In this paper we consider the optimal control problems of linear distributed parameter systems of parabolic type with inequality state constraints. We apply the duality methods and transform the original infinite-dimensional problems to the dual finite-dimensional problems. We solve the following three types of optimal control problem. (1) The determination of input-time functions with minimum energy. (2) The determination of input-distribution functions with minimum norm. (3) The determination of initial states with minimum norm. Several illustrative examples are given with numerical results.