Explicit solution to the singular LQ regulation problem

An explicit solution to the multivariable discrete linear quadratic (LQ) regulation problem is obtained in the limiting singular case where the input weighting matrix tends to zero. Such a solution follows from a suitable spectral factorization of the input spectrum density matrix under the assumption that the system is stabilizable and detectable and that its transfer function matrix is of full rank. The suitable spectral factor is shown to be the product of the system´s minimum-phase image and its unitary interactor matrix. The unitary interactor matrix defined is a special case of the nilpotent interactor matrix defined by M.W. Rogozinski et al. (1987)