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.
Let q ˙ = f ( q , u ) be a C ∞ control system, where q belongs to a finite-dimensional manifold M and u belongs to a control set U . Consider the family F = { f ( ⋅ , u ) ∣ u ∈ U } and assume that every vector field in F is complete. For every f ∈ F and every real t , denote by e t f the flow of f at time t .
The orbit of the control system q ˙ = f ( q , u ) through a point q 0 ∈ M is the subset O q 0 of M defined by
O q 0 = { e t k f k ∘ e t k − 1 f k − 1 ∘ ⋯ ∘ e t 1 f 1 ( q 0 ) ∣ k ∈ N , t 1 , … , t k ∈ R , f 1 , … , f k ∈ F } . RemarksThe 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 ∈ F if and only if − f ∈ 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.
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 ∗ f ( q ) where P ∗ 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 ∘ ⋯ ∘ e t 1 f 1 with k ∈ N , t 1 , … , t k ∈ R and f 1 , … , f k ∈ F .
If all the vector fields of the family F are analytic, then T q O q 0 = L i e q F where L i e q F is the evaluation at q of the Lie algebra generated by F with respect to the Lie bracket of vector fields. Otherwise, the inclusion L i e q F ⊂ T q O q 0 holds true.
If L i e q F = T q M for every q ∈ M and if M is connected, then each orbit is equal to the whole manifold M .
