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
q ˙ = f ( q , u ) {\displaystyle {\ }{\dot {q}}=f(q,u)} be a C ∞ {\displaystyle \ {\mathcal {C}}^{\infty }} control system, where
q {\displaystyle {\ q}} belongs to a finite-dimensional manifold M {\displaystyle \ M} and u {\displaystyle \ u} belongs to a control set U {\displaystyle \ U} . Consider the family F = { f ( ⋅ , u ) ∣ u ∈ U } {\displaystyle {\mathcal {F}}=\{f(\cdot ,u)\mid u\in U\}}
and assume that every vector field in F {\displaystyle {\mathcal {F}}} is complete. For every f ∈ F {\displaystyle f\in {\mathcal {F}}} and every real t {\displaystyle \ t} , denote by e t f {\displaystyle \ e^{tf}} the flow of f {\displaystyle \ f} at time t {\displaystyle \ t} . The orbit of the control system q ˙ = f ( q , u ) {\displaystyle {\ }{\dot {q}}=f(q,u)} through a point q 0 ∈ M {\displaystyle q_{0}\in M} is the subset O q 0 {\displaystyle {\mathcal {O}}_{q_{0}}} of M {\displaystyle \ 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 } . {\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}}\}.}
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 F {\displaystyle {\mathcal {F}}} is symmetric (i.e., f ∈ F {\displaystyle f\in {\mathcal {F}}} if and only if − f ∈ F {\displaystyle -f\in {\mathcal {F}}} ), then orbits and attainable sets coincide. The hypothesis that every vector field of F {\displaystyle {\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 O q 0 {\displaystyle {\mathcal {O}}_{q_{0}}} is an immersed submanifold of M {\displaystyle \ M} . The tangent space to the orbit
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