  In this work we investigate explicit and implicit difference equations and
the corresponding infinite time horizon linear-quadratic optimal control
problem. We derive conditions for feasibility of the optimal control problem as
well as existence and uniqueness of optimal controls under certain weaker
assumptions compared to the standard approaches in the literature which are
using algebraic Riccati equations. To this end, we introduce and analyze a
discrete-time Lur'e equation and a corresponding Kalman-Yakubovich-Popov
inequality. We show that solvability of the Kalman-Yakubovich-Popov inequality
can be characterized via the spectral structure of a certain palindromic matrix
pencil. The deflating subspaces of this pencil are finally used to construct
solutions of the Lur'e equation. The results of this work are transferred from
the continuous-time case. However, many additional technical difficulties arise
in this context.
