Hamiltonian in control theory

Question

In the wikipedia article Hamiltonian (control theory), I don't understand why $H(q,u,p,t)$ doesn't depend on $\dot{q}$. Can someone explain this to me?

Answer 1

This is because in the optimal control problem formulation $\dot{q}$ is a function of $q$, $u$ and $t$. This can be shown by using the fact that $q=x$ and

$$ \dot{x} = f(x,u,t). $$

The Hamiltonian can still be defined in terms of $\dot{q}$, however $\dot{q}$ can also be expressed in terms of $q$, $u$ and $t$. And after substituting in that expression for $\dot{q}$ the Hamiltonian does not explicitly depend on $\dot{q}$.