Orbit (control theory) explained
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.[1] [2] [3]
Definition
Let
be a
control system, where
belongs to a finite-dimensional manifold
and
belongs to a control set
. Consider the family
{lF}=\{f( ⋅ ,u)\midu\inU\}
and assume that every vector field in
is complete.For every
and every real
, denote by
the
flow of
at time
.
The orbit of the control system
through a point
is the subset
of
defined by
\circ
\circ … \circ
(q0)\midk\inN, t1,...,tk\inR, f1,...,fk\in{lF}\}.
- RemarksThe 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.,
if and only if
), 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
.
The tangent space to the orbit
at a point
is the linear subspace of
spanned by the vectors
where
denotes the pushforward of
by
,
belongs to
and
is a diffeomorphism of
of the form
with
and
.
If all the vector fields of the family
are analytic, then
where
is the evaluation at
of the
Lie algebra generated by
with respect to the
Lie bracket of vector fields.Otherwise, the inclusion
holds true.
Corollary (Rashevsky–Chow theorem)
See main article: Chow–Rashevskii theorem. If
for every
and if
is connected, then each orbit is equal to the whole manifold
.
See also
Further reading
- Book: Andrei . Agrachev . Yuri . Sachkov . The Orbit Theorem and its Applications . Control Theory from the Geometric Viewpoint . Berlin . Springer . 2004 . 3-540-21019-9 . 63–80 . https://books.google.com/books?id=wF5kY__YPWgC&pg=PA63 .
Notes and References
- Book: Jurdjevic
, Velimir
. Geometric control theory . . 1997 . xviii+492 . 0-521-49502-4 .
- Sussmann . Héctor J. . Jurdjevic . Velimir . Controllability of nonlinear systems . J. Differential Equations . 12 . 1 . 95–116 . 1972 . 10.1016/0022-0396(72)90007-1. 1972JDE....12...95S . free .
- Sussmann . Héctor J. . Orbits of families of vector fields and integrability of distributions . Trans. Amer. Math. Soc. . 180 . 171–188 . American Mathematical Society . 1973 . 10.2307/1996660 . 1996660. free .
