Is there a solution for $AX+XA^T+XBX=C$ where $X$, $B$ and $C$ are symmetric?



Solution for $AX+XA^T+XBX=C $ where $X$, $B$ and $C$ are symmetric
That equation is called a (continuous-time) algebraic Riccati equation, and there is ample literature on when they are solvable; just look for this search term. For instance, the book *Algebraic Riccati equations* by Lancaster and Rodman, or *Numerical solution of AREs* by Bini, Iannazzo, Meini.


In the generic case there is a finite number of solutions that grows exponentially with the dimension $n$: think about the scalar case, for instance, which already has 2 (possibly complex) solutions if $b \neq 0$. So usually the first point is understanding which solution you need. Typically one seeks the so-called *stabilizing solution* (or the *antistabilizing* one), i.e., the one such that $A^T+BX$ has eigenvalue in the left (resp. right) half-plane.


