Getting the minimum of a mixed functional

Question

I have a functional $T$ defined on the attached picture. The functional always gives non-negative values. So it has a non-negative infinum

I'm trying to figure out whether this infinum is achievable and if yes, what should be the equation on the function that minimizes the functional.

What is confusing is the combination of delta in plain form and under an integral.

So my questions are: 1) how to check whether the infinum is achievable; 2) if it is, how to write an optimal equation for the minimizing function?

Thanks for any suggestions!