The Indefinite LQ-Problem: Existence of a Unique Solution

In this paper we consider the problem under which conditions there is, for a discrete-time system and an arbitrary finite planning horizon, a unique control that minimizes a general quadratic cost functional. The cost functional differs from the usual one considered in optimal control theory in the sense that we do not assume that the considered weight matrices are (semi) positive definite. The system is described by a linear time-invariant recurrence equation and has an exogenous component. For single input, single output systems both necessary and sufficient conditions are derived.