This paper investigates the mathematical programming formulation of semilinear elliptic
optimal control problems with finitely many state constraints, this allows the use of results
in parametric mathematical programming. By applying recent stability results in parametric
mathematical programming, we will obtain some new results on differential stability and
tilt stability for parametric control problems. On the one hand, we derive an explicit upper
estimate for regular subdifferential of marginal function of control problems under basic
parameter perturbations. On the other hand, we establish a characterization of tilt stability
of control problems under tilt parameter perturbations.