Orbit (control theory)

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

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

{\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 ${\mathcal F}$ is symmetric (i.e., $f\in {\mathcal {F}}$ if and only if $-f\in {\mathcal {F}}$ ), then orbits and attainable sets coincide.

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

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

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

Corollary (Rashevsky-Chow theorem)

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

References

↑ Jurdjevic, Velimir (1997). Geometric control theory. Cambridge University Press. pp. xviii+492. ISBN 0-521-49502-4.
↑ 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.
↑ Sussmann, Héctor J. (1973). "Orbits of families of vector fields and integrability of distributions". Trans. Amer. Math. Soc. American Mathematical Society. 180: 171–188. doi:10.2307/1996660. JSTOR 1996660.

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