Virtual Element Method for Control Constrained Dirichlet Boundary Control Problem Governed by the Diffusion Problem

This article develops a conforming virtual element method for a control-constrained Dirichlet boundary optimal control problem governed by the diffusion problem. An energy-based cost functional is used to approximate the control problem which results in a smooth control in contrast to the L-2(Gamma) approach which can lead to a control with discontinuities at the corners (Gong in SIAM J Numer Anal 60:450-474, 2022) . We use virtual element discretization of control, state, and adjoint variables along with a discretize-then-optimize approach to compute the optimal control is used to solve the problem. A new framework for the a priori error analysis is presented, which is optimal up to the regularity of the continuous solution. A primal-dual algorithm is used to solve the Dirichlet optimal control problem, and numerical experiments are conducted to illustrate the theoretical findings on general polygonal meshes.