A Connection between the Adjoint Variables and Value Function for Differential Games

In this paper, we present a deterministic two-player nonzero-sumd ifferential games (NZSDGs) in a finite horizon. The connection between the adjoint varaibles in the maximum principle (MP) and the value function in the dynamic programming principle (DPP) for differentail games is obtained in either case, whether the value function is smooth and nonsmooth. For the smooth case, the connection between the adjoint variables and the derivatives of the value function are equal to each other along optimal trajectories. Furthermore, for the nonsmooth case, this relation is represented in terms of the adjoint variables and the first-order super- and subdifferentials of the value function. We give an example to illustrate the theoretical results.