We consider three optimal control problems, which focus on continuouspath-planning applications, and each problem deals with a specific challenge. First, we introduce a new local factoring technique which can remove numerical artifacts arising from Eikonal equations with non-smooth conditions. Next, we deal with path-planning under an initial uncertainty, which can be removed later at some certainty time, and further discuss methods suitable for different notions of optimality. The third section considers an online-learning path-planning problem with unknown surveillance intensity and develops a Bayesian reinforcement learning method. In addition to the three path-planning problems, a new method of parameterizing neural networks is introduced. It follows the continuous optimal control interpretation of deep learning and uses B-spline basis functions to parameterize. For each problem we use numerical experiments to show the advantages of our proposed methods.