The singular zero-sum differential game with stability using $H_\infty$ control theory

In this paper we consider the time-invariant, finite-dimensional, infinite-horizon, linear quadratic differential game. We will derive sufficient conditions for the existence of (almost) equilibria as well as necessary conditions. Contrary to all classical references we allow for singular weighting on the minimizing player in the cost-criterion. It turns out that this problem has a strong relation with the singular $H_\infty$ problem with state feedback, i.e. the $H_\infty$ problem where the direct feed through matrix from control input to output is not necessarily injective.