The optimal control problem governed by a state equation is addressed. Due to the involvement of the state constraint, a new kind of penalty functionals is constructed for the pay-off functional. Necessary conditions on the optimality for the problem are given and proven.
Wang, G. (2000). Optimal control of parabolic variational inequality involving state constraint. Nonlinear Analysis, Theory, Methods and Applications, 42(5), 789–801. https://doi.org/10.1016/S0362-546X(99)00124-8