Optimal control of elliptic PDEs at points
Provides rigorous error analysis for a specific class of optimal control problems with point observations, which is an incremental contribution to numerical analysis of PDE-constrained optimization.
The paper analyzes and discretizes an elliptic optimal control problem with pointwise state evaluations in the objective, deriving a priori L^2 error estimates for the control using two finite element methods, with numerical results confirming the theory.
We consider an elliptic optimal control problem where the objective functional contains evaluations of the state at a finite number of points. In particular, we use a fidelity term that encourages the state to take certain values at these points, which means our problem is related to ones with state constraints at points. The analysis and numerical analysis differs from when the fidelity is in the $L^2$ norm because we need the state space to embed into the space of continuous functions. In this paper we discretise the problem using two different piecewise linear finite element methods. For each discretisation we use two different approaches to prove a priori $L^2$ error estimates for the control. We discuss the differences between these methods and approaches and present numerical results that agree with our analytical results.