Optimality conditions for state constrained nonlinear control problems

The problem of optimal control of nonlinear control and state constrained control problems, where the state constraint may involve differential operators and the cost functionals may be nonsmooth, is studied. For this class of problems, necessary optimality conditions using techniques from infinite dimensional optimization theory adapted to the framework of control problems are derived. It is shown that the underlying structure admits a considerable relaxation of the classical constraint qualifications. The theory then is applied to examples of various nonlinear elliptic equations and state constraints.