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.