• Images
• Videos
• Documents
• Books
• Articles

Definition

Let   q ˙ = f ( q , u ) {\displaystyle {\ }{\dot {q}}=f(q,u)} be a   C ∞ {\displaystyle \ {\mathcal {C}}^{\infty }} control system, where   q {\displaystyle {\ q}} belongs to a finite-dimensional manifold   M {\displaystyle \ M} and   u {\displaystyle \ u} belongs to a control set   U {\displaystyle \ U} . Consider the family F = { f ( ⋅ , u ) ∣ u ∈ U } {\displaystyle {\mathcal {F}}=\{f(\cdot ,u)\mid u\in U\}} and assume that every vector field in F {\displaystyle {\mathcal {F}}} is complete. For every f ∈ F {\displaystyle f\in {\mathcal {F}}} and every real   t {\displaystyle \ t} , denote by   e t f {\displaystyle \ e^{tf}} the flow of   f {\displaystyle \ f} at time   t {\displaystyle \ t} .

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

O q 0 = { e t k f k ∘ e t k − 1 f k − 1 ∘ ⋯ ∘ e t 1 f 1 ( q 0 ) ∣ k ∈ N ,   t 1 , … , t k ∈ R ,   f 1 , … , f k ∈ F } . {\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 F {\displaystyle {\mathcal {F}}} is symmetric (i.e., f ∈ F {\displaystyle f\in {\mathcal {F}}} if and only if − f ∈ F {\displaystyle -f\in {\mathcal {F}}} ), then orbits and attainable sets coincide.

The hypothesis that every vector field of F {\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 O q 0 {\displaystyle {\mathcal {O}}_{q_{0}}} is an immersed submanifold of   M {\displaystyle \ M} .

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

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

Corollary (Rashevsky–Chow theorem)

Main article: Chow–Rashevskii theorem

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

See also

• Frobenius theorem (differential topology)

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.

References

1. Jurdjevic, Velimir (1997). Geometric control theory. Cambridge University Press. pp. xviii+492. ISBN 0-521-49502-4.[permanent dead link] 0-521-49502-4 ↩

2. Sussmann, Héctor J.; Jurdjevic, Velimir (1972). "Controllability of nonlinear systems". J. Differential Equations. 12 (1): 95–116. Bibcode:1972JDE....12...95S. doi:10.1016/0022-0396(72)90007-1. https://doi.org/10.1016%2F0022-0396%2872%2990007-1 ↩

3. Sussmann, Héctor J. (1973). "Orbits of families of vector fields and integrability of distributions". Trans. Amer. Math. Soc. 180. American Mathematical Society: 171–188. doi:10.2307/1996660. JSTOR 1996660. https://doi.org/10.2307%2F1996660 ↩