Consider the following Lyapunov matrix equation:
$A^T X + XA = -I$,
where superscript $T$ denotes transpose, and $A \in \mathbb{R}^{n \times n}$.
I am aware that if $A$ is Hurwitz stable, then $X$ is unique and positive definite. It also holds that
$X=\int \limits_{0}^{\infty} e^{A^Tt}e^{At} dt$.
Also, $X$ must be symmetric.
The problem with the last equation is that it is extremely computationally costly. I tried coding it in MATLAB and Python, but even for $n=3$ its computation takes forever! (For $n=2$ it is good and correct).
I have read some papers on numerical solution of the Lyapunov matrix equation. My question: Is the mentioned formula useless and impractical?
Thanks a lot.