On the solution of Dolichobrachistochrone differential game via dynamic programming approach

Abstract

This study aims to apply Mirică’s new dynamic programming method, introduced in 2004, to derive the rigorous solution of the famous Dolichobrachistochrone differential game, originally proposed by Isaacs in 1965. In addition, it aims to identify, for the first time, feedback strategies, a novel contribution that offers significant advantages in game theory over other types of strategies. Among them, they promote efficiency through dynamic performance optimization, leading to improved resource utilization and goal attainment. Moreover, their simplicity facilitates implementation and analysis, reducing computational complexity. The essential tool in our approach, involves the use of a certain refinement of Cauchy’s method of characteristics for stratified Hamilton–Jacobi equations, to describe a large class of admissible trajectories and to identify a domain in which the value function exists. As a rigorous criterion for proving the optimality of these admissible feedback strategies, we use the well-known verification Theorem for locally Lipschitz value functions as a sufficient optimality condition.