Asymptotic analysis and open-loop solutions of one class of partial cheap control zero-sum differential games with state and control delays

Abstract

A finite-horizon zero-sum linear-quadratic differential game with delays in the state variable and the players' control variables is considered. The feature of the game is that the cost of some (but, in general, not all) control coordinates of the minimizing player (the minimizer) in the cost functional is much smaller than the cost of the other control coordinates of this player, the cost of the control of the maximizing player (the maximizer) and the state cost. This smallness is expressed by a positive small multiplier (a small parameter) for the square of the weighted L 2 -norm of the corresponding block of the minimizer's vector-valued control in the cost functional. By proper transformations, the originally formulated game is converted to an equivalent zero-sum differential game which does not contain delays any more. The parameter-free open-loop solvability condition of the new (undelayed) game is derived. An asymptotic analysis of the open-loop saddle point solution to this game (as the small parameter tends to zero) is carried out. This analysis yields the boundedness of the game's openloop saddle point and the parameter-free open-loop quasi saddle point of this game. Along with the parameterdependent undelayed game, another finite-horizon zero-sum linear-quadratic differential game (the degenerate game), obtained from the original one by replacing the small minimizer's control cost with zero, is considered. Its open-loop saddle point and value are derived. Relation between solutions of both games is established. Illustrative example is presented.