The Hamilton–Jacobi–Bellman Equation for Differential Games with Composite Distribution of Random Time Horizon

A differential game with random duration is considered. The terminal time of the game is a random variable settled using a composite distribution function. Such a scenario occurs when the operating mode of the system changes over time at the appropriate switching points. On each interval between switchings, the distribution of the terminal time is characterized by its own distribution function. A method for solving such games using dynamic programming is proposed. An example of a non-renewable resource extraction model is given, where a solution of the problem of maximizing the total payoff in closed-loop strategies is found. An analytical view of the optimal control of each player and the optimal trajectory depending on the parameters of the described model is obtained.