Hello, members. I have a problem for the following problem when I derive an optimization algorithm for stochastic singular systems $$S(k+1)=A(k)S(k)A^{T}(k)+R(k)+F(k)S(k+1)F^{T}(k)$$ where $R(k)>=0$ So, how to calculate $S$, is there analytic solution or numerical solution to $S$?

This problem is different from the following one On solution of a recursion with rectangular matrices

Thanks for your help