Orbit (control theory)

Let

$\{\backslash \; \}\backslash dot\; q=f(q,u)$

be a $\backslash \; \{\backslash mathcal\; C\}^\backslash infty$ control system, where

$\{\backslash \; q\}$

belongs to a finite-dimensional manifold $\backslash \; M$ and $\backslash \; u$ belongs to a control set $\backslash \; U$. Consider the family $\{\backslash mathcal\; F\}=\backslash \{f(\backslash cdot,u)\backslash mid\; u\backslash in\; U\backslash \}$

and assume that every vector field in $\{\backslash mathcal\; F\}$ is complete.

For every $f\backslash in\; \{\backslash mathcal\; F\}$ and every real $\backslash \; t$, denote by $\backslash \; e^\{t\; f\}$ the flow of $\backslash \; f$ at time $\backslash \; t$.

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

:$\{\backslash mathcal\; O\}\_\{q\_0\}=\backslash \{e^\{t\_k\; f\_k\}\backslash circ\; e^\{t\_\{k-1\}\; f\_\{k-1\}\}\backslash circ\backslash cdots\backslash circ\; e^\{t\_1\; f\_1\}(q\_0)\backslash mid\; k\backslash in\backslash mathbb\{N\},\backslash \; t\_1,\backslash dots,t\_k\backslash in\backslash mathbb\{R\},\backslash \; f\_1,\backslash dots,f\_k\backslash in\{\backslash mathcal\; F\}\backslash \}.$

;Remarks

The difference between orbits and attainable sets is that, whereas for attainable sets only forward-in-time motions are allowed, both forward and backward motions are permitted for orbits.

In particular, if the family $\{\backslash mathcal\; F\}$ is symmetric (i.e., $f\backslash in\; \{\backslash mathcal\; F\}$ if and only if $-f\backslash in\; \{\backslash mathcal\; F\}$), then orbits and attainable sets coincide.

The hypothesis that every vector field of $\{\backslash mathcal\; F\}$ 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.

Each orbit $\{\backslash mathcal\; O\}\_\{q\_0\}$ is an immersed submanifold of $\backslash \; M$.

The tangent space to the orbit

$\{\backslash mathcal\; O\}\_\{q\_0\}$ at a point $\backslash \; q$ is the linear subspace of $\backslash \; T\_q\; M$ spanned by

the vectors $\backslash \; P\_*\; f(q)$ where $\backslash \; P\_*\; f$ denotes the pushforward of $\backslash \; f$ by $\backslash \; P$, $\backslash \; f$ belongs to $\{\backslash mathcal\; F\}$ and $\backslash \; P$ is a diffeomorphism of $\backslash \; M$ of the form $e^\{t\_k\; f\_k\}\backslash circ\; \backslash cdots\backslash circ\; e^\{t\_1\; f\_1\}$ with $k\backslash in\backslash mathbb\{N\},\backslash \; t\_1,\backslash dots,t\_k\backslash in\backslash mathbb\{R\}$ and $f\_1,\backslash dots,f\_k\backslash in\{\backslash mathcal\; F\}$.

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

Otherwise, the inclusion $\backslash mathrm\{Lie\}\_q\backslash ,\backslash mathcal\{F\}\backslash subset\; T\_q\{\backslash mathcal\; O\}\_\{q\_0\}$ holds true.

{{main|Chow–Rashevskii theorem}}

If $\backslash mathrm\{Lie\}\_q\backslash ,\backslash mathcal\{F\}=\; T\_q\; M$ for every $\backslash \; q\backslash in\; M$ and if $\backslash \; M$ is connected, then each orbit is equal to the whole manifold $\backslash \; M$.

• Frobenius theorem (differential topology)

{{reflist}}

• {{cite book |first1=Andrei |last1=Agrachev |first2=Yuri |last2=Sachkov |chapter=The Orbit Theorem and its Applications |title=Control Theory from the Geometric Viewpoint |location=Berlin |publisher=Springer |year=2004 |isbn=3-540-21019-9 |pages=63–80 |chapter-url=https://books.google.com/books?id=wF5kY__YPWgC&pg=PA63 }}

Category:Control theory