In this paper, we provide a solution to the HJB equation associated with LQR-type problems for fully-actuated left-invariant control systems on compact connected Lie groups. We obtain the corresponding algebraic/differential Riccati equation and in turn generalize some of the results in Berkane and Tayebi (2015) to a broader class of Lie groups. Closed-loop stability results are also derived. (C) 2018 Elsevier Ltd. All rights reserved.