Linear-Quadratic Optimal Controls in Infinite Horizons

This chapter is concerned with stochastic linear-quadratic optimal control problems over an infinite horizon. Existence of an admissible control is non-trivial in this case. To tackle this issue, the notion of $$L^2$$-stabilizability is introduced. The existence of an admissible control for each initial state turns out to be equivalent to the $$L^2$$-stabilizability of the control system, which in turn is equivalent to the existence of a positive solution to an algebraic Riccati equation. Different from finite-horizon problems, the open-loop and closed-loop solvability coincide in the infinite-horizon case, and both can be established by solving for a stabilizing solution to the associated algebraic Riccati equation. As a consequence, every open-loop optimal control admits a closed-loop representation.