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.

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)

Main article: Chow–Rashevskii 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 $\ M$.

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.