Finite Element Approximation of Optimal Control for the Heat Equation with End-Point State Constraints

Abstract

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 ($P_{h\tau}$) 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 ($P_{h\tau}$), in terms of the mesh size and time step.