Orbit (control theory)
Encyclopedia
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.
be a
control system, where
belongs to a finite-dimensional manifold
and 
belongs to a control set 
. Consider the family 
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
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., 
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.
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.
for every 
and if 
is connected, then each orbit is equal to the whole manifold 
.
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...
of a control system used in mathematical control theory
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
is a particular case of the notion of orbit in group theory.
Definition
Letbe a
control system, wherebelongs to a finite-dimensional manifold
and belongs to a control set . Consider the family and assume that every vector field in
is complete.For every
and every real , denote by the flowVector flow
In mathematics, the vector flow refers to a set of closely related concepts of the flow determined by a vector field. These appear in a number of different contexts, including differential topology, Riemannian geometry and Lie group theory...
of
at time .The orbit of the control system
through a point is the subset of defined byRemarks
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., 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 bythe 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 algebraLie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
generated by
with respect to the Lie bracket of vector fieldsLie bracket of vector fields
In the mathematical field of differential topology, the Lie bracket of vector fields, Jacobi–Lie bracket, or commutator of vector fields is a bilinear differential operator which assigns, to any two vector fields X and Y on a smooth manifold M, a third vector field denoted [X, Y]...
.
Otherwise, the inclusion
holds true.Corollary (Rashevsky-Chow theorem)
If for every and if is connected, then each orbit is equal to the whole manifold .