Sufficient conditions for unique global solutions in optimal control of semilinear equations with C1 nonlinearity

We consider a semilinear elliptic optimal control problem possibly subject to control and/or state constraints. Generalizing previous work, presented in Ahmad Ali, Deckelnick and Hinze (2016) we provide a condition which guarantees that a solution of the necessary first order conditions is a global minimum. A similiar result also holds at the discrete level where the corresponding condition can be evaluated explicitly. Our investigations are motivated by G¨unter Leugering, who raised the question whether the problem class considered in Ahmad Ali, Deckelnick and Hinze (2016) can be extended to the nonlinearity φ(s) = s|s|. We develop a corresponding analysis and present several numerical test examples demonstrating its usefulness in practice.