In mathematics, Krener's theorem is a result attributed to
Arthur J. Krener in geometric
control theory
Control theory is a field of mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive ...
about the topological properties of
attainable sets of finite-dimensional control systems. It states that any attainable set of a
bracket-generating system has nonempty interior or, equivalently, that any attainable set has nonempty interior in the topology of the corresponding
orbit
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
. Heuristically, Krener's theorem prohibits attainable sets from being
hairy Hairy may refer to:
* people or animals covered in hairs or fur
* plants covered in trichomes
* insects covered in setae
* people nicknamed "the Hairy"
* Hairy (gene)
The hairy localisation element (HLE) is an RNA element found in the 3' UTR of ...
.
Theorem
Let
be a smooth control system, where
belongs to a finite-dimensional manifold
and
belongs to a control set
. Consider the family of vector fields
.
Let
be the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
generated by
with respect to the
Lie bracket of vector fields
In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields ''X'' and ''Y'' on a smooth ...
.
Given
, if the vector space
is equal to
,
then
belongs to the closure of the interior of the attainable set from
.
Remarks and consequences
Even if
is different from
,
the attainable set from
has nonempty interior in the orbit topology,
as it follows from Krener's theorem applied to the control system restricted to the orbit through
.
When all the vector fields in
are analytic,
if and only if
belongs to the closure of the interior of the attainable set from
. This is a consequence of Krener's theorem and of the
orbit theorem.
As a corollary of Krener's theorem one can prove that if the system is bracket-generating and if the attainable set from
is dense in
, then the attainable set from
is actually equal to
.
References
*
*
*
*{{cite journal
, last =Krener
, first =Arthur J.
, title =A generalization of Chow's theorem and the bang-bang theorem to non-linear control problems
, journal =SIAM J. Control Optim.
, volume =12
, pages =43–52
, date =1974
, doi =10.1137/0312005
Control theory
Theorems in dynamical systems