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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.

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up vote 1 down vote accepted

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.

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Use the standard trick of reducing a higher order ODE to a system of first order ODEs. You should let $v(t)\triangleq\dot u(t)$ be your control, and treat $u(t)$ as another state variable. Then add the equation $\dot u(t) = v(t)$ to your dynamics.

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That would work if $u(t)$ was abs. continuous, but it is piecewise continuous so it can be represented as $u(t) = u^c(t) + u^d(t)$, where $u^c(t)$ is an absolutely continuous part and $u^d(t)$ is a step function. Then $\dot u(t) = \dot u^c(t)$ and if we put $\dot u(t) = v(t)$ we get \begin{align} &\dot x(t) = f(t,x(t),u^c(t) + u^d(t), v(t)) \newline &\dot u^c(t) = v(t) \end{align} I mean $v(t)$ determines only abs. continuous part of $u(t)$ and we still have to choose $u^d(t)$. It would be ok to consider two dim. control $(v(t), u^d(t))$ if $u^d(t)$ wasn't restricted to be step function. – niyazets Feb 8 '12 at 16:37

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