Finally, I get the continuous-time Riccati recursion correctly, which means that what I told is wrong.
I followed the paper, "Connection between continuous and discrete Riccati equations with applications to Kalman filtering" back in 1980s. I modified the procedure to incorporate the different rates of each action.
I am ready to tackle the optimization problem. I think it is better to consider scalar case ($\Pi$ is scarlar) to check its convexity.