LINEAR-QUADRATIC PROBLEMS WITH INDEFINITE COST FOR DISCRETE-TIME-SYSTEMS

This paper deals with the discrete-time, infinite-horizon linear quadr atic problem with indefinite cost criterion. Given a discrete-time lin ear system, an indefinite cost-functional and a linear subspace of the state space, the problem of minimizing the cost-functional over all i nputs that force the state trajectory to converge to the given subspac e is considered. A geometric characterization of the set of all Hermit ian solutions of the discrete-time algebraic Riccati equation is given . This characterization forms the discrete-time counterpart of the wel l-known geometric characterization of the set of all real symmetric so lutions 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. 377-4011. In the se t of all Hermitian solutions of the Riccati equation the solution that leads to the optimal cost for the above-mentioned linear quadratic pr oblem is identified. Finally, necessary and sufficient conditions for the existence of optimal controls are given.