How to solve this Riccati Equation?

[widemanzhao]

Here is a Riccati equation:
X*A + A'*X +X*(gamma*B1*B1' - B2*inv(R)*B2')*X + Q = 0, where A, gamma, B1, B2, R are given.
The function "care" can solve the problem like "A'*X + X*A - X*B*B'*X + Q = 0", the problem is: how can I change the term "(gamma*B1*B1' - B2*inv(R)*B2')" into "B*B'"?
I try to use the function "chol" to decompose the term "(gamma*B1*B1' - B2*inv(R)*B2')" in order to get B*B', but it does not work out with the message "chol : Matrix must be positive definite".
How can I solve that Riccati equation?
Thanks in advance!

 

[lippyka]

One approach to solving this Riccati equation is to use a numerical solver, such as an iterative solver. This can be done by approximating the matrix "(gamma*B1*B1' - B2*inv(R)*B2')" with a simpler matrix, such as a diagonal matrix. Then, the Riccati equation can be written as A'*X + X*A - X*D*D'*X + Q = 0, where D is the diagonal matrix approximation of "(gamma*B1*B1' - B2*inv(R)*B2')". The function "care" can then be used to solve this transformed problem, and the solution can then be used to approximate the solution to the original problem.