Nonnegative solutions of algebraic Riccati equations (Q1362674)

For the Riccati equation (RE) \(XBJB^*X -XA -A^*X -C^*C =0\) one studies the Hermitian solution which is stabilizing for the pair \((A,-BJB^*)\). The approach relies on notions and results from the theory of indefinite inner product spaces and on the Hamiltonian associated to the \(RE\). For the case \(C=0\) a very explicit criterion for nonnegativity of the stabilizing solution in terms of certain Jordan chains of \(A\) is formulated. For the case \(J= I\) and \((A,B)\) controllable the full set of nonnegative solutions is considered. A parametrization of this set is discussed and the inertia of Hermitian solutions is studied. Applications to topics arising from \(LQ\) optimal control like isolatedness, order structure, and stability are given. Finally the discrete-time counterpart of the results are discussed.