Krener's theorem

Krener's theorem

Krener's theorem is a result in geometric control theory 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 interioror, equivalently, that any attainable set has nonempty interior in the topology of the corresponding orbit.Heuristically, Krener's theorem prohibits attainable sets from being hairy.

Theorem

Let { }dot q=f(q,u) be a smooth control system, where { q} belongs to a finite-dimensional manifold M and u belongs to a control set U. Consider the family of vector fields {mathcal F}={f(cdot,u)mid uin U}.

Let mathrm{Lie},mathcal{F} be the Lie algebra generated by {mathcal F} with respect to the Lie bracket of vector fields. Given qin M, if the vector space mathrm{Lie}_q,mathcal{F}={g(q)mid gin mathrm{Lie},mathcal{F}} is equal to T_q M,then q belongs to the closure of the interior of the attainable set from q.

Remarks and consequences

Even if mathrm{Lie}_q,mathcal{F} is different from T_q M,the attainable set from q has nonempty interior in the orbit topology,as it follows from Krener's theorem applied to the control system restricted to the orbit through q.

When all the vector fields in mathcal{F} are analytic, mathrm{Lie}_q,mathcal{F}=T_q M if and only if q belongs to the closure of the interior of the attainable set from q. 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 qin M is dense in M, then the attainable set from qis actually equal to M.

References

*cite book
last = Agrachev
first = Andrei A.
authorlink =
coauthors = Sachkov, Yuri L.
title = Control theory from the geometric viewpoint
publisher = Springer-Verlag
date =2004
location =
pages = xiv+412
url =http://www.springer.com/east/home/math/applications?SGWID=5-10051-22-26872681-0
doi =
id =
isbn = 3-540-21019-9

*cite book
last =Jurdjevic
first =Velimir
authorlink =
coauthors =
title =Geometric control theory
publisher =Cambridge University Press
date =1997
location =
pages =xviii+492
url =http://www.cup.cam.ac.uk/us/catalogue/email.asp?isbn=9780521495028
doi =
id =
isbn = 0-521-49502-4

*cite journal
last =Sussmann
first =Héctor J.
authorlink =
coauthors =Jurdjevic, Velimir
title =Controllability of nonlinear systems
journal =J. Differential Equations
volume =12
issue =
pages =95–116
publisher =
location =
date =1972
url =
doi =
id =
accessdate =

*cite journal
last =Krener
first =Arthur J.
authorlink =
coauthors =
title =A generalization of Chow's theorem and the bang-bang theorem to non-linear control problems
journal =SIAM J. Control Optim.
volume =12
issue =
pages =43–52
publisher =
location =
date =1974
url =
doi =
id =
accessdate =


Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Control theory — For control theory in psychology and sociology, see control theory (sociology) and Perceptual Control Theory. The concept of the feedback loop to control the dynamic behavior of the system: this is negative feedback, because the sensed value is… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”