Pick matrix conditions for sign-definite solutions of the algebraic Riccati equation

Necessary and sufficient conditions for the existence of positive and negative semidefinite solutions of algebraic Riccati equations (AREs) corresponding to linear quadratic problems with an indefinite cost functional are formulated. The tests known from the literature cannot be efficiently carried out, either because they apply only to special cases, or because an infinite number of matrices of unbounded dimension should be checked for positive semidefiniteness. The results presented in this paper, instead, characterize the extremal solutions of the ARE in terms of the finite Pick matrices induced by certain two-variable polynomial matrices associated with the equation.