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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.

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    $\begingroup$ This question may belong to math overflow SE, after reviewing literature I have not found reference to this form of a control problem. $\endgroup$
    – Joce
    Mar 9, 2017 at 8:10
  • $\begingroup$ There is some literature on the problem, under the keyword "parameter identification". E.g. sciencedirect.com.gaelnomade-2.grenet.fr/science/article/pii/… . So I am now of the opinion that this question can pertain to math SE and could provide an answer to my own question if reopened. $\endgroup$
    – Joce
    Mar 14, 2017 at 10:02

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