The aim of this work is to apply the dynamic programming framework developed by S. Mirica in 2004 in order to solve the classical Dolichobrachistochrone differential game, which was originally formulated by R. Isaacs in 1965, in a rigorous manner. A key objective of this study is to identify the optimal feedback strategies for the first time, which is an original contribution that enhances the theoretical landscape of differential games. These strategies offer significant benefits, including the ability to dynamically optimise the system performance, allocate resources more effectively and achieve targets efficiently. Their inherent simplicity also allows for easier implementation and analysis, contributing to reduced computational complexity.

Our method is based on a refined version of Cauchy's method of characteristics, adapted to stratified Hamilton–Jacobi equations. This technique enables the characterisation of a broad class of the optimal trajectories and helps to define the domain of existence of the value function. To demonstrate the effectiveness of the proposed feedback strategies, we apply the well-known Verification Theorem for locally Lipschitz value functions as a sufficient condition for optimality.