Nash Equilibrium Sequence in a Singular Two-Person Linear-Quadratic Differential Game

A finite-horizon two-person non-zero-sum differential game is considered. The dynamics of the game is linear. Each of the players has a quadratic functional on its own disposal, which should be minimized. The case where weight matrices in control costs of one player are singular in both functionals is studied. Hence, the game under the consideration is singular. A novel definition of the Nash equilibrium in this game (a Nash equilibrium sequence) is proposed. The game is solved by application of the regularization method. This method yields a new differential game, which is a regular Nash equilibrium game. Moreover, the new game is a partial cheap control game. An asymptotic analysis of this game is carried out. Based on this analysis, the Nash equilibrium sequence of the pairs of the players’ state-feedback controls in the singular game is constructed. The expressions for the optimal values of the functionals in the singular game are obtained. Illustrative examples are presented.

SOLUTION OF A SINGULAR ZERO-SUM LINEAR-QUADRATIC DIFFERENTIAL GAME BY REGULARIZATION

A linear-quadratic zero-sum singular differential game, where the cost functional does not contain the minimizer's control cost, is considered. Due to the singularity, the game cannot be solved either by applying the MinMax principle of Isaacs, or by using the Bellman–Isaacs equation method. In this paper, the solution of the singular game is obtained by using an auxiliary differential game with the same equation of dynamics and with a similar cost functional augmented by an integral of the square of the minimizer's control multiplied by a small positive weighting coefficient. This auxiliary game is a regular cheap control zero-sum differential game. For the analysis of such a cheap control differential game, in the present paper a singular perturbation technique is applied. Based on this analysis, the minimizing control sequence and the maximizer's optimal strategy in the original (singular) game are derived. Moreover, the existence of the value of the original game is established and its expression is derived. The solution is illustrated by an interception example.