Optimal control of parabolic problems with state constraints: a penalization method for optimality conditions

State constrained optimal control problems governed by parabolic evolution equations, with convex state constraints, are studied. a (first-order) decoupled optimality system is obtained. With a `weak´ assumption the existence of Lagrange multipliers (as measures) is proved, even for nonqualified problems. It is noted that the penalization method considered can be applied to any problem of control with state constraints (for instance, boundary control with Dirichlet boundary condition problems or nonlinear problems) and provides decoupled optimality systems that can be solved with classical multiplier methods