This paper considers the H-x control problem for descriptor systems that possibly have impulsive modes and/or j omega-axis zeros. First, we propose matrix inequalities that give a generalized stability condition and an H-x norm condition for descriptor systems. Using these matrix inequalities, we show that the solvability of a set of matrix inequalities is necessary and sufficient to the existence of a proper controller that satisfies a prescribed H-x norm condition as well as stabilizing the closed-loop system and eliminating all impulsive modes. These inequalities are equivalent to certain linear matrix inequalities, to which we can get solutions whenever they exist using efficient polynomial-time algorithms.