Abstract: An optimal control problem for a semilinear elliptic equation is discussed, where the control appears nonlinearly in the state equation but is not included in the objective functional. The existence of optimal controls is proved by a measurable selection technique. First-order necessary optimality conditions are derived and two types of second-order sufficient optimality conditions are established. A first theorem invokes a well-known assumption on the set of zeros of the switching function. A second relies on coercivity of the second derivative of the reduced objective functional. The results are applied to the convergence of optimal state functions for a finite element discretizion of the control problem.

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