Linear Quadratic Problems with Indefinite Cost for Discrete Time Systems

This paper deals with the discrete-time, infinite-horizon linear quadratic problem with indefinite cost criterion. Given a discrete-time linear system, an indefinite cost-functional and a linear subspace of the state space, the problem of minimizing the cost-functional over all inputs that force the state trajectory to converge to the given subspace is considered. A geometric characterization of the set of all Hermitian solutions of the discrete-time algebraic Riccati equation is given. This characterization forms the discrete-time counterpart of the well-known geometric characterization of the set of all real symmetric solutions of the continuous-time algebraic Riccati equation as developed by Willems [IEEE Trans. Automat. Control, 16 (1971), pp. 621–634] and Coppel [Bull. Austral. Math. Soc., 10 (1974), pp. 377–401]. In the set of all Hermitian solutions of the Riccati equation the solution that leads to the optimal cost for the above-mentioned linear quadratic problem is identified. Finally, necessary and sufficient conditions for the existence of optimal controls are given.