  In this paper, we develop Monte-Carlo based heuristic approaches to
approximate the objective function in long horizon optimal control problems. In
these approaches, to approximate the expectation operator in the objective
function, we evolve the system state over multiple trajectories into the future
while sampling the noise disturbances at each time-step, and find the average
(or weighted average) of the costs along all the trajectories. We call these
methods random sampling - multipath hypothesis propagation or RS-MHP. These
methods (or variants) exist in the literature; however, the literature lacks
results on how well these approximation strategies converge. This paper fills
this knowledge gap to a certain extent. We derive convergence results for the
cost approximation error from the RS-MHP methods and discuss their convergence
(in probability) as the sample size increases. We consider two case studies to
demonstrate the effectiveness of our methods - a) linear quadratic control
problem; b) UAV path optimization problem.
