A discussion on the discrete-time finite-horizon indefinite LQ problem

The aim of this paper is to compare and discuss some old and new results on the discrete-time finite-horizon linear quadratic (LQ) optimal control problem in the case where the quadratic forms in the performance index are not assumed to be positive semidefinite, but only symmetric. We show in particular that the necessary and sufficient conditions presented in most contributions in the literature for the existence of a solution to this problem are in fact only sufficient. Our aim is to investigate this issue further, by addressing some of the most delicate and counterintuitive issues that arise in this context.