I have the following optimal control problem
$$
J=\int\_0^TF(t,y\_1(t),y\_2(t))dt \to \min,
$$
subject to
\begin{align}
 &\dot y\_1(t) = f(t,y\_1(t),y\_2(t)) + g(t)\nu(t),\\
 &\dot y\_2(t) = \nu(t),
\end{align}
where $\nu$ is an impulsive control, and $y\_i$ are phase coordinates. Results on necessary conditions (maximum principle) of this type of problems which I've seen require functions $f$ and $g$ be smooth in $t$. But in my case both of them have unbounded variation in $t$ in any interval (trajectory of Brownian motion as a particular case).


So how this problem can be solved and are there any result on this type of problems?



On impulsive optimal control with functions of not bounded variation
In case someone else is interested, there is a paper by Arutyunov, Karamzin and Pereira, "Pontryagin’s maximum principle for constrained impulsive control problems", which presents a version of maximum principle for problems with non-smooth data. Here is a [link](http://www.sciencedirect.com/science/article/pii/S0362546X11002513) for it on ScineceDirect.


