Consistent Estimator for Constrained Minimal Trajectories

Consider the problem of finding a smooth function joining two points A and B with minimum length constrained to avoid fixed subsets when stochastic measurement errors are present. In this case, the estimator proposed by Dias, Garcia and Zambom (2007) is consistent, in the sense that as the number of observations increases the stochastic trajectory converges to the best deterministic one. Two applications are immediate, searching the optimal a path for an autonomous vehicle while avoiding all fixed obstacles between two points and flight planning to avoid threat or turbulence zones.