Orbit (control theory)

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 uin U} and assume that every vector field in {mathcal F} is complete. For every fin {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_0in 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}}circcdotscirc e^{t_1 f_1}(q_0)mid kinmathbb{N}, t_1,dots,t_kinmathbb{R}, f_1,dots,f_kin{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., fin {mathcal F} if and only if -fin {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 cdotscirc e^{t_1 f_1} with kinmathbb{N}, t_1,dots,t_kinmathbb{R} and f_1,dots,f_kin{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 qin M then each orbit is equal to the whole manifold M.

References

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