Orbit (control theory)
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The notion of
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
of a control system used in mathematical
control theory Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
is a particular case of the notion of orbit in group theory.


Definition

Let \dot q=f(q,u) be a \ ^\infty control system, where belongs to a finite-dimensional manifold \ M and \ u belongs to a control set \ U. Consider the family =\ and assume that every vector field in is complete. For every f\in and every real \ t, denote by \ e^ 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 _ of \ M defined by :_=\. ;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 is symmetric (i.e., f\in if and only if -f\in ), then orbits and attainable sets coincide. The hypothesis that every vector field of 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 _ is an
immersed submanifold In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which p ...
of \ M. The tangent space to the orbit _ 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 The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things. * Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" op ...
of \ f by \ P, \ f belongs to and \ P is a diffeomorphism of \ M of the form e^\circ \cdots\circ e^ with k\in\mathbb,\ t_1,\dots,t_k\in\mathbb and f_1,\dots,f_k\in. If all the vector fields of the family are analytic, then \ T_q_=\mathrm_q\,\mathcal where \mathrm_q\,\mathcal is the evaluation at \ q of the Lie algebra generated by with respect to the Lie bracket of vector fields. Otherwise, the inclusion \mathrm_q\,\mathcal\subset T_q_ holds true.


Corollary (Rashevsky–Chow theorem)

If \mathrm_q\,\mathcal= 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) In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations. In modern geometric te ...


References


Further reading

*{{cite book , first=Andrei , last=Agrachev , first2=Yuri , last2=Sachkov , chapter=The Orbit Theorem and its Applications , title=Control Theory from the Geometric Viewpoint , location=Berlin , publisher=Springer , year=2004 , isbn=3-540-21019-9 , pages=63–80 , chapter-url=https://books.google.com/books?id=wF5kY__YPWgC&pg=PA63 Control theory