I am trying to solve the following equation for $P$
$0=X(t)^{\rm T}\left(Y^{\rm T}P+PY-\gamma P-PZR^{-1}Z^{\rm T}P+S^{\rm T}QS\right)X(t)+\mu^{\rm T}R\mu,$
where $Y\in\mathbb{R}^{n\times n}$, $Z\in\mathbb{R}^{n\times 1}$, $C\in\mathbb{R}^{1\times n}$, $R\in\mathbb{R}$, $P\in\mathbb{R}^{n\times n}$, and $\mu\in\mathbb{R}$. $Y$, $\gamma$, $Z$,$S$,$\mu$, $R$ are known.
we can solve the case
$0=X(t)^{\rm T}\left(Y^{\rm T}P+PY-\gamma P-PZR^{-1}Z^{\rm T}P+S^{\rm T}QS\right)X(t)$ as it is exactly an algebraic Ricatti equation, and can be solved using iterative methods such as Newtons method as depicted in "On an Iterative Technique for Riccati Equation Computations" by D. Klienman http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1098829&tag=1ftp://ftp.math.ucla.edu/pub/camreport/cam05-15.pdf.
but with extra term $\mu^{\rm T}R\mu$, I am stuck. I am trying to figure out what additonal steps do I need to do to solve the equation by using the same iterative newton method with the extra term $\mu^{\rm T}R\mu$ along with the algebraic Ricatti equation. Appreciate any ideas, hints or help :).