Saddle-point equilibrium sequence in one class of singular infinite horizon zero-sum linear-quadratic differential games with state delays

We consider an infinite horizon zero-sum linear-quadratic differential game with state delays in the dynamics. The cost functional of this game does not contain a control cost of the minimizing player (the minimizer), meaning that the considered game is singular. For this game, definitions of the saddle-point equilibrium and the game value are proposed. These saddle-point equilibrium and game value are obtained by a regularization of the singular game. Namely, we associate this game with a new differential game for the same equation of dynamics. The cost functional in the new game is the sum of the original cost functional and an infinite horizon integral of the square of the minimizer's control with a small positive weight coefficient. This new game is regular, and it is a cheap control game. An asymptotic analysis of this cheap control game is carried out. Using this asymptotic analysis, the existence of the saddle-point equilibrium and the value of the original game is established, and their expressions are derived. Illustrative example is presented.