Krener's theorem
# Krener's theorem

In mathematics, Krener's theorem is a result attributed to Arthur J. Krener 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.

## 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 ${\mathcal {F}}=\{f(\cdot ,u)\mid u\in U\}$.

Let $\ \mathrm {Lie} \,{\mathcal {F}}$ be the Lie algebra generated by ${\mathcal {F}}$ with respect to the Lie bracket of vector fields. Given $\ q\in M$, if the vector space $\ \mathrm {Lie} _{q}\,{\mathcal {F}}=\{g(q)\mid g\in \mathrm {Lie} \,{\mathcal {F}}\}$ 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 $\ q\in M$ is dense in $\ M$, then the attainable set from $\ q$ is actually equal to $\ M$.

