Krener's theorem
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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 interior or, 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.
[edit] 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 generated by 
with respect to the Lie bracket of vector fields. Given 
, if the vector space 
is equal to 
, then 
belongs to the closure of the interior of the attainable set from .
[edit] 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 .
[edit] References
- Agrachev, Andrei A.; Sachkov, Yuri L. (2004). Control theory from the geometric viewpoint. Springer-Verlag, xiv+412. ISBN 3-540-21019-9.
- Jurdjevic, Velimir (1997). Geometric control theory. Cambridge University Press, xviii+492. ISBN 0-521-49502-4.
- Sussmann, Héctor J.; Jurdjevic, Velimir (1972). "Controllability of nonlinear systems". J. Differential Equations 12: 95–116.
- Krener, Arthur J. (1974). "A generalization of Chow's theorem and the bang-bang theorem to non-linear control problems". SIAM J. Control Optim. 12: 43–52.
