I'm not sure about simplifying, but you can easily write your objective functional in Bolza form like this:

$$
\begin{align}
&\min_{u(t) \in \Omega(t)} \, J = z(T)^2 + \int_{0}^{T} s(t)u(t)dt \\
s.t. &\frac{dz(t)}{dt} = r(t)u(t),\quad z(0) = 0 \\
& 0 < u(t)< 1
\end{align}
$$