Infinite-horizon linear-quadratic control with end-point state penalty term: The discrete-time case Original Research Article

For a given discrete-time system, consider an infinite-horizon linear-quadratic control problem with positive semidefinite cost criterion and an extra penalty term for the state variable at infinity. Our central result shows that this problem is structurally equivalent to the associated problem, where the state penalty term is required to vanish at infinity, provided only that the latter problem has finite optimal cost everywhere. For this case, the optimal cost is represented by a unique solution of the (possibly singular) algebraic Riccati equation, and if, in addition, the underlying system is left-invertible, then optimal inputs for either problem are implementable as state feedback laws, expressed in terms of the original system coefficients only, even when the control weighting matrix in the cost criterion is not invertible.