Maximum principle for state-constrained control of some semilinear parabolic differential equations (Q1810932)

The authors study the optimal control problem for some semilinear parabolic differential equations, having a local solution only, which may be governed by nonmonotone operators. They transform the original optimal control problems into optimization problems of state and control variables by considering the state equations as mixed constraints involving the above two variables. The controls enter the state systems nonlinearity, so they define new penalty functionals which transform the original optimal control problems into optimization problems which approximate original problems. Then by Ekeland's variational principle they obtain the optimal solution. Finally, they pass to the limit to obtain the maximum principle for the original optimal control problem.