Primaldual pathfollowing methods for constrained minimization problems in function space with low multiplier regularity are introduced and analyzed. Regularity properties of the path are proved. The path structure allows us to define approximating models, which are used for controlling the path parameter in an iterative process for computing a solution of the original problem. The MoreauYosida regularized subproblems of the new pathfollowing technique are solved efficiently by semismooth Newton methods. The overall algorithmic concept is provided, and numerical tests (including a comparison with primaldual pathfollowing interior point methods) for state constrained optimal control problems show the efficiency of the new concept.