Mixed Riccati equation solution

Dear researchers,

I have the following Riccati like differential equations need to be solved. Can anyone give me a hint?

We can assume the system matrices A, B and performance matrices R, Q are time-invariant.w(t) is an external signal related to system disturbance. I understand the first equation is a standard Riccati equation, and the infinite-time solution can be easily obtained. However, in this case, I don't know how to solve the mixed Riccati equations set, especially the second differential equation in the set.The boundary conditions are: