Consider the Lyapunov equation $$ A^TX+XA=−BB^T $$ with Hurwitz matrix A.
We know, its solution $X$ be strictly positive definite if the pair $(B,A)$ is observable.
In this case, is it possible to obtain the lower estimation for $\lambda_\min(X)$ (like well-known $\lambda_\min(X)\ge\frac{\lambda_\min(Q)}{2\|A\|}$ for $-Q\prec0$ in the right-hand side)?