Krener's theorem

Krener's theorem In mathematics, Krener's theorem is a result attributed to Arthur J. Krener in geometric control theory about the topological properties of attainable sets of finite-dimensional control systems. It states that any attainable set of a bracket-generating system has nonempty interior or, equivalentemente, that any attainable set has nonempty interior in the topology of the corresponding orbit. Heuristically, Krener's theorem prohibits attainable sets from being hairy.

Theorem Let {stile di visualizzazione { }{punto {q}}=f(q,tu)} be a smooth control system, dove {stile di visualizzazione { q}} belongs to a finite-dimensional manifold {stile di visualizzazione M} e {stile di visualizzazione u} belongs to a control set {stile di visualizzazione U} . Consider the family of vector fields {stile di visualizzazione {matematico {F}}={f(cdot ,tu)mid uin U}} .

Permettere {displaystyle matematica {Lie} ,{matematico {F}}} be the Lie algebra generated by {stile di visualizzazione {matematico {F}}} with respect to the Lie bracket of vector fields. Given {displaystyle qin M} , if the vector space {displaystyle matematica {Lie} _{q},{matematico {F}}={g(q)mid gin mathrm {Lie} ,{matematico {F}}}} è uguale a {stile di visualizzazione T_{q}M} , poi {stile di visualizzazione q} belongs to the closure of the interior of the attainable set from {stile di visualizzazione q} .

Remarks and consequences Even if {displaystyle matematica {Lie} _{q},{matematico {F}}} is different from {stile di visualizzazione T_{q}M} , the attainable set from {stile di visualizzazione q} has nonempty interior in the orbit topology, as it follows from Krener's theorem applied to the control system restricted to the orbit through {stile di visualizzazione q} .

When all the vector fields in {stile di visualizzazione {matematico {F}}} are analytic, {displaystyle matematica {Lie} _{q},{matematico {F}}=T_{q}M} se e solo se {stile di visualizzazione q} belongs to the closure of the interior of the attainable set from {stile di visualizzazione q} . This is a consequence of Krener's theorem and of the orbit theorem.

As a corollary of Krener's theorem one can prove that if the system is bracket-generating and if the attainable set from {displaystyle qin M} is dense in {stile di visualizzazione M} , then the attainable set from {stile di visualizzazione q} is actually equal to {stile di visualizzazione M} .

References Agrachev, Andrei A.; Sachkov, Yuri L. (2004). Control theory from the geometric viewpoint. Springer-Verlag. pp. xiv+412. ISBN 3-540-21019-9. 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. Krener, Arthur J. (1974). "A generalization of Chow's theorem and the bang-bang theorem to non-linear control problems". SIAM J. Control Optim. 12: 43–52. doi:10.1137/0312005. Categorie: Control theoryTheorems in dynamical systems