# Stochastic Linear Quadratic Optimal Control Problems in Infinite Horizon

arXiv: https://arxiv.org/abs/1610.05021 · HTML: https://mlsanity.com/paper/1610.05021/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Oct 2016.

## Built on by (most cited first, 1 of 1)

- [Convergence of Policy Gradient for Stochastic Linear-Quadratic Control Problem in Infinite Horizon](https://mlsanity.com/paper/2404.11382/) (2024)

## Abstract (arXiv)

This paper is concerned with stochastic linear quadratic (LQ, for short) optimal control problems in an infinite horizon with constant coefficients. It is proved that the non-emptiness of the admissible control set for all initial state is equivalent to the L2-stabilizability of the control system, which in turn is equivalent to the existence of a positive solution to an algebraic Riccati equation (ARE, for short). Different from the finite horizon case, it is shown that both the open-loop and closed-loop solvabilities of the LQ problem are equivalent to the existence of a static stabilizing solution to the associated generalized ARE. Moreover, any open-loop optimal control admits a closed-loop representation. Finally, the one-dimensional case is worked out completely to illustrate the developed theory.