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Optimal control problems for semilinear elliptic equations with control costsin the space of bounded variations are analysed. BV-based optimal controlsfavor piecewise constant, and hence 'simple' controls, with few jumps.Existence of optimal controls, necessary and suffcient optimality conditions offirst and second order are analysed. Special attention is paid on the effect ofthe choice of the vector norm in the definition of the BV-seminorm for theoptimal primal and adjoined variables.

Analysis of optimal control problems of semilinear elliptic equations by BV-functions