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 ${{𝒞}}^{\mathrm{\infty }}$ control system, where $q$ belongs to a finite-dimensional manifold $M$ and $u$ belongs to a control set $U$. Consider the family ${ℱ}=\{f(\cdot ,u)\mid u\in U\}$ and assume that every vector field in ${ℱ}$ is complete. For every $f\in {ℱ}$ 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 ${{𝒪}}_{q_0}$ of $M$ defined by

$$\begin{gathered} {{𝒪}}_{q_0}=\{e^{t_k{f}_k}\circ e^{t_{k-1}{f}_{k-1}}\circ \cdots \circ e^{t_1{f}_1}(q_0)\mid k\in \mathbb{\mathbb{N}}, \\ {t}_1,\dots ,t_k\in \mathbb{ℝ}, \\ {f}_1,\dots ,f_k\in {ℱ}\}. \end{gathered}$$

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 ${ℱ}$ is symmetric (i.e., $f\in {ℱ}$ if and only if $-f\in {ℱ}$), then orbits and attainable sets coincide. The hypothesis that every vector field of ${ℱ}$ 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 ${{𝒪}}_{q_0}$ is an immersed submanifold of $M$. The tangent space to the orbit ${{𝒪}}_{q_0}$ 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 ${ℱ}$ and $P$ is a diffeomorphism of $M$ of the form $e^{t_k{f}_k}\circ \cdots \circ e^{t_1{f}_1}$ with $$\begin{gathered} k\in \mathbb{\mathbb{N}}, \\ {t}_1,\dots ,t_k\in \mathbb{ℝ} \end{gathered}$$ and $f_1,\dots ,f_k\in {ℱ}$. If all the vector fields of the family ${ℱ}$ are analytic, then ${T}_q{{𝒪}}_{q_0}={\mathrm{L}\mathrm{i}\mathrm{e}}_q{ℱ}$ where ${\mathrm{L}\mathrm{i}\mathrm{e}}_q{ℱ}$ is the evaluation at $q$ of the Lie algebra generated by ${ℱ}$ with respect to the Lie bracket of vector fields. Otherwise, the inclusion ${\mathrm{L}\mathrm{i}\mathrm{e}}_q{ℱ}\subset T_q{{𝒪}}_{q_0}$ holds true.

Corollary (Rashevsky–Chow theorem)

Main article: Chow–Rashevskii theorem

If ${\mathrm{L}\mathrm{i}\mathrm{e}}_q{ℱ}=T_{q}M$ 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.