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

The orbit of the control system [math]\displaystyle{ {\ }\dot q=f(q,u) }[/math] through a point [math]\displaystyle{ q_0\in M }[/math] is the subset [math]\displaystyle{ {\mathcal O}_{q_0} }[/math] of [math]\displaystyle{ \ M }[/math] defined by

[math]\displaystyle{ {\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}\}. }[/math]

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

The hypothesis that every vector field of [math]\displaystyle{ {\mathcal F} }[/math] 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 [math]\displaystyle{ {\mathcal O}_{q_0} }[/math] is an immersed submanifold of [math]\displaystyle{ \ M }[/math].

The tangent space to the orbit [math]\displaystyle{ {\mathcal O}_{q_0} }[/math] at a point [math]\displaystyle{ \ q }[/math] is the linear subspace of [math]\displaystyle{ \ T_q M }[/math] spanned by the vectors [math]\displaystyle{ \ P_* f(q) }[/math] where [math]\displaystyle{ \ P_* f }[/math] denotes the pushforward of [math]\displaystyle{ \ f }[/math] by [math]\displaystyle{ \ P }[/math], [math]\displaystyle{ \ f }[/math] belongs to [math]\displaystyle{ {\mathcal F} }[/math] and [math]\displaystyle{ \ P }[/math] is a diffeomorphism of [math]\displaystyle{ \ M }[/math] of the form [math]\displaystyle{ e^{t_k f_k}\circ \cdots\circ e^{t_1 f_1} }[/math] with [math]\displaystyle{ k\in\mathbb{N},\ t_1,\dots,t_k\in\mathbb{R} }[/math] and [math]\displaystyle{ f_1,\dots,f_k\in{\mathcal F} }[/math].

If all the vector fields of the family [math]\displaystyle{ {\mathcal F} }[/math] are analytic, then [math]\displaystyle{ \ T_q{\mathcal O}_{q_0}=\mathrm{Lie}_q\,\mathcal{F} }[/math] where [math]\displaystyle{ \mathrm{Lie}_q\,\mathcal{F} }[/math] is the evaluation at [math]\displaystyle{ \ q }[/math] of the Lie algebra generated by [math]\displaystyle{ {\mathcal F} }[/math] with respect to the Lie bracket of vector fields. Otherwise, the inclusion [math]\displaystyle{ \mathrm{Lie}_q\,\mathcal{F}\subset T_q{\mathcal O}_{q_0} }[/math] holds true.

Corollary (Rashevsky–Chow theorem)

If [math]\displaystyle{ \mathrm{Lie}_q\,\mathcal{F}= T_q M }[/math] for every [math]\displaystyle{ \ q\in M }[/math] and if [math]\displaystyle{ \ M }[/math] is connected, then each orbit is equal to the whole manifold [math]\displaystyle{ \ M }[/math].

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. https://books.google.com/books?id=wF5kY__YPWgC&pg=PA63.