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 $$\ e^{t f}$$ 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


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

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.
 * Remarks

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 $$\ M$$.