A Dynamic Programming Approach for the Brachistochrone Problem

In this paper, we address the brachistochrone problem using DynamicProgramming and an extension that accounts for forbiddenregions or obstacles. The brachistochrone problem seeks the curveof fastest descent under gravity between two points, minimizingtravel time. While traditional approaches derive the cycloidal curveas the solution using calculus of variations, this work employs adiscrete, grid-based Dynamic Programming formulation to approximatethe optimal trajectory. Computational experiments showcasethe method’s flexibility, particularly in adapting to constraints suchas forbidden regions, and its ability to dynamically recalculate paths.Despite some limitations in angular resolution due to discretization,the proposed approach demonstrates robustness and scalability inaddressing constrained trajectory optimization problems. Moreover,this project lays the groundwork for extending the proposedmethodology to more complex scenarios, such as incorporating randomnessinto the trajectory, where the adaptability of this approachcan be effectively leveraged.