You are multiplying 2 matrix variables in the first constraint. That is not allowed in CVX. I will presume that the first constraint is non-convex unless you show otherwise.

Additionally, you need to change the last constraint to lambda_max(A - B1*k) <= g3;
CVX will enforce that the argument of lambda_max is symmetric.

No. gram command cannot be used in CVX (on CVX variables or expressions). Controllability grammians are not my area, so I can’t help with model formulations. I leave it to you to determine whether you can formulate an appropriate LMI, which will be acceptable to CVX.