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

The orbit of the control system $\dot q=f(q,u)$ through a point $q_0\in M$ is the subset $_$ of $\ M$ defined by

:$_=\.$

;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 $$ is symmetric (i.e., $f\in$ if and only if $-f\in$), then orbits and attainable sets coincide.

The hypothesis that every vector field of $$ 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 $_$ is an immersed submanifold of $\ M$.

The tangent space to the orbit
$_$ 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 $$ and $\ P$ is a diffeomorphism of $\ M$ of the form $e^\circ \cdots\circ e^$ with $k\in\mathbb,\ t_1,\dots,t_k\in\mathbb$ and $f_1,\dots,f_k\in$.

If all the vector fields of the family $$ are analytic, then $\ T_q_=\mathrm_q\,\mathcal$ where $\mathrm_q\,\mathcal$ is the evaluation at $\ q$ of the Lie algebra generated by $$ with respect to the Lie bracket of vector fields.
Otherwise, the inclusion $\mathrm_q\,\mathcal\subset T_q_$ holds true.

Corollary (Rashevsky–Chow theorem)

If $\mathrm_q\,\mathcal= T_q M$ for every $\ q\in M$ and if $\ M$ is connected, then each orbit is equal to the whole manifold $\ M$.

See also

*Frobenius theorem (differential topology)

References

Further reading

*{{cite book , first=Andrei , last=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

Control theory