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!