Optimal control for a phase field system with a possibly singular   potential
  In this paper we study a distributed control problem for a phase field system of Caginalp type with logarithmic potential. The main aim of this work would be to force the location of the diffuse interface to be as close as possible to a prescribed set. However, due to the discontinuous character of the cost functional, we have to approximate it by a regular one and, in this case, we solve the associated control problem and derive the related first order necessary optimality conditions. 
Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi, Elisabetta Rocca
math.AP math.OC