An Error Analysis of Second Order Elliptic Optimal Control Problem via Hybrid Higher Order Methods

This paper presents the design and analysis of a Hybrid High-Order (HHO) approximation for a distributed optimal control problem governed by the Poisson equation. We propose three distinct schemes to address unconstrained control problems and two schemes for constrained control problems. For the unconstrained control problem, while standard finite elements achieve a convergence rate of $k+1$ (with $k$ representing the polynomial degree), our approach enhances this rate to $k+2$ by selecting the control from a carefully constructed reconstruction space. For the box-constrained problem, we demonstrate that using lowest-order elements ($\mathbb{P}_0$) yields linear convergence, in contrast to finite element methods (FEM) that require linear elements to achieve comparable results. Furthermore, we derive a cubic convergence rate for control in the variational discretization scheme. Numerical experiments are provided to validate the theoretical findings.