sufficiently large polynomial degree. The
computational complexity of these semidefinite programs can be reduced by
elimination of the control variables. Disconnectedness proofs are synthesized
for example systems.
