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I am currently working on optimal control problems where the control function belongs to the Sobolev space $ H^1(\Omega) $ and the objective functional is of the type $ J(u,y)=\int_\Omega L(x,y)dx+ \frac{\alpha}{2}\|u\|^2_{H^1(\Omega)} $, where $y$ is a solution of a partial differential equation. Most of the literature I have encountered primarily deals with control functions in $ L^2(\Omega) $ (that is, with a tracking term like $\frac{\alpha}{2}\|u\|^2_{L^2(\Omega)} $).

I am particularly interested in understanding the second-order optimality conditions for these types of problems. Specifically, I am looking for references or papers where the authors address second-order conditions for controls in $ H^1(\Omega) $.

If anyone knows of any books, articles, or papers that cover this topic, could you please share them?

Thank you very much for your assistance!

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  • $\begingroup$ Do you have constraints on $u$? $\endgroup$
    – gerw
    Commented May 30 at 10:39
  • $\begingroup$ @gerw Yes, I do. Pointwise box constraints in $L^\infty(\Omega}$. $\endgroup$
    – MATAKA
    Commented May 31 at 21:22

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If I understand your comment correctly, you are minimizing over a set $$ U := \{ u \in H^1(\Omega) \mid a \le u \le b \} $$ for some $a, b \in L^\infty(\Omega)$. Such a set is polyhedric in the sense of Mignot. This also allows for nice second-order conditions, see, e.g., Section 3.2.3 in the book "Perturbation analysis of optimization problems" by Bonnans and Shapiro.

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  • $\begingroup$ Exactly. Thanks for the reference. $\endgroup$
    – MATAKA
    Commented Jun 6 at 12:50

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