# Open-Loop and Closed-Loop Solvabilities for Stochastic Linear Quadratic Optimal Control Problems

arXiv: https://arxiv.org/abs/1508.02163 · HTML: https://mlsanity.com/paper/1508.02163/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Aug 2015.

## Built on by (most cited first, 3 of 3)

- [Convergence of policy gradient methods for finite-horizon exploratory linear-quadratic control problems](https://mlsanity.com/paper/2211.00617/) (2022)
- [Comba: Improving Bilinear RNNs with Closed-loop Control](https://mlsanity.com/paper/2506.02475/) (2025)
- [Convergence of Proximal Policy Gradient Method for Problems with Control Dependent Diffusion Coefficients](https://mlsanity.com/paper/2505.18379/) (2025)

## Abstract (arXiv)

This paper is concerned with a stochastic linear quadratic (LQ, for short) optimal control problem. The notions of open-loop and closed-loop solvabilities are introduced. A simple example shows that these two solvabilities are different. Closed-loop solvability is established by means of solvability of the corresponding Riccati equation, which is implied by the uniform convexity of the quadratic cost functional. Conditions ensuring the convexity of the cost functional are discussed, including the issue that how negative the control weighting matrix-valued function R(s) can be. Finiteness of the LQ problem is characterized by the convergence of the solutions to a family of Riccati equations. Then, a minimizing sequence, whose convergence is equivalent to the open-loop solvability of the problem, is constructed. Finally, an illustrative example is presented.