A Galerkin Approach to the Generalized Karush–Kuhn–Tucker Conditions for the Solution of an Elliptic Distributed Optimal Control Problem with Pointwise State and Control Constraints

Abstract We develop a convergence analysis for the simplest finite element method for a model linear-quadratic elliptic distributed optimal control problem with pointwise control and state constraints under minimal assumptions on the constraint functions. We then derive the generalized Karush–Kuhn–Tucker conditions for the solution of the optimal control problem from the convergence results of the finite element method and the Karush–Kuhn–Tucker conditions for the solutions of the discrete problems.