Linear Quadratic Problems with Indefinite Cost for Discrete Time Systems

摘要

The discrete time infinite horizon linear quadratic problem with indefinite costcriterion is discussed. 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 geometric characterization of the set of all real symmetric solutions of the continuous time algebraic Riccati equation. 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 problems is identified. Necessary and sufficient conditions for the existence of optimal controls are given.