Optimal control of elliptic PDEs at points

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 that our problem is related to ones with state constraints at points. The analysis and numerical analysis differs from when the fidelity is in the L2-norm because we need the state space to embed into the space of continuous functions. In this paper, we discretize the problem using two different piecewise linear finite element methods. For each discretization we use two different approaches to prove a prioriL2-error estimates for the control. We discuss the differences between these methods and approaches, and present numerical results that agree with our analytical results.

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