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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?

roi_saumon
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  • Can you explain why you came to that conclusion after looking at that page? Because to me it seems that page does say that the Hamiltonian depends on $q$ (namely $x=q$ and $\lambda=p$). – Kwin van der Veen Jan 05 '18 at 06:53
  • Or are you referring to that $L(q,u,t)$ does not depend on $\dot{q}$? – Kwin van der Veen Jan 05 '18 at 08:48
  • Oh sorry, I did a mistake. I wanted t say $\dot{q}$. I edited the question. Thanks. What I don't understand is why H(q,u,p,t) doesn't depend on $\dot{q}$ even though I understand L doesn't depend on $\dot{q}$ – roi_saumon Jan 05 '18 at 13:48

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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 $\dot{q}$.

Kwin van der Veen
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