  This paper addresses an open problem in the area of linear quadratic optimal
control. We consider the regular, infinite-horizon,
stability-modulo-a-subspace, indefinite linear quadratic problem under the
assumption that the dynamics are stabilizable. Our result generalizes previous
works dealing with the same problem in the case of controllable dynamics. We
explicitly characterize the unique solution of the algebraic Riccati equation
that gives the optimal cost and optimal feedback control, as well as necessary
and sufficient conditions for the existence of optimal controls.
