  We study stochastic motion planning problems which involve a controlled
process, with possibly discontinuous sample paths, visiting certain subsets of
the state-space while avoiding others in a sequential fashion. For this
purpose, we first introduce two basic notions of motion planning, and then
establish a connection to a class of stochastic optimal control problems
concerned with sequential stopping times. A weak dynamic programming principle
(DPP) is then proposed, which characterizes the set of initial states that
admit a control enabling the process to execute the desired maneuver with
probability no less than some pre-specified value. The proposed DPP comprises
auxiliary value functions defined in terms of discontinuous payoff functions. A
concrete instance of the use of this novel DPP in the case of diffusion
processes is also presented. In this case, we establish that the aforementioned
set of initial states can be characterized as the level set of a discontinuous
viscosity solution to a sequence of partial differential equations, for which
the first one has a known boundary condition, while the boundary conditions of
the subsequent ones are determined by the solutions to the preceding steps.
Finally, the generality and flexibility of the theoretical results are
illustrated on an example involving biological switches.
