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

${𝒪}_{q_0}=\{e^{t_k{f}_k}\circ e^{t_{k-1}{f}_{k-1}}\circ \cdots \circ e^{t_1{f}_1}(q_0)\mid k\in \mathbb{N},t_1,\dots ,t_k\in ℝ,f_1,\dots ,f_k\in ℱ\}.$

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 $ℱ$ 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 $P_{\ast }f(q)$ where $P_{\ast }f$ 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 $k\in \mathbb{N},t_1,\dots ,t_k\in ℝ$ and $f_1,\dots ,f_k\in ℱ$.

If all the vector fields of the family $ℱ$ are analytic, then $T_q{𝒪}_{q_0}={\mathrm{L}\mathrm{i}\mathrm{e}}_qℱ$ where ${\mathrm{L}\mathrm{i}\mathrm{e}}_qℱ$ is the evaluation at $q$ of the Lie algebra generated by $ℱ$ with respect to the Lie bracket of vector fields. Otherwise, the inclusion ${\mathrm{L}\mathrm{i}\mathrm{e}}_qℱ\subset T_q{𝒪}_{q_0}$ holds true.

Corollary (Rashevsky–Chow theorem)

Template:Main If ${\mathrm{L}\mathrm{i}\mathrm{e}}_qℱ=T_{q}M$ 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

• Template:Cite book