Let $T>0$ be fixed and let $C:\mathbb (0, T)\times(0, T)\longrightarrow[0, +\infty[$ a function regular as much as you want (say at least differentiable w.r.t. the second time variable and w.r.t. the first if necessary). Theoretically $C(t, \tau)$ has to represent a cost to reach a certain target at a starting time $t$, with a crossing time $\tau$ and with an arrival time given by $t+\tau$. What is the derivative of $C$ w.r.t. the arrival time $t+\tau$?
Another point that can be helpful. In the model assumptions, there is the possibility to think to $C$ as the sum of two costs, say $C_1$ and $C_2$, with $C_1$ depending only on the crossing time variable $\tau$ (so $C_1(\tau)$), and $C_2$ depending on the starting time variable $t$ and the arrival time $t+\tau$ (so $C_2(t, t+\tau)$) with the derivative w.r.t. the arrival time variable not depending on the starting time one (say $\partial_{t+\tau}C_2(t, t+\tau)=\tilde C(t+\tau)$).
Can someone help me?
Thank you