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

Abstract

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.

Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence

“…It is important to note the following. Although the regularization approach to study of the singular zero-sum game in the present paper is similar to the approach in the works [12,13,33], the main results of the present paper, as well as their derivation, differ considerably from the ones of these works. Namely, in the works [12,33], the finite horizon games were considered in wide classes of the players' admissible feedback controls.…”

“…Integrating this inequality with respect to t from 0 to +∞ and using the equations (3), (7) and (13), and the inclusions z…”

“…[4,9,10,24] and references therein). However, to the best of our knowledge, such an approach to rigorous analysis of a singular zero-sum linear-quadratic differential game was applied only in these three papers [12,13,33].…”

“…In [36], the existence of an almost equilibria of a singular zero-sum game was established using a Riccati matrix inequality. In [13], using a regularization approach, a minimizing sequence of state-feedback controls was designed and an upper value was obtained in a singular zero-sum game. In [41], an asymptotic (with respect to time) Nash equilibrium was designed in a two-person singular non zero-sum game.…”

“…The exact saddle-point equilibrium in the regular infinite horizon zero-sum linear-quadratic differential game exists in a restricted class of the players' admissible feedback controls [18]. In the short conference paper [13], the singular infinite horizon game was considered in the wide class of the minimizer's admissible feedback controls. Due to the above mentioned feature of the regular infinite horizon game, only the minimizing control sequence was designed but neither an optimal maximizer's feedback control nor a saddle point.…”