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 $q$is actually equal to $M$.

References

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