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] Contenu 1 Définition 2 Orbit theorem (Nagano–Sussmann) 3 Corollaire (Rashevsky–Chow theorem) 4 Voir également 5 Références 6 Further reading Definition Let {style d'affichage { }{point {q}}=f(q,tu)} être un {style d'affichage {mathématique {C}}^{infime }} control system, où {style d'affichage { q}} belongs to a finite-dimensional manifold {style d'affichage M} et {style d'affichage u} belongs to a control set {style d'affichage U} . Consider the family {style d'affichage {mathématique {F}}={F(cdot ,tu)mid uin U}} and assume that every vector field in {style d'affichage {mathématique {F}}} is complete. Pour chaque {displaystyle fin {mathématique {F}}} and every real {style d'affichage t} , denote by {style d'affichage e^{tf}} the flow of {style d'affichage f} at time {style d'affichage t} .

The orbit of the control system {style d'affichage { }{point {q}}=f(q,tu)} through a point {style d'affichage q_{0}in M} is the subset {style d'affichage {mathématique {O}}_{q_{0}}} de {style d'affichage M} Défini par {style d'affichage {mathématique {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},des points ,t_{k}en mathbb {R} , F_{1},des points ,F_{k}dans {mathématique {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. En particulier, if the family {style d'affichage {mathématique {F}}} is symmetric (c'est à dire., {displaystyle fin {mathématique {F}}} si et seulement si {displaystyle -fin {mathématique {F}}} ), then orbits and attainable sets coincide.

The hypothesis that every vector field of {style d'affichage {mathématique {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 {style d'affichage {mathématique {O}}_{q_{0}}} is an immersed submanifold of {style d'affichage M} .

The tangent space to the orbit {style d'affichage {mathématique {O}}_{q_{0}}} at a point {style d'affichage q} is the linear subspace of {style d'affichage T_{q}M} spanned by the vectors {style d'affichage P_{*}F(q)} où {style d'affichage P_{*}F} denotes the pushforward of {style d'affichage f} par {style d'affichage P} , {style d'affichage f} belongs to {style d'affichage {mathématique {F}}} et {style d'affichage P} is a diffeomorphism of {style d'affichage M} of the form {style d'affichage e^{t_{k}F_{k}}circ cdots circ e^{t_{1}F_{1}}} avec {style d'affichage kin mathbb {N} , t_{1},des points ,t_{k}en mathbb {R} } et {style d'affichage f_{1},des points ,F_{k}dans {mathématique {F}}} .

If all the vector fields of the family {style d'affichage {mathématique {F}}} are analytic, alors {style d'affichage T_{q}{mathématique {O}}_{q_{0}}= mathrm {Lie} _{q},{mathématique {F}}} où {style d'affichage mathrm {Lie} _{q},{mathématique {F}}} is the evaluation at {style d'affichage q} of the Lie algebra generated by {style d'affichage {mathématique {F}}} with respect to the Lie bracket of vector fields. Autrement, the inclusion {style d'affichage mathrm {Lie} _{q},{mathématique {F}}subset T_{q}{mathématique {O}}_{q_{0}}} holds true.

Corollaire (Rashevsky–Chow theorem) Main article: Chow–Rashevskii theorem If {style d'affichage mathrm {Lie} _{q},{mathématique {F}}=T_{q}M} pour chaque {displaystyle qin M} et si {style d'affichage M} est connecté, then each orbit is equal to the whole manifold {style d'affichage M} .

See also Frobenius theorem (differential topology) References ^ Jurdjevic, Velimir (1997). Geometric control theory. la presse de l'Universite de Cambridge. 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. est ce que je: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. Société mathématique américaine. 180: 171–188. est ce que je:10.2307/1996660. JSTOR 1996660. Further reading Agrachev, Andrei; Sachkov, Youri (2004). "The Orbit Theorem and its Applications". Control Theory from the Geometric Viewpoint. Berlin: Springer. pp. 63–80. ISBN 3-540-21019-9. Catégories: Control theory