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 of finite-dimensional control systems. It states that any attainable set of a 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 ${\displaystyle {\ }{\dot {q}}=f(q,u)}$ be a smooth control system, where ${\displaystyle {\ q}}$ belongs to a finite-dimensional manifold ${\displaystyle \ M}$ and ${\displaystyle \ u}$ belongs to a control set ${\displaystyle \ U}$. Consider the family of vector fields ${\displaystyle {\mathcal {F}}=\{f(\cdot ,u)\mid u\in U\}}$.

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

Remarks and consequences[]

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

When all the vector fields in ${\displaystyle \ {\mathcal {F}}}$ are analytic, ${\displaystyle \ \mathrm {Lie} _{q}\,{\mathcal {F}}=T_{q}M}$ if and only if ${\displaystyle \ q}$ belongs to the closure of the interior of the attainable set from ${\displaystyle \ q}$. 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 ${\displaystyle \ q\in M}$ is dense in ${\displaystyle \ M}$, then the attainable set from ${\displaystyle \ q}$ is actually equal to ${\displaystyle \ M}$.

References[]

• Agrachev, Andrei A.; Sachkov, Yuri L. (2004). Control theory from the geometric viewpoint. Springer-Verlag. pp. xiv+412. ISBN 3-540-21019-9.
• Jurdjevic, Velimir (1997). Geometric control theory. Cambridge University Press. pp. xviii+492. ISBN 0-521-49502-4.^{[permanent dead link]}
• Sussmann, Héctor J.; Jurdjevic, Velimir (1972). "Controllability of nonlinear systems". J. Differential Equations. 12 (1): 95–116. doi:10.1016/0022-0396(72)90007-1.
• 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. doi:10.1137/0312005.