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Stability results are given for a class of feedback systems arising from the regulation of time-invariant, discrete-time linear systems using optimal infinite-horizon control laws. The class is characterized by joint constraints on the state and the control and a general nonlinear cost function. It is shown that weak conditions on the cost function and the constraints are sufficient to guarantee asymptotic stability of the optimal feedback systems. Prior results, which concern the linear quadratic regulator problem, are included as a special case. The proofs make no use of discrete-time Riccati equations and linearity of the feedback law, hence, they are intrinsically different from past proofs.