Pontryagin’s Maximum Principle for Non-cooperative Differential Games with Continuous Updating

Abstract

The paper is devoted to the optimality conditions as determined by Pontryagin’s maximum principle for a non-cooperative differential game with continuous updating. Here it is assumed that at each time instant players have or use information about the game structure defined for the closed time interval with a fixed duration. The major difficulty in such a setting is how to define players’ behavior as the time evolves. Current time continuously evolves with an updating interval. As a solution for a non-cooperative game model, we adopt an open-loop Nash equilibrium within a setting of continuous updating. Theoretical results are demonstrated on an advertising game model, both initial and continuous updating versions are considered. A comparison of non-cooperative strategies and trajectories for both cases are presented.