Suppose all of the eigenvalues of $A$ locate strictly on the right half plane. $(A,B)$ is controllable, $H$ is symmetric and strictly positive definite.
I wonder is there a optimal solution $H^*$ for the following problem:
$\inf \|H\|_2$
$s.t. A-BB^TH $ is stable, i.e. all of its eigenvalues are strictly located on the left half plane.