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The notion of orbit of a control system used in mathematical control theory is a particular case of the notion of orbit in group theory.
## Definition

Let
$\{\; \}dot\; q=f(q,u)$
be a $\{mathcal\; C\}^infty$ control system, where
$\{\; q\}$
belongs to a finite-dimensional manifold $M$ and $u$ belongs to a control set $U$. Consider the family $\{mathcal\; F\}=\{f(cdot,u)mid\; uin\; U\}$
and assume that every vector field in $\{mathcal\; F\}$ is complete.
For every $fin\; \{mathcal\; F\}$ and every real $t$, denote by $e^\{t\; f\}$ the flow of $f$ at time $t$. ## Orbit theorem (Nagano-Sussmann)

Each orbit $\{mathcal\; O\}\_\{q\_0\}$ is an immersed submanifold of $M$. ## Corollary (Rashevsky-Chow theorem)

If $mathrm\{Lie\}\_q,mathcal\{F\}=\; T\_q\; M$ for every $qin\; M$ then each orbit is equal to the whole manifold $M$.## References

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The orbit of the control system $\{\; \}dot\; q=f(q,u)$ through a point $q\_0in\; M$ is the subset $\{mathcal\; O\}\_\{q\_0\}$ of $M$ defined by

- $\{mathcal\; O\}\_\{q\_0\}=\{e^\{t\_k\; f\_k\}circ\; e^\{t\_\{k-1\}\; f\_\{k-1\}\}circcdotscirc\; e^\{t\_1\; f\_1\}(q\_0)mid\; kinmathbb\{N\},\; t\_1,dots,t\_kinmathbb\{R\},\; f\_1,dots,f\_kin\{mathcal\; F\}\}.$Remarks

The hypothesis that every vector field of $\{mathcal\; F\}$ is complete simplifies the notations but can be dropped. In this case one has to replace flows of vector fields by local versions of them.

The tangent space to the orbit $\{mathcal\; O\}\_\{q\_0\}$ at a point $q$ is the linear subspace of $T\_q\; M$ spanned by the vectors $P\_*\; f(q)$ where $P\_*\; f$ denotes the pushforward of $f$ by $P$, $f$ belongs to $\{mathcal\; F\}$ and $P$ is a diffeomorphism of $M$ of the form $e^\{t\_k\; f\_k\}circ\; cdotscirc\; e^\{t\_1\; f\_1\}$ with $kinmathbb\{N\},\; t\_1,dots,t\_kinmathbb\{R\}$ and $f\_1,dots,f\_kin\{mathcal\; F\}$.

If all the vector fields of the family $\{mathcal\; F\}$ are analytic, then $T\_q\{mathcal\; O\}\_\{q\_0\}=mathrm\{Lie\}\_q,mathcal\{F\}$ where $mathrm\{Lie\}\_q,mathcal\{F\}$ is the evaluation at $q$ of the Lie algebra generated by $\{mathcal\; F\}$ with respect to the Lie bracket of vector fields. Otherwise, the inclusion $mathrm\{Lie\}\_q,mathcal\{F\}subset\; T\_q\{mathcal\; O\}\_\{q\_0\}$ holds true.

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Last updated on Tuesday June 10, 2008 at 00:36:50 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Tuesday June 10, 2008 at 00:36:50 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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