The discrete algebraic Riccati equation and linear matrix inequality

In this paper we study the discrete time algebraic Riccati equation and its connection to the discrete time linear matrix inequality. We show that in general only a subset of the set of rank-minimizing solutions of the linear matrix inequality correspond to the solutions of the associated algebraic Riccati equation, and study under what conditions these sets are equal. In this process we also derive very weak assumptions under which a Riccati equation has a solution