AbstractIn this paper we consider abstract control‐approximation problems in Hilbert spaces, related dual problems and equivalent problems which arise from optimality conditions. We give the connections between these problems and study several properties of them, particularly properties concerning the subdifferentiability of the cost functionals. Furthermore, we derive estimates for the solutions and the optimal functional values. The results of this paper are above all useful for solving the control‐approximation problems, by the aid of the dual problems, directly or in connection with regularization meth