We know that the Lyapunov sequence is:
$$P_{n+1}=AP_{n}A^*+R_0$$
$$P_0=\Pi_0$$
or in summation form:
$$P_{n+1}=A^n\Pi_0(A^*)^n+\sum_{k=0}^n A^kR_0(A^*)^k$$
$A$ is a stable matrix and $R_0$ is positive definite matrix. In the case which $n\rightarrow\infty$ the sequence equal to the solution of Lyapunov equation:
$$P=APA^*+R_0 $$
Now, if the $R_0$ varies through the sequence:
$$P_{n+1}=AP_{n}A^*+R_n$$
$$P_0=\Pi_0$$
or in summation form:
$$P_{n+1}=A^k\Pi_0(A^*)^k+\sum_{k=0}^n A^kR_{n-k}(A^*)^k$$
In this case, is there any anwer to the summation when $n\rightarrow\infty$?
Or is there any limit which the summation bounded with it?
And if these answers exsit, do they usable for varying stable matrix ($A_n$)?
Note: By "vary" I mean that the matrix elements fluctuate in an interval. For example for $R_n$, the diagonal elements vary through an interval $[\sigma_{min}^2,\sigma_{max}^2]$.
Lyapunov sequence with time variant coefficients.
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linear-algebra
sequences-and-series
convergence
stability-theory