The role of terminal cost/reward in finite-horizon, discrete-time LQ optimal control

The optimal control problem for time-invariant linear systems with quadratic cost is considered for arbitrary, i.e., non-necessarily positive semidefinite, terminal cost matrices. A classification of such matrices is proposed, based on the maximum horizon for which there is a finite minimum cost for all initial states. When such an horizon is infinite, the classification is further refined, based on the asymptotic behavior of the optimal control law. A number of characterizations and other properties of the proposed classification are derived. In the study of the asymptotic behavior, a characterization is given of those matrices A such that the image of AtS0 converges in the gap metric for any subspace S0.