Optimization of Integral over Decreasing Functions with Varying Integration Limit

Question

I need to minimize the following $$ \int_0^{T(b)} \left(e^{-x \: t}a \cdot b(t) - e^{-y\:t} c \right)\: dt $$ over the space of weakly decreasing functions (so I have the constraint that $\dot{b}(t)\leq 0$), where the upper integration limit $T(b)$ is given by the solution to $$ \int_0^{T(b)}\left( e^{-x \: t} f \cdot b(t) - e^{-y \: t} g \: c\right) \:dt = V.$$ I did not know whether there is an extension of the standard calculus of variation approach for varying integration limits.