The study of optimal control problems governed by partial differential equations (PDEs) has been a central topic in applied mathematics and scientific computing. Such problems involve determining a control variable that minimizes a given cost functional subject to PDE constraints. In this talk, we focus on finite element approximations for a class of optimal control problems governed by second-order linear elliptic PDEs. The discussion includes both a priori and a posteriori error analysis, along with a detailed investigation of the convergence and quasi-optimality properties of adaptive finite element methods (AFEM).

The talk will be presented in two parts. The first part of the talk focuses on the adaptive convergence of conforming AFEM for Dirichlet boundary control problems governed by PDEs in divergence form. We begin with an energy-space-based approach to the problem governed by the Poisson equation. Using techniques analogous to the Aubin–Nitsche duality argument, we establish the key result of quasi-orthogonality under mild additional regularity assumptions on certain PDEs and a smallness condition on the initial mesh size. Building on this, we prove both the convergence and quasi-optimality of the conforming AFEM.We then extend this framework to a more general setting, where the constraint PDE is a second-order general linear elliptic PDE. Within this generalized framework, convergence and quasi-optimality of the conforming AFEM are achieved without imposing any additional regularity assumptions or mesh-size restrictions, thereby generalizing the earlier results. In the second part of the talk, we consider optimal control problems where the constraint PDE is a second order elliptic PDE in non-divergence form. We first discuss a distributed control problem and subsequently a Dirichlet boundary control problem, along with their respective formulations. To approximate the solutions, we develop a quadratic C0 interior penalty method in each setting. Under minimal regularity assumptions on the optimal state, costate, and control, we establish both a priori and a posteriori error estimates. While the constants in a priori and reliability estimates for the distributed control problem depend on the domain geometry, the corresponding estimates for the Dirichlet control problem are geometry independent, achieved through a suitable choice of the discrete bilinear form and refined estimates of the enriching operator.