Spectral theory of the linear-quadratic optimal control problem: Discrete-time single-input case

The single-input discrete-time linear-quadratic optimal control problem, together with the classical time-domain and frequency domain conditions, is reviewed and treated from a novel point of view. We show that when the quadratic cost is not positive semidefinite, the problem of the boundedness of the infimum in both the zero and the free terminal state cases is related to the positivity of a self-adjoint Hilbert space operator. The structure of the spectrum of the operator in question, in both cases, is investigated, leading to a clarification of the linkage between the time-domain and the frequency-domain conditions for boundedness. More over, it is shown that the spectra of the operators in question reflect the relevant properties of the control problem.