Optimal control problems for semilinear elliptic equations with control constraints and pointwise state constraints are studied. Several theoretical results are derived, which are necessary to carry out a numerical analysis for this class of control problems. In particular, sufficient second-order optimality conditions, some new regularity results on optimal controls and a sufficient condition for the uniqueness of the Lagrange multiplier associated with the state constraints are presented.

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... ematics Subject Classification. 49J20, 49K20, 35J65. In [3, 5] , error estimates for approximated locally optimal controls were shown for problems with semilinear elliptic equation and finitely many state constraints. Recently, in [17] , higher order error estimates were established for a similar setting with control vectors instead of control functions. A study of these three papers reveals that second-order sufficient conditions at (locally) optimal controls are indispensable to obtain results on convergence or approximation of optimal controls. This is due to the nonconvex character of the problems with nonlinear equations. It is meanwhile known that second-order sufficient optimality conditions are fairly delicate under the presence of state constraints. In [9], second-order sufficient conditions were established, which are, in some sense, closest to associated necessary ones and admit a form similar to the theory of nonlinear programming in finite-dimensional spaces. Here, we briefly discuss this result and show its equivalence to an earlier form stated in [8] that was quite difficult to explain.