Finite Element Approximations of Optimal Controls for the Heat Equation with End-point State Constraints

This study presents a new finite element approximation for an optimal control problem (P ) governed by the heat equation and with end-point state constraints. The state constraint set S is assumed to have an empty interior in the state space. We begin with building a new penalty functional where the penalty parameter is an algebraic combination of the mesh size and the time step. Based on it, we establish a discrete optimal control problem (Phτ ) without state constraints. With the help of Pontryagin’s maximum principle and by suitably choosing the above-mentioned combination, we successfully derive error estimate between optimal controls of problems (P ) and (Phτ ), in terms of the mesh size and time step.