Optimal control problem with spike source and “split” state

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.