Nonlinear Control Systems: Topological Properties of Trajectory Space

Sarychev, Andrey

doi:10.1007/978-1-4612-3214-8_32

Cited by 7 publications

(6 citation statements)

References 12 publications

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“…al. [16][17][18], however it is much more closely aligned with the work of Sarychev [19,20]. Indeed, the arguments presented in this paper follow very closely those of Sarychev.…”

Section: Introductionsupporting

confidence: 85%

“…If true, that theorem would largely contain the results of Theorem 1 of the present work. However, Sarychev's proof in [20] only covers symmetric conic polysystems, which are systems without drift, suggesting that the wording of the theorem was a simple mis-statement. Moreover, Montgomery [21,22] has constructed a counterexample on SO(3) based on Little's [23] work on nondegenerate curves on S 2 which demonstrates that the conclusions of Sarychev's theorem do not hold for all conic polysystems.…”

Section: Introductionmentioning

confidence: 99%

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Dynamic homotopy and landscape dynamical set topology in quantum control

Dominy

Rabitz

2012

Journal of Mathematical Physics

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We examine the topology of the subset of controls taking a given initial state to a given final state in quantum control, where "state" may mean a pure state |ψ , an ensemble density matrix ρ, or a unitary propagator U (0, T ). The analysis consists in showing that the endpoint map acting on control space is a Hurewicz fibration for a large class of affine control systems with vector controls.Exploiting the resulting fibration sequence and the long exact sequence of basepoint-preserving homotopy classes of maps, we show that the indicated subset of controls is homotopy equivalent to the loopspace of the state manifold. This not only allows us to understand the connectedness of "dynamical sets" realized as preimages of subsets of the state space through this endpoint map, but also provides a wealth of additional topological information about such subsets of control space.

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“…al. [16][17][18], however it is much more closely aligned with the work of Sarychev [19,20]. Indeed, the arguments presented in this paper follow very closely those of Sarychev.…”

Section: Introductionsupporting

confidence: 85%

Section: Introductionmentioning

confidence: 99%

Dynamic homotopy and landscape dynamical set topology in quantum control

Dominy

Rabitz

2012

Journal of Mathematical Physics

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“…An explicit construction of approximating sequences is given by Liu in [5] (see also preliminary work by Sussmann and Liu [10]), giving a method for approximate motion planning. Related results can also be found in [4] and [7]. These results can be readily extended to systems with nonzero drift if the brackets of control vector fields only still generate the whole tangent space.…”

Section: Introductionmentioning

confidence: 61%

On the curves that may be approached by trajectories of a smooth control affine system

Pomet

1999

Systems & Control Letters

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In this paper, we give a characterization of the set of curves that may be approached by trajectories of a smooth control-a ne nonlinear system, in the topology of uniform convergence. This characterization is in terms of the drift vector field and the distribution spanned by the Lie algebra generated by the control vector fields. The characterization is valid on open sets where this distribution has constant rank. These results also characterize the support of di↵usion processes with smooth coe cients.

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“…O conteúdo desse capítulo está intimamente relacionado com os trabalhos de Sarychev. De fato, Sarychev considerou em [13], [14] as propriedades homotópicas do espaço de trajetórias de sistemas em variedades Riemannianas completamente controláveis e, em particular, sistemas que não possuem campo drift (i.e., sistema simétrico). Mais precisamente, ele provou uma equivalência homotópica entre o espaço de trajetórias munido com a topologia de convergência uniforme e o espaço de laços da variedade Riemanniana adjacente.…”

Section: Dedicamos O Capítulo 5 Para Estudar As Propriedade Universais Do Espaço γ(σ X)unclassified

“…Para uma exposição mais detalhada veja [21]. O trabalho em [5] onde a homotopia monotônica (ou dinâmica) foi considerada na área "Quantum control" e também está relacionado com [2], [9] e [10], porém é muito mais alinhado com os trabalhos de Sarychev em [13], [14]. Foi provado [5], de uma maneira mais geral, uma equivalência homotópica entre a fibra da aplicação de ponto final e o espaço de laços da variedade (espaço estado).…”

Section: Introductionunclassified

Recobrimento monotônico de sistemas de controle

Lopes¹

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In this work, we deal with monotonic homotopy between trajectories of a control system Σ on a manifold M . This is an apropriate variant of usual homotopy, where two trajectories are considered to be homotopic if they can be deformed to each other in a continuous way through trajectories. We introduce regularity for controls and consider monotonic homotopy between trajectories generated by regular controls. In particular, we present an example of a system having homotopic trajectories which are not monotonically homotopic.The main goal was to understand the construction for monotonic homotopy of the universal covering space and, in particular, the differentiable manifold structure on the set Γ(Σ, x) of monotonic homotopy classes of trajectories starting at x ∈ M . As a consequence of that result, we obtain a local diffeomorphism which permits lifting of Σ to another system Σ in Γ(Σ, x). To consider universal properties of Γ(Σ, x) we take a covering π : N → A R (Σ, x) in the sense that N is a differentiable manifold provided with a control system Σ and π is a local diffeomorphism mapping Σ to Σ. Comparing the trajectories of Σ and Σ we construct a lifting mapping f : Γ(Σ, x) → N that relates Σ e Σ.Finally, we take into account the particular class of symmetric systems, for which both coverings Γ(Σ, x) and M coincide.

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Nonlinear Control Systems: Topological Properties of Trajectory Space

Cited by 7 publications

References 12 publications

Dynamic homotopy and landscape dynamical set topology in quantum control

Dynamic homotopy and landscape dynamical set topology in quantum control

On the curves that may be approached by trajectories of a smooth control affine system

Recobrimento monotônico de sistemas de controle

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