WeconsiderafamilyofoptimalcontrolproblemswherethecontrolvariableisgivenbyaboundaryconditionofNeumanntype.Thisfamilyisgovernedbyparabolicvariationalinequalitiesofthesecondkind.Weprovethestrongconvergenceoftheoptimalcontrolandstatesystemsassociatedtothisfamilytoasimilaroptimalcontrolproblem.ThisworksolvestheopenproblemleftbytheauthorsinIFIPTC7CSMO2011.