- Given a matrix $A$ and a symmetric positive definite matrix $Y$, find a symmetric positive definite matrix $X$ which solves $$ X + AXA'+A^2X\left(A'\right)^2=Y $$ (This differs from the algebraic Riccati equations)
- Is the solution $X$ (if any) unique?
Any help is appreciated.
Your equation is a linear one and, consequently, has nothing to do with a Riccati equation.
Here $UX=Y$ where $U=I_{n^2}+A\otimes A+A^2\otimes A^2$ (when we stack matrix in vector row by row). cf. http://en.wikipedia.org/wiki/Kronecker_product
If $spectrum(A)=(\lambda_i)_i$, then $spectrum(U)=(1+\lambda_i\lambda_j+(\lambda_i\lambda_j)^2)_{i,j}$. We assume that $A$ is generic ; then $U$ is invertible and there is a sole solution $X=U^{-1}Y$. Morover if $X$ is a solution, then $X^T$ is also a solution (since $Y$ is symmetric) ; thus $X$ is symmetric. Yet $X$ is not necessarily $\geq 0$ as this instance shows: $A=\begin{pmatrix}0&10&4\\0&-8&-6\\3&-4&0\end{pmatrix}$ and $Y=diag(1,0,0)$ ; then the solution $X$ has a $<0$ eigenvalue $\approx -0.0012$. Note that $Y$ is only $\geq 0$, but, by a continuity reasoning , we obtain also a contradiction when $Y>0$.