In this paper, we consider an H-infinity problem using unstable weighting functions. Such weights are used to achieve asymptotically disturbance attenuation and/or asymptotically reference tracking, and they cannot be handled by the standard H-infinity control theory. For this problem, we present a necessary and sufficient condition to exist a solution using Riccati inequalities. This condition is derived under no unnecessary assumption to enlarge the applicability to various control problems. Moreover, introducing the assumption that the image of the zero Vectors of G(12) and G(21) are unique, we obtain a LMI condition in order to check the solvability numerically. A numerical example is presented to demonstrate how to calculate a controller.