I want to test the controllability of an LTI system constrained by convex control constraint set in the following two steps: i) check the rank of controllability matrix C(A,B); ii) check that zero is in the interior of the control constraint set.
Check this link: link. springer.com/content/pdf/10.1007%2F978-1-4419-1549-8_9.pdf
You can use pole placement to set the eigen values where you wish in the left half plane and heck for Routh Hurwitz Criteria of Stability. That is a sufficient method that guarantees convergence of LTI Systems.