Singular Finite-Horizon Zero-Sum Differential Game

Abstract

AbstractIn this chapter, a finite-horizon zero-sum linear-quadratic differential game, which cannot be solved by application of the first-order solvability conditions, is considered. Thus, this game is singular. Its singularity is due to the singularity of the weight matrix in the control cost of a minimizing player in the game’s functional. For this game, novel definitions of a saddle-point equilibrium (a saddle-point equilibrium sequence) and a game value are introduced. Regularization method is proposed for obtaining these saddle-point equilibrium sequence and game value. This method consists in an approximate replacement of the original singular game with an auxiliary regular finite-horizon zero-sum linear-quadratic differential game depending on a small positive parameter. Thus, the first-order solvability conditions are applicable for this new game. Asymptotic analysis (with respect to the small parameter) of the Riccati matrix differential equation, arising in these conditions, yields the solution (the saddle-point equilibrium sequence and the game value) to the original singular game.