I faced to a bit weird control problem, that is minimize cost functional
\begin{equation}
 J(u) = \int\_0^Tg(t,x(t),u(t),\dot u(t))dt
\end{equation}
subject to 
\begin{equation}
 \dot x(t) = f(t,x(t),u(t),\dot u(t)), \quad x(0) = x\_0
\end{equation}
where $u(t)$ is control, a piecewise continuous function. And it differs from ordinary control problem in presence of $\dot u(t)$ term. 


I'd be happy if somebody gave me a hint how this problem could be solved or maybe reduced to an ordinary control problem.



Optimal control problem with control derivative.
Can be reduced to impulse optimal control problem. In the following system
\begin{align} 
 &\dot x(t) = f(t,x(t),u(t), v(t)) \newline 
 &\dot u(t) = \nu(t) 
\end{align}
where $\nu(t) = v(t) + \sum\limits\_i c\_i \delta(t-\tau\_i)$ is an impulse control, and 
$u(t)$ should be considered as another variable. 


For further investigation literature on impulse control theory should be looked up. Unfortunately I can't give any references on books in English. But as I know yet there is good theory only for the cases when $f$ is linear in control.


