I am just beginning with optimal control. My purpose is to solve control problems of the form: $$ \min_u \|y^u-z\|^2 + \|u\|^2 $$ subject to $y^u$ solution of $$ y - u \nabla^2 y = f $$ in a bounded domain, with $\nabla^2$ the Laplace operator, plus appropriate BC on $y$ (say Dirichlet homogeneous). The control parameter $u$ is thus the inverse of the square of the wavenumber of this Helmholtz equation.
Most of the example problems I find are subject to models that can be written as $A y = B u + f$. This is not the case of my problem, and I am at odds to use the abstract frameworks in this case without an example which would have a similar cross coupling between state and control parameters.
References with similar problems or the keyword referring to this type of optimal control problem would be appreciated. Possibly it has not been studied as an optimal control problem.