Definitions

# Orbit (control theory)

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

The orbit of the control system $\left\{ \right\}dot q=f\left(q,u\right)$ through a point $q_0in M$ is the subset $\left\{mathcal O\right\}_\left\{q_0\right\}$ of $M$ defined by

$\left\{mathcal O\right\}_\left\{q_0\right\}=\left\{e^\left\{t_k f_k\right\}circ e^\left\{t_\left\{k-1\right\} f_\left\{k-1\right\}\right\}circcdotscirc e^\left\{t_1 f_1\right\}\left(q_0\right)mid kinmathbb\left\{N\right\}, t_1,dots,t_kinmathbb\left\{R\right\}, f_1,dots,f_kin\left\{mathcal F\right\}\right\}.$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 $\left\{mathcal F\right\}$ is symmetric (i.e., $fin \left\{mathcal F\right\}$ if and only if $-fin \left\{mathcal F\right\}$), then orbits and attainable sets coincide.

The hypothesis that every vector field of $\left\{mathcal F\right\}$ 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.

## Orbit theorem (Nagano-Sussmann)

Each orbit $\left\{mathcal O\right\}_\left\{q_0\right\}$ is an immersed submanifold of $M$.

The tangent space to the orbit $\left\{mathcal O\right\}_\left\{q_0\right\}$ at a point $q$ is the linear subspace of $T_q M$ spanned by the vectors $P_* f\left(q\right)$ where $P_* f$ denotes the pushforward of $f$ by $P$, $f$ belongs to $\left\{mathcal F\right\}$ and $P$ is a diffeomorphism of $M$ of the form $e^\left\{t_k f_k\right\}circ cdotscirc e^\left\{t_1 f_1\right\}$ with $kinmathbb\left\{N\right\}, t_1,dots,t_kinmathbb\left\{R\right\}$ and $f_1,dots,f_kin\left\{mathcal F\right\}$.

If all the vector fields of the family $\left\{mathcal F\right\}$ are analytic, then $T_q\left\{mathcal O\right\}_\left\{q_0\right\}=mathrm\left\{Lie\right\}_q,mathcal\left\{F\right\}$ where $mathrm\left\{Lie\right\}_q,mathcal\left\{F\right\}$ is the evaluation at $q$ of the Lie algebra generated by $\left\{mathcal F\right\}$ with respect to the Lie bracket of vector fields. Otherwise, the inclusion $mathrm\left\{Lie\right\}_q,mathcal\left\{F\right\}subset T_q\left\{mathcal O\right\}_\left\{q_0\right\}$ holds true.

## Corollary (Rashevsky-Chow theorem)

If $mathrm\left\{Lie\right\}_q,mathcal\left\{F\right\}= T_q M$ for every $qin M$ then each orbit is equal to the whole manifold $M$.

## References

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