I am facing a differential equation - with boundary condition $v(T)$ given - without an analytic solution but still need to understand how the solution is affected by a change of the function's value.
$$ v(t) = v(T) - \int_t^T \dot{v}(s) ds $$
All I know is $\dot{v}(t)$ and $v(T)$. If I consider a change in a particular $v(s)$, I am interested in the effect on $v(t)$ for all $t<s$. If I am not mistaken this should be given by
$$ \frac{d v(t)}{d v(s)} = - \int_t^s \frac{d\dot{v}(\tau)}{dv(\tau)}\frac{dv(\tau)}{dv(s)} d\tau.$$
The first part under the integral $\frac{d\dot{v}(\tau)}{dv(\tau)}$, I can quantify. However, the second is then again the same as the left-hand side. Is there a way to simplify this or a result that may apply to this and help studying this change?
Hint:
From the look of the equation, it tells us that
$$\int_t^s \dot{v}(s) ds$$
is an integral of acceleration that subracts from a given $v(T)$.