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.[1][2][3] Contents 1 Definition 2 Orbit theorem (Nagano–Sussmann) 3 Corollary (Rashevsky–Chow theorem) 4 See also 5 References 6 Further reading Definition Let {displaystyle { }{dot {q}}=f(q,u)} be a {displaystyle {mathcal {C}}^{infty }} control system, where {displaystyle { q}} belongs to a finite-dimensional manifold {displaystyle M} and {displaystyle u} belongs to a control set {displaystyle U} . Consider the family {displaystyle {mathcal {F}}={f(cdot ,u)mid uin U}} and assume that every vector field in {displaystyle {mathcal {F}}} is complete. For every {displaystyle fin {mathcal {F}}} and every real {displaystyle t} , denote by {displaystyle e^{tf}} the flow of {displaystyle f} at time {displaystyle t} .

The orbit of the control system {displaystyle { }{dot {q}}=f(q,u)} through a point {displaystyle q_{0}in M} is the subset {displaystyle {mathcal {O}}_{q_{0}}} of {displaystyle M} defined by {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 kin 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 {displaystyle {mathcal {F}}} is symmetric (i.e., {displaystyle fin {mathcal {F}}} if and only if {displaystyle -fin {mathcal {F}}} ), then orbits and attainable sets coincide.

The hypothesis that every vector field of {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 {displaystyle {mathcal {O}}_{q_{0}}} is an immersed submanifold of {displaystyle M} .

The tangent space to the orbit {displaystyle {mathcal {O}}_{q_{0}}} at a point {displaystyle q} is the linear subspace of {displaystyle T_{q}M} spanned by the vectors {displaystyle P_{*}f(q)} where {displaystyle P_{*}f} denotes the pushforward of {displaystyle f} by {displaystyle P} , {displaystyle f} belongs to {displaystyle {mathcal {F}}} and {displaystyle P} is a diffeomorphism of {displaystyle M} of the form {displaystyle e^{t_{k}f_{k}}circ cdots circ e^{t_{1}f_{1}}} with {displaystyle kin mathbb {N} , t_{1},dots ,t_{k}in mathbb {R} } and {displaystyle f_{1},dots ,f_{k}in {mathcal {F}}} .

If all the vector fields of the family {displaystyle {mathcal {F}}} are analytic, then {displaystyle T_{q}{mathcal {O}}_{q_{0}}=mathrm {Lie} _{q},{mathcal {F}}} where {displaystyle mathrm {Lie} _{q},{mathcal {F}}} is the evaluation at {displaystyle q} of the Lie algebra generated by {displaystyle {mathcal {F}}} with respect to the Lie bracket of vector fields. Otherwise, the inclusion {displaystyle mathrm {Lie} _{q},{mathcal {F}}subset T_{q}{mathcal {O}}_{q_{0}}} holds true.

Corollary (Rashevsky–Chow theorem) Main article: Chow–Rashevskii theorem If {displaystyle mathrm {Lie} _{q},{mathcal {F}}=T_{q}M} for every {displaystyle qin M} and if {displaystyle M} is connected, then each orbit is equal to the whole manifold {displaystyle M} .

See also Frobenius theorem (differential topology) References ^ Jurdjevic, Velimir (1997). Geometric control theory. Cambridge University Press. pp. xviii+492. ISBN 0-521-49502-4.[permanent dead link] ^ Sussmann, Héctor J.; Jurdjevic, Velimir (1972). "Controllability of nonlinear systems". J. Differential Equations. 12 (1): 95–116. doi:10.1016/0022-0396(72)90007-1. ^ Sussmann, Héctor J. (1973). "Orbits of families of vector fields and integrability of distributions". Trans. Amer. Math. Soc. American Mathematical Society. 180: 171–188. doi:10.2307/1996660. JSTOR 1996660. 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. Categories: Control theory

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