For $p \in \mathbb{R}$, consider the following problem: \begin{equation} \label{1} \begin{cases} \operatorname{div}(a \nabla u ) = p\delta_{x_0} \quad \text{in } \Omega \\ u=0 \quad \text{on } \partial \Omega ; \end{cases} \end{equation} under the assumption that $a \in L^\infty$ is constant in some neighbourhood of $x_0$, i.e. $a(\mathbf{x})= a_0 \text{Id}$ for $\mathbf{x} \in B=B_{r_0}(x_0)$, $a_0 \in \mathbb{R}$, we can look for a solution in the form $$ u(x) = \psi(x) + K(x-x_0), $$ where $K(\cdot)$ is the fundamental solution (up to the constants $a_0,p$) of the Laplace operator and $\psi \in H^1(\Omega)$ satisfies a classical, well-posed, Neumann problem with data depending on $K|_{\Omega \setminus B}$. Note that the solution $u$ is not quite regular globally since it reads the singularity of $K$ at $x_0$.

Nevertheless, we can set up a control problem "away" from $x_0$ with the number $p$ as control and the quadratic tracking cost functional $$ \min_{p} \left( \frac{1}{2} \| u(p) - u_{d} \|_{0, \Omega \setminus B}^2 + \frac{1}{2} |p|^2 \right), $$ for some desired state $u_d \in L^2$, $u(p)$ being the solution of the above problem (in the above sense!) corresponding to the control $p$.

I see some problems arising while trying to formulate go-to results like necessary optimality conditions: it is not clear what should be a suitable adjoint problem, since a weak formulation is only available for $\psi=\psi_p$, but the state $u$ also depends on $K=K_p$, making $u(p)$ not a trivial translation of $\psi$. Moreover, the choice of the $L^2(\Omega \setminus B)$ norm in the optimization was made to somehow regularize $u$, on the other hand:

  • Is the control problem still meaningful, as we are trying - in principle - to approximate a global a priori chosen desired state taking into account only the behavior away from a fixed point?
  • Working with integrals in $\Omega \setminus B$ rather than $\Omega$ gives rise to unwanted boundary terms in integrations by parts.

Are there any references for optimal control problems of this kind?

Note: I know that it is possible to set up a global weak formulation for this type of Dirac-source problems (see reference) using sharp functional analysis results on weighted spaces, but this is not known to be possible for larger classes of operators, like those I have to deal with in my research. Therefore, this is a model example and the "split" solution is most likely the only option.

Reference: Allendes, Alejandro, et al. "An a posteriori error analysis for an optimal control problem with point sources." ESAIM: Mathematical Modelling and Numerical Analysis 52.5 (2018): 1617-1650.


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