Nash equilibrium in a singular two-person linear-quadratic differential game: A regularization approach

A finite horizon two-person non zero-sum linear dynamics differential game is considered. Each player has its own quadratic cost functional to be minimized. The case where control costs of one player (the “singular” player) do not appear in both cost functionals is treated. Thus the game under consideration is singular. A new definition of the Nash equilibrium in such a game is proposed. Then, the game is solved by the regularization approach, yielding a new differential game. The new game is a regular Nash equilibrium game, and it is a cheap control game. Based on an asymptotic analysis of this game, a Nash equilibrium sequence of the pairs of the players' feedback controls in the original game is designed, and the expressions for the optimal values of the cost functionals are derived. An illustrative example is presented.