I'm from econ. That is I'm not conceptually familiar with the underlying math of dynamical systems. When I usually deal with dynamic systems it's of the form $$ \max_u\int{F(t,x(t),u(t))}\\ \text{s.t.}~\dot{x}(t)=f(t,x(t),u(t))\\ x(0)=x_0 $$
where $x(t)\in\mathbb{R}^n$ is the state and $u(t)\in\mathbb{R}^m$ the control vector respectively.
However, I try to model a system where the state actually depends on the control, i.e. $x(t,u(t))$. Does it change the analysis substantially? Or does it work the usual way $$ H=F(t,x(t,u(t)),u(t))+\lambda(t)f(t,x(t,u(t)),u(t)) $$ with FOCs $$ H_u=0\quad\text{and}\quad H_x=-\dot{\lambda} $$ Appreciate help or a good read advice.
Usually the state always depends on the input. Otherwise, changing the input would have no effect on the state! We control the states of the systems with the control inputs! This is the case because $x(t) = x(t_0) + \int_{t_0}^t f(t,u(t),x(t) dt$ and the RHS of this equation is dependend on u(t)!
You have to be aware that $H_u$ are the paritial and not the total derivatives, that means you consider x and t constant when calculating $\frac{\partial H}{\partial u}$
You also have to be aware that the solution might not be the optimal one. For some dynamic systems it however be optimal! See therfore: http://en.wikipedia.org/wiki/Pontryagin%27s_minimum_principle
What do you mean by "FOCs"?