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

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

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

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