I've just a short question (at least I think so):
I want to study optimal control problems of the form:
$\min_{\omega\in L^2(\Omega)} J(\omega)+\frac{\epsilon}{2} \|\omega\|^2_{L^2{\Omega}}$
s.t.
$\omega\geq 0 ~a.e.$, $\|\omega\|_{L^1(\Omega)}\leq 1$
where $\Omega$ is an open and bounded subset of $R^d$ and $J(\omega)$ is a convex functional with $J(\omega)\geq 0$ for admissible $\omega$. I'm especially interessted in the structure of the (unique) optimal solution in dependance of $\epsilon$.
My question(just for curiosity) : Are there any works on control problems, where control constraints in a weaker norm are given( like in this case, controls from $L^2(\Omega)$ and control constraints in the $L^1$ Norm)? I'm looking for this quite some time,but couldn't find anything. Thanks a lot.