The general discrete-time linear-quadratic control problem

The General Discrete-Time Linear-Quadratic Control Problem, together with the classical time-domain and frequency-domain conditions, is reviewed and treated from a novel point of view. When the quadratic cost is allowed to be non positive semi-definite, an important problem is to determine whether or not the cost admits a lower bound. We show that this can be related to the positivity of a self-adjoint Hilbert space operator. This point of view clarifies many features that have remained obscure in the literature. A new frequency-domain technique and a fast algorithm to check the positivity of the operator in question are outlined.