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

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*cite journal
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*cite journal
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