In this study we focus on a zero-sum linear-quadratic differential game. One of the main features of such a game is that the weight matrix of the minimizer’s control cost in the cost functional is singular. Due to this singularity, the game cannot be solved either by applying the Isaacs Min-Max principle, [1], or the Bellman-Isaacs equation approach, [2]. In [3] such a game was analyzed with so-called regularization approach in the case where the weighting matrix of the minimizer’s control cost equals zero. In [4], the game was studied and analyzed in which the weight matrix of the minimizer’s control cost has appropriate diagonal singular form also using the regularization approach. In the present work we introduce a slightly more general case of the weight matrix of the minimizer’s control cost than in [4]. This means that only a part of coordinates of the minimizer’s control is singular, while the rest of coordinates are regular. As application we introduce a pursuit-evasion differential game and we propose two gradient methods, the Arrow-Hurwicz-Uzawa and the Korpelevich methods, for solving this game. We present numerical illustrations demonstrating the iterative procedures performances.
