Stability for Parametric Control Problems of PDEs via Generalized Differentiation

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.