The notion of orbit of a control system used in mathematical control theory is a particular case of the notion of orbit in group theory.^[1][2][3]

Definition

Let ${\displaystyle {\ }{\dot {q}}=f(q,u)}$ be a ${\displaystyle \ {\mathcal {C}}^{\infty }}$ 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 ${\displaystyle {\mathcal {F}}=\{f(\cdot ,u)\mid u\in U\}}$ and assume that every vector field in ${\displaystyle {\mathcal {F}}}$ is complete. For every ${\displaystyle f\in {\mathcal {F}}}$ and every real ${\displaystyle \ t}$, denote by ${\displaystyle \ e^{tf}}$ the flow of ${\displaystyle \ f}$ at time ${\displaystyle \ t}$.

The orbit of the control system ${\displaystyle {\ }{\dot {q}}=f(q,u)}$ through a point ${\displaystyle q_{0}\in M}$ is the subset ${\displaystyle {\mathcal {O}}_{q_{0}}}$ of ${\displaystyle \ M}$ defined by

${\displaystyle {\mathcal {O}}_{q_{0}}=\{e^{t_{k}f_{k}}\circ e^{t_{k-1}f_{k-1}}\circ \cdots \circ e^{t_{1}f_{1}}(q_{0})\mid k\in \mathbb {N} ,\ t_{1},\dots ,t_{k}\in \mathbb {R} ,\ f_{1},\dots ,f_{k}\in {\mathcal {F}}\}.}$

Remarks

The difference between orbits and attainable sets is that, whereas for attainable sets only forward-in-time motions are allowed, both forward and backward motions are permitted for orbits. In particular, if the family ${\displaystyle {\mathcal {F}}}$ is symmetric (i.e., ${\displaystyle f\in {\mathcal {F}}}$ if and only if ${\displaystyle -f\in {\mathcal {F}}}$), then orbits and attainable sets coincide.

The hypothesis that every vector field of ${\displaystyle {\mathcal {F}}}$ is complete simplifies the notations but can be dropped. In this case one has to replace flows of vector fields by local versions of them.

Orbit theorem (Nagano–Sussmann)

Each orbit ${\displaystyle {\mathcal {O}}_{q_{0}}}$ is an immersed submanifold of ${\displaystyle \ M}$.

The tangent space to the orbit ${\displaystyle {\mathcal {O}}_{q_{0}}}$ at a point ${\displaystyle \ q}$ is the linear subspace of ${\displaystyle \ T_{q}M}$ spanned by the vectors ${\displaystyle \ P_{*}f(q)}$ where ${\displaystyle \ P_{*}f}$ denotes the pushforward of ${\displaystyle \ f}$ by ${\displaystyle \ P}$, ${\displaystyle \ f}$ belongs to ${\displaystyle {\mathcal {F}}}$ and ${\displaystyle \ P}$ is a diffeomorphism of ${\displaystyle \ M}$ of the form ${\displaystyle e^{t_{k}f_{k}}\circ \cdots \circ e^{t_{1}f_{1}}}$ with ${\displaystyle k\in \mathbb {N} ,\ t_{1},\dots ,t_{k}\in \mathbb {R} }$ and ${\displaystyle f_{1},\dots ,f_{k}\in {\mathcal {F}}}$.

If all the vector fields of the family ${\displaystyle {\mathcal {F}}}$ are analytic, then ${\displaystyle \ T_{q}{\mathcal {O}}_{q_{0}}=\mathrm {Lie} _{q}\,{\mathcal {F}}}$ where ${\displaystyle \mathrm {Lie} _{q}\,{\mathcal {F}}}$ is the evaluation at ${\displaystyle \ q}$ of the Lie algebra generated by ${\displaystyle {\mathcal {F}}}$ with respect to the Lie bracket of vector fields. Otherwise, the inclusion ${\displaystyle \mathrm {Lie} _{q}\,{\mathcal {F}}\subset T_{q}{\mathcal {O}}_{q_{0}}}$ holds true.

Corollary (Rashevsky–Chow theorem)

If ${\displaystyle \mathrm {Lie} _{q}\,{\mathcal {F}}=T_{q}M}$ for every ${\displaystyle \ q\in M}$ and if ${\displaystyle \ M}$ is connected, then each orbit is equal to the whole manifold ${\displaystyle \ M}$.

See also

• Frobenius theorem (differential topology)

References

Further reading

• Agrachev, Andrei; Sachkov, Yuri (2004). "The Orbit Theorem and its Applications". Control Theory from the Geometric Viewpoint. Berlin: Springer. pp. 63–80. ISBN 3-540-21019-9.