Solution of a singular infinite horizon zero-sum linear-quadratic differential game: A regularization approach

An infinite horizon zero-sum linear-quadratic differential game is considered. The case where the cost functional does not contain a minimizer´s control cost is treated. Thus the game under consideration is singular. This game is associated with a new differential game for the same equation of dynamics. The cost functional in this new game is the sum of the original cost functional and an infinite horizon integral of the square of the minimizer control with a small positive weight coefficient. The new game is regular. Moreover, it is a cheap control game. Using the solvability conditions, the solution of this game is reduced to solution of an algebraic matrix Riccati equation, perturbed by a small parameter. Based on an asymptotic solution of this equation, the finiteness of the upper value in the original game is established. An expression of this value is derived. A minimizing sequence of feedback controls in the original game also is designed. Illustrative example is presented.