The value-function of an infinite-horizon linear-quadratic problem Original Research Article

For a discrete-time infinite-horizon linear-quadratic optimal control problem, under the assumption of the nonemptyness of the set of the admissible processes, we prove the existence and the uniqueness of an optimal process, we prove that the value-function is a quadratic function of the initial state, and we characterize the matrix of this quadratic value-function among the solutions of an algebraic Riccati equation by using a fixed-point theorem.