The optimal control problem for time-invariant linear systems with quadratic cost is considered forarbitrary, i.e., non-necessarily positive semidefinite, terminal cost matrices. A classification of such matricesis 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 ofthe optimal control law. A number of characterizations and other properties of the proposed classificationare derived. In the study of the asymptotic behavior, a characterization is given of those matrices A such thatthe image of AtS0 converges in the gap metric for any subspace S0.
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